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1、Advanced Imaging System Design Using Zemax | OpticStudioDiffraction, MTF, and Image Quality2 - 2 DiffractionTo this point, we have used terms like diffraction limit and MTF without defining themIn this section, we will cover diffraction in imaging systems in some depth and considerFraunhofer diffrac

2、tion Huygens-Fresnel diffractionThe effects of diffraction on image formationThe effects of spatial coherence on diffraction image formation2 - 3 Rays and WavesThe term “wavefront” means a surface of constant phase A point source emits spherical waves. If we propagate to the entrance pupil, we can d

3、efine a spherical wavefront, differentiate the wavefront to give us rays, and then trace the rays through the optical system.But how do we get the output wavefront? 2 - 4 Output WavefrontWe trace the rays through the system to the image plane, and then propagate back to the exit pupil locationWe com

4、pute the path length of each ray, and subtract out the chief ray path lengthWe then integrate the path lengths to compute the wavefront of the optical system2 - 5 Output WavefrontRemember that the wavefront is really the “wavefront difference”, so a “flat wavefront” is perfectly spherical in a focal

5、 system2 - 6 Output WavefrontThe output wavefront has two important differences compared to the inputThe output may be aberrated, so it is not perfectly flatIf you have enough degrees of freedom, you could make the wavefront perfectly flatThe output wavefront is truncated to the size of the exit pup

6、ilThis introduces diffraction, which causes an uncorrectable blurring of the imageHence the “diffraction limit” represents an ultimate limit of the imaging quality of a lens, even if it is geometrically perfect and produces a perfectly flat wavefront2 - 7 What is Diffraction?Diffraction is the gener

7、al term applied to the propagation of energy in an optical beam, including the effects of finite apertures and coherenceIf a wave is radiated in free space it travels at constant velocity without change until it strikes a surface. If the wave is partially blocked by an aperture or barrier the direct

8、ion of travel will change. The bending of the optical path is called diffraction.Diffraction is a form of interference. Waves impinging on one part of the aperture interfere with those from another part to produce a diffraction pattern.It is the finite size of the aperture which causes the diffracti

9、on effects. An un-constricted wave does not produce a diffraction pattern.2 - 8 Diffraction2 - 9 Huygens WaveletsChristiaan Huygens was among the first to consider light to be a wave phenomenaIn Huygens-Fresnel diffraction theory, a wavefront is considered to be made up of an infinite number of poin

10、t sources. Each of the point sources acts as a secondary emitter of radiation. As a wavefront passes through an aperture, some of the wavefront is cut off and the shape of the wavefront is changed at the edges of the aperture.Since the entire wavefront initially originated from the same point source

11、, all of the secondary wavefronts are in phase (coherent), and in regions of overlap the wavefronts display interference effects2 - 10 Huygens WaveletsMost diffraction effects can be visualized easily by considering the beam at any point to be a collection of point source radiatorsThese Huygens wave

12、lets radiate from every point, and have amplitude and phaseAll Huygens point sources are assumed to be coherentThis means each point has a specific phase relationship with all other points that does not change over time2 - 11 Diffraction & Image FormationThe Huygens model yields great insight for no

13、n-plane waves, such as those wavefronts which form imagesOnce a beam is clipped by an aperture, the Huygens wavelets propagate only from the remaining area of the beam After passing through a finite aperture, the beam spreads over all space and es (in principle) infinite in extent, although most of

14、the energy is not deviatedAdvanced Imaging System Design Using Zemax | OpticStudioFraunhofer Diffraction 2 - 13 Fraunhofer DiffractionFraunhofer diffraction theory is an important special case of Huygens-Fresnel theory, specifically important for imaging systemsThe spot formed by an imaging system i

15、s given by the Fourier transform of the wavefront as measured in the exit pupil of the systemWhenever you see Fast Fourier Transform (FFT) analysis features, these use Fraunhofer diffraction theoryFFT is a slight misnomer, as often autocorrelation methods are usedMore on this laterSince the pupil ha

16、s a hard edge, its Fourier transform cannot have a hard edge, and it ripples out to infinityFor a circular pupil, the resulting image is described by a Bessel function2 - 14 The Airy DiskIf the exit pupil of an optical system is circular, unaberrated and uniformly illuminated, then the resulting ima

17、ge intensity has the shape of a Bessel function. The bright central spot of this pattern is commonly called the Airy disk”. 2 - 15 Bessel FunctionThe Bessel function has several important properties:There is a bright central core which contains 84% of the total energy in the imageThere is a dark rin

18、g, called the first Airy ring (in optics), surrounding the central coreThe first dark ring has a radius of 1.22*l*F/#The rest of the pattern consists of alternating bright and dark ringsAt the radial location of the second dark ring, 91% of the total energy is enclosed; at the third dark ring, 93.8%

19、 of the energy is enclosed, etc.2 - 16 Encircled Energy2 - 17 Point Spread FunctionThe term Point Spread Function (PSF) is used to indicate that, because of diffraction, a point source can never be focused to a point image, even if the lens is geometrically perfectIn principle, the spot diagram, foo

20、tprint diagram, etc. are “point spread functions” also. But because geometric optics can predict an infinitely small size, the term PSF is usually taken to mean a diffraction calculation computed at the image surface only 2 - 18 Airy Disk SizeFor an optical system operating at some wavelength l, the

21、 radial size of the Airy disk (the distance to the first dark ring) is given by 1.22*l*F/# (approximately of the F/#, in the visible)So the smallest resolvable spot is defined by the wavelength and F/# of the lens onlySo if you know the wavelength and the pixel size on your detector, you can compute

22、 the minimum F/# neededThat still leaves the question of how big the lens aperture needs to beWhy choose a big F/3 lens instead of a small F/3 lens?2 - 19 ResolutionIf an optical system images two close point objects, then their images will also be close. Due to diffraction effects, the images may a

23、ctually overlap, possibly forming only one image.Resolution defines how close the objects can be and still be detectable as separate images. There are several factors involved in determining the resolution of an optical system, including if the objects are coherent with each other or are illuminated

24、 with coherent light, or partially coherent light, the properties of the detector, etc.One important resolution definition is the Rayleigh criteria. This defines two incoherent images as resolved when the images are separated by their Airy disk radius.2 - 20 Angular Resolution Two points are conside

25、red to be resolvable if the central maximum of one points PSF lands on the first dark ring of the secondIn other words, the two PSFs are an Airy radius away from each otherThis occurs at an angle given bySo the spot size depends on the wavelength and F/#, but the resolution depends on the wavelength

26、 and entrance pupil diameterSo a big F/3 lens can resolve finer detail than a small F/3 lens, even though the spot sizes are the same, because the image plane is further away2 - 21 ResolutionThe “fineness of detail” of an image is described by the spatial frequencies it containsSince the PSF is the

27、Fourier transform of the pupil function, spatial frequency (measured in cycles/mm) or angular frequency (cycles/mrad) is a natural way to describe the behavior of a Fourier imaging systemAn input scene can be Fourier-transformed into spatial frequency space, multipled by a transfer function, and thi

28、s yields the output spatial frequencies, which can be inverse Fourier transformed to yield the final imageWe can therefore define the Optical Transfer Function of a lens in the same way as we could with an electronic amplifier or any linear system2 - 22 Optical & Modulation Transfer FunctionsThe OTF

29、 is a complex relationship of the ratio of the amplitude of a given spatial frequency in image space to that of the same spatial frequency component in object spaceThe MTF (modulation transfer function) is the modulus of the OTF. It is a measure of the contrast in the image of a sinusoidal intensity

30、 distribution.2 - 23 Spatial FrequencySpatial frequency, measured in cycles/mm, is a convenient way to describe the fineness of detail in an imageImagine imaging a sinusoidal or bar-chart pattern through an optical systemThe frequency of the pattern is its spatial frequencyWe refer to the number of

31、cycles/mm in the image plane, not object plane2 - 24 Modulation Transfer FunctionA modulation of 100% indicates perfect contrast. If the maximum modulation is lower than 100% (which also means a minimum modulation greater than 0% ), the fringes are less sharp.These plots show the resolution of a sys

32、tem with a square-wave pattern at 11 cycles/mm and 58 cycles/mm2 - 25 The MTF Plot The MTF plot shows all spatial frequencies at once2 - 26 MTF PlotThe effects of diffraction limit the maximum spatial resolution to the value given by:F/# is the working F-number. This expression shows that faster len

33、ses can resolve (ignoring aberrations) a finer sinusoidal image than a slower lens.Even if an optical system has no aberrations, the resolution is still limited by the final F/#The MTF falls off as a function of spatial frequency2 - 27 MTF PlotThe normalized value for the MTF is a maximum, M = 1.0,

34、at zero spatial frequency; M drops to zero at nmax, the cutoff frequency. The drop off for a diffraction-limited system is not quite linear:2 - 28 Adding AberrationsWhat happens as you start to add aberrations to a diffraction limited optical system?System performance gets closer and closer to that

35、predicted by pure ray tracingDiffraction calculations are only needed once the ray tracing predicts better imaging than you could actually seeBecause diffraction introduces an uncorrectable blurHow close do you need to be?2 - 29 Strehl RatioThe Strehl ratio is a method of relating the quality of an

36、aberrated image to that of an unaberrated image formed by an optical system with the same F/#:An approximation to the Strehl ratio based on the wavefront error is:This approximation is valid for systems having Strehl ratios greater than 0.1 is the wavefront error2 - 30 MTF and Wavefront ErrorThe MTF

37、 improves as the RMS wavefront error decreases. The MTF is closely related to the Strehl ratio, since both are functions of the PSF. The Strehl ratio is related to the wavefront error: As with the Strehl ratio, the MTF approaches its maximum value indicating full contrast as the wavefront error appr

38、oaches 0Generally, a system is considered to be near diffraction-limited ifThe peak-to valley OPD is less than 1/4 of a waveThe Strehl Ratio is greater than 0.82 - 31 Domains of ValiditySo if diffraction computations are more accurate, why not use them all the time?Two very good reasons:Diffraction

39、equivalents are slower, sometimes dramatically so. Some computations, such as MTF optimization, are 10 to 500 times slower than geometric equivalents, and not fundamentally better!Diffraction effects require the discrete sampling of the phase in the exit pupil. For systems with modest to large amoun

40、ts of aberration, say 2-20 waves, this requires a lot of sample points to meet the Nyquist criterion. Large sampling grids are slow and consume huge amounts of memory.No real gain because the geometric computations are perfectly accurate in this domain2 - 32 SummaryDiffraction imposes a fundamental

41、limitation on the imaging properties of a lens, even if it suffers no aberrations at allThe point spread function shows the uncorrectable blur caused by the finite aperture of the lensFraunhofer diffraction theory shows there is a Fourier relationship between the wavefront produced in the exit pupil

42、, and the image formedSpatial frequencies describe the “fineness of detail” in an imageModulation transfer function (MTF) shows how image contrast varies with spatial frequencyAdvanced Imaging System Design Using Zemax | OpticStudioComputing MTF2 - 34 Computing MTFIn systems well defined by Fraunhof

43、er theory, there is a Fourier relationship between the wavefront in the pupil and the Point Spread FunctionHowever, you can also compute MTF as an autocorrelation of the pupil functionThe autocorrelation theorem links the Fourier transform of a function in one domain to its auto-correlation in the o

44、therWhich to use?2 - 35 Computing MTFIn the MTF analysis feature, we want to show all spatial frequencies supported by the lens at onceWere most interested in the shape of the curveThe explicit FFT method gives us this directlyWhen we optimize or tolerance, we only need data at a few (often only one

45、) spatial frequencyThe autocorrelation method gives us this much more quickly, and with fewer rays tracedThe MTF* operands support a GRID switch that allows you to choose the FFT or autocorrelation methodThe autocorrelation method is vastly superior however2 - 36 ComparisonUsing the double Gauss sam

46、ple file provided with Zemax, and a macro to compute the polychromatic MTF at 50 cy/mm with both settings of the GRID control gives this:2 - 37 ComparisonAt the edge of the field of view we get this2 - 38 MTF OptimizationNote that generally speaking, 1% convergence is adequate for the purposes of op

47、timization and tolerancingExperimental methods of measuring MTF are generally not repeatable below 0.1% in any caseExtreme precision is not required for good optimization results; three significant figures is usually adequateFor optimization, we just need to know if the merit function gets bigger or

48、 smaller as a result of some change However, both algorithms will converge to arbitrary precision with adequate sampling, and the fast algorithm will do so many orders of magnitude faster when extreme precision is needed2 - 39 Geometric MTFGeometric MTF is a neat feature for use in systems with seve

49、ral waves of aberrationIt uses the geometric spot diagram data instead of the PSFThe primary advantage to using the geometric MTF is higher accuracy in the presence of significant aberrations, compared to the diffraction calculationThe geometric MTF is very accurate for systems with large aberration

50、s, where the diffraction MTF calculation would need enormous sampling to converge The geometric calculation is typically factors of 100 or more faster in this regime The Geometric MTF also has a handy Scale by Diffraction Limit switch that prevents unfeasibly good resultsAdvanced Imaging System Desi

51、gn Using Zemax | OpticStudioDouble Gauss Design2 - 41 Double GaussThe double Gauss is a traditional SLR-type camera lensWe will design one, and specifically discuss how to optimize for the required MTFHere is a preview of what the final design will look like:2 - 42 Design GoalsF/3, 75 mm focal lengt

52、h, F,d,CDesign for 35 mm film: a 24x36 formatSet image height to 21.6 mmDistortion less than 1%2 mm edge/center minimum, 12 mm maximum glass thicknessBack focus distance at least 40 mm for mirror clearance24mm36mm43.2mmThe field is defined by a circle whose diameter is the diagonal of a single frame

53、 of film2 - 43 MTF GoalsMTF 80% at 30 cy/mm, 60% at 50 cy/mm. This is about the response of professional-grade film.Note that 20% is about the limit at which humans can distinguish contrast by eye, so you should rarely ever be required to design to a target less than thisOff axis may be worse, but “

54、best obtainable”Lets say performance may drop to 50% at 30 cy/mm, 40% at 50 cy/mmThis kind of camera is typically used with the subject in the center of the field, so requiring best performance there makes most sense2 - 44 Starting PointLoad SamplesShort coursesc_dbga1.zmx2 - 45 Why Start With a Sam

55、ple?The double Gauss is a well known design variant, and we do not want to “reinvent the wheel”The goal of this exercise is really to address MTF optimization, and some other system performance issues, rather than starting from flat pieces of glassSo we have generated a starting point that cuts out

56、some workLook at the file12 surfaces, 6 lenses comprising of 2 doublets and 2 singletsThe stop is on a surface in the centerEPD is 25mmSurface 11 has a F/3 solve to constrain the EFL2 - 46 Add FieldsAdd three fields of paraxial image height 0, 15.1, 21.62 - 47 Add WavelengthsAdd F, d, C wavelengths2

57、 - 48 Edge Thickness MarginsRequire the semi-diameters to be at least 1 mm larger than required to pass the rays, so there is a mounting area2 - 49 Starting PointUse Optimize Quick Focus2 - 50 Merit Function ConstructionIt is way too early in the design process to target MTFMTF improves as the wavef

58、ront error decreases, so our initial optimization should be for wavefront errorIt is difficult to optimize for MTF if the desired frequencies lie beyond the first minimum of the MTF, as the MTF has to get worse before it can get better2 - 51 Merit FunctionSince we want great MTF, we will use an RMS

59、wavefront default merit function with reasonable glass and air thickness constraints. Go to Optimize and select the Optimization Wizard.2 - 52 Merit FunctionAdd DIMX on field point 3, target = 1% to control distortionAdd CTGT surface = 11 target = 40 This targets the back focal length to be greater

60、than 40 mmVariablesAll remaining radii (except STOp surface) and all thicknessesStarting MF is hugeOptimize!2 - 53 Initial DesignMF reduces to around 0.3!The first cut looks pretty good!2 - 54 AnalysisWhat are limiting aberrations? Check boundary conditions to ensure not in violation. Check distorti