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1、Advanced Imaging System Design Using Zemax | OpticStudioLaser Beam Propagation3 - 2 Goals for the DayTo introduce participants to the Physical Optics Propagation capabilities of Zemax. A spatial filter and a fiber coupler will be among the examples designed in this course.Topics to be covered:How ra

2、ys and wavefront propagation differDefining the input beamPropagating the beam in an optical systemQuantitative beam analysisInterfacing with BPM and FDTD programs3 - 3 Laser Beam PropagationLaser beam propagation is an entire field of study in itselfRay-tracing gives approximate answers when lookin

3、g at laser beamsZemax has tools to look at laser beam propagationGaussian beamsSkew Gaussian beamsFull physical optics propagation (POP)Before we look at POP, we will examine Gaussian beamsGaussian Beams are useful for “well-behaved”, TEM00 beams, no aberrations from optics, no aperturesPhysical Opt

4、ics Propagation needed when beams not TEM00, significant aberrations, aperturesAdvanced Imaging System Design Using Zemax | OpticStudioGaussian Beams3 - 5 Gaussian BeamsGaussian beams are an alternative to ray-tracing!Purely paraxial, no aberrationsGaussian beams are defined by any two of these thre

5、e parametersWavelengthBeam waistDivergenceOften used to study TEM00 laser beam propagationSometimes M2 factor used to study beams that are not TEM00, but this is not useful as a defining parameterAlthough the paraxial Gaussian beam calculation does support itPOP is far superior when beams are not si

6、mple GaussianAnd also when they are!3 - 6 TheoryTheory covered in detail in numerous places:Joseph Goodman, Introduction to Fourier OpticsGeorge Lawrence, “Optical Modeling” in Applied Optics and Optical EngineeringFollowing pages follow Lawrence3 - 7 Gaussian BeamsdivergencewaistzR is Rayleigh rang

7、e2*waist3 - 8 Properties of Gaussian BeamsWell-behaved, stable lasers generate beams that are very nearly Gaussian. A Gaussian beam has several interesting properties:It is analytically propagatable,It is its own Fourier transform,The function is ideally smoothCan be differentiated to any order with

8、out discontinuity,The tightest focus position is not at paraxial focusTo the accuracy of scalar diffraction theory, the complex amplitude of a Gaussian beam propagates according to3 - 9 Properties of Gaussian BeamsA Gaussian Beam is characterized by a well-defined minimum beam size at the waist, 0,

9、obtained when z=0The definition of a Gaussian spherical wave adds the quadratic phase factorR is the radius of the phasefront. At the waist R=infinity and the phase factor disappears to give the top equation3 - 10 More Properties.The phase radius R(z) has the formwhere zR is the Rayleigh distance gi

10、ven by The phase radius R(z) varies fromInfinity at z=0A minimum at z=zRInfinity again when z goes to infinity3 - 11 Beam SizeThe beam size at any z is given byso the beam size has increased by 2 at the Rayleigh distanceAt large z the beam size increases linearlyFinally there is a piston phase term,

11、 (z) called the Gouy shift:which varies from -/2 to +/2 as z goes from - to + 3 - 12 Rays and Gaussian BeamsRays and Gaussian beams have some fundamental differencesThe most basic difference is that rays always travel in a straight lineA collimated beam never changes sizeA diverging beam retains the

12、 same divergence everywhereGaussian beams diffract as they propagate, so the beam changes size and effective divergence as the beam propagatesWithin the Rayleigh range, the beam size changes very slowlyA long way outside the Rayleigh range, the beam size changes linearly with propagation distance3 -

13、 13 Rays and Gaussian BeamsFor propagation from a plane inside the Rayleigh range to another plane inside the Rayleigh range, a collimated ray bundle is only an approximate description of how the laser propagatesFor propagation from inside the Rayleigh range to a plane well outside the Rayleigh rang

14、e, a point source emitting rays with the same divergence angle is a good description of how the laser propagatesFor propagation from outside the Rayleigh range to near the beam waist, the rays will focus to a point instead of forming a waistFor propagation from a plane outside the Rayleigh range, to

15、 another plane outside the Rayleigh range, again a divergence angle yields rays with the approximate behavior of the laser beam3 - 14 Effect of LensesWhen a Gaussian beam of size (z) and phase radius R(z) hits a thin lens of power :The beam size after the lens is unalteredThe phase radius is transfo

16、rmed byThe transformed phase radius is still the same shape, so no aberrations have been added3 - 15 Rays and Gaussian BeamsAnother fundamental difference between rays and Gaussian beams is that a Gaussian beam waist is always imaged to another Gaussian beam waist, but rays can be imaged to a pointO

17、f course, diffraction means that an optical system can never produce a point image, but thats a different process. Gaussian beams can NOT be imaged to points, no matter what (imaginary) optical system we use. Rays, at least in principle, can be imaged to points.Advanced Imaging System Design Using Z

18、emax | OpticStudioAn Example with Gaussian Beams3 - 17 ExampleA He-Ne laser beam has the following specification, measured at the output port of the lasers body:Wavelength 0.633 micronsBeam diameter 2.5 mmBeam divergence 0.175 mradThis specification is taken from a CVI Melles Griot He-Ne laser10 mm

19、after the laser is a 5 mm thick N-BK7 lens. The image plane is 50 mm beyond the lensProblem:Design the lens that gives the smallest Gaussian spot size at the image plane3 - 18 Note!The output port of the laser is not the beam waistThe beam waist is usually some large distance (meters) behind the out

20、put faceSome resonator designs do place the waist at or near the output port, but they are not in general the sameIn the Gaussian beam calculation, the starting beam waist is always positioned relative to surface 1It does not have to be on surface 1, just positioned relative to it3 - 19 Step 1: Defi

21、ne the LaserGaussian beams are not ray-traced!Set EPD=6 mm just so we can see the lens, set wavelength = 0.633Beam Waist is located relative to surface 1In this example we will set surface 1 to be at the beam waistSurface 2 defines output face of laser beamSet surface 1 thickness to 100 mm initially

22、Waist size is computed from wavelength and divergence angle or can be computed interactively3 - 20 Interactive CalculatorThe beam waist is 1.15 mm. Check with the calculator (AnalysisPhysical OpticsParaxial Gaussian Beam) that this gives the correct divergence at surface 2:3 - 21 Waist LocationSurfa

23、ce 1 is the beam waist, and its radius is 1.15 (sets divergence of 0.175 mrad)Surface 2 is the laser beam output, and has radius 1.25What is the thickness of surface 1?Easy way:Make thickness of surface 1 variableOptimize using GBPS to make Gaussian beam waist on surface 2 =1.25Separation is about 2

24、.8 meters!Surface 2 is now effectively the output face of the laser!N.B. Can make surface 1 the output face if we set “S1toW” = -2800 mm in GBPS operand3 - 22 Set UpEPD = 6 mm, wavelength 1 = 0.633 microns3 - 23 Optimize!Merit function is effectively zero:Now Add the LensRemove the variable from the

25、 thickness of surface 1!Add a 5 mm thick N-BK7 lens 10 mm from surface 2, 50 mm thickness to the image plane, make radii variable3 - 24 3 - 25 Merit FunctionChange MF to GBPS on surface 5, target 0Optimize!Gaussian beam waist is around 8.3 microns3 - 26 What Else Can We Analyze?Ummthats it!Gaussian

26、beam calculations limited to Gaussian parameters like beam waist, phase radius, Rayleigh range, etc. in spherical or cylindrical systemsCannot discuss MTF, PSF, etc. of a Gaussian beam (PSF of a Gaussian is just a Gaussian)Only real use is incorporating beam divergencePOP allows more detailed analys

27、isSave file as Gaussian.zmx for later useNote focal plane is not at paraxial focus (49.854 mm), best spot focus (49.663) or best wavefront (49.698)3 - 27 Diversion!There is an interesting side-note discussion we can have about optimization hereFirst, note that each person in the room may have differ

28、ent values of the radii of the two surfaces, but the same beam waist.Why? Because we have two variables, and only one targetThere is no aberration considered in the paraxial Gaussian beam calculation, so for any R1 there is an R2 that yields a minimum beam size, and there are multiple R1/R2 combinat

29、ions that yield the smallest beam size, because beam size is very slowly varying around the waist3 - 28 Beam Size is Slow Near the OptimumOn the Analyze Tab, go tothe Universal Plot 1D button and select New Universal Plot 1D:3 - 29 Beam SizeBeam size varies slowly inside the Rayleigh range, and VERY

30、 slowly near the waistIt does not go to zero: it minimizes around 8.3 mIs there something better to optimize on? Look at the phase radius of curvature. It should be infinite at the exact beam waist:3 - 30 Phase Radius of CurvatureZemax has the GBPR operand to compute the phase radius of curvatureBut

31、, never optimize a value to infinity: instead optimize its reciprocal to zeroUse a zero-weight on the GBPR, and then use the RECI operand: 3 - 31 Again, Universal PlotA well-defined minimum results:3 - 32 Re-optimize:Beam size does not change significantly, but the phase radius does3 - 33 Aberration

32、sJust as a taster, lets set up the POP calculation: Analysis Tab Physical Optics3 - 34 POP Settings3 - 35 POP Settings3 - 36 Results!3 - 37 ConsequenceThe paraxial Gaussian beam calculation does not consider any aberration of the beam, and POP does. Well consider POP in more detail soon.However, the

33、re is an important point here. Phase is generally a lot more sensitive than beam size. When we get into POP, we will spend a lot of time looking at the phase of beams.Save file as Gaussian.zmx for later use.3 - 38 When to Use Gaussian Beams?TEM00 laser beams onlySmall divergence conesM2 factor is no

34、t useful as a beam definitionMore soonSimple systems with no aberrationsBeam size, phase radius of curvature are only diagnostics neededSkew-Gaussian is more sophisticatedPhysical Optics Propagation gives comprehensive treatmentAdvanced Imaging System Design Using Zemax | OpticStudioSkew Gaussian Be

35、ams3 - 40 Skew Gaussian BeamsSkew Gaussian Beams may enter an optical system at any surface from any field position, and may travel through the optical system off axis The Skew Gaussian Beam parameters are computed using real rays and account for astigmatism but not higher order aberrationsExample:T

36、he laser beam enters our lens at 5 degreesWhat is the beam waist at the image plane now, and is it still circular?3 - 41 Skew GaussianMake surface 2, the laser output port, the stop surfaceAdd a field point, +5 degrees in yN.B. Layout shows surfaces 2-53 - 42 Skew GaussianOn the Analyze Tab, select

37、Gaussian Beams Skew Gaussian BeamCan specify waist in x, yCheck results for on-axis:3 - 43 Off-axisOn-axis results agree well with Paraxial Gaussian Beam dataNow look at 5 degree fieldBeam has size 13 microns in y, 9.9 in x!Can optimize on skew data as wellAdvanced Imaging System Design Using Zemax

38、| OpticStudioPhysical Optics3 - 45 Physical Optics PropagationOverviewGeometrical vs. Physical OpticsDiffraction Propagation MethodsThe Pilot BeamPropagation Through Optical SurfacesPolarizationExamplesAdvanced Imaging System Design Using Zemax | OpticStudioOverview of POP3 - 47 IntroductionPhysical

39、 optics is the modelling of optical systems by propagating wavefrontsThe beam is represented by an array of discretely sampled complex amplitude Ae-j at each point in the arrayThe entire array is propagated through the space between surfacesAt each optical surface, a transfer function is computed th

40、at transfers the beam from one side of the surface to the otherAllows detailed study of coherent beam propagation, includingGaussian and higher order multi-mode beamsBeams along any field position (called skew beams in POP)Amplitude, phase and intensity can be calculated at any surfaceEffects of fin

41、ite apertures, including lens apertures, spatial filtering and arbitrary apertures placed in the beam, can be modelled3 - 48 Comparison with RaysIn the ray-based model, we assume a spherical wavefront is produced by a point sourceWe differentiate the wavefront to get raysWe propagate the rays throug

42、h the system, to the exit pupilWe integrate the rays at the pupil to compute the output wavefrontIn POP, we always propagate the wavefront itselfAn option exists to use rays to propagate between a range of surfaces, which is useful where rays are a good modelSince POP propagates wavefronts, it is mu

43、ch more computationally expensive than using raysUse it only when it matters3 - 49 Return to the GaussianBefore we get into theory, lets repeat the simple test of POP we did as part of the singlet lens design example (Gaussian.zmx)We will go into more detail than we previously did on exactly whats h

44、appening, but not yet go into full detailWe will still set up the POP calculation and just do as were toldWe will get some important principles of operation, however3 - 50 Our First POP RunOn the Analyze Tab, select Physical OpticsClick on Settings:3 - 51 POP Settings 1: The General TabThis tab defi

45、nes where the beam propagation starts and ends, wavelength, and the field point along which the POP beam propagatesPOP beams are always centered on the chief ray of the specified fieldSetup like so:Start on surface 1 (the beam waist)Surface to beam = 0 (surface 1 is the beam waist)End Surface is the

46、 Image surfaceWavelength, field = 1Other parameters are uncheckedNote we could start the propagation on surface 2, with a Surface to Beam value of -2796 3 - 52 POP Settings 2: Beam DefinitionX, Y sampling = 128Beam Type = Gaussian WaistWaist X, Y = 1.15THEN press Automatic to set X-, Y-WidthSet tota

47、l power = 1 Watt3 - 53 POP Beam DefinitionThe beam is defined as a two-dimensional array of complex amplitude dataSo we have amplitude and phase at each point in the arrayThe observables in a real experiment are irradiance and phaseWe are defining the beam so that its irradiance pattern has a specif

48、ied 1/e2 widthThe calculations are always performed on complex amplitude fields3 - 54 POP Settings 3: Display Save Output beam as Gaussian.zbfSave beam on all surfaces should be checkedZoom In = 4xOther data is left as default3 - 55 POP Settings 4: Fiber CouplingThis tab is only used when we go on t

49、o look at fiber couplingLeave Compute Fiber Coupling Integral UNCHECKEDPress OK to launch the POP calculation 3 - 56 ResultThe result shows a Gaussian irradiance on the final surface3 - 57 DetailText at bottom shows important informationThe pilot beam is a best-fit Gaussian to the selected beam (whi

50、ch is a Gaussian in this case, but need not be)Conceptually similar to the Skew Gaussian calculation, but actually computed in the beam coordinate system not the surface coordinate systemActs as a guide to the calculationMore soon!3 - 58 Beam File ViewerLets us see the saved beam data on any surface

51、. Analyze Beam File ViewerUseful because POP is often a big calculation, so compute once, view many3 - 59 Beam File ViewerSelect the beam on surface 1 (Gaussian_0001.zbf)Useful keyboard shortcutRight cursor key increases the surface numberLeft cursor key decreases the surface numberAllows us to step

52、 through surface by surface without opening the SettingsOther choices:View irradiance or phaseView false color, grey scale, slices, etc.Zoom in, scale with logs, etc.3 - 60 AnalysisThe irradiance data shows a nice Gaussian on all surfacesBut irradiance is not as sensitive as phase, remember?Set the

53、BFV to show phase, cross-section in x, zoom factor 43 - 61 Surface 1: At the WaistThe phase radius of curvature is infinite, as it should be:3 - 62 Surface 2: The Output PortNote phase radius of curvature and Gouy shift:3 - 63 Surfaces 3 and 4: The LensThe power of the surfaces changes the phase rad

54、ius of curvature, and adds aberrations that are hard to see with all the bulk power presentNote also what the phase is measured relative to 3 - 64 Surface 5: The ImageA slight phase ripple is added due to focus and spherical aberration in the lens, but the beam profile is still Gaussian in irradianc

55、e:3 - 65 POP Propagation ReportGo back to the POP window (not the Beam File Viewer) and press the Text menu itemOn the resulting Text listing, open Settings and select Data as “Prop Report”This is only available on the text listing of the POP data! 3 - 66 POP Propagation ReportThe Propagation Report

56、 gives details of the calculation, including error messagesWe mend you drive POP via the Propagation Report, and use the Beam File Viewer to display the data graphically3 - 67 SummaryWe have propagated the Gaussian Beam using POPAlthough we have not discussed how POP works yet!Results consistent wit

57、h paraxial, skew Gaussian because the lens is low powerSome amount of aberration apparentWill look at M2 in later stages of this courseThe POP Propagation Report gives full details of the calculation and gives error messages if neededWill now go on to look at POP in more detailAdvanced Imaging Syste

58、m Design Using Zemax | OpticStudioThe POP Calculation3 - 69 Geometrical vs. Physical OpticsGeometrical Optics is the modeling of optics by ray tracingRays are imaginary lines normal to wavefrontsRay properties:Rays propagate along straight linesRays dont interfere with one anotherRays either pass an

59、 aperture or they dontWavefront properties:Wavefronts must be propagated as a single entityWavefronts coherently self-interfere as they propagate3 - 70 Single Step ApproximationSimplifying AssumptionAll important diffraction effects occur going from the exit pupil to the imageRay distribution in the

60、 exit pupil, with transmitted amplitude and accumulated OPD, is used to compute complex amplitude wavefrontCompute FFT to calculate PSF, OTF, MTF, etc. at “focus”Works well for a wide range of systems, but not, e.g.At intermediate focusEspecially with clipping optics, like spatial filtersWhen effect