Modulation Transfert Function

Transcription

1 Modulation Transfert Function

2 Summary Reminders : coherent illumination Incoherent illumination Measurement of the : Sine-wave and square-wave targets Some examples

3 Reminders : coherent illumination We assume that the coherent PSF (la Réponse Percussionnelle Cohérente (RPC) in french) is space - invariant in the small zone of p is the pupil function. The coherent PSF is given by : PSFcoh r coh( σ ) interest of the field. r ( ) ~ p, with R distance pupil - paraxial image λr reference sphere R λ R PSFcoh r iπ r σ ( ) e d r p( λrσ ) radius of the We define the coherent modulation transfert function as the Fourier transform of the coherent PSF: 3

4 is space - invariant in the small part of The incoherent PSF is :PSF Incoherent illumination In this course and most of the time, the illumination is incoherent. We assume that the incoherent PSF (la Réponse Percussionnelle Incohérente (RPI) in french) with PSF OTF( σ ) OTF( σ ) ( r ) λr the field of ikw ( ρ ) ( r ) dx dy et p( ρ) p( ρ ) e ( r ) [ ( ) ( )] p ρ p ρ [ ( ) ( )] p ρ p ρ ~~ * r pp λr ~~ * pp d ρ ( ρ ) interest. [ is the autocorrelation] PSFcoh PSFcoh PSFcoh PSFcoh We define the Optical Transfert Function (OTF) as the Fourier transform of R PSF e iπ r σ As the PSF depends on the Fourier transform of the pupil function, we get : λrσ d r * *( r ) ( r ) d r the PSF: 4

5 ( ) ( ) p ρ OTF σ FT ( ) PSFcoh r FT ( ) PSF r 5

6 The function OTF We define the modulus of OTF e iφ ( ) ( ) σ σ One can show that : Properties ( ) ( ) ( ) iφ ( σ σ is complex : OTF σ OTF σ e ) ( σ ) is the Phase Transfer Function. - the function - ( ) ( ) σ ( ) σ the OTF as the Modulation Transfer Function : - the aberrations always degrade the : p ( ) d ρ OTF R ( ) ( ) ik W ( ρ ) W ( ρ λrσ ρ p ρ λrσ e ) is even, R p ( ) ρ d ρ R p ( ) ( ) ρ p ρ λrσ R p ( ) ρ d ρ Instrument with aberrations Perfect instrument d ρ 6

7 Filtering of the spatial frequencies ( ) ( ) is the objet, I r O r We know that : I ~ Hence: I is the image, g y is the transverse magnification. g y ( ) r ( ) r O PSF r ( ) ~ ( ) ( ) σ g O g σ OTF σ y y 7

8 Perfect instrument with a circular pupil Assume that the pupil is circular (radius R Then : p ). PSF ( r ) Hence: πr P J ( λr) r R π λr r RP π λ R P Airy disc y (m) ( ) arccos σ σ σ σ π σ c σ c σ c RP ON with σ c λr λ / F λn The optical system is a low - pass filter for the spatial frequencies x (m) FTM frequence spatiale Spatial frequencies 8

9 Summary ( ) ( ) ( ) p ρ OTF σ σ W RMS FT FT resolution, contrasts ( ) PSFcoh r ( ) PSF r Strehl ratio, encircled energy, resolution 9

10 Measurement of the σ Contrast in the objet Emax Emin E Emin ( ) Contrast in the image max E + E E < max + E max + min min Contrast of the image Contrast of the objet ( σ ) < Important : the PSF has to be space-invariant across the target. Hence, this measurement has to be done for all the points of the field

11 Square-wave vs sine-wave targets Square-wave targets are more practical Still we measure the reduction of contrast in the image with increasing spatial frequencies The result is the Contrast Transfer Function (CTF)

12 Square-wave targets Spatial frequency Number of line pairs per mm (lp/mm) Modern optical engineering, W. Smith

13 Analogy with the Bode diagrams R Bode Plot: Gain Response V i C V o -4 H(jω) - V V + jrcω o i H Transfer function ( jω) H(jω) ( ) Bode Plot: Phase Response ω (rad/sec) 3

14 Square-wave vs sine-wave targets Square-wave target-> CTF CTF ( ν ) ( ν ) ( 3ν ) ( 5ν ) ( 7 ) 4 ν π π 4 ( ν ) CTF( ν ) CTF + 3 ( 3ν ) CTF( 5ν ) CTF( 7ν ) Sine-wave target-> Handbook Thierry of Lépine Optics, - Optical vol I, OSA design 4

15 Modulation and resolution Which system is the best? Modern optical engineering, W. Smith 5

16 Defocus W λ 4 W λ W diffraction 3λλ 4 W λ f (cyc/mm) 6

17 Defocus : radial target x -4 x v (m) v (m) u (m) x -4 u (m) 3λ with W 4 x -4 7

18 Spherical aberration (best focus) W PV λ 4 W PV λ diffraction W PV λ f (cyc/mm) 8

19 Obstruction obstructio n (diameter) diffraction,5 obstructio n (diameter),5 obstructio n (diameter), f (cyc/mm) 9