Chapter 6 Aberrations

Transcription

1 EE90F Chapter 6 Aberrations As we have seen, spherical lenses only obey Gaussian lens law in the paraxial approxiation. Deviations fro this ideal are called aberrations. F Rays toward the edge of the pupil (even parallel to the axis) violate the paraxial condition on the incidence angle at the first surface. They focus closer (for a biconvex lens) than F. No truly sharp focus occurs. The least blurred spot (sallest disc) is called circle of least confusion, or best focus. This for of syetric aberration is spherical aberration. There are any fors of aberration. Coa: Variation of agnification with aperture. Rays passing through edge portions of the pupil are iaged at a different height than those passing through the center. Aberrations_post.f Chapter 6

2 EE90F Map of Rays in Pupil Iage Plane H A A E A B C G C P B D F D F H A B C D C D G E B Astigatis tangential rays tangential iage sagital iage sagital rays Aberrations_post.f Chapter 6

3 EE90F In astigatis the tangential and sagittal iages do not coincide. There are line iages with a circle of least confusion in the iddle. Field Curvature Object plane Lens Positive lenses give inward curvature negative lenses give backward curvature. Iage points lie on a curved surface, not a plane Distortion: Field dependent agnification Barrel distortion Pincushion distortion Wave Front Aberration In a wave-optics picture, the thin lens is represented by phase delay. φ( x, y) k x + y k ( x, y) (6.) Which gives Gaussian iaging. Aberrations odify φ. A spherical lens only gives this φ in the paraxial approxiation. We collect these phase errors at the exit pupil and use the to define a generalized pupil function: W(x,y) : path length error (6.) Aberrations_post.f Chapter 6

4 EE90F ideal wave front aberrated wavefront Expressed in this way, the priary aberrations are written as follows, with coordinate in the pupil, and h : the iage height ρ: noralized radial Spherical aberration: Coa: Astigatis: Field Curvature: A d ρ h Distortion: A t h 3 ρcos θ Monochroatic Aberrations: All of the preceding discussion refers to aberrations that do not depend on wavelength. Chroatic Aberrations: Dependance of wavefront on wavelength. Consider the siple thin lens equation: (6.3) The index n is generally λ dependent, n( λ), so f is λ dependent. Violet iage red iage Change in iage distance: longitudinal chroatic aberration Aberrations_post.f Chapter 6

5 EE90F Change in agnification: lateral color. Lateral color is usually ore noticeable Achroat: lens designed to cancel chroatic aberration. Lens Design: The general proble of lens design involves cancelling aberrations Aberration depends on the lens index, as well as the surface radii. Coplex lens systes can iniize aberrations Siple singlet case: For a given desired focal length, there is freedo to choose one of the radii for a singlet. The spherical aberration and coa depend on the particular choice, so these aberrations can be iniized by the design for. This is illustrated in the following diagra: design: 00 focal length ± 7 field. Spherical aberration coa R 5R 50 R R 50 R 5 R 6.7 Optiu for (nearly plano-convex) Fourier-space treatent of aberrations Recall that the ideal effect of the lens is to ipart a quadratic phase shift, ultiplied by the aperture function. The aplitude ipulse response: h(u,v) is a scaled Fourier transfor of the pupil function. This can be generalized to the Fourier transfor of P(x,y). huv (, ) λ P( x, y) exp j π ( ux+ vy) z o z λz i i dx dy Pxy (, ) exp[ jkw( xy, )] exp j π ( ux+ vy) λ z o z λz i i dx dy (6.4) (6.5) The aplitude transfer function is still the Fourier transfor of h, so Aberrations_post.f Chapter 6

6 EE90F Hf ( x ) P( λz i f x, λz i f y ) P( λz i f x, λz i f y ) exp[ jkw( λz i f x, λz i f y )] (6.6) Aberrations introduce a pure phase distortion within an unaffected passband: P( λz i f x, λz i f y ) The effects with coherent illuination are described by H. For incoherent illuination - we ust consider the optical transfer function: H( f x ). Recall that the optical transfer function for a diffraction liited case is the noralized overlap area of displaced pupils. Define A( f x ) as the overlap area of A and (6.7) Then for a diffraction liited syste, the optical transfer function is H( f x ) Af ( x ) dx dy dxdy A( 00, ) (6.8) With aberrations, the optical transfer function becoes: H( f x ) λz jk W x i f x λz y i f, y λz Wx i f x λz y i f, y exp dxdy Af ( x ) dxdy A( 00, ) (6.9) The aberrations can only decrease the odulation transfer function. It is easy to show that: H( f x ) aberrations H( f x ) diffraction liited (6.0) Aberrations_post.f Chapter 6

7 EE90F These aberrations do not reduce the absolute cutoff frequency, but they do reduce the contrast at soe frequencies. For a diffraction liited syste, the optical transfer function is positive definite. With aberrations, the optical transfer function can be negative in certain frequency bands, resulting in a contrast reversal. Calculation of the optical transfer function generally requires nuerical calculation. An analytical solution is possible for a siple focal error. We need to assue a square aperture. The diffraction liited OTF for a square aperture with a side w is: H( f x ) Λ f x f f o Λ y f o ; fo w : cutoff of coherent syste λz i Λ( x).5 f o f o With a focal error, the phase at the pupil is quadratic, but with a radius of. It does not converge exactly to the iage plane. z a z i ideal phase φ( xy, ) π ( x + y ) λz i actual phase φ a ( x, y) π ( x + y ) λz a so Wxy (, ) At the edge of the aperture along the x or y axis Ww0 (, ) W ( x + y ) z a z i w z a z i (6.) (6.) (6.3) (6.4) Aberrations_post.f Chapter 6

8 EE90F W is the easure of the agnitude of error. The analytical solution for the OTF is then: H( f x ) Λ f x f f o Λ y 8W f o sinc f λ f o x 8W f o sinc f λ f o y f o f x f y (6.5) Paraetrized in H W λ, the nuber of waves of the path length difference: W λ Higher order aberrations Most coon systes are rotationally syetric with a circular pupil. We can use polar coordinates for pupil points: Wxy (, ) Wr ( cosθ, rsin. The aberration function W depends on the object height, h (or iage height h' ) and the pupil coordinates ( r,. If we use a noralized radius, and suppress the explicit dependence on iage height h', A d : defocus A t : tilt W( ρ, A s ρ 4 + A c ρ 3 cosθ + A a ρ cos θ + A d ρ + A t ρcos θ (6.6) The field curvature and distortion are equivalent to the field dependent focus and tilt errors, respectively. A coonly used figure of erit: the Strehl ratio h( 00, ) S h( 00, ) aberrated aberration-free aberration free aberrated Define ρ r w ; Φρθ (, ) π W( ρ, λ Aberrations_post.f Chapter 6

9 EE90F S π exp[ jφ( ρθ, )]ρdρdθ π (6.7) For sall aberrations, expand exp( jφ) + jφ - Φ Then S π Φ + jφ ρdρdθ π (6.8) π Φ ρdρdθ π π π Φρdρdθ (6.9) Define the wavefront variance π S - Φ ρdρdθ π Φρdρdθ σ W π π (6.0) π σ W - [ W( ρ, W π av ] ρdρdθ (6.) π - W ( ρ, ρdρdθ π W av (6.) with Thus, we can write π W av - W( ρ, ρdρdθ π (6.3) (6.4) The Strehl ratio depends only on the idea of balanced aberrations. σ φ Exaple: we can balance spherical aberration with defocus - not on the details of aberration function. This leads us to Aberrations_post.f Chapter 6

10 EE90F Φρ ( ) A s ρ 4 + A d ρ We choose just the right aount of defocus to iniize the wavefront variance: (6.5), (6.6) which leads to. A d A s Balanced aberration: An aberration of a certain order in a power expansion in pupil coordinates is ixed with lower order aberrations to iniize. σ φ Zernike circle polynoials See Handout: Mahajan, Appl. Opt. 33, 84 (994). For further detail: Mahajan, Aberration theory ade siple SPIE press, 99 Balanced Orthonoral over a unit circle Coplete set n φ( ρ, kw( ρ, { ε ( n + )R n ( ρ) [ Cn cosθ + S n sinθ] } n 0 0 (6.7) (6.8) where n and are positive integers, with n 0, and always even. The radial function : Wavefront variance: ( n ) R n ( ρ) ( ) s ( n s)! s! [( n + ) s]! [( n ) s]! ρ n s s 0 (6.9) W ( ρ, [ C n + S n ] n (6.30) The Zernike polynoials are typically given an ordering. However, be careful! The ordering is not universally agreed to. Different texts and even different lens design and optical iaging analysis software use their own ordering convention. Aberrations_post.f Chapter 6

11 EE90F W( ρ, a j Z j ( ρ, j The table in the handout gives one of the possible orderings {j, n, } Z evenj ( n + )R n ( ρ) cosθ 0 Z oddj ( n + )R n ( ρ) sinθ 0 Z j n + R n ( ρ) 0 (6.3) (6.3) (6.33) (6.34) Point spread functions for Zernike s (circular pupils) We write the aberrated pupil function inside the unit circle as P( ρ, e iφ( ρ, ρ r w Expand φ( ρ, φ( ρ, kw( ρ, in Zernike polynoials phase aberration (6.35) We wish to calculate the aberrated point spread function: [object radial coordinates: (δ, Ψ)] Take the Fourier transfor of P( ρ, : π h( δ, Ψ) e jφ r, θ ( ) e jπrδcos( θ Ψ) If the aberration is sall φ «π, e jφ + jφ + rdr dθ δ Ψ where + jφ α' n R cosθ n n sinθ, (6.36) α' n jα n n 0 (6.37) α' oo + jα oo n 0 π h( δ, Ψ) α' n R n ( ρ) cosθ jπrδ θ Ψ e sinθ n cos( ) rdr dθ (6.38) (6.39) Aberrations_post.f Chapter 6

12 EE90F It can be shown: (see Born and Wolf) h( δ, Ψ) α' n πe n i π - + ( ) n J n + ( πδ) cosψ πδ sinψ The point spread functions for Zernike aberrations are related to higher order Bessel functions. (6.40) Aberrations_post.f Chapter 6