UDC 621.373.8:539.1.073

PHYSICS

Yu. N. DENISYUK, D. I. STASELKO

ON THE POSSIBILITY OF OBTAINING HOLOGRAMS USING A REFERENCE BEAM WHOSE WAVELENGTH DIFFERS FROM THE WAVELENGTH OF THE RADIATION SCATTERED BY THE OBJECT

(Presented by Academician I. V. Obreimov, January 9, 1967)

One of the basic conditions for obtaining a hologram by the Gabor method (¹) is the strict phase coherence of the oscillations of the waves interfering in its plane. This circumstance limits the range of applications of the holographic method to the case of objects that are located close together and remain stationary during the exposure.

Below we discuss the possibility of obtaining holograms in the case when the condition of phase coherence is not fulfilled. In our consideration we shall use a method of reasoning analogous to (¹), which is applicable to a plane hologram of any type. We define the wave functions of the radiation scattered by the object and of the reference beam by the expressions

\[ \Psi_0 = a_0(r)l \exp [ik_0L_0(r)] \exp [i\omega_0t]; \tag{1} \]

\[ \Psi_s = a_s \exp [ik_sL_s(r)] \exp [i\omega_st], \tag{2} \]

where \(r\) is the radius vector of an arbitrary point of the hologram; \(k_0 = 2\pi/\lambda_0\); \(k_s = 2\pi/\lambda_s\); \(\lambda_0\) and \(\lambda_s\) are the wavelengths of the radiation scattered by the object and of the reference beam. The eikonals \(L_0\) and \(L_s\) are assumed not to depend on the wavelength, which for narrow spectral ranges outside absorption lines is apparently permissible. The intensity distribution of the total wave field \(\Psi_h\) in some plane \(P\) is found by adding (1) and (2) and multiplying the result by the complex conjugate quantity:

\[ \Psi_h\Psi_h^* = a_0^2(r) + a_s^2 + 2a_0(r)a_s \cos [(\omega_0-\omega_s)t + k_0L_0(r) - k_sL_s(r)] . \tag{3} \]

Expression (3) describes traveling intensity waves, whose spatial configuration is determined by the function \(k_0L_0(r) - k_sL_s(r)\). With direct recording on a photographic plate, such a traveling interference pattern forms a uniform background. However, this pattern can be preserved, for example, by placing in front of the photographic plate an inertia-free optical shutter whose transmission coefficient is proportional to the field intensity at some reference point of the photographic plate with coordinate \(r_0\). In this case the reference point must be located outside the optical shutter.* The radiation-intensity distribution on the photographic plate in this case can be found by multiplying expression (3) by the value of the intensity at the point \(r_0\):

\[ \begin{aligned} \Psi_{hT}\Psi_{hT}^{*} &= C_0\Psi_h\Psi_h^*(r)\Psi_h\Psi_h^*(r_0) = C_1 + C_2 \cos [(\omega_0-\omega_s)t + \varphi_1(r_0)] \\ &\quad + C_3 \cos [(\omega_0-\omega_s)t + \varphi_1(r)] + C_4 \cos [2(\omega_0-\omega_s)t + \varphi_2(r,r_0)] \\ &\quad + C_5 \cos \{ k_0[L_0(r)-L_0(r_0)] - k_s[L_s(r)-L_s(r_0)] \}, \end{aligned} \tag{4} \]

where \(C_0\)–\(C_5\) and \(\varphi_1, \varphi_2\) are certain constants. The first through fourth terms describe a constant background, a background harmonically varying in time, and traveling waves of various structure. The fifth term of (4) describes

* The optical shutter mentioned is necessary only when recording the hologram directly on a photographic plate. With photoelectric registration, operations (4) and (6) can be carried out in electronic circuits.

a stable interference pattern. However, a change in the quantities \(k_0\) and \(k_s\) during the exposure will lead to smearing of the photographic image of the interference pattern far from the reference point \(r_0\), whereas near it the photographic image will remain sharp. The condition for preserving the interference pattern over the entire photographic plate can be written as follows:

\[ \{\Delta k_0[L_0(r)-L_0(r_0)]-\Delta k_s[L_s(r)-L_s(r_0)]\}_{\max}\leq \pi, \tag{5} \]

where \(\Delta k_0\) and \(\Delta k_s\) are the increments of \(k_0\) and \(k_s\) during the exposure.

If we assume that the amplitude transmission coefficient \(T_h\) of the exposed and developed photographic plate is proportional to the intensity of the radiation exposing it, and take into account that the radiation described by the first terms of (4), for exposure times \(t_{\mathrm{exp}}\gg 2\pi/(\omega_0-\omega_s)\), forms a uniform background, we obtain

\[ T_h=T_0+ca_0(r)\exp\{i\{k_0[L_0(r)-L_0(r_0)]-k_s[L_s(r)-L_s(r_0)]\}\}+ \]
\[ +ca_0(r)\exp\{-i\{k_0[L_0(r)-L_0(r_0)]-k_s[L_s(r)-L_s(r_0)]\}\}. \tag{6} \]

Let, during reconstruction, radiation from the reference source be incident on the hologram. The value of the wave function behind the hologram is found by multiplying (2) by (6):

\[ \Psi_{\mathrm{p}}=T_h\Psi_s=T_0a_s\exp[ik_sL_s(r)]\exp[i\omega_s t] +ca_sa_0(r)\exp\{i[k_sL_s(r_0)-k_0L_0(r_0)]\}\exp[ik_0L_0(r)]\exp[i\omega_s t] \]
\[ +ca_sa_0(r)\exp\{i[k_0L_0(r_0)-k_sL_s(r_0)]\}\exp\{-i[k_0L_0(r)-k_s2L_s(r)]\}\exp[i\omega_s t]. \tag{7} \]

The second and third terms of (7) are almost completely analogous to the terms that describe the virtual and real images in the corresponding expression of Gabor’s holographic method. The difference consists in the appearance of an inessential constant depending on the choice of \(r_0\), and also in the fact that the amplitude-phase part of these expressions coincides with the amplitude-phase part of the radiation scattered by the object (1), while the time part coincides with the time part of the wave function of the reference beam (2). Introducing a new value of the eikonal

\[ L'_0(r)=\frac{k_0}{k_s}L_0(r) \]

and combining all constants, the wave corresponding to the virtual image of the object can be written in the form

\[ \Psi_{\mathrm{m}}=Aa_0(r)\exp[ik_sL'_0(r)]\exp[i\omega_s t]. \tag{8} \]

Comparing (8) and (1), we note that in (8) the eikonal and the wavelength have been replaced. Replacement of the eikonal in the case \(\lambda_s-\lambda_0\ll \lambda_s\lambda_0\) indicates a slight displacement and change in the dimensions of the object image, while the change in wavelength will slightly alter the spectral composition of the reconstructed image.

In conclusion, let us dwell on some questions connected with the practical realization of this method. The difference in wavelengths of the signal and reference beams may be caused by a Doppler shift in recording a moving object, by changes in the spectral composition of the laser radiation when the beams mentioned are formed from successive pulses of one and the same source, by a difference in the wavelengths of two lasers, etc. Wavelength shifts due to the listed causes are quantities of the order of hundredths of an angstrom (see, for example, \((^2)\)). In the visible range this corresponds to beats with frequency \(\omega_0-\omega_s\) of the order of hundreds of megahertz, which is within the capabilities of techniques for reception and modulation of radiation.

Received
22 XII 1966

CITED LITERATURE

1. D. Gabor, Proc. Roy. Soc., A197, 454 (1949).
2. M. Hercher, Appl. Phys. Let., 7, 39 (1965).