第六章像質品估（Image Evaluation）

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榔错马
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1 6.1 Image Evaluation Spot size Analysis wavefront Analysis 6.4 Point spread functions fiber coupling 6.6 Energy Distribution 6.7 Transfer functions object 6. Geometrical V.S. diffraction evaluation 6-1

2 Schrodinger Helmholtz ( x, y, z) + k u( x, y, z) = v u 6.1 u(x,y,z) k= / ray Fermat stationary value geometric wavefront 6.3 Spot diagrams and wavefront Spot diagrams Spot diagram

3 6.1 Petzval lens OSLO pupil ray intersection density OSLO entrance pupil spot diagram operating conditions calculate menu setup spd\wavefront spreadsheet aperture division APDIV 6. APDIV= APDIV=16. OSLO direction cosine 6-3

4 exit pupil Abbe Abbe exit pupil entrance pupil pupil aberrations OSLO Six image space spot diagrams Zernike 6. OSLO x y reference sphere through-focus MTF(modulation transfer function) OSLO Gaussian apodization ration x y 6-4

5 1 1/e entrance beam radius x y 6.3. Spot size Analysis Gaussian apodization function moment average values x p(x) centroid ( x) x = xp dx 6. x ( x) = x p dx 6.3 σ = = x ( x x ) p( x) x dx 6.4 standard deviation x n w i DX i,dy i x y 1 = w 1 = w w = n i= 1 n i= 1 w DX i w DY i n w i i= 1 i i

6 1 σ x = w 1 σ y = w n i= 1 n i= 1 w w i i ( DX x ) ( DY y ) i i 6.7 σ x σ y root-mean square, RMS radial RMS spot size σ = σ + σ 6.8 r x y OSLO Calculate>>spot size Analysis r OSLO RMS x, y r GEO RMS X GEO RMS Y GEO RMS R centroid x y CENTX CENTY y 6.3 CENTX= DIFFR LIMIT Airy Airy Disk 6.3 CENTX CENTY 6-6

7 6.3.3 wavefront Analysis optical path difference; OPD wavefront aberration OPD OSLO real exit pupil General Operating Condition OSLO RMS diffraction focus OSLO fidelity distortion Calculate>>Wavefront Analysis

8 6. RSY, RSX, RSZ y RSX= nominal location RSZ= interferometric testing Zernike 1 order Zernike Zernike Zernike Zernike n i Zernike Z i (, ) ( θ ) = ( n w ρ, c i z i ρ, θ ) 6.9 i= c i OSLO Six Zernike Zernike Zernike 6.4 Point spread functions point spread function impulse response ( ) exit pupil A(x,y) w y pupil function (x,y) 6-8

9 ( x, y) = A( x, y) exp[ ikw( x, y) ] ρ 6.1 k=π/λ (x,y)= Kirchhoff u(x',y') u i λ ( x', y' ) = ρ( x, y) A exp R' ( ikk' ) da 6.11 A R' (x,y) (x',y') u ( x', y' ) { ik[ R+ ( x', y' )]} i exp π = ρ( x, y) exp i ( xx' + yy' ) dxdy λm R λr R 6.1 R ( x', y' ) MR z 6.1 Hopkins Yzuel Optical Acta 17, (197) A (x,y)= 6.1 x' y' ν x =, ν y λr λr u(x',y') squared modulus ( x', y' ) u( x', y' ) PSF =

10 transfer function 6.1 FFT FFT point-by-point Fourier kernel integral N NxN Strehl OSLO 6.1 ν y y = ' N N λ R 1 ν y = 6.14 N y y λr y' = 6.15 N y D M y=d/m λ λrm M y' = = 6.16 ND NA N ( ) D λ NA NA = n M/N k fill factor 6.16 M N 6-1

11 = N y' = M 6.17 NA 6.17 M λ M N aliasing Whittaker-Shannon ν max 1 y ' 6.18 ν max transfer function OTF NA ν = OTF PSF y' λ λ 6.18 y' M 4ΝΑ ( λm ) λ ( ) = ΝΑ Ν 4 ΝΑ ( ) M =N/ M M N M sampling intervals squared modules a r'=(x' +y' ) / ( b) J 1 Bessel J1 PSF ( r') = 6.19 b sπar' π b = = NAr' 6. λr λ NA a NA = nsinu = N 6.19 Airy J 1 (πx) x=1. k Airy.61λ r' Airy = 6.1 NA b=1. r' Airy DIFFR LIMIT 6-11

12 6.4. OSLO 6.19 infinite object conjugate File>>New 1mm image numerical aperture OSLO 6.19 Calculate>>Spot Size Analysis Calculate>>Spread Function 6.4 DIFFR LIMZT= r' Airy =.61*.58756mm/.=.179mm

13 figure of merit Strehl Strehl Strehl variance of the wavefront OSLO PSF PSF line spread functions and knife edge distributions line spread function, LSF x y knife edge - x' knife edge distribution, KED KED x' ( x' ) = LSF( x ) ~ dx ~ 6.3 irradiance point spread function 6-13

14 6.5 fiber coupling fiber coupling efficiency u(x',y') (x',y') η normalized overlap integral ( x', y' ) Ψ *( x', y' ) u dx' dy' η = 6.4 u dx' dy' ( x', y' ) u * ( x', y' ) dx' dy' Ψ( x', y' ) Ψ * ( x', y' ) T T = ηη * = η 6.5 T power 6.4 x',y' OSLO u(x',y') Schwarz T 1 T=1 (x',y')=ku(x',y') k (x',y') gradient index step Ψ ( x', y' ) x' + y' = exp r r' = exp r Gaussian 6.6 r r 1/e cladding 1 n core a 3 cladding n cladding n core n cladding n cladding <<1 6-14

15 Ψ step index ( r' ) ur' J a, J ( u) = wr' K a, K ( w) r' a r' > a 6.7 r'=(x' +y' ) J Bessel K Hankel Modified Hankel function u w π ν = a 6.8 λ n core n cladding u w ( 1 ) v u = ( 4 +ν ) 4 w u = ν 6.3 Gloge Appl. Opt. 1, 5-59 (1971) OSLO Six CCL ball lenses y 3 x 1 w 1 λ θ = tan 6.31 πw.83 m y w y =.458µm x w x =1.498µm w =(w x w y ) 1/ =.88µm ( ) 6 nominal paraxial magnification m = 5.88 Melles-Groit 1mm

16 6.5 Melles-Groit 6.7 Melles-Groit aperture stop Gaussian apodization 3 [ ] ( ) mm [] tan( 1 ) = =. mm 1 th tan 3 = =.74 th

17 6.6-6 (.75*8.99) 1/ =5µm diffraction pattern Gaussian shape µm

18 6.8 6µm Energy Distribution spread function 8 spot size square pixels energy distribution x',y' r' ' π ( x', y' ) dx' dy' = PSF( r', θ ') PSF r' dr' dθ ' =

19 r' r' r' OSLO 6.3 discrete Fourier transform Parseval 6.3 image patch size OSLO radial energy distribution, RED a encircled energy RED a a π ( ) = PSF( r', ') θ r' dr' dθ ' 6.33 s ensquared energy, SQE s / s / () s = PSF( x', y' ) SQE dx' dy' 6.34 s / s / OSLO 6.19 Rayleigh RED a π π ( ) = 1 J NAa J NAa 1 λ λ Airy 91 Airy 6.35 J 1 (πx) x=3.38 r'=1.619λ NA=1.619*.58756mm.48mm 1µm.1 calculate>>energy Distribution>>Diffraction Spreadsheet 6-19

20 6.9.18mm.33mm.48mm 1 e. x y EBR e 4mm

21 Airy lobes ( ) ( ) ( ) PSF r' = exp r' π w w w RED a = 1 exp a w ( ) ( ) 6.7 Transfer functions 6-1

22 6.7.1 optical transfer function, OTF ν x ν y OTF (, ν ) PSF ( x', y' ) exp[ iπ ( ν x' + ν y' )] x y ν x y = 6.36 PSF ( x', y' ) dx' dy' dx' dy' OTF(,)=1 6.1 autocorrelation OSLO 6.19 cutoff frequency ν NA ν = 6.37 λ modulus modulation transfer function, MTF phase transfer function, PTF MTF modulation PTF 6.1 modulation M M E max min = 6.38 E max E + E MTF PTF object min M image MTF = 6.39 M 6-

23 g PTF = π = πgν 6.4 p 6.1 OTF o(x,y) O(ν x,ν y ) I(ν x,ν y ) ( ν, ν ) = OTF( ν, ν ) O( ν, ν ) Ι 6.41 x y x y x y OTF singlet File>>New 1mm f 1 1mm (*1)=5mm paraxial axial ray angle solve -1 (*FNB)=-.5 axial ray height solve 6-3

24 .5mm Schott F µm

25 MTF 5cycle mm 5cycles mm peak-to-valley OPD default switchover 3λ ( )

26 6.14 MTF 5 MTF 3 MTF MTF 6-6

27 6-7

28 [1] OSLO Optical Reference version 5 (Sinclair Optics, 1996) Chapter 6. [] OSLO Optical Reference version 6.1 (Lambda Research Corp., 1) [3] Wagner Tomlinson Appl. Opt. 1, (198) 6-8

29 1. catalog lens triplet lens spot diagram point spread function. ball lens 15 µm( ) 6 µm( ) 3. MatLab MTF 6-9

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