Imaging Metrics. Frequency response Coherent systems Incoherent systems MTF OTF Strehl ratio Other Zemax Metrics. ECE 5616 Curtis

Transcription

1 Imaging Metrics Frequenc response Coherent sstems Incoherent sstems MTF OTF Strehl ratio Other Zema Metrics

2 Where we are going with this Use linear sstems concept of transfer function to characterize sstem qualit. What we will find is that: coherent sstems are linear in Field Incoherent sstem are linear in Intensit Thus the transfer function for these respective sstems are in those domains and are therefore different.

3 Generalized imaging sstems Infinite paraial sstem ( ) E,? ( ) E, Entrance pupil Goodman, Fourier Optics (first edition), Chapter 6 Eit pupil Consider an ideal, infinite aperture imaging sstem. The electric field at the output plane must be related to the input field b 1 E, M M M (, ) = E Our goal in this section is to understand how the image field (for coherent light) and intensit (for incoherent light) differs from this perfect cop in the presence of diffraction and aberrations.

4 E Generalized imaging sstems Finite paraial sstem We can define a function h that describes the response in the output plane, for an infinitel small ecitation in the input plane,. Since we know that Mawell s equations are linear, the response to a general input must be the sum of the responses to ever point in the input plane: (, ) = h (, ;, ) E ( ) d d, Now let us assume that the imaging sstem is shift invariant. For eample, there is no vignetting. The superposition integral can now be written E (, ) = h ( M, M ) E ( ) d d, Note that this is in the form of a convolution integral. This reduces to the infinite paraial case if 1 h, M (, ) = δ ( ) We know from linear sstems theor that a convolution in real space can be represented b a product in Fourier space, so it must be possible to describe the shift-invariant imaging sstem via ( k, k ) = H ( k, k ) E ( k k ) E, where H is the Fourier transform of h and k = 2π f is spatial frequenc.

5 Frequenc response of a diffraction-limited sstem This result can be formall derived from Fourier optics, but we can argue our wa there from what we know about paraial optics. k Entrance pupil, P ent (,) =1 inside, 0 outside pupil Ras that pass pupil thus satisf P(,)=1 or H k = = 2 π λ 2 π λ λ t ( f, f ) = P ( λ t f, λ t f ) = P ( λ t f, λ t f ) θ -t ent eit θ f sin t k z θ

6 Coherent diffraction-limited f Coherent transfer function imaging Impulse response f 0 1 f r 0 H 0 H(f,f ) P ent, D π r 0 = or r = λ t 2 0 λ NA h(,) D 1 ( f, f ) = P ( λ t f, λ t f ) h (, ) = F { H ( f, f )} ( ) = Circ λ t f r = Circ D Let s find the first null of the impulse response: ent The cutoff frequenc (highest spatial frequenc that passes) is D f 0 = = 2 λ t r NA λ = 2 J 1 [ 2 π r D / ( 2 λ t )] π r D / ( 2 λ t ) 2

7 Coherence of light To understand the concept of coherence, define the mutual coherence function which is the cross-correlation of the electric field at different spatial locations r r r r G, ( ) * r r, τ E ( r, t ) E ( r t + τ ) 1, where < > indicates ensemble average but can be considered an infinitetime average in most cases. When E is well-correlated with itself across the entire object (G is a constant) the light is referred to as coherent. At the other etreme, when even a small change of position r 1 -r 2 >λ causes the correlation to sharpl decrease (G approaches zero), the light is completel incoherent. There are man interesting and important intermediate cases, but we will limit ourselves to just the two etremes. Saleh and Teich, Chapter 10

8 Coherence of light The intensit is related to G b r r r * r r I, ( r) = G( r, r,0) E ( r, t) E( r t) Eamples of coherent light: 1. Lasers 2. Lamp after spatial filtering. This leads to the common but not totall accurate definition of spatial coherence as light that appears to emerge from a single point. 3. Epanded laser beam passed through a static diffuser (which is a violation of the previous rule ). Incoherent light: virtuall anthing else (lamps, stars, black-bod, laser passed through randoml time-varing spatial diffuser) Saleh and Teich, Chapter 10

9 Imaging with incoherent light ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = 2 2 * * *,,,,,, E E h h d d d d t r E t r E r I r r r Substitute our impulse-response equation into the definition of intensit and interchange the orders of integration: M where and we are assuming quasi-monochromatic light. For incoherent light, the mutual coherence function is nearl zero outside a region roughl one wavelength in diameter. We can approimate this as a delta function: ( ) ( ) ( ) ( ) * 1 1,,,, I E E δ κ

10 Imaging with incoherent light which reduces the integral to r I 2 r ( r ) = κ h (, ) I ( r ) d d which sas that incoherent sstems are also linear in intensit and that the impulse response is the abs magnitude squared of the coherent impulse response. Using a FT rule again, we can write the optical transfer function ( k, k ) = H ( k, k ) I ( k k ) I, H ( k, k ) = F h(, ) { } 2 = H( k, k ) H( k, k )

11 Properties of the OTF f Coherent transfer function H(f,f ) 1 f 0 f Auto-correlation operation ( k, k ) H ( k k ) H, 0 f f Area of overlap between displaced copies of pupil function Normall normalize this to total area of pupil f Optical transfer function (incoherent). H ( f, f ) 2 f 0 f OTF = Modulation transfer function

12 Eample for circular pupil Diffraction limited sstem For coherent case, H ( f, f ) = P( λz f, λz f ) i i z i is distance from eit pupil to image plane For incoherent case, Calculate overlap area (4B) (normalized) Which gives.. From Goodman, Introduction to Fourier Optics, pp

13 Eample for circular pupil Diffraction limited sstem Where p o is the coherent cutoff frequenc From Goodman, Introduction to Fourier Optics, pp

14 Generalized pupil function Imaging with aberrations Previous analsis used pupil function that was either 0 (light blocked) or 1 (light passed); this was the onl non-ideal element in the otherwise perfect paraial optical imaging sstem. Now consider adding aberrations to the pupil as wavefront error. P (, ) = P (, ) ep [ jkw ( )] eit eit, All the previous results carr over. It can be shown (see Goodman) that aberrations never increase the MTF (modulus of the OTF). OTF vs. waves of defocus (Goodman)

15 Practical Issues of MTF/OTF Input (Object) Output(Image) The ratio of the image modulation to the object modulation at all spatial frequencies is defined as the modulation transfer function (MTF). The optical transfer function is comple amplitude (intensit) function with a phase term. The amp/int part is call the MTF and phase transfer function is called the PTF.

16 MTF/PTF If the PSF is asmmetrical (onl off ais for rotational smmetric and centered sstems) then the input/output sinusoid with have a phase shift. (coma etc) This phase shift versus spatial frequenc is the PTF

17 Contrast Ratio and MTF/OTF A: Indicates good resolving power and good contrast. B: Indicates good contrast but poor resolving power. C: Indicates good resolving power but poor contrast. (Zeiss tech note)

18 MTF/OTF Must use for whole sstem all lenses etc. The MTF of the sstem is product of MTF of each components. Some detectors have poor MTF so the detectors MTF has to be considered as well.

19 Strehl Ratio A measure of coherent image qualit 1 Strehl ratio is the on-ais intensit in the presence of aberrations relative to the on-ais intensit w/o aberrations. Strehl ratio = ( π ω ) 2 P-V OPD RMS OPD SR Energ in Air Energ in rings % 16% λ/ λ % 17% λ/ λ % 20% λ/ λ % 32% λ/ λ % 60% 3 λ/ λ % 80% λ 0.29 λ % 90% * SR does not correlate well with image qualit at these low levels. I U r@0.6 lnad RMS OPD = P-V OPD/3.5 to P-V OPD/5 depending on the aberration tpe. SR also gives fiber coupling efficienc in man cases AStrehl ratio The relationship between SR and RMS wavefront error (ω) is Smith, Modern Optical Engineering, Chapter 11 Eact for defocus, close for most.

20 Eample Sstem in Zema NEWPORT PAC016 Field points in (0,1.4,2) Wavelength 550nm Magnification = -1.0 Entrance Pupil=3mm

21 Modulus of OTF FFT Method

22 Different Calculations Available in Zema Hugens MTF computes an FFT of the Hugens PSF. Since the transform is done on the PSF in image space coordinates, the tangential response corresponds to spatial frequencies in the direction in local image surface coordinates, and the sagittal response corresponds to spatial frequencies in the direction. The Hugens MTF also has no dependence on the location of ras in the paraial pupils. The MTF can therefore be computed for man non-sequential sstems using ports where reference ras required b other diffraction algorithms would not make it through, or for sstems where pupils or images formed b multiple nonsequential sub-apertures are overlapped. Sstems with etreme eit pupil distortion, such as ver fast off-ais reflectors, are also handled correctl with the Hugens technique. FFT MTF computation is based upon an FFT of the pupil data. The resulting MTF is the modulation as a function of spatial frequenc for a sine wave (can select Square wave option) For focal sstems, the cutoff frequenc at an one wavelength is given b one over the wavelength times the working F/#. This ields accurate MTF data even for sstems with anamorphic and chromatic distortion. For afocal sstems, the cutoff frequenc is the eit pupil diameter divided b the wavelength. Geometric MTF is a useful approimation to the diffraction MTF if the sstem is not close to the diffraction limit. The primar advantage to using the geometric MTF is for sstems which have too man waves of aberration to permit accurate calculation of the diffraction MTF. The geometric MTF is also ver accurate at low spatial frequencies for sstems with large aberrations.

23 Ra Fans for 3 Image Points Scale is 100 micron ma The data being plotted is the difference between the ra intercept coordinate and the chief ra intercept coordinate. The tangential fan is the plot of the difference between the ra or coordinate and the chief ra or coordinate at the primar wavelength (TRA), as a function of the pupil coordinate (can change to )

24 Optical Path Difference The vertical ais scale is given at the bottom of the graph. The data being plotted is the optical path difference, or OPD, which is the difference between the optical path length of the ra and the optical path length of the chief ra. Usuall, the calculation is referenced back to the difference between the ra path lengths at the sstem eit pupil. The horizontal scale for each graph is the normalized entrance pupil coordinate

25 Spot Diagrams

26 Spot Diagrams Through Focus

27 Encircled Energ Two method available. Use Diff for near diffraction limited sstems and geometrical when not.

28 Point Spread Function The FFT PSF computes the intensit of the diffraction image formed b the optical sstem for a single point source in the field. The intensit is computed on an imaginar plane which is centered on and lies perpendicular to the incident chief ra at the reference wavelength. Because the imaginar plane lies normal to the chief ra, and not the image surface, the FFT PSF computes overl optimistic (a smaller PSF) results when the chief ra angle of incidence is not zero. This is often the case for sstems with tilted image surfaces, wide angle sstems, sstems with aberrated eit pupils, or sstems far from the telecentric condition. The other main assumption the FFT method makes is that the image surface lies in the far field of the optical beam. This means the computed PSF is onl accurate if the image surface is fairl close to the geometric focus for all ras; or put another wa, that the transverse ra aberrations are not too large.

29 Wave Front Map

30 Interferometer Picture

31 RMS Wavefront Error versus Field

32 RMS Spot Radius vs Field

33 Strehl Ratio

34 Aberration Summar

35 Field Curvature and Distortion Under Miscellaneous Can also get Grid Diagram

36 Reading W. Smith Modern Optical Engineering Chapter 15