Letters to the Editor: Image contrast in the coherent apodized optical system

O ptica A p p l i c a t a , Vol. X I V , No. 2, 1984

Image contrast in the coherent apodized optical system4

An n a Ma g ie r a, Ka z im ie r z Pie t r a s z k ie w ic z

In stitu te of P h ysics, Technical U niversity of W rocław, W ybrzeże W yspiańskiego 27, 50-370 W rocław, Poland.

I t has been shown th at th e introduction of an am plitude phase apodizer into a coherent aberration optical system imaging a periodical amplitude or phase object results in th e change of th e contrast which, in turn, depends on the test m odulation depth, and on th e shape of the am plitude part of th e function describing th e apodizing filter. The change of contrast has been exam ined w ith respect to th e function of apodizing filter as well as to the system aber­ ration for am plitude apodizers of th e types

[ l / 2 ( l + r*)]*, ( 1 - r T , ( l - № for p = 1 , 2 , 3 , 4.

Let us assume th at in th e ex it pupil of a coherent optical system there is an am plitude-phase apodizer of th e transm ittance

A { r ) = t (r ) ei<l,(r\ 0 < r ^ l . (1)

If we adm it w ave aberrations in th e optical system W ( x , y ) , then th e total phase change in th e pupil will equal

W ( x , y) = W ( x , y ) + & ( r ) , r = V x i + y 2. (2)

Assume th a t in th e object space of an optical system there is a test of the am ­ plitude transm ittance

H ( x , y ) = a + b co s(2nfxx ) . (3)

M ichelson’s contrast of th e te st equals * ( / * )

2 ab

a 2 + b2 ' (4)

Energy contrast in th e image is [1]

* ' ( / * )

2 a b t { 0 ) t ( s ) ( r W (s) + W ( —s)

--- cos{ K I — ---

—---a 2t 2(0) + b2t2(s) \ |_ 2 (5)

where s = A/xE //a ( f g — cutoff frequency, R — reference sphere radius, f x — spatial frequencies).

2 7 4 A . Ma g ie r a, K . Pie t r a s z k ie w ic z

Contrast change in th e image with respect to the object is t(s) D ( L ) = K' (f x) _ K (fx) (1 + m2) 1 + m --- cos < k , t*(s) \ i 2(0) W(s) + TF( — s) ~ 2 —TT(0)

]}

(m = bja — test m odulation depth).

The phase shift appearing in th e image will have the form O ( L ) = *

IF(s) — TF( - * )

(6 )

(7)

(k — 2jt/A, A — light wavelength).

The introduction of an apodizer causes th e change of contrast. Moreover due to th e introduction of the apodizer the change of contrast depends on th e test m odulation depth. For low-contrast object (m ->0), when t (0)-+0, con ­ trast will be strongly improved. For high-contrast object the apodizer m ay weaken th e contrast of the object.

For a phase te st of the transm ittance H ( x , y ) ~ 1 + i m sin x

the change of contrast with respect to the object equals

D { f x) = t(s) H 0) (1 + m2) 1 + m s ¿2(s) f 2 ( 0 ) . (7 r i F ( s ) + T F ( - s ) Sin K ---l L 2 (8)

From Equation (8) it results th at for low -contrast object, a t th-a-O, w hen TF(0) = 7t¡2 and TF(s) = 0, th e change of th e contrast is th e strongest one. In functions describing th e fall contrast for am plitude (6) and phase (8) tests, two parts m ay be distinguished, nam ely, a part depending solely on th e shape of apodizing function t(r) and a part which depends on th e w ave aberration of the system W { x , y). Let D t denote th e first part of the function, it will am ount

[1] to D t t(s) H 0) (1 + m 2) 1 + m2t 2(s) · i2(0)

For test of a small m odulation depth (m-> 0) th e run of th e function is given by th e formula

D.,m-»0 *(*) ¿(0) ·

Letter to the E d ito r 275

[Tl1«rJl] t r

MD-ftlW2)] m -0

Fig. 1. The effect of am plitude apodi zation t(r) on th e im age con trast o f am plitude te st for: t(r) = 1 /2 (1 + r2) (a), t(r) = [1/2(1 + r2)]2 (b), t(r) H r ) · [t I It 2)] m « 1 = [ 1 /2 ( 1 + r 2) p , p = 1, 2, 3, 4; m = 0 (c), t(r) = [1/2(1 + r 2) P ,

276 A . Ma g ie r a, К . Pie t r a s z k ie w ic z

F ig. 2. The effect of ap odization t (r) on th e im age contrast of am plitude test for : t ( r ) = 1 — ra (a), t(r) = (1 -r * )* (h), i(r) = (1 - r 2) P , p = 1, 2, 3, 4; m = 0 (c), f(r) = (1 - τ ψ , p = 1, 2, 3, 4; m = 1 (d)

Letter to the Editor 277

F ig . 3. T h e e ffe c t o f a p o d iz a tio n t(r) on t h e im a g e co n tra st o f am p litu d e te s t fo r: f(r) = 1 — |r| (a), t(r) = (1 - |r|)a (b), f(r) = (1 - |r|)P , p = 1, 2, 3, 4: m = 0 (c), t (r) = (1 - |r|)J>, p = 1 , 2 , 3, 4: to = 1 (d)

278 A . Ma g ie r a, K . Pie t r a s z k ie w ic z

Fig. 4. Contrast change D ( s ) for am plitude test in th e op tical system w ith aberrations:

W = 0.5 Ar2, Ar2, 2Ar2 apodized w ith th e fun ction t(r) = 1/2 ( 1 + r 2), m = 0 (a), t(r) =

[1/2(1 + r 2)]2, m = 0 (b)

F ig . 5. C ontrast ch a n g e D ( s ) fo r a m p litu d e t e s t in th e o p tic a l sy ste m w ith ab erration s W (r) = 0.5Ar4, Ar4, 2Ar4 a p o d iz e d w ith t h e fu n c tio n : t(r) = 1 /2 (1 + r 2), m = 0 (a), t(r) = [ 1 /2 (1 + + r2)]2, m = 0 (b)

Let ter to the E ditor 279

Fig. 6. Contrast cliange D ( s ) for am plitude te st in th e op tical system with aberrations TV (r) = 0.5 Xr2, Xr2, 2Xr2 apodized w ith th e fun ction s: t(r) = 1 - r 2, to= 0 (a), i(r) = ( 1 - r 2)2, m = 0 (b)

F ig . 7. C ontrast ch a n g e D ( s ) fo r a m p litu d e t e s t in t h e o p tic a l sy ste m w ith a b erration s W (r) “ · 0.5Ar4, Ar4, 2Ar4 a p o d iz ed w ith t h e fu n c tio n s : t(r) = 1 — r 2, to = 0 (a), l(r) = (1 — r 2)2,

2 80 A . Ma g ie r a, K . Pie t r a s z k ie w ic z

Fig. 8. Contrast change D { s ) for am plitude test in th e op tical system w ith aberrations IF(r) = 0.5Xr2, Ar2, 2Ar2 apodized w ith th e functions: t(r) = 1 - \r\, m = 0 (a), t(r) = (1 - |r])2,

m «· 0 (b)

F ig . 9. C ontrast ch a n g e Z)(s) for a m p litu d e te s t in th e o p tic a l s y s te m w ith ab erra tio n s I F (r) = O.oAr4, Xr*, 2Ar4 a p o d iz ed w ith th e fu n c tio n s: t(r) = 1 — |r|, m = 0 (a), t(r) = (1 — |r|)*,

Leiter to the E d ito r ₂₈₁

Fig. 10. Contrast change D ( s ) for phase te st in th e optical system w ith aberrations TF(r) = 0.5Ar2, Ar2, 2Ar2 apodized w ith th e fun ction s: t{r) = 1/2(1 + r2), m = 0 (a) , t ( r ) = [1/2(1 + + r2)]2, m = 0 (b)

F ig . 11. C on trast ch an ge D ( s ) for p h a se t e s t in t h e o p tic a l s y s te m w ith a b err a tio n s W (r) = 0.5Ar4, Ar4, 2Ar4 a p o d iz ed w ith t h e fu n c tio n s t(r) = 1 / 2 ( 1 + r2) ,m * 0 (a), t( r ) = [ 1 /2 (1 + + r2)]2, m = 0 (b)

282 A . Ma g ie r a, K . Pie t r a s z k ie w ic z

Fig. 12. Contrast change D ( s ) for phase test in th e op tical system w ith aberrations W( r )

= 0.5Ar2, Ar2, 2Ar2 apodized w ith th e fun ction s: t(r) = 1 —r2, to = 0 (a), t(r) = (1 —r2)2, to = 0 (b)

F ig . 13. C ontrast ch a n g e D ( s ) for p h a se t e s t in th e o p tic a l s y s te m w ith a b erration s W (r) = 0.5Ar4, Ar4, 2AT4 a p o d iz ed writh t h e fu n c tio n s : t(r) = 1 — r2, m = 0 (a), t(r) — (1 — r2)2,

Letter to the E d i to r

Fig. 14. Contrast change D( s ) for phase te st in th e optical system w ith aberrations W (r) = 0.5Ar2, Ar2, 2Ar2 apodized w ith th e fun ction s: t(r) «= 1 — |r|, m = 0 (a), t ( r ) = (1 — |r|)2, m =■ 0 (b)

F ig . 15. C ontrast c h a n g e D l {s) for p h a se t e s t in th e o p tic a l s y s te m w ith a b erra tio n s W (r) => 0.5Ar4, Ar1, 2Ar4 a p o d iz e d w ith t h e fu n c tio n s: t(r) = 1 — |r |, m — 0 (a), t(r) = (1 — |r |)2, m = 0 (b)

284 A. Ma g ie r a, K . Pie t r a s z k ie w ic z

Figures 1 -3 present the functions D,(s) for th e apodizers [1/2(1 + r 2)]p , (1 —i·8)®, (1 — |r|)p, where p = 1, 2, 3, 4. For low-contrast object th e contrast increases w ith p for apodizer [ 1 /2 ( 1 + r2)]p (Fig. la - c ) , it however, decreases for high-contrast objects (Fig. la , b, d). For apodizers (1 —r2)p, (1 — \r\)p decreases both for low-contrast and high-contrast objects (Figs. 2,3). In Figs. 4-15 there are shown th e change of contrast D ( s ) in optical system s with apodizers l / 2 ( l + r2), [1 /2 (1 + r)2)2, (1 — r2), (1 —r2)2, 1 — |r|, (1 — |r|)2 for am plitude test (Figs. 4-9) and phase test (Figs. 10-15) with aberrations T7(r) — 0.5Ar2, Ar2, 2Ar2 (Figs. 4, 6, 8, 10, 12, 14) and w ith aberrations W (r) = 0.5 Ar4, Ar4, 2Ar4 (Figs. 5, 7, 9, 11, 1 3 ,1 5 ). I t follows from these figures th at the introduction of the aberration de­ teriorates th e contrast in the case of the am plitude test (Figs. 4-9) for p = 1, 2 and that in th e case of the phase test (Figs. 10-15) this contrast is improved for p — 1 and p = 2.

The change of contrast in incoherent aberration optical system w ith am ­ plitude-phase apodizers has been also described in papers [2, 3].

This work was carried on under the Iiesearch Project M.R. 1.5.

References

[1] Ma g ie r a A ., Pie t r a s z k ie w ic z K ., Optik 63 (1983), 305. [2] Ja is w a h l A. K ., Bh o g r a E. K ., O ptica A cta 12 (1973), 965.

[3 ] Ma g ie r a A ., Za ją c M ., Optica A pp licata 11 (1981), 29.