<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lesson21intro" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">Introduction to Wave Plates</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson21_intro">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson21_intro" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
Next we discuss in detail the interaction of polarized light with interfaces and objects. For instance, how reflection from a perfect conductor affects the polarization state. </p><p>
We also see how to generate different polarization states through the use of polarizers and wave plates. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L31v1" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31v1: Polarized Waves Incident on Perfect Conductor</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v1" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L31v1: Polarized Waves Incident on Perfect Conductor</h3>
<div
id="video_L31v1"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v1/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v1/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:6sGBF6oGcr8", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v1/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v1/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L31v1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L31v1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L31v1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/7fbacb558131b0088ac2db1c8b2b2049/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L31v1.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_19_ex07_0" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31Q1: Easy Axis</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_07">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_07" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_19_07" class="problems-wrapper" role="group"
aria-labelledby="lect_19_07-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_07" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_07/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_19_07-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_07-problem-progress" tabindex="-1">
Easy Axis and Polarization Matrix
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_07-problem-progress"></div>
<div class="problem">
<div>
<p>
The figure below shows a polarization grating, where the grating itself consists of narrow conducting strips. One axis will effectively "block" the light, while the other axis will permit the light to be transmitted easily&#8212;hence, the latter is known as the "easy" axis. For the sake of this problem, let's call the other axis the "hard" axis. </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/4c55ced5b96efe8f9b18974a760bc3b9/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_19_07.png" width="165"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
Select TWO of the following statements that accurately characterize the different axes of a polarizer. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_19_07_2_1">
<fieldset aria-describedby="status_lect_19_07_2_1">
<div class="field">
<input type="checkbox" name="input_lect_19_07_2_1[]" id="input_lect_19_07_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_19_07_2_1-choice_0-label" for="input_lect_19_07_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_19_07_2_1">
<text>a) the hard axis is perpendicular to the grating lines, and it blocks light because the spacing between the grating lines is much smaller than the amplitude of light, so light cannot fit through</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_19_07_2_1[]" id="input_lect_19_07_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_19_07_2_1-choice_1-label" for="input_lect_19_07_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_19_07_2_1">
<text>b) the hard axis is parallel to the grating lines, and it blocks light because it acts like a perfect conductor, which reflects the electric field completely</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_19_07_2_1[]" id="input_lect_19_07_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_19_07_2_1-choice_2-label" for="input_lect_19_07_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_19_07_2_1">
<text>c) the easy axis is parallel to the grating lines, and it transmits light because the electric field can fit through the grating along this direction</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_19_07_2_1[]" id="input_lect_19_07_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_19_07_2_1-choice_3-label" for="input_lect_19_07_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_19_07_2_1">
<text>d) the easy axis is perpendicular to the grating lines, and it transmits light because the grating lines are too narrow along this direction (compared to the wavelength of light) to act as an effective conductor</text>
</label>
</div>
<span id="answer_lect_19_07_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_19_07_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_19_07_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Easy Axis and Polarization Matrix" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_19_07" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_19_07">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_07-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_07-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_07-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L31v2" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31v2: Polarizers</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v2">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v2" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L31v2: Polarizers</h3>
<div
id="video_L31v2"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v2/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v2/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:HBNLyOLSKSM", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v2/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v2/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L31v2"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L31v2">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L31v2">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/e1233bb57856a052d6bb0782c8f1e04d/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L31v2.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_19_ex07_1" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31Q2: Polarization Matrix I</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_08">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_08" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_19_08" class="problems-wrapper" role="group"
aria-labelledby="lect_19_08-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_08" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_08/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_19_08-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_08-problem-progress" tabindex="-1">
Easy Axis and Polarization Matrix I
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_08-problem-progress"></div>
<div class="problem">
<div>
<p>
Now we have formalized how to represent the ACT of polarization mathematically&#8212;through the polarization operator [mathjaxinline]\mathrm{\textbf{P}}[/mathjaxinline]. </p>
<p>
This operator transforms the polarization state of a wave from [mathjaxinline]\mathrm{\textbf{Z}} \, \rightarrow \, \mathrm{\textbf{Z'}}[/mathjaxinline] through the following operation: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{P}}_{\theta }\cdot \mathrm{\textbf{Z}}=\mathrm{\textbf{Z'}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
The operator [mathjaxinline]\mathrm{\textbf{P}}[/mathjaxinline] can be defined in the following way: </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{P}}_{\theta } = \begin{pmatrix} \cos ^{2}\theta &amp; &amp; \cos \theta \sin \theta \\ \cos \theta \sin \theta &amp; &amp; \sin ^{2}\theta \\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]\theta[/mathjaxinline] is the angle of the polarizer relative to the [mathjaxinline]x[/mathjaxinline]-axis (with counterclockwise angles defined to be positive). Thus, the following operation for [mathjaxinline]\theta =0[/mathjaxinline] should preserve the polarization state: </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{P}}_{0}\cdot \mathrm{\textbf{Z}} = \begin{pmatrix} 1 &amp; &amp; 0\\ 0 &amp; &amp; 0\\ \end{pmatrix}\cdot \begin{pmatrix} 1\\ 0\\ \end{pmatrix}= \begin{pmatrix} 1\\ 0\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
For each of the following polarization states, [mathjaxinline]\mathrm{\textbf{Z}}[/mathjaxinline], determine the final polarization state after the wave passes through a polarizer of a given angle. </p>
<p><b class="bfseries">(Part a)</b> A wave in the initial state [mathjaxinline]\mathrm{\textbf{Z}}[/mathjaxinline] passes through a polarizer of angle [mathjaxinline]\theta =\frac{\pi }{2}[/mathjaxinline]. Determine [mathjaxinline]\mathrm{\textbf{Z'}}[/mathjaxinline] </p>
<table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{Z}} = \begin{pmatrix} 1\\ 1\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p style="display:inline">[mathjaxinline]\mathrm{\textbf{Z'}}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_19_08_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/3c6b846f79f362682b770fc1967d7a10/asset-v1:MITx+8.03x+1T2020+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_lect_19_08_2_1" id="input_lect_19_08_2_1" aria-describedby="status_lect_19_08_2_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_lect_19_08_2_1"/>
<span class="status unanswered" id="status_lect_19_08_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_19_08_2_1" class="answer"/>
<div id="display_lect_19_08_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_19_08_2_1_dynamath" name="input_lect_19_08_2_1_dynamath"/>
</div>
</div></div>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p><b class="bfseries">(Part b)</b> A wave in the initial state [mathjaxinline]\mathrm{\textbf{Z}}[/mathjaxinline] passes through a polarizer of angle [mathjaxinline]\theta =\frac{3\pi }{4}[/mathjaxinline]. Determine [mathjaxinline]\mathrm{\textbf{Z'}}[/mathjaxinline] </p>
<table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{Z}} = \begin{pmatrix} 1\\ 0\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p style="display:inline">[mathjaxinline]\mathrm{\textbf{Z'}}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lect_19_08_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/3c6b846f79f362682b770fc1967d7a10/asset-v1:MITx+8.03x+1T2020+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_lect_19_08_3_1" id="input_lect_19_08_3_1" aria-describedby="status_lect_19_08_3_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_lect_19_08_3_1"/>
<span class="status unanswered" id="status_lect_19_08_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_19_08_3_1" class="answer"/>
<div id="display_lect_19_08_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_19_08_3_1_dynamath" name="input_lect_19_08_3_1_dynamath"/>
</div>
</div></div>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_19_08_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Easy Axis and Polarization Matrix I" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_19_08" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_19_08">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_08-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_08-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_08-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_19_ex07_2" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31Q3: Polarization Matrix II [WITH SIMULATION]</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09a" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_19_09a" class="problems-wrapper" role="group"
aria-labelledby="lect_19_09a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_19_09a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09a-problem-progress" tabindex="-1">
Easy Axis and Polarization Matrix II - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09a-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider light that is initially polarized in the state [mathjaxinline]\mathrm{\textbf{Z}}_{0}[/mathjaxinline], which travels through two polarizers, [mathjaxinline]\mathrm{\textbf{P}}_{\theta _1}[/mathjaxinline] and [mathjaxinline]\mathrm{\textbf{P}}_{\theta _2}[/mathjaxinline]. We can write the state of the light after it passes through each polarizer in the following way: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{P}}_{\theta _1}\cdot \mathrm{\textbf{Z}}_{0}=\mathrm{\textbf{Z}}_{1}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{P}}_{\theta _2}\cdot \mathrm{\textbf{Z}}_{1}=\mathrm{\textbf{Z}}_{2}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Let's say these polarizers are oriented at angles [mathjaxinline]\theta _{1}[/mathjaxinline] and [mathjaxinline]\theta _{2}[/mathjaxinline] with respect to the [mathjaxinline]x[/mathjaxinline]-axis, and the light is initially linearly polarized along the [mathjaxinline]x[/mathjaxinline]-axis: </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{\textbf{Z}}_{0} = \begin{pmatrix} 1\\ 0\\ \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Find an expression for the final electric field vector [mathjaxinline]\mathrm{\textbf{Z}}_{2}[/mathjaxinline]. Express your answer using <code>theta_1</code> for [mathjaxinline]\theta _1[/mathjaxinline] and <code>theta_2</code> for [mathjaxinline]\theta _2[/mathjaxinline]. </p>
<p style="display:inline">[mathjaxinline]\mathrm{\textbf{Z}}_{2}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_19_09a_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/3c6b846f79f362682b770fc1967d7a10/asset-v1:MITx+8.03x+1T2020+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_lect_19_09a_2_1" id="input_lect_19_09a_2_1" aria-describedby="status_lect_19_09a_2_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_lect_19_09a_2_1"/>
<span class="status unanswered" id="status_lect_19_09a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_19_09a_2_1" class="answer"/>
<div id="display_lect_19_09a_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_19_09a_2_1_dynamath" name="input_lect_19_09a_2_1_dynamath"/>
</div>
</div></div>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_19_09a_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Easy Axis and Polarization Matrix II - part a" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_19_09a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_19_09a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_09a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_09a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_09a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09b" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_19_09b" class="problems-wrapper" role="group"
aria-labelledby="lect_19_09b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_19_09b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09b-problem-progress" tabindex="-1">
Easy Axis and Polarization Matrix II - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_19_09b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> For the case when [mathjaxinline]\theta _{2}=\frac{\pi }{2}[/mathjaxinline], what value of [mathjaxinline]\theta _{1}[/mathjaxinline] will maximize [mathjaxinline]|\mathrm{\textbf{Z}}_{2}|[/mathjaxinline]? Express your answer in degrees. </p>
<p>
<p style="display:inline">[mathjaxinline]\theta _{1} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_19_09b_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_19_09b_2_1" id="input_lect_19_09b_2_1" aria-describedby="status_lect_19_09b_2_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_19_09b_2_1"/>
<span class="status unanswered" id="status_lect_19_09b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_19_09b_2_1" class="answer"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_19_09b_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Easy Axis and Polarization Matrix II - part b" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_19_09b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_19_09b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_09b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_09b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_19_09b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L31v3" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31v3: Single Photon Source</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v3">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v3" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L31v3: Single Photon Source</h3>
<div
id="video_L31v3"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v3/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v3/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:KnSks3ZJ35Y", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v3/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v3/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L31v3"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L31v3">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L31v3">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/f3da56a038ed73d1052f941c7c7b6acf/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L31v3.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_20_ex02" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31Q4: Behavior of Single Photons</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_02">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_02" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_20_02" class="problems-wrapper" role="group"
aria-labelledby="lect_20_02-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_02" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_02/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_20_02-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_02-problem-progress" tabindex="-1">
Behavior of Single Photons
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_02-problem-progress"></div>
<div class="problem">
<div>
<p>
In the thought experiment from the preceding video segment, unpolarized light is transmitted through two polarizing filters, one photon at a time. The filters have different angles. </p>
<p><b class="bfseries">(Part a)</b> Let's assume that a single photon cannot be described as a superposition of two different polarization states at the same time. When a single photon becomes polarized along one direction after transmission through the first polarizer, what would we expect to measure after it is transmitted through the second filter, based on this reasoning? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_20_02_2_1">
<fieldset aria-describedby="status_lect_20_02_2_1">
<div class="field">
<input type="radio" name="input_lect_20_02_2_1" id="input_lect_20_02_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_20_02_2_1-choice_1-label" for="input_lect_20_02_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_20_02_2_1">
<text> a) zero intensity</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_20_02_2_1" id="input_lect_20_02_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_20_02_2_1-choice_2-label" for="input_lect_20_02_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_20_02_2_1">
<text> b) some intensity, but less than before it is transmitted through filter 2</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_20_02_2_1" id="input_lect_20_02_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_20_02_2_1-choice_3-label" for="input_lect_20_02_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_20_02_2_1">
<text> c) the same intensity as before it is transmitted through filter 2</text>
</label>
</div>
<span id="answer_lect_20_02_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_20_02_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_20_02_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part b)</b> The observation shown in the video is not what is expected based on the reasoning in Part (a). What underlying assumption(s) made in Part (a) are incorrect? Select ALL that apply. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_20_02_3_1">
<fieldset aria-describedby="status_lect_20_02_3_1">
<div class="field">
<input type="checkbox" name="input_lect_20_02_3_1[]" id="input_lect_20_02_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_20_02_3_1-choice_0-label" for="input_lect_20_02_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_20_02_3_1">
<text>a) it is possible to emit photons one at a time</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_20_02_3_1[]" id="input_lect_20_02_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_20_02_3_1-choice_1-label" for="input_lect_20_02_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_20_02_3_1">
<text>b) a single photon cannot be described as a superposition of polarization states</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_20_02_3_1[]" id="input_lect_20_02_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_20_02_3_1-choice_2-label" for="input_lect_20_02_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_20_02_3_1">
<text>c) unpolarized light consists of many photons with different polarization states</text>
</label>
</div>
<span id="answer_lect_20_02_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_20_02_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_20_02_solution_2"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Behavior of Single Photons" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_20_02" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_20_02">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_02-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_02-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_02-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L31v4" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31v4: Creating Circularly Polarized Waves Usig Wave Plates</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v4">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v4" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L31v4: Creating Circularly Polarized Waves Usig Wave Plates</h3>
<div
id="video_L31v4"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v4/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v4/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:WIFXvKHvR-A", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v4/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v4/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L31v4"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L31v4">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L31v4">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/7c9d5f1ea27666dd45572ccf77e5b89e/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L31v4.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_20_ex04" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31Q5: Phase Shift from Wave Plate</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04a" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_20_04a" class="problems-wrapper" role="group"
aria-labelledby="lect_20_04a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_20_04a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04a-problem-progress" tabindex="-1">
Phase Shift from Wave Plate- part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04a-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider a medium with two different indices of refraction in the [mathjaxinline]\hat{x}[/mathjaxinline] and [mathjaxinline]\hat{y}[/mathjaxinline] directions, [mathjaxinline]n_{x}[/mathjaxinline] and [mathjaxinline]n_{y}[/mathjaxinline]. For simplicity, let's define the difference in the indices to be [mathjaxinline]\Delta n \equiv n_{x} - n_{y}[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Calculate the phase difference, [mathjaxinline]\Delta \phi[/mathjaxinline], accumulated by red light that travels a distance [mathjaxinline]100~ \mathrm{nm}[/mathjaxinline] in a medium where [mathjaxinline]\Delta n = 0.5[/mathjaxinline]. The light has a vacuum wavelength [mathjaxinline]\lambda _{\mathrm{vac}}=650~ \mathrm{nm}[/mathjaxinline]. Report your result as a number in units of degrees. </p>
<p>
<p style="display:inline">[mathjaxinline]\Delta \phi =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_20_04a_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_20_04a_2_1" id="input_lect_20_04a_2_1" aria-describedby="status_lect_20_04a_2_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_20_04a_2_1"/>
<span class="status unanswered" id="status_lect_20_04a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_20_04a_2_1" class="answer"/>
</div>
</div></div>
<p style="display:inline">[mathjaxinline]\mathrm{deg}[/mathjaxinline] </p>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_20_04a_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Phase Shift from Wave Plate- part a " />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_20_04a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_20_04a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_04a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_04a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_04a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04b" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_20_04b" class="problems-wrapper" role="group"
aria-labelledby="lect_20_04b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_20_04b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04b-problem-progress" tabindex="-1">
Phase Shift from Wave Plate - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_04b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> Calculate the phase difference, [mathjaxinline]\Delta \phi[/mathjaxinline], accumulated by green light that travels a distance [mathjaxinline]135~ \mathrm{nm}[/mathjaxinline] in a medium where [mathjaxinline]\Delta n = 1[/mathjaxinline]. The light has a vacuum wavelength [mathjaxinline]\lambda _{\mathrm{vac}}=540~ \mathrm{nm}[/mathjaxinline]. Report your result as a number in units of degrees. </p>
<p>
<p style="display:inline">[mathjaxinline]\Delta \phi =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_20_04b_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_20_04b_2_1" id="input_lect_20_04b_2_1" aria-describedby="status_lect_20_04b_2_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_20_04b_2_1"/>
<span class="status unanswered" id="status_lect_20_04b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_20_04b_2_1" class="answer"/>
</div>
</div></div>
<p style="display:inline">[mathjaxinline]\mathrm{deg}[/mathjaxinline] </p>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_20_04b_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Phase Shift from Wave Plate - part b" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_20_04b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_20_04b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_04b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_04b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_04b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L31v5" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31v5: Quarter Wave Plate</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v5">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v5" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L31v5: Quarter Wave Plate</h3>
<div
id="video_L31v5"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v5/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v5/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:AiFAcUwCqYw", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v5/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v5/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L31v5"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L31v5">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L31v5">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/462c20f872ab19319ad9ade5ae2da867/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L31v5.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_20_ex05" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31Q6: Polarization States</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05a" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_20_05a" class="problems-wrapper" role="group"
aria-labelledby="lect_20_05a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_20_05a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05a-problem-progress" tabindex="-1">
Polarization States - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05a-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part a)</b> Consider the wave plate that we examined previously, with [mathjaxinline]\Delta n = 0.5[/mathjaxinline]. Calculate the thickness, [mathjaxinline]l[/mathjaxinline], that is required to circularly polarize red light that is initially linearly polarized at [mathjaxinline]45^{\circ }[/mathjaxinline] (vacuum wavelength [mathjaxinline]\lambda _{\mathrm{vac}}=650~ \mathrm{nm}[/mathjaxinline]). Report your result in units of [mathjaxinline]\mathrm{nm}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]l =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_20_05a_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_20_05a_2_1" id="input_lect_20_05a_2_1" aria-describedby="status_lect_20_05a_2_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_20_05a_2_1"/>
<span class="status unanswered" id="status_lect_20_05a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_20_05a_2_1" class="answer"/>
</div>
</div></div>
<p style="display:inline">[mathjaxinline]\mathrm{nm}[/mathjaxinline] </p>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_20_05a_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Polarization States - part a" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_20_05a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_20_05a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_05a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_05a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_05a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05b" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_20_05b" class="problems-wrapper" role="group"
aria-labelledby="lect_20_05b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_20_05b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05b-problem-progress" tabindex="-1">
Polarization States - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_20_05b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> Circularly polarized light of vacuum wavelength [mathjaxinline]\lambda _{\mathrm{vac}}=460~ \mathrm{nm}[/mathjaxinline] passes through a wave plate with [mathjaxinline]\Delta n = 1[/mathjaxinline], and thickness [mathjaxinline]l=230~ \mathrm{nm}[/mathjaxinline]. What is the polarization state of the light when it is transmitted through this wave plate. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input ">
<select name="input_lect_20_05b_2_1" id="input_lect_20_05b_2_1" aria-describedby="status_lect_20_05b_2_1">
<option value="option_lect_20_05b_2_1_dummy_default">Select an option</option>
<option value="a) linear"> a) linear</option>
<option value="b) circular"> b) circular</option>
<option value="c) elliptical"> c) elliptical</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_20_05b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_lect_20_05b_2_1"/>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_20_05b_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Polarization States - part b" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_20_05b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_20_05b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_05b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_05b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_20_05b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L31v6" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">L31v6: Polarizer Demo</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v6">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L31v6" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="dd33aa2aed2311eeb4281299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L31v6: Polarizer Demo</h3>
<div
id="video_L31v6"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v6/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v6/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:Ae1xZX7bjoo", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v6/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L31v6/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L31v6"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L31v6">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L31v6">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/e7439c600718b74e0fbf980c43429634/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L31v6.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>