(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 5836814, 96881] NotebookOptionsPosition[ 5823487, 96638] NotebookOutlinePosition[ 5823986, 96656] CellTagsIndexPosition[ 5823943, 96653] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["3.029 Spring 2022\[LineSeparator]Lecture 24 - 05/02/2022", "Subtitle", CellChangeTimes->{{3.8525512993398438`*^9, 3.8525513206118402`*^9}, { 3.852652054138073*^9, 3.8526520591301193`*^9}, {3.853194369726288*^9, 3.8531943739664793`*^9}, 3.8531971130005827`*^9, {3.853361889945813*^9, 3.853361893353859*^9}, {3.854455749568426*^9, 3.8544557604776297`*^9}, { 3.8545621484661007`*^9, 3.8545621501697083`*^9}, {3.85499650158072*^9, 3.854996504060568*^9}, {3.855176893175762*^9, 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relate to diffraction and imaging?", "Item", CellChangeTimes->{{3.8604327787831783`*^9, 3.860432796611432*^9}},ExpressionUUID->"22007a46-32be-4c18-94dc-\ d571e4c8e318"], Cell["Or more importantly to a Fourier Transform?", "Subitem", CellChangeTimes->{{3.8604327787831783`*^9, 3.860432809132885*^9}},ExpressionUUID->"3e5a0355-689e-4bde-b798-\ dedb8221da62"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "First, note that rays from a ", StyleBox["parallel ", FontSlant->"Italic"], "beam, all get converged (hens the lens\[CloseCurlyQuote] name) to a single \ point on the ", StyleBox["focal", FontSlant->"Italic"], " plane" }], "Item", CellChangeTimes->{{3.860432810694888*^9, 3.860432945455059*^9}, { 3.86043297877514*^9, 3.8604329804376297`*^9}},ExpressionUUID->"c36516bb-60db-4a84-a903-\ a3ddc998bf4f"], Cell["\<\ if the rays forming the plane wave hit the lens \[OpenCurlyDoubleQuote]face \ on\[CloseCurlyDoubleQuote], the focused point lies on the optical axis \ (horizontal x-axis above)\ \>", "Subitem", CellChangeTimes->{{3.860432810694888*^9, 3.860432945455059*^9}, { 3.86043297877514*^9, 3.860433080824541*^9}},ExpressionUUID->"92d9f1e0-ae36-4b70-8348-\ b1a50767ee5d"], Cell["\<\ if the rays arrive at an angle however, they will get focused on a point \ displaced from the optical axis, but still at the same focal plane\ \>", "Subitem", CellChangeTimes->{{3.860432810694888*^9, 3.860432945455059*^9}, { 3.86043297877514*^9, 3.8604331363652554`*^9}},ExpressionUUID->"40d953b2-fa3f-4e16-b7a7-\ 064b060de0c9"], Cell[CellGroupData[{ Cell["\<\ As such, the lens \[OpenCurlyDoubleQuote]transforms\[CloseCurlyDoubleQuote] \ the plane wave\[CloseCurlyQuote]s incident angles to displacement positions \ on the focal plane\ \>", "Subitem", CellChangeTimes->{{3.860432810694888*^9, 3.860432945455059*^9}, { 3.86043297877514*^9, 3.860433199994115*^9}, 3.8604333253225594`*^9},ExpressionUUID->"7ea6611c-0c14-4730-bc60-\ a27c320489a4"], Cell[BoxData[ RowBox[{"Import", "[", RowBox[{ "\"\\"", ",", "\"\\"", ",", RowBox[{"Paneled", "->", "False"}], ",", RowBox[{"AnimationRunning", "->", "False"}]}], "]"}]], "Input", CellChangeTimes->{{3.860433564990685*^9, 3.860433600109212*^9}}, CellLabel->"In[25]:=",ExpressionUUID->"263548ee-1bb3-41aa-a45c-e402754aef37"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Second, recall that a ", StyleBox["monochromatic", FontSlant->"Italic"], " light beam can be represented as a sum of many plane waves traveling at \ different angles" }], "Item", CellChangeTimes->{{3.860432810694888*^9, 3.860432969183193*^9}, { 3.860433231653421*^9, 3.8604332729447317`*^9}},ExpressionUUID->"f18e6ec1-8ae5-4316-a2e5-\ 75d68b655430"], Cell[CellGroupData[{ Cell["\<\ A complicated wave made up of many plane wave components hitting the lense \ will thus by focused on various different displacement spots on the focal \ plane\ \>", "Subitem", CellChangeTimes->{{3.860432810694888*^9, 3.860432969183193*^9}, { 3.860433231653421*^9, 3.860433318011591*^9}},ExpressionUUID->"458a6553-2154-4a4a-b3c1-\ c3e89957692e"], Cell["\<\ These are the light wave\[CloseCurlyQuote]s \[OpenCurlyDoubleQuote]spatial \ frequencies\[CloseCurlyDoubleQuote] and are effectively the components of the \ 2-D Fourier transform \ \>", "Subsubitem", CellChangeTimes->{{3.860432810694888*^9, 3.860432969183193*^9}, { 3.860433231653421*^9, 3.8604333637787113`*^9}},ExpressionUUID->"75dc4525-d48a-46fd-a41f-\ 1869cfe5511c"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Aperture Diffraction", "Subsection", CellChangeTimes->{{3.860433666950091*^9, 3.860433671781927*^9}},ExpressionUUID->"bf2e558e-97d3-4da3-9188-\ d167fb0b699b"], Cell[CellGroupData[{ Cell[TextData[{ "More generally, when a beam\[CloseCurlyQuote]s (not necessarily made of \ light) path is blocked by an object, the wave gets ", StyleBox["scattered ", FontSlant->"Italic"], "around the object" }], "Item", CellChangeTimes->{{3.8604336733129387`*^9, 3.860433724646599*^9}, { 3.8604337669274273`*^9, 3.860433773457664*^9}},ExpressionUUID->"d971d0eb-9133-40d2-b7ae-\ 8daa8e7a011c"], Cell["\<\ these scattered wave-fronts interfere constructively/destructively to form \ bright/dark bands respectively\ \>", "Subitem", CellChangeTimes->{{3.8604336733129387`*^9, 3.860433724646599*^9}, { 3.8604337669274273`*^9, 3.860433872137992*^9}},ExpressionUUID->"00487e4f-c8a2-42f3-ba28-\ f01e471f17ab"], Cell["this is known as a diffraction pattern", "Subitem", CellChangeTimes->{{3.8604336733129387`*^9, 3.860433724646599*^9}, { 3.8604337669274273`*^9, 3.8604338850635033`*^9}},ExpressionUUID->"c8e31e6d-461e-46b7-9223-\ df959b92e008"] }, Open ]], Cell[TextData[{ "Since we\[CloseCurlyQuote]ve seen diffraction patterns can be modeled using \ Fourier Transforms, let\[CloseCurlyQuote]s investigate diffraction off some \ common objects, or ", StyleBox["apertures", FontSlant->"Italic"] }], "Item", CellChangeTimes->{{3.8604338875068502`*^9, 3.860433896976344*^9}, { 3.860433936640503*^9, 3.8604340082495337`*^9}},ExpressionUUID->"00c7fa54-fad7-4e56-a350-\ bbfe7dfe5e28"], Cell[CellGroupData[{ Cell["\<\ To do this, we\[CloseCurlyQuote]ll use the built-in Discrete Fourier \ Transform (DFT) in Mathematica, Fourier\ \>", "Item", CellChangeTimes->{{3.8604340315091887`*^9, 3.8604340768187523`*^9}},ExpressionUUID->"3b4612bc-7477-43cf-86f8-\ 677f52c715f8"], Cell[CellGroupData[{ Cell["\<\ Fun fact: this formula is also important in science and engineering, a \ particularly-clever implementation for this algorithm is called the Fast \ Fourier Transform\ \>", "Subitem", CellChangeTimes->{{3.8604340315091887`*^9, 3.860434128523447*^9}},ExpressionUUID->"1ccb1118-2774-442e-8f28-\ 30984d23a4b5"], Cell["\<\ with one popular library to compute it called FFTW (Fastest Fourier Transform \ in the West), and was developed by Prof. Steven G. Johnson at MIT\ \>", "Subsubitem", CellChangeTimes->{{3.8604340315091887`*^9, 3.8604342042994947`*^9}},ExpressionUUID->"d5a78d26-8b76-40a5-abfc-\ 44547ffd511b"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Fourier however places the largest frequencies at the beginning of a 1D \ diffraction pattern, or equivalently at the corners of a 2D diffraction \ pattern\ \>", "Subitem", CellChangeTimes->{{3.860434255635284*^9, 3.860434258779916*^9}, { 3.860434471111899*^9, 3.860434529287546*^9}},ExpressionUUID->"a167b7dd-a603-4fbe-905a-\ 9a91b824b685"], Cell[CellGroupData[{ Cell["\<\ as such we need to center the diffraction pattern by shifting it\ \>", "Subsubitem", CellChangeTimes->{{3.860434255635284*^9, 3.860434258779916*^9}, { 3.860434471111899*^9, 3.860434560439794*^9}},ExpressionUUID->"219833da-6d25-4f61-b9b1-\ bbeda4618bd4"], Cell[BoxData[{ RowBox[{ RowBox[{"fftshift", "[", RowBox[{"data_", "?", "ArrayQ"}], "]"}], ":=", RowBox[{"RotateRight", "[", RowBox[{"data", ",", RowBox[{"Quotient", 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Cell[TextData[{ "A parallel electron beam scatters off a sample, and can be focused and \ magnified using ", StyleBox["electro-magnetic", FontSlant->"Italic"], " lenses " }], "Subitem", CellChangeTimes->{{3.860435671128119*^9, 3.860435801485835*^9}},ExpressionUUID->"d7a1d991-6a92-42b8-bb6e-\ 37b6d3eb6707"], Cell["\<\ However, since the wavelength of electrons is much smaller than that of \ visible light, the resolution limit is much better than with an optical \ microscope\ \>", "Subitem", CellChangeTimes->{{3.860435671128119*^9, 3.860435827012848*^9}},ExpressionUUID->"363711de-b211-4682-a71b-\ 33894901d6b0"] }, Open ]], Cell["\<\ Let\[CloseCurlyQuote]s consider a simple \[OpenCurlyDoubleQuote]Gaussian\ \[CloseCurlyDoubleQuote] magnetic lens, i.e. a magnetic field with \ distribution\ \>", "Item", CellChangeTimes->{{3.8604360283935328`*^9, 3.860436044799436*^9}, { 3.8604360779749613`*^9, 3.860436082479765*^9}},ExpressionUUID->"8c4d76d6-4b21-4ff3-9b3d-\ 79009bc6c408"], Cell[BoxData[ RowBox[{"\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox[ StyleBox["B", "TI"], StyleBox["z", "TI"]], RowBox[{"(", StyleBox["z", "TI"], ")"}], "\[LongEqual]", SubscriptBox[ StyleBox["B", "TI"], "0"], "exp", RowBox[{"(", FractionBox[ RowBox[{"-", SuperscriptBox[ RowBox[{"(", RowBox[{ StyleBox["z", "TI"], "-", SubscriptBox[ StyleBox["z", "TI"], "0"]}], ")"}], "2"]}], SuperscriptBox["\[Sigma]", "2"]], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "B_z(z) = B_0 \ \\exp\\left(\\frac{-\\left(z-z_0\\right)^2}{\\sigma^2}\\right)", "state" -> "Boxes"|>, "TeXAssistantTemplate"]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8604361363901243`*^9, 3.8604361387027903`*^9}},ExpressionUUID->"190e2958-fd2c-4e15-8c05-\ 2e4cfb0ae05e"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ SubscriptBox["z", "0"], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "8f2a5473-06eb-405d-89a3-d96cc9b22a1d"], " is the center of the distribution and \[Sigma] is the distribution\ \[CloseCurlyQuote]s Full-Width-Half-Maximum (FWHM) divided by 2.355" }], "Subitem", CellChangeTimes->{{3.860439963703656*^9, 3.86043996491827*^9}, { 3.860440007044835*^9, 3.8604400748478937`*^9}},ExpressionUUID->"0c535d75-d1a5-4d03-9d91-\ e16d0ed33f80"], Cell["\<\ An electron beam passing through this magnetic field will get deflected off \ its optical axis according to the path-equation\ \>", "Item", CellChangeTimes->{{3.860436166579097*^9, 3.860436213569297*^9}},ExpressionUUID->"00189c47-08a8-48a0-bad3-\ c18e461804fc"], Cell[BoxData[ RowBox[{"\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox[ StyleBox["d", "TI"], "2"], StyleBox["r", "TI"]}], RowBox[{ StyleBox["d", "TI"], SuperscriptBox[ StyleBox["z", "TI"], "2"]}]], "\[LongEqual]", "\[LongEqual]", "-", FractionBox[ StyleBox["e", "TI"], RowBox[{"8", SubscriptBox[ StyleBox["m", "TI"], "0"], StyleBox["U", "TI"]}]], SubsuperscriptBox[ StyleBox["B", "TI"], StyleBox["z", "TI"], "2"], StyleBox["r", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "\\frac{d^2 r}{dz^2}==-\\frac{e}{8 m_0 U}B^2_z r", "state" -> "Boxes"|>, "TeXAssistantTemplate"]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8604361363901243`*^9, 3.8604361387027903`*^9}, 3.860436265404317*^9},ExpressionUUID->"d811b609-8ae2-43fa-9a77-\ 802c7d367316"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["U", "TI"], TraditionalForm], "errors" -> {}, "input" -> "U", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "c5443349-e263-4e5c-aa75-2cb6fec22c8f"], " is the accelerating voltage, ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["e", "TI"], TraditionalForm], "errors" -> {}, "input" -> "e", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "ce112a95-35e7-45c2-9ded-7e976a91cefd"], " is the electron charge, and ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ SubscriptBox[ StyleBox["m", "TI"], "0"], TraditionalForm], "errors" -> {}, "input" -> "m_0", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "67d09b79-dfb1-4cbb-9f87-19818fbbdabc"], " is the electron\[CloseCurlyQuote]s rest mass" }], "Subitem", CellChangeTimes->{{3.8604400765225677`*^9, 3.860440119631857*^9}, { 3.8604401817852163`*^9, 3.860440215048793*^9}},ExpressionUUID->"a1094fe8-315c-4f30-a656-\ 4a9e5b02ac10"], Cell[CellGroupData[{ Cell["This is a simple ODE we can solve numerically", "Item", CellChangeTimes->{{3.8604402465729113`*^9, 3.860440254137434*^9}, { 3.860440400842874*^9, 3.860440401258881*^9}, {3.860440554924182*^9, 3.860440554924944*^9}},ExpressionUUID->"d97f051c-ce9f-48e8-b71d-\ 37ff3cb2b8fe"], Cell[TextData[{ "We\[CloseCurlyQuote]ll consider a magnetic field of strength ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["B", "0"], "=", RowBox[{"1.8", "T"}]}], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> 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Pattern[$CellContext`rayPosition, Blank[]], Pattern[$CellContext`rayVelocity, Blank[]]] := \ ($CellContext`multiLensPathEquation[$CellContext`acceleratingVoltage, \ $CellContext`lensesDetails, $CellContext`rayPosition, \ $CellContext`rayVelocity] = With[{$CellContext`prefactor = Apply[Plus, Apply[ $CellContext`lensPrefactor[$CellContext`acceleratingVoltage, \ $CellContext`z], $CellContext`lensesDetails, {1}]]}, NDSolveValue[{ Derivative[ 2][$CellContext`r][$CellContext`z] + $CellContext`prefactor \ $CellContext`r[$CellContext`z] == 0, $CellContext`r[-0.1] == $CellContext`rayPosition, Derivative[ 1][$CellContext`r][-0.1] == $CellContext`rayVelocity}, \ $CellContext`r, {$CellContext`z, -0.1, 0.5}]]), $CellContext`lensPrefactor[ Pattern[$CellContext`acceleratingVoltage, Blank[]], Pattern[$CellContext`z, Blank[]]][ Pattern[$CellContext`lensStrength, Blank[]], Pattern[$CellContext`lensFWHM, Blank[]], Pattern[$CellContext`position, Blank[]]] := With[{$CellContext`e = QuantityMagnitude["ElementaryCharge", "SIBase"], $CellContext`m0 = QuantityMagnitude["ElectronMass", "SIBase"]}, ($CellContext`e/( 8 $CellContext`m0 $CellContext`acceleratingVoltage)) \ ($CellContext`lensStrength Exp[-(($CellContext`z - \ $CellContext`position)^2/($CellContext`lensFWHM/ 2.355)^2)])^2], $CellContext`relativisticElectrons[ Pattern[$CellContext`acceleratingVoltage, Blank[]]] = (1 + 1.9569511835738732479114`9.211300968617545*^-6 \ $CellContext`acceleratingVoltage) $CellContext`acceleratingVoltage, \ $CellContext`axisFlip = ReplaceAll[#, {Alternatives[ Pattern[$CellContext`x, Blank[Line]], Pattern[$CellContext`x, Blank[GraphicsComplex]]] :> MapAt[Reverse[#, 2]& , $CellContext`x, 1], Pattern[$CellContext`x, PlotRange -> Blank[]] :> Reverse[$CellContext`x, 2]}]& }; Typeset`initDone$$ = True), SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], 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