Optica Applicata, Vol. XII, No. 3 - 4 , 1982
Letters to the Editor
Holographic imaging of a sinusoidal test*
Jerzy Nowak
I n s titu t e o f P hysics, Technical U niversity of Wrocław, Wybrsete Wyspiańskiego 27, 50-370 Wrocław, Poland.
The purpose of t h i s work i s to examine the influence of p a r tic u la r ab e rra tio n s on the image c o n tra s t of an o b je ct given in th e form of an amplitude sin u so id al t e s t . As i t i s w ell known in a coherent system a tr a n s f e r function fo r both the c b n tra st and phase cannot be defined in a way an alo g ical to th a t used in incoherent o p tic a l systems. Under c e r ta in assum ptions, however, i t i s p o ssib le to define the degradation of c o n tra st in the image w ith re sp e c t to th a t p resent in the o b je c t.
Let the amplitude transm ittance of th e t e s t be defined by the formula
t ( x 1f y.j) « a + b cos 2lt(vx x1 + Vy y ^ » a > b , ( l ) where v Vy - s p a tia l frequencies in x^ and y^ d ir e c tio n s , re sp e c tiv e ly .
The lig h t in te n s ity may be ca lc u la te d from the formula
I = a2 + b2 cos2 2Tl(Vx x^ + y^ ) + 2ab c o s u < " ,* 1 <*> The o b je ct c o n tra s t i s defined as follows»
I max - I . min 2ab , -
n
v = i — n — 3 — · ( 3 ^>
max min a + b
In the paper [1] i t has been shown th a t when the object space i s assumed to be r e s tr ic te d so th a t th e wave a b e rra tio n be co n stan t, the lig h t in te n s ity in the image plane is determined by the formula
o o * ( ^ 2 I' = a2 + b2 --5--- cos P (0 ,0 ) 2TtmR - ( V x ' + V y ') + © + 2ab P( » *(0,0) *
472
J . NOWAK
f f , s » ( v * . ) ♦ » ( - v - > 11 r i | k2[W(0,0) - --- M y— g---y--- J j cob[2uC V x x* + Vy y #) + e j . ( 4 ) where 0 = 2 w m R 1 Vx M \ B0
•BsJ
r
l V x X 2 mR1 V v X 2 mB1 £ę - ^ R R c + k„ w( * v « y v ) “ - y v ) 2 * y. . ( 5 ) (6)XR' yR’ ZR “ coordinates of the reference wave source, x , y , z - coordinates of the rec o n stru ctin g wave source, c c c
R^, R^, Re , R* - resp ectiv e d istan c es of the object wave source, reference wave source, rec o n stru ctin g wave source and image wave source, kg * 2 It/Xg, \ g - wavelength of the lig h t rec o n stru ctin g the hologram, ji * A. g A j , \ 1 - wavelength of the lig h t recording the hologram, m - c o e f fic ie n t determining the hologram sc a le ,
tf(x, y ) - wave a b e rra tio n .
Then the c o n tra s t in the image i s expressed as follows
V' 2ab f T , yv ) + ” yV >11 ,
&2 + fe2 CO8| k2[ ^ 0,0> "
^
2
^
J * V yV>'
( 7 ) where P(*v » y^) - pup il function defined assP(V V
1 in the p u p il,
,0 beyond the p u p il.
The drop in c o n tra st may be expressed as follow s:
, v ' [ f , . yu >+ “yu> V* « cos k2 [w(0,0)
---*----F(xv ’ yv >* ( e )
The wave ab e rra tio n in the region of I I I order ab e rra tio n i s determined by the expres sion [2] , (*2 + y2)2 *3 + *y2 x2y ♦ y3 x2 W (x ,y ) --- S1 + — F — 32x * — 2— 32y ' t S3x 8 y 2 ’ ’ " T S3y ■ xyS3xy·
(9J
Letters to the Editor 473
By in s e rtin g the resp e ctiv e terms of ( 9) to ( 8) i t i s p ossible to examine the in f lu ence of p a r tic u la r a b e rra tio n on c o n tra s t. For the sake of s im p lic ity , l e t us assume
spective drops in c o n tra s t caused by sp h e ric al a b e rra tio n coma and astigm atism axe
I t may be seen th a t coma does not influence the c o n tra st degradation, but obviously i t a f f e c ts the imaging q u a lity by causing a phase s h if t which i s shown in th e formula ( 5 ) . However, in the case of a simple sin u so id al t e s t used as an ob ject th is phase s h a ft i s of no p r a c tic a l importance.
The f a c t th a t only two ab e rra tio n s decide about the imaging q u a lity allows the re co n stru c tio n of sin u so id al o bject with the lig h t of wavelength d if fe r e n t from th a t of th $ lig h t used f o r rec o rd in g , provided th a t the hologram scale remains unchanged (m = * f ) . Thus, f o r in sta n c e , i f 3 R^ i t su ffic e s to f u l f i l l the conditions [2, 3]*
Our co n sid eratio n s re f e r rin g to an a r b itr a r y but only single frequency a re v alid fo r a lim ite d range, thus the conclusions should not be gen eralised to an a r b itr a r y o b je c t. This i s , however, a problem which - though to a lower degree - occurs always in ttyd coherent o p tic a l imaging when the imaging q u a lity of a complex o bject i s evaluated by te s tin g the image q u a lity of a p o in t-o b je c t.
(10)
(
1 1)
References
d h NOWAK J , , ZAJ$0 M., PIETRASZKIEWICZ K ., Optik 61 (1982), 147. [ 2 \ CHAMPAGNE E .B ., J.Opt.Soc.Am. £7 (1967), 51.
[31* MEIER R.W., J.Opt.Soc.Am. 21 (1965), 987.
