1. 1 OPTICS Part II SOLO HERMELIN Updated: 16.01.10http://www.solohermelin.com

2. 2 Table of Content SOLO OPTICS Maxwells Equations Boundary Conditions Electromagnatic Wave Equations Monochromatic Planar Wave Equations Spherical Waveforms Cylindrical Waveforms Energy and Momentum Electrical Dipole (Hertzian Dipole) Radiation Reflections and Refractions Laws Development Using the Electromagnetic Approach IR Radiometric Quantities Physical Laws of Radiometry Geometrical Optics Foundation of Geometrical Optics Derivation of Eikonal Equation The Light Rays and the Intensity Law of Geometrical Optics The Three Laws of Geometrical Optics Fermats Principle (1657) O P T I C S P a r t I 3. 3 Table of Content (continue) SOLO OPTICS Plane-Parallel Plate Prisms Lens Definitions Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermats Principle Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snells Law Derivation of Lens Makers Formula First Order, Paraxial or Gaussian Optics Ray Tracing Matrix Formulation O P T I C S P a r t I 4. 4 Table of Content (continue) SOLO OPTICS Optical Diffraction Fresnel Huygens Diffraction Theory Complementary Apertures. Babinet Principle Rayleigh-Sommerfeld Diffraction Formula Extensions of Fresnel-Kirchhoff Diffraction Theory Phase Approximations Fresnel (Near-Field) Approximation Phase Approximations Fraunhofer (Near-Field) Approximation Fresnel and Fraunhofer Diffraction Approximations Fraunhofer Diffraction and the Fourier Transform Fraunhofer Diffraction Approximations Examples Resolution of Optical Systems Optical Transfer Function (OTF) Point Spread Function (PSF) Modulation Transfer Function (MTF) Phase Transfer Function (PTF) Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function Other Metrics that define Image Quality Srahl Ratio Other Metrics that define Image Quality - Pickering Scale Other Metrics that define Image Quality Atmospheric Turbulence Fresnel Diffraction Approximations Examples 5. 5 SOLO OPTICS Continue from OPTICS Part I 6. 6 Optical DiffractionSOLO Augustin Jean Fresnel 1788-1827 In 1818 Fresnel, by using Huygens concept of secondary wavelets and Youngs explanation of interface, developed the diffraction theory of scalar waves. P 0P Q 1x 0x 1y 0y Fr Sr r O ' Screen Image plane Source plane 0O 1O Sn1 - Screen Aperture Sn1 - normal to Screen FrPP 0 SrQP 0 rQP From a source P0 at a distance from a aperture a spherical wavelet propagates toward the aperture: Srktj S source Q e r A tU ' ' According to Huygens Principle second wavelets will start at the aperture and will add at the image point P. dre rr A Kdre r U KtU rrktj S sourcerkttjQ P S 2/2/' ',', where: ',K obliquity or inclination factor SSS nrnr 11cos&11cos' 11 0',0 max0',0 K K Obliquity factor and /2 phase were introduced by Fresnel to explain experiences results. Fresnel Diffraction Formula Fresnel took in consideration the phase of each wavelet to obtain: Fresnel Huygens Diffraction Theory Fresnel Kirchoff Diffraction Formula See full development in P.P. Diffraction Table of Content 7. 7 SOLO Fresnel-Kirchhoff Diffraction Theory In 1882 Gustav Kirchhoff, using mathematical foundation, succeeded to show that the amplitude and phases ascribed to the wavelets by Fresnel, by enhancing the Huyghens Principle, were a consequence of the wave nature of light. HBED & For an Homogeneous, Linear and Isotropic Medium where are constant scalars, we have , t E t D H t t H t B E ED HB Since we have also tt t D H t B E For Source less Medium 0& 0 2 2 2 DED EEE t E E 02 2 2 t E E Maxwell Equations are eJ t D HA mBGM )( mJ t B EF eDGE James C. Maxwell (1831-1879) Gustav Robert Kirchhoff 1824-1887 Optical Diffraction 8. 8 SOLO Fresnel-Kirchhoff Diffraction Theory 0 1 2 2 2 2 U tv Scalar Differential Wave Equation For a monochromatic wave of frequency f ( = 2f ) a solution is: tjPjPUPtPUtPU expexpRecos, Define the phasor PjPUPU exp U v U tv 2 2 2 2 2 1 2 2 v f v k 022 UkPhasor Scalar Differential Wave Equation This is the Scalar Helmholtz Differential Equation Hermann von Helmholtz 1821-1894 Boundary Conditions for the Helmholtz Differential Equation: Dirichlet (U given on the boundary) Neumann (dU/dn given on the boundary) Johann Peter Gustav Lejeune Dirichlet 1805-1859 Franz Neumann 1798-1895 1 0 11 2 2 2 2 2 2 2 2 2 vE tvt E v E Vector Differential Wave Equation Optical Diffraction 9. 9 To find the solution of the Scalar Helmholtz Differential Equation we need to use the following: Scalar Greens Identity SV dSGUUGdVGUUG 22 Greens Function SF SF FS rr rrkj rrG exp ; This Greens Function is a particular solution of the following Helmholtz Non-homogeneous Differential Equation: SFFSFSS rrrrGkrrG 4;; 22 SOLO Fresnel-Kirchhoff Diffraction Theory provided that and are continuous in volume V UUU 2 ,, GGG 2 ,, Free-Space Greens Function n i iSS 1 iS nS dV dSnS 1 V Fr Sr F 0r SF rrr PositionSourcerS PositionFieldrF 022 Uk Scalar Helmholtz Differential Equation Optical Diffraction 10. 10 SOLO Scalar Greens Identities SV dSGUUGdVGUUG 22 Let start from the Gauss Divergence Theorem SV dSAdVA Karl Friederich Gauss 1777-1855 where is any vector field (function of position and time) continuous and differentiable in the volume V bounded by the enclosed surface S. Let define . A UGA UGUGUGA 2 Then S Gauss VV dSUGdVUGUGdVUG 2 S Gauss VV dSGUdVGUUGdVGU 2 Subtracting the second equation from the first we obtain First Greens Identity Second Greens Identity We have GEORGE GREEN 1793-1841 Fresnel-Kirchhoff Diffraction Theory To find a general solution of the Scalar Helmoltz Differential Equation we need to use the n i iSS 1 iS nS dV dSn 1 V Fr Sr F 0r SF rrr If we interchange with we obtainG U Optical Diffraction 11. 11 Integral Theorem of Helmholtz and Kirchhoff F V sF V SS rUdVUrrUGkUGkdVGUUG 442222 Using: SFFSFSS rrrrGkrrG 4;; 22 n i iSS 1 iS nS dV dSnS 1 V Fr Sr F 0r SF rrr PositionSourcerS PositionFieldrF SOLO Fresnel-Kirchhoff Diffraction Theory 0,22 SFS rrUk From the left side of the Second Scalar Greens Identity we have: SS SS dS n G U n U GdSGUUG SF SF FS rr rrkj rrG exp ;Using: we obtain: S SF SF SF SF F dS rr rrkj n U n U rr rrkj rU expexp 4 1 This is the Integral Theorem of Helmholtz and Kirchhoff that enables to calculate as function of the values of and on the enclosed surface S.nU /UU Note: This Theorem was developed first by H. von Helmholtz in acoustics. Hermann von Helmholtz 1821-1894 Gustav Robert Kirchhoff 1824-1887 From the right side of the Second Scalar Greens Identity, using we have:dS n U dSnUdSU SSS 1 Scalar Helmholtz Differential Equation Optical Diffraction 12. 12 Sommerfeld Radiation Conditions SOLO Fresnel-Kirchhoff Diffraction Theory SS S F dS n G U n U G dS n G U n U GrU 1 4 1 4 1 P Fr Sr r 1S S R Screen Aperture d Sn1 Sn1 since the condition that the previous integral be finite is: R Rkj rrG SFS exp ; Consider the surface of integration SSS 1 1S - on the screen S - hemisphere with radius R Gkj R Rkj R kj n G exp1 dRUkj n U GdS n G U n U G S 2 1 exp limlim R Rkj RGR RR 0lim Ukj n U R R This is Sommerfeld Radiation Conditions - on the aperture Optical Diffraction 13. 13 is known as optical disturbance. Being a scalar quantity, it cannot accurately represent an electromagnetic field. However, the square of this scalar quantity can be regarded as a measure of the irradiance at a given point. U Sommerfeld Radiation Conditions (continue) SOLO Fresnel-Kirchhoff Diffraction Theory SS S F dS n G U n U G dS n G U n U GrU 1 4 1 4 1 0lim Ukj n U R R This is Sommerfeld Radiation Conditions This implies that: 0 4 1 S dS n G U n U G and the Integral of Helmholtz and Kirchhoff becomes: 1 4 1 S F dS n G U n U GrU P Fr Sr r 1S S R Screen Aperture d Sn1 Sn1 0P Q Arnold Johannes Wilhelm Sommerfeld 1868 - 1951 Optical Diffraction 14. 14 The Kirchhoff Boundary Conditions SOLO Fresnel-Kirchhoff Diffraction Theory Kirchhoff assumed the following boundary conditions: dS n G U n U GrU F 4 1 1. The field distribution and its derivative , across the aperture , are the same as in the absence of the screen. U nU / 2. On the shadowed part of the screen and0 1 S U 0/ 1 S nU The Integral of Helmholtz and Kirchhoff becomes: The field at point P is the superposition of the aperture values 0 U 0/ nU Note: Moreover, mathematically the condition implies0/&0 11 SS nUU 0/&0 nUU However, if the dimensions of the aperture are large relative to the wavelength , the integral agrees well with the experiment. P Fr Sr r 1S S R Screen Aperture 0P 0,0 1 1 S S n U U n U U , Q Sn1 Sn1 S Kirchhoff boundary conditions are not physical since the presence of the screen changes field values on the aperture and on the screen. Optical Diffraction 15. 15 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory dS n G U n U GrU F 4 1 The Integral of Helmholtz and Kirchhoff: Assume that the aperture is illuminated by a single spherical wave: S Ssource S r rkjA rU exp SS S Ssource S S S Ssource SSSS S nr r rkjA r kj n r rkjA nrU n rU 11 exp1 1 exp 1 SF SF FS rr rrkj rrG exp ; S S SF SF S rrr SFSS FS nr r rkj r kj n rr rrkj nrrG n rrG SF 11 exp1 1 exp 1, , P Fr Sr r 1S S R Screen Aperture 0P 0,0 1 1 S S n U U n U U , Q Sn1 Sn1 S Optical Diffraction 16. 16 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory dS n G U n U GrU F 4 1 The Integral of Helmholtz and Kirchhoff: Assume that the aperture is illuminated by a single spherical wave, and: Srr, SS S SsourceS nr r rkjA j n rU 11 exp2 r rkj rrG FS exp ; S FS nr r rkj j n rrG 11 exp2, Srr k 1 , 12 dS nrnr rr rrkjA jrU SSS s ssource F 2 1111 exp S Ssource S r rkjA rU exp P Fr Sr r 1S S R Screen Aperture 0P 0,0 1 1 S S n U U n U U , Q Sn1 Sn1 S Optical Diffraction 17. 17 Fresnel-Kirchhoff Diffraction Formula SOLO Fresnel-Kirchhoff Diffraction Theory dSK rr rrkj A dS nrnr rr rrkjA jrU S s s source SSS s ssource F , 2 exp 2 1111 exp SSSS S SSS S nrnr nrnr K 11cos&11cos 2 coscos 2 1111 , 1. Obliquity or Inclination Factor: 0,0&10,0 SS KK 2. Additional phase /2 3. The amplitude is scaled by the factor 1/ (not found in Fresnel derivation) P Fr Sr r 1S S R Screen Aperture 0P 0,0 1 1 S S n U U n U U , Q Sn1 Sn1 S We recovered Fresnel Diffraction Formula with: Optical Diffraction 18. 18 Reciprocity Theorem of Helmholtz SOLO Fresnel-Kirchhoff Diffraction Theory dSK rr rrkj A dS nrnr rr rrkjA jrU S s s source SSS s ssource F , 2 exp 2 1111 exp We can see that the Fresnel-Kirchhoff Diffraction Formula is symmetrical with respect to r and rS, i.e. point source and observation point. Therefore we can interchange them and obtain the same relation. This result is called Reciprocity Theorem of Helmholtz. P Fr Sr r 1S S R Screen Aperture 0P 0,0 1 1 S S n U U n U U , Q Sn1 Sn1 S Hermann von Helmholtz 1821-1894 Note: This is similar to Lorentzs Reciprocity Theorem in Electromagnetism. Optical Diffraction 19. 19 Huygens-Fresnel Principle SOLO Fresnel-Kirchhoff Diffraction Theory dSK rr rrkj A rU S s s source F , 2 exp The Fresnel Diffraction Formula can be rewritten as: dS r rkj QVrU F exp where: s s S source r rkj K A QV 2 exp , The interpretation of this formula is that each point of a wavefront can be considered as the center of a secondary spherical wave, and those secondary spherical waves interfere to result in the total field, is known as the Huygens-Fresnel Principle. P Fr Sr r 1S S R Screen Aperture 0P 0,0 1 1 S S n U U n U U , Q Sn1 Sn1 S Table of Content Optical Diffraction 20. 20 SOLO Consider a diffracting aperture . Suppose that the aperture is divided into two portions 1 and 2 such that = 1 + 2. The two aperture 1 and 2 are said to be complementary. Complementary Apertures. Babinet Principle From the Fresnel Diffraction Formula: 21 dS r rkj QVdS r rkj QV dS r rkj QVrU F expexp exp P Fr 1Sr 1 r 2 1S Screen Apertures 0P 1 Q Sn1 2Sr 2r 1 2Q We can see that the result is the added effect of all complimentary apertures. This is known as Babinet Principle. The result can be very helpful when is a very complicated aperture, that can be decomposed in a few simple apertures. Table of Content Optical Diffraction 21. 21 SOLO The Kirchhoff Diffraction Formula is an approximation since for zero field and normal derivative on any finite surface the field is zero everywhere. This was pointed out by Poincare in 1892 and by Sommerfeld in 1894. The first rigorous solution of a diffraction problem was given by Sommerfeld in 1896 for a two-dimensional case of a planar wave incident on an infinitesimally thin, perfectly conducting half plane. This solution is not given here. Arnold Johannes Wilhelm Sommerfeld 1868 - 1951 Jules Henri Poincar 1854-1912 Sommerfeld, A. : Mathematische Theorie der Diffraction, Math. Ann., 47:317, 1896 translated in english as Optics, Lectures on Theoretical Physics, vol. IV, Academic Press Inc., New York, 1954 Rayleigh-Sommerfeld Diffraction Formula Optical Diffraction 22. 22 SOLO Rayleigh-Sommerfeld Diffraction Formula Let start from the Helmholtz and Kirchhoff Integral: P Fr Sr r 1S S R Screen Aperture 0P 0,0 1 1 S S n U U n U U , Q Sn1 Sn1 S 0'PFr' SSSSS nnrrr 112' 1 4 1 S F dS n G U n U GrU Suppose that the Scalar Green Function is generated not only by P0 located at , but also by a point P0 located symmetric relative to the screen at SSSSS nnrrr 112' Sr G SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,_ SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp , or: We have 11 ,, ' SSFSSF rrrr SSSFSSSF nrrnrr 1'1 11 ,, 0 1, S G 011 exp1 2 11 ,, _ S S S nr r rkj r kj n G 0 1, S n G 0 exp 2 1 1 , , S S r rkj G Optical Diffraction 23. 23 SOLO Rayleigh-Sommerfeld Diffraction Formula 1. Start from the Helmholtz and Kirchhoff Integral: P Fr Sr r 1S S R Screen Aperture 0P 0,0 1 1 S S n U U n U U , Q Sn1 Sn1 S 0'PFr' SSSSS nnrrr 112' 1 4 1 S F dS n G U n U GrU SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp ,_ Choose 0 1, S G 011 exp1 2 11 ,, _ S S S nr r rkj r kj n G On the shadowed part of the screen and0 1 S U 0/ 1 S nU dSnr r rkj rU j dS n G UrU SS k r kj F 11 exp 4 1 /2 1 _ This is Rayleigh-Sommerfeld Diffraction Formula of the first kind SF rrr Arnold Johannes Wilhelm Sommerfeld 1868 - 1951 S Ssource S r rkjA UrU exp John William Strutt Lord Rayleigh (1842-1919) dSnr r rkj r rkjAj rU S S Ssource F 11 expexp we obtain: Optical Diffraction 24. 24 SOLO Rayleigh-Sommerfeld Diffraction Formula 2. Start from the Helmholtz and Kirchhoff Integral: P Fr Sr r 1S S R Screen Aperture 0P 0,0 1 1 S S n U U n U U , Q Sn1 Sn1 S 0'PFr' SSSSS nnrrr 112' 1 4 1 S F dS n G U n U GrU SF SF SF SF SF rr rrkj rr rrkj rrG ' 'expexp , Choose On the shadowed part of the screen and0 1 S U 0/ 1 S nU SF rrr 0 1, S n G 0 exp 2 1 1 , , S S r rkj G dS n U r rkj dS n U GrU F exp 2 1 4 1 S Ssource S r rkjA UrU exp SS S SsourceS nr r rkjA j n rU 11 exp2 dSnr r rkj r rkjAj rU SS S Ssource F 11 expexp For we obtain: This is Rayleigh-Sommerfeld Diffraction Formula of the second kind Table of Content Optical Diffraction 25. 25 P 0P Q 1x 0x 1y 0y Sr' Sr r O S Screen Image plane Source plane 0O 1O Sn1 - Screen Aperture Sn1 - normal to Screen 1r 0r SSS rn cos11 cos11 rnS z Sn1 'r Fr FrPP 0 SrQP 0 rQP SrOP '0 '1 rOO SOLO dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp Start with Fresnel-Kirchhoff (or Rayleigh-Sommerfeld) Diffraction Formula 1. If the inclination factor is nearly constant over the aperture constKK S , Extensions of Fresnel-Kirchhoff Diffraction Theory dS r rkj r rkjAKj rU S Ssource F expexp dS r rkj rU Kj rU SF exp 2. Replace the incident point source wavefront with a general waveform S S r rkjexp Sinc rU 3. Characterize the aperture by a transfer function to model amplitude or phase changes due to optic system dS r rkj rrU j rU SSF exp Table of Content Optical Diffraction 26. 26 SOLO Phase Approximations Fresnel (Near-Field) Approximation Fresnel Approximation or Near Field Approximation can be used when aperture dimensions are comparable to distance to source rS or image r. dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp Start with Fresnel-Kirchhoff Diffraction Formula If the inclination factor is nearly constant over the aperture constKK S , dS r rkj rU Kj dS r rkj r rkjAKj rU S S Ssource F expexpexp P Q 1x 1y r O Screen Image plane 1O Sn1 1r z Sn1 'r rQP '1 rOO P 0P Q 1x 0x 1y 0y Sr' Sr r O S Screen Image plane Source plane 0O 1O Sn1 - Screen Aperture Sn1 - normal to Screen 1r 0r SSS rn cos11 cos11 rnS z Sn1 'r Fr 1 ''2 ' ' 1''2' ' 2 2 1 2 1 2 11 2/1 2 2 1 2/1 11 0 1 2 1 r r k r r r r r rrrrrrr rrr x x '2 exp ' 'expexp 2 1 r r kj r rkj r rkj 2 max 2 1 2 1 ' '2 exp ' 'exp rrk dS r r kjrU r rkjKj rU SF Augustin Jean Fresnel 1788-1827 Optical Diffraction 27. 27 SOLO Phase Approximations Fraunhofer (Near-Field) Approximation Fraunhofer Approximation or Far Field Approximation can be used when aperture dimensions are very small comparable to distance to source rS or image r. dS r rkj r rkjAj rU S S Ssource F 2 coscosexpexp Start with Fresnel-Kirchhoff Diffraction Formula If the inclination factor is nearly constant over the aperture constKK S , dS r rkj rU Kj dS r rkj r rkjAKj rU S S Ssource F expexpexp P Q 1x 1y r O Screen Image plane 1O Sn1 1r z Sn1 'r rQP '1 rOO P 0P Q 1x 0x 1y 0y Sr' Sr r O S Screen Image plane Source plane 0O 1O Sn1 - Screen Aperture Sn1 - normal to Screen 1r 0r SSS rn cos11 cos11 rnS z Sn1 'r Fr ' exp ' 'expexp 1 r r kj r rkj r rkj 2 max 22 1 1 ' 2 ' exp ' 'exp rr k dS r r kjrU r rkjKj rU SF 1 '2' ' '2' ' ' 2 1''2' ' 2 22 11 2 22 11 2 11 2/1 2 2 1 2 1 2/1 11 0 1 2 1 r rk r r r r r r r r r rr rrrrrrr rrr x x Optical Diffraction 28. 28 0P Q 0x 0y Sr' Sr O S ScreenSource plane 0O Sn10r SrQP 0 S rOP '0 SOLO Fresnel and Fraunhofer Diffraction Approximations Fresnel Approximations at the Source S S SS S S xx x SS S S SSS SS r r rr r r rr r r rrr rr '2 '1 '2' ' ' '' ' 21' '2' ' 2 282 11 2/12 2 2/122 2 S S S S S S S r r kjrkj r rkj r rkj '2 '1 exp'1exp ' 'expexp 2 2 S S r rkj ' 'exp Srkj '1exp S S r r kj '2 '1 exp 2 2 Spherical wave centered at P0. Lowest order approximation to the phase of a spherical wavefront Planar wave propagating in directionSr'1 P 0P Q 1x 0x 1y 0y Sr' Sr r O S Screen Image plane Source plane 0O 1O Sn1 - Screen Aperture Sn1 - normal to Screen 1r 0r SSS rn cos11 cos11 rnS z Sn1 'r Fr Optical Diffraction 29. 29 SOLO Fresnel and Fraunhofer Diffraction Approximations P Q 1x 1y r O Screen Image plane 1O Sn1 1r z Sn1 'r rQP '1 rOO ''2 ' ' 2 1' '2' ' 1 22 1 2 11 2/1 2 2 1 2 1 2/1 11 0 1 2 1 r r r r r r rr r rrrrrr rrr x x ' exp '2 exp '2 exp ' 'expexp 1 22 1 r r kj r kj r r kj r rkj r rkj Fresnel Approximations at the Image plane P 0P Q 1x 0x 1y 0y Sr' Sr r O S Screen Image plane Source plane 0O 1O Sn1 - Screen Aperture Sn1 - normal to Screen 1r 0r SSS rn cos11 cos11 rnS z Sn1 'r Fr ' 'exp r rkj '1exp rkj '2 '1 exp 2 2 r r kj Spherical wave centered at O. Lowest order approximation to the phase of a spherical wavefront Planar wave propagating in direction'1r Optical Diffraction 30. 30 SOLO Fresnel and Fraunhofer Diffraction Approximations (1st way) dS r rkj r rkjAj rU SK S S Ssource F , 2 coscosexpexp Fresnel Approximation dS r rrr r r kjrrkjrrkj rr rrkjKAj rU S S S S Ssource F '2 '1 '2 '1 exp'1'1exp'1exp '' ''exp 2 1 2 1 2 2 1 Fraunhofer Approximation 1 '2 '1 '2 '12 2 1 2 1 2 2 S S k r rrr r r or S MAX rr ',' 2 dSrrkjrrkj rr rrkjKAj rU S S Ssource F '1'1exp'1exp '' ''exp 1 If 1 '2 '1 '2 '1 exp 2 1 2 1 2 2 S S r rrr r r kj we obtain Augustin Jean Fresnel 1788-1827 constKK S , Start with '1'1 rrq S P 0P Q 1x 0x 1y 0y Sr' Sr r O S Screen Image plane Source plane 0O 1O Sn1 - Screen Aperture Sn1 - normal to Screen 1r 0r SSS rn cos11 cos11 rnS z Sn1 'r Fr FrPP 0 SrQP 0 rQP SrOP '0 '1 rOO Optical Diffraction 31. 31 SOLO Fresnel and Fraunhofer Diffraction Approximations (2nd way) Fresnel Approximation dS r rr kjrrU r rkjj rU SSF '2 exp ' 'exp 11 Fraunhofer Approximation '1 '2 2 max 2 1 22 1 2 r r r r k If we obtain Augustin Jean Fresnel 1788-1827 Start with dS r rkj rrU j rU SSF exp - aperture optical transfer function Sr - disturbance at the aperture SrU dS r r kjrrU r rkjj rU SSF ' exp ' 'exp 1 ' exp '2 exp ' 'expexp 1 2 1 2 r r kj r r kj r rkj r rkj P Q 1x 1y r O Screen Image plane 1O Sn1 1r z Sn1 'r rQP '1 rOO 1' rrr 2 11 2/1 2 11 2/1 11 0 1 2 '2 1' ' 1''2' r rr r r rr rrrrrrr dS r r kj r r kjrrU r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1 2 '2' ' ' 2 1''2' 22 11 2 112/1 2 2 1 2 1 2/1 11 0 1 2 r r r r r r rr rrrrrrr x x Optical Diffraction 32. 32 SOLO Fresnel and Fraunhofer Diffraction Approximations Augustin Jean Fresnel 1788-1827 1x 1y max D Screen 1O 1r z 2 D R Fresnel Region Fraunhofer Region 2 D R R O '1 '2 2 max 2 1 22 1 2 r r r r k Fraunhofer Approximation If Optical Diffraction 33. 33 SOLO Fraunhofer Diffraction and the Fourier Transform dS r r kjrrU r rkjj rU SSF ' exp ' 'exp 1 11 1 ' 2 ' yx rr r k ddyx r jrrU r rkjj rU SSF 11 ' 2 exp ' 'exp The integral is the two dimensional Fourier Transform of the field within the aperture SS rrU fFTddkkjfkkF yxyx exp, 2 1 :, 2 SSF rrUFT r rkjj rU 2 2 ' 'exp Therefore P 0P Q 1x 0x 1y 0y Sr' Sr r O S Screen Image plane Source plane 0O 1O Sn1 - Screen Aperture Sn1 - normal to Screen 1r 0r SSS rn cos11 cos11 rnS z Sn1 'r Fr FrPP 0 SrQP 0 rQP SrOP '0 '1 rOO Two Dimensional Fourier Transform Optical Diffraction 34. 34 SOLO Fraunhofer Diffraction Approximations Examples Rectangular Aperture P Q 1x 1y r O Screen Image plane 1O Sn1 1r z Sn1 'r 1 2 1 2 1 1 1 1 11 0 2 11 2 10 ' 2 exp ' 2 exp '2 'exp ' exp ' exp '2 exp ' 'exp dy r jdx r j r rkjUkj dd r y kj r x kj r r kj r rkjUj rU k F elsevere U rrU SS 0 & 21110 For a Rectangular Aperture Therefore dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1 11 11 1 1 1111 1 ' 2 ' 2 sin 2 ' 2 ' 2 exp ' 2 exp ' 2 exp 1 1 x r x r x r j x r jx r j dx r j 11 11 1 1 1111 1 ' 2 ' 2 sin 2 ' 2 ' 2 exp ' 2 exp ' 2 exp 1 1 y r y r y r j y r jy r j dy r j 11 11 11 11 4/ 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp2 y r y r x r x r r rkjUkj rU A F Optical Diffraction 35. 35 SOLO Fraunhofer Diffraction Approximations Examples Rectangular Aperture (continue 1) Since U stands for scalar field intensity (E or H), the irradiance I is given by where < > is the time average and * is the complex conjugate. 11 11 11 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp8 y r y r x r x r Ar rkjUkj rU F FFF rUrUrI ~ Therefore 2 11 11 2 2 11 11 2 ' 2 ' 2 sin ' 2 ' 2 sin 0 y r y r x r x r IrI F I (0) is the irradiance at O1 (x1 = y1 = 0). Hecht pg.466 Optical Diffraction 36. 36 SOLO Fraunhofer Diffraction Approximations Examples Single Slit Aperture Let substitute in the rectangular aperture 1 0 where < > is the time average and * is the complex conjugate. 11 11 11 11 0 ' 2 ' 2 sin ' 2 ' 2 sin ' 'exp8 y r y r x r x r Ar rkjUkj rU F FFF rUrUrI ~ Therefore 2 11 11 2 ' 2 ' 2 sin 0 y r y r IrI F I (0) is the irradiance at O1 (x1 = y1 = 0). to obtain the single (vertical) slit diffraction 11 11 0 ' 2 ' 2 sin ' 'exp2 y r y r Ar rkjUkj rU FSLITSINGLE Since U stands for scalar field intensity (E or H), the irradiance I is given by Hecht, pg. 453 Hecht, pg. 456 Optical Diffraction 37. 37 SOLO Fraunhofer Diffraction Approximations Examples Single Slit Aperture (continue) 2 11 11 2 ' 2 ' 2 sin 0 y r y r IrI F I (0) is the irradiance at O1 (x1 = y1 = 0). Hecht, pg. 456 Hecht 455 Define: 11 ' 2 : y r 2 2 sin 0 II The extremum of I () is obtained from: 0 sincossin2 0 3 I d Id The results are given by: minimum,3,2,0sin maximum tan The solutions can be obtained graphically as shown in the figure and are: ,4707.3,4590.2,4303.1 Optical Diffraction 38. 38 SOLO Fraunhofer Diffraction Approximations Examples Double Slit Aperture d r x kjd r x kj r r kj r rkjUj rU ba ba ba ba F 2/ 2/ 1 2/ 2/ 1 2 10 ' exp ' exp '2 exp ' 'exp P Q 1x 1y r O Screen Image plane 1O Sn1 1r z Sn1 'r 1 b b a dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1 ax r j bx r bx r b x r j bax r jbax r j dx r j ba ba 1 1 1 1 112/ 2/ 1 ' exp ' ' sin 1 ' 2 ' exp ' exp ' 2 exp ax r j bx r bx r b x r j bax r jbax r j dx r j ba ba 1 1 1 1 112/ 2/ 1 ' exp ' ' sin 1 ' 2 ' exp ' exp ' 2 exp ax r bx r bx r br r kj r rkjUj rU F 1 1 12 10 ' cos ' ' sin 2 '2 exp ' 'exp ax r bx r bx r IrI F 1 2 2 1 1 2 ' cos ' ' sin 0 FFF rUrUrI ~ elsevere babababaU rrU SS 0 2/2/&2/2/0 Optical Diffraction 39. 39 SOLO Fraunhofer Diffraction Approximations Examples Double Slit Aperture (continue -= 1) Hecht p.458 2 2 2 12 2 1 12 cos sin 0 ' cos ' ' sin 0 I a r x b r x b r x IrI F P Q 1x 1y r O Screen Image plane 1O Sn1 1r z Sn1 'r 1 b b a The factor (sin / )2 that was previously found as the distribution function for a single slit is here the envelope for the interference fringes given by the term cos2. Bright fringes occur for = 0, ,2, The angular separation between fringes is = . Optical Diffraction 40. 40 Hecht 459 SOLO Fraunhofer Diffraction Approximations Examples Double Slit Aperture (continue 2) Optical Diffraction 41. 41 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture P 1y r Image plane 1O 1r Q O Screen Sn1 'r b b a a b b a b a The Aperture consists of a large number N of identical parallel slits of width b and separation a. 1 0 2/ 2/ 1 2 10 ' exp '2 exp ' 'exp N k bak bak F d r x kj r r kj r rkjUj rU dS r r kjrrU r r kj r rkjj rU SSF ' exp '2 exp ' 'exp 1 2 1 akx r j bx r bx r b x r j b kax r j b kax r j dx r j bka bka 1 1 1 1 112/ 2/ 1 ' 2 exp ' ' sin 1 ' 2 2' 2 exp 2' 2 exp ' 2 exp ax r aNx r bx r bx r br r kj r rkjUj ax r j aNx r j bx r bx r br r kj r rkjUj akx r j bx r bx r br r kj r rkjUj rU N k F 1 1 1 12 10 1 1 1 12 10 1 0 1 1 12 10 ' sin ' sin ' ' sin 1 '2 exp ' 'exp ' 2 exp1 ' 2 exp1 ' ' sin 1 '2 exp ' 'exp ' 2 exp ' ' sin 1 '2 exp ' 'exp elsevere NkbkabkaU rrU SS 0 1,,1,02/2/0 Optical Diffraction 42. 42 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue 1) P 1y r Image plane 1O 1r Q O Screen Sn1 'r b b a a b b a b a The Aperture consists of a large number N of identical parallel slits of width b and separation a. ax r aNx r bx r bx r br r kj r rkjUj rU F 1 1 1 12 10 ' sin ' sin ' ' sin 1 '2 exp ' 'exp 22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F FFF rUrUrI ~ Optical Diffraction 43. 43 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue 2) Hecht p.462 Hecht p.463 22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F Optical Diffraction 44. 44 SOLO Fraunhofer Diffraction Approximations Examples Multiple Slit Aperture (continue 2) 22 2 2 2 1 22 1 2 2 1 1 2 sin sinsin 0 ' sin ' sin ' ' sin 0 N N I ax r N aNx r bx r bx r IrI F Sears p.222 Hecht p. 462 Sears p.236 Interference Irradiation for 1, 2, 3 and 4 slits as function of observation angle. Diffraction Pattern for 1, 2, 3, 4 and 5 slits. Optical Diffraction 45. 45 SOLO Resolution of Optical Systems According to Huygens-Fresnel Principle, a differential area dS, within an optical Aperture, may be envisioned as being covered with coherent secondary point sources. z y Z Y R q sincos yz sincos qYqZ Differential area dS, coordinates Image , coordinates dddS dSe r E dE rktiA The spherical wave that propagates from dS to Image is where 22/122/1222 /1/21 RZzYyRRZzYyRzZyYXr 2/1222 ZYXR RkaqJkaqRe R E ddee R E dSee R E dEE RktiA a RkpqiRktiA Aperture RzZyYkiRktiA Aperture // 1 0 2 0 cos// The spherical wave at Image, for a Circular Aperture, is Optical Diffraction 46. 46 SOLO Resolution of Optical Systems z y Z Y R q RkaqJkaqRe R E E RktiA // 1 where 2 0 cos 2 dve i uJ vuvmi m m Bessel Function (of the first kind) E. Hecht, Optics The spherical wave at Image, for a Circular Aperture, is Optical Diffraction 47. 47 SOLO Resolution of Optical Systems z y Z Y R q Irradiance EEHEHESI EH 2 1 2 1 2 1 2 1 2 1 2 22 / /2 0 / /22 2 1 Rkaq RkaqJ I Rkaq RkaqJ R aE EEI A Daaak RkaquuJ n Rq 22.1 2 22.1 2 83.383.3 sin83.3/0 sin/ 11 D nnn 44.22 E. Hecht, Optics Circular Aperture Optical Diffraction 48. 48 SOLO Resolution of Optical Systems z y Z Y R qDistribution of Energy in the Diffraction Pattern at the Focus of a Perfect Circular Lens E. Hecht, Optics Ring f/(f#) Peak Energy in ring Illumination (%) Central max 0 1 83.9 1st dark ring 1.22 0 1st bright ring 1.64 0.017 7.1 2nd dark 2.24 0 2nd bright 2.66 0.0041 2.8 3rd dark 3.24 0 3rd bright 3.70 0.0016 1.5 4th dark 4.24 0 4th bright 4.74 0.00078 1.0 5th dark 5.24 0 Optical Diffraction 49. 49 SOLO Fraunhofer Diffraction Approximations Examples Circular Aperture Hecht p.469 Optical Diffraction 50. 50 SOLO Resolution of Optical SystemsAiry Rings In 1835, Sir George Biddell Airy, developed the formula for diffraction pattern, of an image of a point source in an aberration-free optical system, using the wave theory. E. Hecht, Optics Optical Diffraction 51. 51 SOLO Resolution of Optical Systems E. Hecht, Optics Optical Diffraction 52. 52 Rayleighs Criterion (1902) The images are said to be just resolved when the center of one Airy Disk falls on the first minimum of the Airy pattern of the other image. The minimum resolvable angular separation or angular limit is: D nnn 44.22 Sparrows Criterion At the Rayleighs limit there is a central minimum Or saddle point between adjacent peaks. Decreasing the distance between the two point sources cause the central dip to grow shallower and ultimately to disappear. The angular separation corresponding to that configuration is the Sparrows Limit. SOLO Resolution of Optical Systems Optical Diffraction 53. 53 Resolution Diffraction Limit Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error,University of California, Berkley Optical Diffraction 54. 54 Diffraction limit to resolution of two close point-object images: best resolution is possible when the two are of near equal, optimum intensity. As the two PSF merge closer, the intensity deep between them rapidly diminishes. At the center separation of half the Airy disc diameter - 1.22/D radians (138/D in arc seconds, for =0.55 and the aperture diameter D in mm), known as Rayleigh limit - the deep is at nearly 3/4 of the peak intensity. Reducing the separation to /D (113.4/D in arc seconds for D in mm, or 4.466/D for D in inches, both for =0.55) brings the intensity deep only ~4% bellow the peak. This is the conventional diffraction resolution limit, nearly identical to the empirical double star resolution limit, known as Dawes' limit. With even slight further reduction in the separation, the contrast deep disappears, and the two spurious discs merge together. The separation at which the intensity flattens at the top is called Sparrow's limit, given by 107/D for D in mm, and 4.2/D for D in inches (=0.55). Optical Diffraction 55. 55 SOLO Fresnel Diffraction Approximations Examples Rectangular Aperture dS r rr kjrrU r rkjj rU SSF '2 exp ' 'exp 11 define Augustin Jean Fresnel 1788-1827 P Q 1x 1y r O Screen Image plane 1O Sn1 1r z Sn1 'r 1 2 1 2 2 1 2 1 ' 2 2 exp ' 2 2 exp '2 'exp '2 exp '2 exp ' 'exp 2 1 2 10 2 2 1 2 10 d r y jd r x j r rkjUkj dd r y kj r x kj r rkjUj rU k F elsevere U rrU SS 0 & 21110 For a Rectangular Aperture d r d r x ' 2 ' 2 : 2 12 2 1 2 1 2 2 1 2 exp 2 ' ' 2 2 exp dj r d r x j 212111 ' 2 & ' 2 x r x r Therefore d r d r y ' 2 ' 2 : 2 12 212111 ' 2 & ' 2 y r y r 2 1 2 1 2 2 1 2 exp 2 ' ' 2 2 exp dj r d r y j Optical Diffraction 56. 56 SOLO Fresnel Diffraction Approximations Examples Rectangular Aperture (continue 1) Augustin Jean Fresnel 1788-1827 P Q 1x 1y r O Screen Image plane 1O Sn1 1r z Sn1 'r 1 2 1 2 2 1 2 1 2 1 2 1 220 2 1 2 10 2 exp 2 exp 2 'exp ' 2 2 exp ' 2 2 exp ' 'exp djdj rkjUj d r y jd r x j r rkjUj rU F Define Fresnel Integrals 0 2 0 2 2 sin: 2 cos: dS dC SjCdj 0 2 2 exp 2 1 2 1 2 'exp0 SjCSjC rkjUj rU F Using the Fresnel Integrals we can write 5.0 SC Optical Diffraction 57. 57 SOLO Fresnel Diffraction Approximations Examples Rectangular Aperture (continue 2) Augustin Jean Fresnel 1788-1827 Hecht p.499 Optical Diffraction 58. 58 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 P Q 1x 1y r O Screen Image plane 1O Sn1 1r z Sn1 'r 1 2 1 2 Optical Diffraction 59. 59 SOLO Fresnel Diffraction Approximations Examples Cornu Spiral Fresnel Integrals are defined as uu duuuSduuuC 0 2 0 2 2 sin:& 2 cos: uSjuCduuj u 0 2 2 exp 5.0 SC Marie Alfred Cornu professor at the cole Polytechnique in Paris established a graphical approach, for calculating intensities in Fresnel diffraction integrals. The Cornu Spiral is defined as the plot of S (u) versus C (u) duuSd duuCd 2 2 2 sin 2 cos duSdCd 22 Therefore u may be thought as measuring arc length along the spiral. Mthode nouvelle pour la discussion des problmes de diffraction dans le cas dune onde cylindrique, J.Phys.3 (1874), 5-15,44-52 Optical Diffraction 60. 60 SOLO Fresnel Diffraction Approximations Examples Cornu Spiral (continue 1) uu duuuSduuuC 0 2 0 2 2 sin:& 2 cos: uSjuCduuj u 0 2 2 exp 5.0 SC The Cornu Spiral is defined as the plot of S (u) versus C (u) duuSdduuCd 22 2 sin& 2 cos duSdCd 22 2 2 2 2 tan 2 cos 2 sin u u u Cd Sd Therefore every point on the curve makes the angle with the real ( C ) axis. 2 2 u The radius of curvature of Cornu Spiral is The tangent vector of Cornu Spiral is SuCuT 1 2 sin1 2 cos 22 u SuCuu udTdSdCdTd 1 1 2 cos1 2 sin 1 / 1 / 1 22 22 showing that the curve spirals toward the limit points. 2 1 2 2 cos u u duu 2 1 2 2 sin u u duu 2 1 2 2 exp u u duuj Table of Content Optical Diffraction 61. 61 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Circular Aperture Hecht p.491 Hecht p.492 Optical Diffraction 62. 62 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Circular Obstacles Optical Diffraction 63. 63 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Fresnel Zone Plate Optical Diffraction 64. 64 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Fresnel Diffraction by a Slit Hecht p.504 a Fresnel Diffraction Hecht p.504 b Optical Diffraction 65. 65 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Semi-Infinite Opaque Screen Hecht p.506 a Hecht p.506 Optical Diffraction 66. 66 SOLO Fresnel Diffraction Approximations Examples Augustin Jean Fresnel 1788-1827 Semi-Infinite Opaque Screen Hecht p.506 a Hecht p.507 Optical Diffraction 67. 67 SOLO Optics Optical Transfer Function (OTF) 68. 68 Point Spread Function (PSF) The Point Spread Function, or PSF, is the image that an Optical System forms of a Point Source. The PSF is the most fundamental object, and forms the basis for any complex object. PSF is the analogous to Impulse Response Function in electronics. 2 , yxPFTPSF The PSF for a perfect optical system (with no aberration) is the Airy disc, which is the Fraunhofer diffraction pattern for a circular pupil. SOLO Optics 69. 69 Point Spread Function (PSF) As the pupil size gets larger, the Airy disc gets smaller. SOLO Optics 70. 70 Convolution yxIyxOyxPSF ,,, yxIyxOFTyxPSFFTFT ,,,1 Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error,University of California, Berkley SOLO Optics 71. 71 Modulation Transfer Function (MTF) Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error,University of California, Berkley SOLO Optics 72. 72 Modulation Transfer Function (MTF) The Modulation Transfer Function (MTF) indicates the ability of an Optical System to reproduce various levels of details (spatial frequencies) from the object to image. Its units are the ratio of image contrast over the object contrast as a function of spatial frequency. 3.57 a fcutoff Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error,University of California, Berkley SOLO Optics 73. 73 Modulation Transfer Function (MTF) Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error,University of California, Berkley SOLO Optics 74. 74 Phase Transfer Function (PTF) PTF contains information about asymmetry in PSF PTF contains information about contrast reversals (spurious resolution) Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error,University of California, Berkley SOLO Optics 75. 75 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function The Point Spread Function (PSF) is the Fourier Transform (FT) of the pupil function The Modulation Transfer Function (MTF) is the amplitude component of the FT of the PSF The Phase Transfer Function (PTF) is the phase component of the FT of the PSF The Optical Transfer Function (OTF) composed of MTF and PTF can also be computed as the autocorrelation of the pupil function. yxWi eyxPFTyxPSF , 2 ,, iiyx yxPSFFTAmplitudeffMTF ,, iiyx yxPSFFTPhaseffPTF ,, yxyxyx ffPTFiffMTFffOTF ,exp,, SOLO Optics 76. 76 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function yxWi eFTyxPSF , 2 , iiyx yxPSFFTAmplitudeffMTF ,, iiyx yxPSFFTPhaseffPTF ,, Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error, University of California, Berkley SOLO Optics Ideal Optical System 77. 77 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function yxWi eFTyxPSF , 2 , iiyx yxPSFFTAmplitudeffMTF ,, iiyx yxPSFFTPhaseffPTF ,, Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error, University of California, Berkley SOLO Optics Real Optical System 78. 78 Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function yxWi eFTyxPSF , 2 , iiyx yxPSFFTAmplitudeffMTF ,, iiyx yxPSFFTPhaseffPTF ,, Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error, University of California, Berkley SOLO Optics Real Optical System 79. 79 FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled. (a) the effect of 1/4 and 1/2 wave P-V wavefront error of defocus on the PSF intensity distribution (left) and image contrast (right). Doubling the error nearly halves the peak diffraction intensity, but the average contrast loss nearly triples (evident from the peak PSF intensity). (b) 1/4 and 1/2 wave P-V of spherical aberration. While the peak PSF intensity change is nearly identical to that of defocus, wider energy spread away from the disc results in more of an effect at mid- to high- frequency range. Central disc at 1/2 wave P-V becomes larger, and less well defined. The 1/2 wave curve indicates ~20% lower actual cutoff frequency in field conditions. http://www.telescope-optics.net/ SOLO Optics 80. 80 FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled. (c) 0.42 and 0.84 wave P-V wavefront error of coma. Both, intensity distribution (PSF) and contrast transfer change with the orientation angle, due to the asymmetric character of aberration. The worst effect is along the axis of aberration (red), or length-wise with respect to the blur (0 and orientation angle), and the least is in the orientation perpendicular to it (green). (d) 0.37 and 0.74 wave P-V of astigmatism. Due to the tighter energy spread, there is less of a contrast loss with larger, but more with small details, compared to previous wavefront errors. Contrast is best along the axis of aberration (red), falling to the minimum (green) at every 45 (/4), and raising back to its peak at every 90. The PSF is deceiving here: since it is for a linear angular orientation, the energy spread is lowest for the contrast minima. http://www.telescope-optics.net/ SOLO Optics 81. 81 (e) Turned down edge effect on the PSF and MTF. The P-V errors for 95% zone are 2.5 and 5 waves as needed for the initial 0.80 Strehl (the RMS is similarly out of proportion). Lost energy is more evenly spread out, and the central disc becomes enlarged. Odd but expected TE property - due to the relatively small area of the wavefront affected - is that further increase beyond 0.80 Strehl error level does almost no additional damage. f) The effect of ~1/14 and ~1/7 wave RMS wavefront error of roughness, resulting in the peak intensity and contrast drop similar to those with other aberrations. Due to the random nature of the aberration, its nominal P-V wavefront error can vary significantly for a given RMS error and image quality level. Shown is the medium- scale roughness ("primary ripple" or "dog biscuit", in amateur mirror makers' jargon) effect. (g) 0.37 and 0.74 wave P-V of wavefront error caused by pinching having the typical 3-sided symmetry (trefoil). The aberration is radially asymmetric, with the degree of pattern deformation varying between the maxima (red MTF line, for the pupil angle =0, 2/3, 4/3), and minima (green line, for =/3, , 5/3); (the blue line is for a perfect aperture). Other forms do occur, with or without some form of symmetry. FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled. 82. 82 h) 0.7 and 1.4 wave P-V wavefront error caused by tube currents starting at the upper 30% of the tube radius. The energy spreads mainly in the orientation of wavefront deformation (red PSF line, to the left). Similarly to the TE, further increase in the nominal error beyond a certain level has relatively small effect Contrast and resolution for the orthogonal to it pattern orientation are as good as perfect (green MTF line). (i) Near-average PSF/MTF effect of ~1/14 and ~1/7 wave RMS wavefront error of atmospheric turbulence. The atmosphere caused error fluctuates constantly, and so do image contrast and resolution level. Larger seeing errors (1/7 wave RMS is rather common with medium-to-large apertures) result in a drop of contrast in the mid- and high- frequency range to near-zero level. FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled. http://www.telescope-optics.net/ SOLO Optics 83. 83 Other Metrics that define Image Quality Strehl Ratio Strehl, Karl 1895, Aplanatische und fehlerhafte Abbildung im Fernrohr, Zeitschrift fr Instrumentenkunde 15 (Oct.), 362-370. Dr. Karl Strehl 1864 -1940 One of the most frequently used optical terms in both, professional and amateur circles is the Strehl ratio. It is the simplest meaningful way of expressing the effect of wavefront aberrations on image quality. By definition, Strehl ratio - introduced by Dr. Karl Strehl at the end of 19th century - is the ratio of peak diffraction intensities of an aberrated vs. perfect wavefront. The ratio indicates image quality in presence of wavefront aberrations; often times, it is used to define the maximum acceptable level of wavefront aberration for general observing - so-called diffraction-limited level - conventionally set at 0.80 Strehl. SOLO Optics 84. 84 The Strehl ratio is the ratio of the irradiance at the center of the reference sphere to the irradiance in the absence of aberration. Irradiance is the square of the complex field amplitude u 0 E E Strehl 2 uE dxdyyxWjUu )),(2exp(0 Other Metrics that define Image Quality Strehl Ratio Expectation Notation dxdy dxdyyxu uu ),( SOLO Optics 85. 85 Derivation of Strehl Approximation 2 0 21 W E E Strehl ),(2 0 yxWj eUu 22 0 ),(2 2 1 ),(21 yxWyxWjUu 2 0 2 00 ),(2 2 1 ),(2 yxWUyxWUjUu series expansion 2 0 22 0 2 0 ),(2),(2 yxWEyxWEEE multiply by complex conjugate 2222 ),(),(),(),( yxWyxWyxWyxWW wavefront variance: SOLO Optics 86. 86 2 0 21 W E E Strehl 22 2 ),(),( yxWyxWWWW where W is the wavefront variance: 2 2 W eStrehl Another approximation for the Strehl ratio is Strehl Approximation Diffraction Limit 8.0Strehl A system is diffraction-limited when the Strehl ratio is greater than or equal to 0.8 Marchals criterion: This implies that the rms wavefront error is less than /13.3 or that the total wavefront error is less than about /4. SOLO Optics 87. 87 Other Metrics that define Image Quality Strehl Ratio dl eye H H RatioStrehl Austin Roorda, Review of Basic Principles in Optics, Wavefront and Wavefront Error,University of California, Berkley SOLO Optics 88. 88 Other Metrics that define Image Quality Strehl Ratio 2m n Crms when rms is small 2 2 2 1 rmsStrehl SOLO Optics 89. 89 Other Metrics that define Image Quality FIGURE 34: Pickering's seeing scale uses 10 levels to categorize seeing quality, with the level 1 being the worst and level 10 near-perfect. Its seeing description corresponding to the numerical seeing levels are: 1-2 "very poor", 3 "poor to very poor", 4 "poor", 5 "fair", 6 "fair to good" 7 "good", 8 "good to excellent", 9 "excellent" and 10 "perfect". Diffraction-limited seeing error level (~0.8 Strehl) is between 8 and 9. Pickering 1 Pickering 2 Pickering 3 Pickering 4 Pickering 5 Pickering 6 Pickering 7 Pickering 9 Pickering 10Pickering 8 William H. Pickering (1858-1938) SOLO Optics 90. 90 Other Metrics that define Image Quality FIGURE: Illustration of a point source (stellar) image degradation caused by atmospheric turbulence. The left column shows best possible average seeing error in 2 arc seconds seeing (ro~70mm @ 550nm) for four aperture sizes. The errors are generated according to Eq.53-54, with the 2" aperture error having only the roughness component (Eq.54), and larger apertures having the tilt component added at a rate of 20% for every next level of the aperture size, as a rough approximation of its increasing contribution to the total error (the way it is handled by the human eye is pretty much uncharted territory). The two columns to the right show one possible range of error fluctuation, between half and double the average error. The best possible average RMS seeing error is approximately 0.05, 0.1, 0.2 and 0.4 wave, from top to bottom (the effect would be identical if the aperture was kept constant, and ro reduced). The smallest aperture is nearly unaffected most of the time. The 4" is already mainly bellow "diffraction-limited", while the 8" has very little chance of ever reaching it, even for brief periods of time. The 16" is, evidently, affected the most. The D/ro ratio for its x2 error level is over 10, resulting in clearly developed speckle structure. Note that the magnification shown is over 1000x per inch of aperture, or roughly 10 to 50 times more than practical limits for 2"-16" aperture range, respectively. At given nominal magnification, actual (apparent) blur size would be smaller inversely to the aperture size. It would bring the x2 blur in the 16" close to that in 2" aperture (but it is obvious how a further deterioration in seeing quality would affect the 16" more). Eugne Michel Antoniadi (1870 , 1944) The scale, invented by Eugne Antoniadi, a Greek astronomer, is on a 5 point system, with one being the best seeing conditions and 5 being worst. The actual definitions are as follows: I. Perfect seeing, without a quiver. II. Slight quivering of the image with moments of calm lasting several seconds. III. Moderate seeing with larger air tremors that blur the image. IV. Poor seeing, constant troublesome undulations of the image. V. Very bad seeing, hardly stable enough to allow a rough sketch to be made. Image Degradation Caused by Atmospheric Turbulence SOLO Optics 91. 91 iiyx yxPSFFTAmplitudeffMTF ,, yxWi eFTyxPSF , 2 , Point Spread Function SOLO Optics 92. 92 yxWi eFTyxPSF , 2 , iiyx yxPSFFTAmplitudeffMTF ,, Point Spread Function SOLO Optics 93. 93 iiyx yxPSFFTAmplitudeffMTF ,, yxWi eFTyxPSF , 2 , Point Spread Function SOLO Optics 94. 94 SOLO converging beam = spherical wavefront parallel beam = plane wavefront Image Plane Ideal Optics ideal wavefront parallel beam = plane wavefront Image Plane Non-ideal Optics defocused wavefront ideal wavefrontparallel beam = plane wavefront Image Plane Non-ideal Optics aberrated beam = iregular wavefront diverging beam = spherical wavefront aberrated beam = irregular wavefront Image Plane Non-ideal Optics ideal wavefront Optical Aberration See full development in P.P. Optical Aberration 95. 95 SOLO converging beam = spherical wavefront parallel beam = plane wavefront Image Plane Ideal Optics P' Optical Aberration converging beam = spherical wavefront Image Plane Ideal Optics diverging beam = spherical wavefront P P' An Ideal Optical System can be defined by one of the three different and equivalent ways: All the rays emerging from a point source P, situated at a finite or infinite distance from the Optical System, will intersect at a common point P, on the Image Plane. 1 All the rays emerging from a point source P will travel the same Optical Path to reach the image point P. 2 The wavefront of light, focused by the Optical System on the Image Plane, has a perfectly spherical shape, with the center at the Image point P. 3 Ideal Optical System 96. 96 SOLO ideal wavefrontparallel beam = plane wavefront Image Plane Non-ideal Optics aberrated beam = iregular wavefront diverging beam = spherical wavefront aberrated beam = irregular wavefront Image Plane Non-ideal Optics ideal wavefront Optical Aberration Real Optical System An Aberrated Optical System can be defined by one of the three different and equivalent ways: The rays emerging from a point source P, situated at a finite or infinite distance from the Optical System, do not intersect at a common point P, on the Image Plane. 1 The rays emerging from a point source P will not travel the same Optical Path to reach the Image Plane 2 The wavefront of light, focused by the Optical System on the Image Plane, is not spherical. 3 97. 97 Optical Aberration W (x,y) is the path deviation between the distorted and reference Wavefront. SOLO Optical Aberration 98. 98 SOLO Optical Aberration Display of Optical Aberration W (x,y) Rays Deviation1 Optical Path Length Difference2 wavefront shape W (x,y)3 x y x x y yxW , y yxW , Red circle denotes the pupile margin. Arrows shows how each ray is deviated as it emerges from the pupil plane. Each of the vectors indicates the the local slope of W (x,y). The aberration W (x,y) is represented in x,y plane by color contours. Wavefront Error Optical Distance Errors Ray Errors The Wavefront error agrees with Optical Path Length Difference, But has opposite sign because a long (short) optical path causes phase retardation (advancement). Aberration Type: Negative vertical coma Reference 99. 99 SOLO Optical Aberration Display of Optical Aberration W (x,y) Advanced phase