<description/> <author>Andrei Tokmakoff</author> <company/> <keywords/> <revisedBy>Andrei Tokmakoff</revisedBy> </userData> <identityInfo> <revision>1</revision> <documentID>D1B9BA29-FF8C-4765-9391-28CBAEA23201</documentID> <versionID>B87CFA29-400F-4290-A6DA-165F4767DEF2</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 1"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> 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<currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="455" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="203" left="24" top="18.75" width="318.75" height="14.25" align-x="24" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="12"> <inlineAttr line-through="false">Gaussian-stochastic model for the absorption lineshape</inlineAttr> </f> </text> </region> <region region-id="204" left="24" top="38.25" width="500.25" height="39" align-x="24" align-y="48" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> This worksheet plots the frequency correlation function C<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">w</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">w</inlineAttr> </f>, dephasing<f family="Arial" charset="0"> function F, and absorption lineshape</f> <f family="Arial" charset="0"> <sp/> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">s</inlineAttr> </f> <f family="Arial" charset="0">.<sp count="2"/>The</f> <f family="Arial" charset="0">parameters that determine the lineshape are the width of the frequency distribution </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">D</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> and the frequency correlation </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">time </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Symbol" charset="2" size="10">t</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">c</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Defining </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">k</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10"> = </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Symbol" charset="2" size="10">D</f> </inlineAttr> <f family="Symbol" charset="2">t</f> c <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, we will investigate the</inlineAttr> </f> fast (<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Symbol" charset="2" size="10">k</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">>1)</inlineAttr> </f>, intermediate (<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">k</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">=1)</inlineAttr> </f> and slow (<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">k</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <1) </inlineAttr> </f>modulation cases. </text> </region> <region region-id="247" left="24" top="92.25" width="94.5" height="12" align-x="24" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> First define variables. </text> </region> <region region-id="246" left="126" top="92.25" width="213.75" height="12" align-x="126" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Range variables for the time and frequency axes : </text> </region> <region region-id="245" left="348" top="87" width="39.75" height="18.75" align-x="366" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ma</ml:id> <ml:apply> <ml:pow/> <ml:real>2</ml:real> <ml:real>12</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="244" left="396" top="93" width="57" height="12.75" align-x="405" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">ma</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="243" left="468" top="87" width="61.5" height="18.75" align-x="479.25" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ii</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:real>2</ml:real> <ml:real>13</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="277" left="300" top="116.25" width="40.5" height="12" align-x="300" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> time grid: </text> </region> <region region-id="274" left="348" top="117" width="36.75" height="18" align-x="360" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">i</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="250" left="24" top="146.25" width="403.5" height="12" align-x="24" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> We will perform all calculations for a fixed value of <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">D</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> and calculate for three correlation times</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">:</inlineAttr> </f> </text> </region> <region region-id="273" left="462" top="147" width="44.25" height="12.75" align-x="472.5" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">z</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:sequence> <ml:real>2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="316" left="54" top="177" width="27.75" height="12.75" align-x="69" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Δ</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="314" left="120" top="173.25" width="24" height="72" align-x="125.25" align-y="192" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:sequence> <ml:real>0.2</ml:real> <ml:real>1</ml:real> <ml:real>10</ml:real> </ml:sequence> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="289" left="246" top="195" width="48.75" height="18.75" align-x="264" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">κ</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:id xml:space="preserve">Δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="288" left="330" top="178.5" width="49.5" height="45.75" align-x="342" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">κ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="3" cols="1"> <ml:real>0.2</ml:real> <ml:real>1</ml:real> <ml:real>10</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="10"/> </region> <region region-id="259" left="24" top="260.25" width="230.25" height="12" align-x="24" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Define the lineshape function and dephasing function: </text> </region> <region region-id="319" left="72" top="290.25" width="168" height="30" align-x="107.25" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">g</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:boundVars> </ml:function> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">Δ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:parens> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="302" left="258" top="297" width="98.25" height="15" align-x="293.25" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">F</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="305" left="24" top="332.25" width="237.75" height="12" align-x="24" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Evaluate the correlation function and lineshape function </text> </region> <region region-id="318" left="72" top="345" width="78.75" height="39" align-x="103.5" align-y="366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="fast">Cω</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="309" left="186" top="345" width="83.25" height="39" align-x="222" align-y="366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="slow">Cω</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="310" left="288" top="345" width="81" height="39" align-x="321.75" align-y="366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="mid">Cω</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="447" left="78" top="405" width="39" height="18.75" align-x="104.25" align-y="414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="fast">F</ml:id> <ml:id xml:space="preserve">ii</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="448" left="138" top="405" width="43.5" height="18.75" align-x="168.75" align-y="414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="slow">F</ml:id> <ml:id xml:space="preserve">ii</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="449" left="204" top="405" width="41.25" height="18.75" align-x="232.5" align-y="414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="mid">F</ml:id> <ml:id xml:space="preserve">ii</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="444" left="72" top="441" width="66.75" height="18.75" align-x="96" align-y="450" 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xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="slow">F</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">F</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="446" left="222" top="441" width="69" height="18.75" align-x="248.25" align-y="450" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="mid">F</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">F</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="326" left="0" top="504" width="6000" height="6" align-x="0" align-y="504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="322" left="336" top="524.25" width="19.5" height="57.75" align-x="336" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="10"> <inlineAttr line-through="false">fast</inlineAttr> </f> <f family="Arial" charset="0" size="8"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr line-through="false">mid</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr line-through="false">slow</inlineAttr> </f> </text> </region> <region region-id="323" left="156" top="549" width="31.5" height="12.75" align-x="169.5" align-y="558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Δ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="324" left="204" top="532.5" width="51" height="45.75" align-x="217.5" align-y="558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="3" cols="1"> <ml:real>0.2</ml:real> <ml:real>1</ml:real> <ml:real>10</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="325" left="270" top="532.5" width="49.5" height="45.75" align-x="282" align-y="558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">κ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="3" cols="1"> <ml:real>0.2</ml:real> <ml:real>1</ml:real> <ml:real>10</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="171" left="36" top="588" width="375.75" height="177.75" align-x="36" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="25"/> <rendering item-idref="26"/> </region> <region region-id="329" left="42" top="768" width="368.25" height="177.75" align-x="42" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="27"/> <rendering item-idref="28"/> </region> <region region-id="401" left="0" top="954" width="6000" height="6" align-x="0" align-y="954" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="337" left="18" top="968.25" width="447.75" height="12" align-x="18" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Now Fourier transform the dipole correlation function<f family="Arial" charset="0"> C</f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">m</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">m</inlineAttr> </f> and use the real part to obtain the lineshape.<sp count="2"/> </text> </region> <region region-id="427" left="48" top="1005" width="73.5" height="18.75" align-x="78.75" align-y="1014" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="fast">Cμ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">F</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="428" left="156" top="1005" width="78" height="18.75" align-x="191.25" align-y="1014" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="slow">Cμ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">F</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="429" left="276" top="1005" width="75.75" height="18.75" align-x="309" align-y="1014" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="mid">Cμ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">F</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="430" left="48" top="1029" width="72.75" height="15" align-x="69.75" align-y="1038" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="fast">S</ml:id> <ml:apply> <ml:id xml:space="preserve">cfft</ml:id> <ml:id xml:space="preserve" subscript="fast">Cμ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="431" left="156" top="1029" width="81.75" height="15" align-x="182.25" align-y="1038" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define 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z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Some manipulations to wrap and normalize the Fourier transform: </text> </region> <region region-id="423" left="90" top="1086" width="97.5" height="32.25" align-x="112.5" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="fast">σ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">stack</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="fast">S</ml:id> <ml:id xml:space="preserve" subscript="fast">S</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">max</ml:id> <ml:apply> <ml:id xml:space="preserve">Re</ml:id> <ml:id xml:space="preserve" subscript="fast">S</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="424" left="210" top="1086" width="111" height="32.25" align-x="237" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="slow">σ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">stack</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="slow">S</ml:id> <ml:id xml:space="preserve" subscript="slow">S</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">max</ml:id> <ml:apply> <ml:id xml:space="preserve">Re</ml:id> <ml:id xml:space="preserve" subscript="slow">S</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="425" left="348" top="1086" width="104.25" height="32.25" align-x="372.75" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="mid">σ</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">stack</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="mid">S</ml:id> <ml:id xml:space="preserve" subscript="mid">S</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">max</ml:id> <ml:apply> <ml:id xml:space="preserve">Re</ml:id> <ml:id xml:space="preserve" subscript="mid">S</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="450" left="24" top="1148.25" width="233.25" height="12" align-x="24" align-y="1158" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Determine the frequency axis for the Fourier transform </text> </region> <region region-id="451" left="90" top="1172.25" width="61.5" height="33" align-x="113.25" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">freq</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="452" left="168" top="1179" width="55.5" height="18" align-x="190.5" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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</calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="455" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="203" left="24" top="18.75" width="318.75" height="14.25" align-x="24" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="12"> <inlineAttr line-through="false">Gaussian-stochastic model for the absorption lineshape</inlineAttr> </f> </text> </region> <region region-id="204" left="24" top="38.25" width="500.25" height="39" align-x="24" align-y="48" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> This worksheet plots the frequency correlation function C<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">w</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">w</inlineAttr> </f>, dephasing<f family="Arial" charset="0"> function F, and absorption lineshape</f> <f family="Arial" charset="0"> <sp/> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">s</inlineAttr> </f> <f family="Arial" charset="0">.<sp count="2"/>The</f> <f family="Arial" charset="0">parameters that determine the lineshape are the width of the frequency distribution </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">D</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> and the frequency correlation </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">time </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Symbol" charset="2" size="10">t</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">c</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Defining </inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">k</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="10"> = </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Symbol" charset="2" size="10">D</f> </inlineAttr> <f family="Symbol" charset="2">t</f> c <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, we will investigate the</inlineAttr> </f> fast (<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Symbol" charset="2" size="10">k</f> </inlineAttr> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">>1)</inlineAttr> </f>, intermediate (<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">k</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">=1)</inlineAttr> </f> and slow (<f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">k</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <1) </inlineAttr> </f>modulation cases. </text> </region> <region region-id="247" left="24" top="92.25" width="94.5" height="12" align-x="24" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> First define variables. </text> </region> <region region-id="246" left="126" top="92.25" width="213.75" height="12" align-x="126" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Range variables for the time and frequency axes : </text> </region> <region region-id="245" left="348" top="87" width="39.75" height="18.75" align-x="366" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ma</ml:id> <ml:apply> <ml:pow/> <ml:real>2</ml:real> <ml:real>12</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="244" left="396" top="93" width="57" height="12.75" align-x="405" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">ma</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="243" left="468" top="87" width="61.5" height="18.75" align-x="479.25" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ii</ml:id> <ml:range> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:real>2</ml:real> <ml:real>13</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="277" left="300" top="116.25" width="40.5" height="12" align-x="300" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> time grid: </text> </region> <region region-id="274" left="348" top="117" width="36.75" height="18" align-x="360" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">i</ml:id> <ml:real>0.1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="250" left="24" top="146.25" width="403.5" height="12" align-x="24" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> We will perform all calculations for a fixed value of <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">D</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> and calculate for three correlation times</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">:</inlineAttr> </f> </text> </region> <region region-id="273" left="462" top="147" width="44.25" height="12.75" align-x="472.5" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">z</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:sequence> <ml:real>2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="316" left="54" top="177" width="27.75" height="12.75" align-x="69" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Δ</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="314" left="120" top="173.25" width="24" height="72" align-x="125.25" align-y="192" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:sequence> <ml:real>0.2</ml:real> <ml:real>1</ml:real> <ml:real>10</ml:real> </ml:sequence> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="289" left="246" top="195" width="48.75" height="18.75" align-x="264" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">κ</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:id xml:space="preserve">Δ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="288" left="330" top="178.5" width="49.5" height="45.75" align-x="342" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">κ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="3" cols="1"> <ml:real>0.2</ml:real> <ml:real>1</ml:real> <ml:real>10</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="10"/> </region> <region region-id="259" left="24" top="260.25" width="230.25" height="12" align-x="24" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Define the lineshape function and dephasing function: </text> </region> <region region-id="319" left="72" top="290.25" width="168" height="30" align-x="107.25" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">g</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:boundVars> </ml:function> <ml:parens> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">Δ</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> </ml:parens> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="302" left="258" top="297" width="98.25" height="15" align-x="293.25" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">F</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve" subscript="c">τ</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="305" left="24" top="332.25" width="237.75" height="12" align-x="24" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Evaluate the correlation function and lineshape function </text> </region> <region region-id="318" left="72" top="345" width="78.75" height="39" align-x="103.5" align-y="366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="fast">Cω</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="309" left="186" top="345" width="83.25" height="39" align-x="222" align-y="366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="slow">Cω</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="310" left="288" top="345" width="81" height="39" align-x="321.75" align-y="366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="mid">Cω</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">exp</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="447" left="78" top="405" width="39" height="18.75" align-x="104.25" align-y="414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="fast">F</ml:id> <ml:id xml:space="preserve">ii</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="448" left="138" top="405" width="43.5" height="18.75" align-x="168.75" align-y="414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="slow">F</ml:id> <ml:id xml:space="preserve">ii</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="449" left="204" top="405" width="41.25" height="18.75" align-x="232.5" align-y="414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="mid">F</ml:id> <ml:id xml:space="preserve">ii</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="444" left="72" top="441" width="66.75" height="18.75" align-x="96" align-y="450" 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xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="slow">F</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">F</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="446" left="222" top="441" width="69" height="18.75" align-x="248.25" align-y="450" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="mid">F</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">F</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="326" left="0" top="504" width="6000" height="6" align-x="0" align-y="504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="322" left="336" top="524.25" width="19.5" height="57.75" align-x="336" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="10"> <inlineAttr line-through="false">fast</inlineAttr> </f> <f family="Arial" charset="0" size="8"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr line-through="false">mid</inlineAttr> </f> <f family="Arial" charset="0" size="10"> <inlineAttr line-through="false">slow</inlineAttr> </f> </text> </region> <region region-id="323" left="156" top="549" width="31.5" height="12.75" align-x="169.5" align-y="558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Δ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="324" left="204" top="532.5" width="51" height="45.75" align-x="217.5" align-y="558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="3" cols="1"> <ml:real>0.2</ml:real> <ml:real>1</ml:real> <ml:real>10</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="325" left="270" top="532.5" width="49.5" height="45.75" align-x="282" align-y="558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">κ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="3" cols="1"> <ml:real>0.2</ml:real> <ml:real>1</ml:real> <ml:real>10</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="171" left="36" top="588" width="375.75" height="177.75" align-x="36" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="25"/> <rendering item-idref="26"/> </region> <region region-id="329" left="42" top="768" width="368.25" height="177.75" align-x="42" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="27"/> <rendering item-idref="28"/> </region> <region region-id="401" left="0" top="954" width="6000" height="6" align-x="0" align-y="954" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="337" left="18" top="968.25" width="447.75" height="12" align-x="18" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Now Fourier transform the dipole correlation function<f family="Arial" charset="0"> C</f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">m</inlineAttr> </f> <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">m</inlineAttr> </f> and use the real part to obtain the lineshape.<sp count="2"/> </text> </region> <region region-id="427" left="48" top="1005" width="73.5" height="18.75" align-x="78.75" align-y="1014" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="fast">Cμ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">F</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="c">τ</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="428" left="156" top="1005" width="78" height="18.75" align-x="191.25" align-y="1014" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="slow">Cμ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">F</ml:id> 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