Volume
16, number
March 1976
OPTICS COMMUNICATIONS
3
THEDIFFRACTlONEFFICIENCYOFHOLOGRAMGRATlNGS RECORDEDINANABSORF'TIVEMEDIUM T.
KUBOTA
Institute of Industrial Science, Received
24 December
University of Tokyo, 7-22-1, Roppongi, Minatoku, Tokyo, Japan
1975
The profile of the hologram grating is described considerating the absorption of the recording medium and its diffraction efficiency is calculated for special cases using the coupled wave theory. The maximum diffraction efficiency obtainable is not affected by the absorption of the recording medium in the case of transmission holograms, while it decreases with increasing absorption in the case of absorptive reflection holograms.
Light is absorbed in a photographic recording medium and suffers a progressive attentuation as it is propagated through the medium. The intensity transmittance of a 649F plate with emulsion thickness 15 pm is 42% at 6328 .& and 33% at 4880 A [I], and that of a dichromated gelatin plate with the same thickness is 10% or less. The contrast of the interference fringes recorded within such a medium is, therefore, modulated along a direction perpendicular to the surface of the medium. The analysis of such a hologram grating has been carried out by Uchida [2]. He assumed that the modulation term of the grating decreased exponentially along the direction perpendicular to the grating vector and that the bias term was kept constant. The bias term, however, is usually affected by the absorption. In this letter, taking into account the absorption of the recording medium, the profile of the hologram grating is described and its diffraction efficiency is calculated for special cases using the coupled wave theory developed by Kogelnik [3,4]. First we consider the propagation of the light wave in a homogeneous recording medium having an absorption constant cy. Fig. 1 shows the geometry to be analyzed. The incident wave is assumed to polarize perpendicular to the plane of incidence. A solution of the wave equation corresponding to plane wave propagation through the medium in the x-z plane has the form
u (x,2) = T(z)
exp (- ik - r) ,
(1)
where the amplitude factor T(z) accounts for the slower variation of amplitude incurred as the wave progresses through the medium, and k and r are the propagation vector and the position vector, respectively. T(z) has the form T(Z) = C exp (-az
set 0) ,
(2)
or T(z) = C exp [cl@-z)
set U] ,
(3)
Fig. 1. Geometry for the wave propagation through the recording medium. 0 is the angle of incidence in the medium, D is the thickness of the medium.
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3
according to whether the wave is propagated to the right (0 < fl
(4)
where A(z) = CT exp (-2az B(z)=2C,C,exp
set 0 l) + C2, exp ((2az [--otz(sectll
tsecti2)],
set H2), (5)
for O
set 02],
B(z) = 2C, C, exp (CID set B 2) X exp [-az(sec
0 l + set O,)]
(6)
for 0 < 0 1 < 7r/2, rr/2 < H2 < 3n/2 (case II), and the grating vector K is oriented perpendicular to the fringe planes. Here C, and C, are the constant amplitudes, and 0, and ti2 are the angles of the two waves with respect to the z-axis. The cases for I and II correspond to transmission holograms and reflection holograms, respectively. The intensity distribution of the fringes is converted to the spatial variation of the absorption constant and/or the refractive index of the hologram grating by the development process. Assuming that the variations of the absorption constant o(z) and the refractive index n(z) are linearly proportional to the exposure, these quantities are expressed as follows: (u(z) = CQ(Z) + cyl (z) cos (K * r) , II(Z) = u. + II 1(z) cos (K - r) ,
(7)
where au(z) is the bias term of the variation of the absorption constant and is proportional to .4(z), and oI(z) and RI(Z) are the modulation terms of the variation of the absorption constant and the refractive index, respectively, and both are proportional to f?(z). The average refractive index ?z,, can be assumed to be constant because it is considerably larger than the re348
March
OPTICS <‘OMMUNlCATIONS
1976
fractive index change produced by the development process. /rt = 0 and ol = 0 correspond to absorption gratings and phase gratings, respectively. For the special case of an unslanted hologram grating and Cl = C,, the visibility of the fringes for the absorptive transmission hologram equals to 1 independently of the values of z because set 0 t = set 0, in eq. (5) and the modulation term for the reflection hologram becomes constant independently of the values of z because set 01 = -set 0, in eq. (6). These properties of the hologram grating differ from those of the grating model used by Uchida. The diffraction of the light within the thick hologram gratings whose absorption constant and refractive index are given by eq. (7) can be analyzed by the coupled wave theory extended by Kogelnik. In the case of Bragg incidence and unslanted hologram gratings, the following coupled wave equations with respect to the amplitude of the incident wave R(z) and the diffracted wave S(z) are obtained with slight modifications from the uniform case [3]. d&z) z
cos 0 + au(z) R(z) = ik(z)S(z)
Wz) _ + dz
(8) iK(z) R(z) .
cos 6, + q,(z) S(z) =
where the upper sign holds for the transmission hologram and the lower sign for the reflection hologram. K(Z)
=
(7?/h)
HI
(Z)-ial(Z
.
and the relations cos 0 1 = lcos S2 / = cm 0 are used. Here X is the wavelength in free space. In order to examine how the values of the absorption constant o affect the properties of the diffracted wave, some numerical results for the diffraction efficiency are shown The discussions are restricted to the case for the absorption hologram and Ct = C2 = 1. I. Trarlsmission holograms. Applying the boundary conditions R(O)- 1 and S(0) = 0 to eq. (8), the amplitude of the diffracted wave S(D) is obtained as S(D) = m-exp(-X)
sinh(X/2)
,
(9)
where x = exp(-&)sinhh -~ 6
_
aDsecH
(IO)
Here 6 = CUDset 0, and L?‘isau(z) at o( = 0. The abso-
Volume
16, number
3
March 1976
OPTICS COMMUNICATIONS
0.
Fig. 2. Absolute value of the diffracted amplitude versus X for the absorptive transmission hologram. X is proportional to the resultant optical density.
lute value of S(D) is plotted in fig. 2 as a function of X. The diffraction efficiency is obtained by squaring IS(D)]. Because X is proportional to the resultant optical density of the hologram grating, fig. 2 indicates that the diffraction efficiency is determined uniquely by a quantity depending only on the resultant optical density. Therefore, the diffraction efficiency can reach the maximum value of 3.7% at any value of CLThis result is different from the result derived by Udhida in which the maximum diffraction efficiency decreases rapidly with increasing values of 6. 2. Reflection holograms. The bias term au(z) shows the maximum values at z = 0 and z = D and the minimum value at z = D/2. When the absorption of the light in the recording medium is small, that is, 6 5 1, Q(Z) may be regarded as a constant G. If Z is set to be equal to the average value of cyO(z), we can write (y=_exp (-6) sinhd Z. 6
where
[ 1 + dp
a = (sinh6)/6
.
(13)
The absolute value of S(0) is plotted in fig. 3 as a function of X. The diffraction efficiency is obtained by squaring lS(O)l. a is the inverse of the visibility of the fringes. Because the visibility of the fringes is a function of 6, the diffraction efficiency becomes lower with increasing values of 6. The maximum diffraction efficiency is 7.2% for 6 = 0 (no absorption), 6.6% for 6 = 0.5, and 4.5% for 6 = 1 .O. For phase hologram gratings, the maximum diffraction efficiency for both transmission and reflection holograms can reach 100% at any value of CL The author is greatly indebted to Professor T. Ose for his helpful advice and encouragement during the reported investigation.
(11) References
crI(z) becomes constant as mentioned before. Applying the boundary conditions R(0) = 1 and S(D) = 0 to eq. (8), the amplitude of the diffracted wave S(0) is obtained as S(0) = -1/{2a
Fig. 3. Absolute value of the diffracted amplitude versus X for the absorptive reflection hologram with various values of a. a is the inverse of the visibility of the fringes and is a function of 6.
coth (fi$?
X)] },
[l]
R.J. Collier, C.B. Burckhardt, and L.H. Lin, Optical Holography (Academic, New York, 1971), p. 281. [2] N. Uchida, J. Opt. Sot. Amer. 63 (1973) 280. [3] H. Kogelnik, in: Proc. Symp. Modern Optics, J. Fox (Polytechnic Press, Brooklyn, 1967), pp. 605-617. [4] H. Kogelnik, Bell Syst. Tech. J. 48 (1969) 2909.
(12)
349