Third-order aberration coefficients o f the Fraunhofer
hologram formed at spherical surface o f the recording medium
Eu g e n iu s z Ja g o s z e w s k i
Institute of Physics, Technical University of Wroclaw, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland.
1 . Introduction
The aberrations of reconstructed images in holography are given as a function of the object, reference and reconstruction coordinates, as well as the wavelength ratio and the scale factor of hologram [1-6]. In the case of a non-plane hologram, for example, in a hologram with the basis of spherical shape, the aberrations depend also upon the curvature of its surface [7-15]. In this paper the expres sion for third-order aberrations in the wavefronts reconstructed from Fraun hofer hologram made at spherical surface are derived, and the influence of the hologram curvature on the aberrations is shown,
Like in paper [16] let us consider first an object point P 0 in Cartesian co ordinate system whose origin is at the vertex of the spherical hologram surface with a centre on «-axis. When a spherical wave emerges from the object, reference or reconstruction point source, then its phase within the spherical surface of the recording medium (hologram) related to the phase at the vertex is given by
~ ^i(ro -^o)> 0 G( x , y , z ) = k2(rc - B c),
respectively, Aq = 2^/1, and k2 = 2n ß 2 being the wavenumber. If the points P ( x , y , z ) of the spherical surface are described in spherical coordinates: P (q, 0 , <p), then the expansion of the square roots in the phase expression yields
( ß f ® » <p) = Ć sin2 ( - j j ( l + - 2 Q sin 0 (.% cos <P+ yN sin ę>)J
- If [10e4sii1' (I) I1+
t
) -
Sain’ (f) I1+
t
)
x (¡rN cos<p -f yN sin cp) + 4 q2 sin2 0 (ash cos2 cp + ifN sin2 <p (1)
+<%2An sin 2cp) + 8 (a& + 2/n) ( l + -y-J Q2 sin2 | - 4 (a& + )
where IT = O, K, 0, I for object, reference, reconstruction and image wave, respectively. By adding the phases of all wavefronts the participating in record ing and reconstruction processes, the phase of the two reconstructed waves can be calculated from the equation
k2Ai\ — k2{vi — J2j) — k2ArCi^zk1(Ar0 — ArR) — i>AB (2)
where 0>AB represents the wavefront aberrations, defined as the phase difference between the Gaussian reference sphere and the actual wavefront created during illumination of the hologram surface. Assuming (#n +2/n)M ^ and x, for the Fraunhofer hologram (Fig. 1), the first approximation of Eq. (2) pro vides (#ab = 0) the paraxial imaging equations
Hologram
Pig. 1. Formation of a Fraunhofer hologram at a spherical surface
zi = *0»
sincq = sinac ± lM(sinao _ sinaR), (3)
shift = sin ft ± /i (sin ft-sh ift*)
where fi = and aN, f)N are the angles between «-axis and suitable directions of ray projection at (x, «)- and (y, «)-plane, respectively.
2 . Third-order aberration coefficients
By the subsequent approximation of Eq. (2) and when the above conditions are fulfilled, we have 0>AB ^ 0 and the third-order aberration coefficients [16] for a Fraunhofer hologram formed on the spherical surface are given by
8 = ■ 1 1 ”13 7F ZG z l
e \
1 ZG —” 2“) + - T ı\
1
“ i l Q \ z c Z1 , n — sinac sin tq + ( sin a c Sin cq \VX »2 ZG o z i
e
l ~c ZI ) ’ sin f t sin ft + 1 / s h i f t sin ft \ — "C ZIe 2e
l z c « I / W A-x sin2ac Zc sin2ax J *1sin2/?c Sin2/?!
zc *1
sinac sin/?c sin ax sin/?!
A xv = - - >
F —— (sin2ac + sin2/?c ) ----— (sin2ai + s in 2/?i)
zc zl — (sin2ac + sin2/?c Q (4)
— sin2ai —sin2/?i),
B x — (sin2 ac + sin2 /?c ) sin ac — (sin2 aj + sin2 /?j) sin ax, J)y — (sin2ac + sin2/?c )sin/?c —(sin2ai + sin2j?i)sinj?i.
As was shown in paper [16], only a spherical aberration, coma and field curvature depend on the curvature of the hologram surface. In a special case, by using the paraxial imaging Eqs. (3), we get the aberration coefficients in simple form, namely:
8 = 0 , Cx = — (— + — | (sinac —sinai), zc \zc Q I = — (— + — ) (sin/?c -s in /? i), \zo e l A x = — (sin2ac —sin2ai), zc A V = — (sin2l? o -s m 2/?i), zc
A = — (sin ac sin/?c —sinai sin/?i), zc
F — (— + — |(sin2ac + s in 2/?c —sin2ai —sin2/?i); \«c e l
the distortion coefficients being the same as in Eqs. (4). We see that the spherical aberration always disappears, since for the Fraunhofer hologram zI = zc . We can also see that when a reconstruction point source is situated at a distance zc — — q, coma and field curvature will be eliminated simultaneously. Thus, when a Fraun hofer hologram is formed on a spherical surface of the recording medium for the two images the spherical aberration, coma and field curvature can be made to disappear simultaneously by using the reconstruction beam point source. In this situation, all the aberrations do not depend on the wavelength ratio. 8 — Optica Applicata XV/1/85
3 . Lensless Fraunhofer hologram
In the case where the reference-beam source on the optical axis is at infinity and the object at the finite distance from the surface of the recording medium, a lensless Fraunhofer hologram (Fig. 2) is obtained. Thompson [17], Jagosze
w s k iand Pa w l u k [18] employed this method to measure the size and shape of dynamic aerosol particles, but at the plane surface. A Fraunhofer hologram formed by illuminating an object surface with plane coherent wave, when the recording surface is placed in a far-field distance from the object.
Hologram
Fig. 2. Formation of a lensless Fraunhofer hologram at a spherical surface
In this case, due to small skewness (ac , /?c ) of the reconstruction beam the paraxial Eqs. (3) take the forms
1 1 /j,
z l Zq z q ’
sin a! = sin ac ± p sin aG, sin/?! = Sin/?c ±jMSin/?0,
and the third-order aberration coefficients [16] are as follows:
„ 1 1 n 2 / 1 1 fi\ Zq ZI Zq Q \ Zq Zj; Zq /
cr.
sin ac zc sinßc smaj fi sin a0 ~2 i ~2 Cc aI Sin/?! V2 ZI /xsin/?o ± -H H -4 -)» zo l + ¿ 1 Q 1 sin ac \ zc + t ( sin/?c zc sinax ^ jMSinaQ ^ Z1 Sin/?i ± *o /«sinßon/?o \ A „ = sin2a0 zc sin2(?c s m 2 a i ^ / i S m 2a 0 zi Sin2/?i±
zo MSin2/?o (5) (6)A xi/ sinac sin/?c
zc
sinaxsin/?! _J_ /MSina0 sin/?0
F ---- (sin2ac + sin2/?c ) --- (sin2aI + s in 2/?I) ± ---1 1 u
ZG ZI ZQ
^»111
+ — [sin2 ac + sin2 /?0 — sin2 ar — sin2/?! ± [i (sin2 aQ + sin2
e
'Po)l,Dx = (sin2a0 + sin2/?c )sinac —(sin2aI + sin2/?I)sinaI ± /i(s in 2a0 (6) + sin2 ^0) sin a0 ,
Dv = (sin2ac + sin2/?c )sin/?c —(sin2aI + sin2/SI)sin/?I ± H (sin2 aQ + sin2 /?0) sin /30 .
In contradistinction to the Fraunhofer hologram formed by a lens, the aber rations of the lensless Fraunhofer hologram depend also on the wavelength ratio. The aberrations of the Fraunhofer hologram can be also eliminated by the plane, and partially by the spherical illuminating wave. If n — 1 and zc = zQ, then one image is placed at the finite distance (z1 = z0/2) from the hologram, and the other one at infinity (zx — oo). Thus, for the primary image we have:
„ 2 / 3 2 \
8 — --- ¿ -I---1---1 > *o \*o e l
Gx = - — ( — + — I (sin ac + sin a0), zo \zo e l
Cv = - — (— + —) (sin/90 + sin/?0), zo \zo e l
A x = — —- [(sinac + sina0)2 + 2 sin a c sina0], zo
A y = [(sin/9c + sin/3o)2+2sin/3c sin/30], (7)
"O
A Xy --- [sin p0 (sin a0 + 2 sinac ) + sin /Sc (sin ac + 2 sin aQ) ], zo
F = [(sinac + sina0)2 + (sin/?c + sin/30)2] zo
— 2 (--- h — ) (sin ac sin aQ +sin/9c sin/S0),
\zo e l
Dx = —sinac (3sin2a0 + sin2/90)~ s in a 0 (3sin2ac + sin 2/3c ) — 2 sin /?c sin 0O (sin a0 + sin a0) ,
By = —sin/?c (3sin2/?0 + sin2a0) — sin/?0 (3sin2/?c + sin2ac ) — 2 sin ac sin a0(sin /S0 + sin /S0) .
And for the secondary image 8
=
0,
„ 1 / 1 1\ . Cx = — --- (sinac - s m a 0), zo \ zo 61 Gv = — ( — - —I (sin pc - sin p0 ), zo \ zo 61A x = ---- (sin2 ac — sin2 a0) , zo
A y = — (sin2/3c - s in 2/?0),
A xv = ---- (sin ac sin Pq — sin a0 sin /?0) , zo
(8)
F = ---- (sin2ac + sin2/90) H--- (sinac sina0 + sin/?0sin/?0)
z0 6
+ —) (sin2a0 + sin2/S0). \ ^o 6 I
Coefficients J)x, Dv are the same as in expressions (7). 4 . Final remarks
Considerations over the quality of reconstructed images from the Fraunhofer holograms lead to conclusions that all the aberrations can be eliminated by using a plane illuminating wave during the reconstruction process. By using a spherical illuminating wave only some aberrations can be eliminated, the more so, when the hologram is formed at the nonplane surface.
References
[1] Me ie r W. W ., J. Opt. Soc. Am. 55 (1965), 987.
[2] Ch a m p a g n e E. B., J. Opt. Soc. Am. 57 (1967), 51.
[3] La t t a J. N., Appl. Opt. 10 (1971), 599.
[4] Ibidem, p. 609. [5] Ibidem, p. 2098.
[6] Miles J. F., Optica Acta 19 (1972), 165.
[7] Mu s t a f in K. S., Optika i Spektr. 37 (1974), 1158.
[8] We l f o r d W. T., Opt. Commun. 8 (1973), 239. [9] Ibidem, 15 (1975), 46.
[10] We l f o r d W . T., J. Phot. Sc. 23 (1975), 84.
[11] Sw e a t tW. C., J. Opt. Soc. Am. 67 (1977), 803.
[13] Ibidem, p. 106.
[14] No w a k J., Optica Applicata 10 (1980), 245.
[15] Pe n g Ke-o u, Imaging by Curved Holographic Optical Elements, Doctor’s Thesis, Delft
University, Dutch Efficiency Bureau-Pijnacker, 1983.
[16] Ja g o s z e w s k iE ., The influence of the hologram surface curvature on the holographic imag ing quality, Optik, in press.
[17] Th o m p so nB. J., Wa k d J. H., Zin k yW ., Appl. Opt. 6 (1967), 519.
[18] Ja g o s z e w s k i E ., Pa w l u k T., Optica Applicata 10 (1980), 399.
Received August 30, 1984, in revised form October 8, 1984
