Dual-Frequency Hologram Formation in the Short-Wavelength Range

V. A. Berdonosov, V. I. Gorbunov, A. K. Stoyanov, L. K. Yanisova

Heterodyning of laser radiation has made it possible to obtain good results in a number of practical applications of laser technology [1,2]. When coherent radiations are mixed, a beat signal is formed, whose frequency, amplitude, and phase make it possible to obtain information on the passage of these radiations through some medium.

It is known that holography makes it possible to record and then reconstruct amplitude-phase information about an object, in contrast to other known methods of image formation. It is also known that, in order to obtain an image of an object located in a turbid medium, radiation in the visible range is not always suitable. In this case radiation is used for which the given medium is transparent. Holograms formed by radiation lying outside the visible part of the spectrum have the feature that, in order to reconstruct the image, it is necessary to change their scale by \(\lambda_1/\lambda_2\) times, where \(\lambda_1\) is the wavelength used at the reconstruction stage and \(\lambda_2\) at the recording stage [3].

In this work a variant is considered for recording such holograms that makes it possible to reconstruct a visible image without changing their scale. This is achieved by heterodyning short-wavelength laser radiation at the stage of hologram formation.

The simultaneous use of radiation of two wavelengths has been discussed in a number of works on holographic interferometry [4,5], where this technique was used to change the sensitivity of the arrangement. For solving the problem of scale transformation, heterodyning was proposed in acoustic holography [6], where it was carried out at the stage of reconstructing the visible image.

Let us consider obtaining a hologram of a point using the mixing of two frequencies of the short-wavelength range at the stage of its recording (Fig. 1).

From point 1 to point \(P\) there arrives the wave

\[ \varphi_1 = e^{i\mathbf{k}_1\cdot\mathbf{r}_1 - i\omega_1 t} + e^{i\mathbf{k}_2\cdot\mathbf{r}_2 - i\omega_2 t}, \]

where \(\mathbf{k}_1\) and \(\mathbf{k}_2\) are wave numbers, and \(\omega_1\) and \(\omega_2\) are frequencies. From point 2 to point \(P\) there arrives a wave formed by components with the same wavelengths,

\[ \varphi_2 = e^{i\mathbf{k}_1\cdot\mathbf{r}_2 - i\omega_1 t} + e^{i\mathbf{k}_2\cdot\mathbf{r}_2 - i\omega_2 t}. \]

Here \(r_1\) and \(r_2\) are the distances from points 1 and 2 to point \(P\).

Interference of the complex fronts \(\varphi_1\) and \(\varphi_2\) makes it possible to obtain amplitude-phase information encoded in variations of the amplitude transmittance coefficient of the hologram, in the first approximation…

Fig. 1. Scheme for obtaining a hologram of a point.

proportional to the recorded intensity \(J\). It is easy to show that it is equal to

\[ I=\langle |\varphi_1+\varphi_2|^2\rangle =J_0+A\cdot \cos \frac{\Delta k\cdot \Delta r}{2}\cdot \cos \frac{(k_1+k_2)\Delta r}{2}, \tag{1} \]

where the angle brackets \(\langle\ \rangle\) denote the operation of time averaging; \(J_0\) and \(A\) are constants; \(\Delta k=k_1-k_2\), \(\Delta r=r_2-r_1\). A plot of the intensity variation is shown in Fig. 2. It is not difficult to see that in the distribution \(J\) there are two spatial frequencies: the carrier

\[ \nu_1 \sim \frac{1}{\lambda_1}+\frac{1}{\lambda_2} \]

and the envelope

\[ \nu_2 \sim \frac{1}{\lambda_1}-\frac{1}{\lambda_2} = \frac{\lambda_2-\lambda_1}{\lambda_1\lambda_2}. \]

Obviously, the quantities \(\lambda_1\) and \(\lambda_2\) can be chosen so that the dimensions of the envelope \(\nu_2\) are commensurate with a wavelength lying in the visible part of the spectrum. Then, when the hologram is illuminated with light of wavelength

\[ \lambda_0 \sim \frac{\lambda_1\lambda_2}{\lambda_2-\lambda_1}, \]

its diffraction will be determined by variations of the amplitude-transmission coefficient, which depend on the envelope \(\nu_2\).

Let us consider in more detail the properties of such holograms.

Fig. 2. Plot of the distribution of the recorded intensity.

Recording holograms

Let the formation take place in the short-wavelength range according to the Fourier-holography scheme [7] with two point apertures. The choice of this scheme was dictated only by the convenience of recording in this range. In principle, the analysis can also be carried out for other holography schemes. The diffracted field \(\psi(\vec{x})\) in the plane \(\vec{x}\), removed from the object, is equal to

\[ \psi(\vec{x})=\int \Gamma(\vec{\xi})\, \frac{\partial G(\vec{x},\vec{\xi})}{\partial n}\,d\vec{\xi}, \tag{2} \]

where \(\Gamma(\vec{\xi})\) is the boundary condition for the waves in the object plane \(\vec{\xi}\);

\[ \frac{\partial G(\vec{x}, \vec{\xi})}{\partial n} \]
is the derivative of the Green’s function along the normal to the wave front.

The boundary condition for waves with wave number \(k_1\) and unit amplitude is

\[ \Gamma_1(\vec{\xi})=\delta(\vec{\xi}-\vec{\xi}_0)+D(\vec{\xi}) \tag{3} \]

and with wave number \(k_2\)

\[ \Gamma_2(\vec{\xi})=\delta(\vec{\xi}+\vec{\xi}_0)+D(\vec{\xi}). \tag{4} \]

Expression (3) describes the coordinate part of the perturbed wave under the assumption that the object is a Rayleigh scatterer. Substituting the boundary conditions (3) and (4) into (2) and using the usual paraxial approximations for the Green’s function, we obtain for the perturbed waves the expressions:

\[ \Psi_1(\vec{x},t)= \left(C_1\exp\{i\cdot k_1\cdot z_1\}\int [\delta(\vec{\xi}-\vec{\xi}_0)+D(\vec{\xi})] \exp\left\{\frac{i\cdot k_1(\vec{x}-\vec{\xi})^2}{2z_1}\right\} \times d\vec{\xi}\right)e^{-i\cdot\omega_1\cdot t}, \tag{5} \]

\[ \Psi_2(\vec{x},t)= \left(C_2\exp\{i\cdot k_2\cdot z_1\}\int [\delta(\vec{\xi}+\vec{\xi}_0)+D(\vec{\xi})] \exp\left\{\frac{i\cdot k_2(\vec{x}-\vec{\xi})^2}{2z_1}\right\} \times d\vec{\xi}\right)e^{i\cdot\omega_2\cdot t}. \tag{6} \]

In what follows we shall consider the formation of holograms in the far zone of the object, i.e., we impose the condition

\[ \frac{k_1|\vec{\xi}|_{\max}^2}{2z_1}\ll 1;\qquad \frac{k_1|\vec{\xi}|_{\max}^2}{2z_1}\ll 1. \]

Then the reference wave will be written as

\[ \Psi(\vec{x},t)= C_1\exp\left\{ ik_1z_1+ik_1\frac{|\vec{x}-\vec{\xi}_0|^2}{2z_1} \right\}e^{-i\cdot\omega_1\cdot t} + C_2\exp\left\{ i\cdot k_2\cdot z_1+ \frac{i\cdot k_2|\vec{x}+\vec{\xi}_0|^2}{2z_1} \right\}e^{-i\cdot\omega_2\cdot t}, \tag{7} \]

and the object wave as

\[ \Psi_n(\vec{x},t)= \left[ C_1\exp\left\{ i\cdot k_1\cdot z_1+\frac{i\cdot k_1|\vec{x}|^2}{2z_1} \right\} \int D(\vec{\xi})\exp \left\{ \frac{-ik_1\cdot\vec{x}\,\vec{\xi}}{z_1} \right\}d\vec{\xi} \right]e^{-i\cdot\omega_1\cdot t} + \left[ C_2\exp\left\{ i\cdot k_2\cdot z_1+\frac{ik_2|\vec{x}|^2}{2z_1} \right\} \times \int D(\vec{\xi})\exp \left\{ \frac{-i\cdot k_2\vec{x}\,\vec{\xi}}{z_1} \right\}d\vec{\xi} \right]e^{-i\cdot\omega_2\cdot t}, \tag{8} \]

where

\[ C_1\approx-\frac{ik_1}{2\pi z_1};\qquad C_2\approx-\frac{ik_2}{2\pi z_1}. \]

The intensity recorded by the hologram is equal to

\[ I(\vec{x})=\left\langle\left|\Psi_0(\vec{x},t)+\Psi_n(\vec{x},t)\right|^2\right\rangle = \left\langle|\Psi_0|^2\right\rangle+ \left\langle|\Psi_n|^2\right\rangle+ \left\langle\Psi_0^*\Psi_n\right\rangle+ \left\langle\Psi_0\Psi_n^*\right\rangle . \tag{9} \]

Averaging the first two terms gives the component responsible for the appearance of the background \(I_0\). The information of interest to us is contained in the cross terms. Let us consider in more detail the term \(\left\langle\Psi_0^*\Psi_n\right\rangle\). Carrying out simple but cumbersome calculations, we obtain

\[ \begin{aligned} \left\langle\Psi_0^*\Psi_n\right\rangle ={}& |B_0|^2 \int D(\vec{\xi}) \exp \times \\ &\times \left\{ \frac{ i k_1 (y-y_1)+ i k_2 (y-y_2) - i (k_1+k_2)\dfrac{\vec{x}\vec{\xi}}{z_1} }{2} \right\} \times \\ &\times \cos\left[ - k_1(y-y_1)+k_2(y-y_2) +\Delta k\,\frac{\vec{x}\vec{\xi}}{z_1} \right]\,d\vec{\xi} + \\ &+2|B_0|^2 \int D(\vec{\xi})\exp\left\{ i k_1 y-i k_1 y_1 -\frac{i\Delta k\,\vec{x}\vec{\xi}}{2z_1} -\frac{i(k_1+k_2)}{2z_1}\vec{x}\vec{\xi} \right\} \times \\ &\times d\vec{\xi} + |B_0|^2 \int D(\vec{\xi})\exp\left\{ \frac{i\Delta k y-i(k_1y_1-k_2y_2)}{2} -\frac{i\Delta k\,\vec{x}\vec{\xi}}{2z_1} \right\} \times \\ &\times \cos\left( \frac{-k_1(y-y_1)-k_2(y-y_2)}{2} +\frac{\vec{x}\vec{\xi}(k_1+k_2)}{2z_1} \right)\,d\vec{\xi}. \end{aligned} \tag{10} \]

The notation is

\[ y_1=z_1+\frac{|\vec{x}-\vec{\xi}_0|^2}{2z_1}; \qquad y_2=z_1+\frac{|\vec{x}+\vec{\xi}_0|^2}{2z_1}; \]

\[ \Delta k=k_1-k_2; \qquad y=z_1+\frac{|\vec{x}|^2}{2z_1}; \qquad |B_0|^2=k_1^2+k_2^2, \tag{11} \]

As is known, the image is reconstructed by means of the terms

\[ \left\langle\Psi_0^*\Psi_n\right\rangle \quad\text{and}\quad \left\langle\Psi_n^*\Psi_0\right\rangle . \]

Image Reconstruction

Suppose the corresponding processing of the hologram has been carried out and the transmittance coefficient is \(T(\vec{x})\sim I(\vec{x})\). Let us illuminate the obtained hologram with a normally incident plane light wave of unit amplitude (Fig. 4). Then in the image plane \(\vec{a}\) the amplitude of the transmitted wave \(\Phi(\vec{a})\) is equal to

\[ \Phi(\vec{a}) \simeq C\exp\{i k z\} \int I(\vec{x})\exp\left\{ \frac{i k|\vec{x}-\vec{a}|^2}{2z} \right\}\,d\vec{x}, \tag{12} \]

where

\[ C\simeq -\,\frac{i k}{2\pi z}. \]

Fig. 3. Scheme of hologram formation: 1 and 2 are point apertures; \(D(\vec{\xi})\) is the transmittance function of a plane object; \(\vec{\xi}\) is a vector in the object plane; \(\vec{x}\) is a vector in the hologram plane; \(k_1=\dfrac{2\pi}{\lambda_1}\) and \(k_2=\dfrac{2\pi}{\lambda_2}\) are the wave numbers of normally incident plane waves.

Fig. 4. Scheme of image reconstruction: \(\vec{\alpha}\) is a vector in the image plane; \(\vec{x}\) is a vector in the hologram plane; \(k=\dfrac{2\pi}{\lambda}\) is the wave number of the normally incident plane light wave.

One way to eliminate the influence of the quadratic term

\[ \exp\left\{\frac{i\cdot k\cdot |\vec{x}|^2}{2z}\right\} \]

in (12) is to reconstruct the image in the far zone of the hologram, for which the inequality

\[ \frac{|\vec{x}|^2_{\max}}{\lambda\cdot z}\ll 1 \tag{13} \]

must be satisfied.

If the parameters of recording and reconstruction are such that the equality

\[ \frac{\Delta k}{2z_1}=\frac{k}{z}, \tag{14} \]

is satisfied, then, carrying out for \(\langle \Psi_0 \Psi_n^*\rangle\) calculations similar to those used in obtaining \(\langle \Psi_0^* \Psi_n\rangle\), and substituting all the necessary expressions into (12), we obtain the following image structure:

\[ \begin{aligned} \Phi(\vec{\alpha})=\Phi_0(\vec{\alpha})+\Bigg\{& B_1\left[ D\left(-\frac{\vec{\alpha}}{m_1}-\vec{\xi}_0\right) + D^*\left(-\frac{\vec{\alpha}}{m_1}-\vec{\xi}_0\right) \right] \\ &- B_2\left[ D\left(-\frac{\vec{\alpha}}{m_1}-\vec{\xi}_0\right) + D^*\left(\frac{\vec{\alpha}}{m_1}-\vec{\xi}_0\right) \right] + BD\left(-\frac{\vec{\alpha}}{m_2}+\vec{\xi}_0\right) \\ &- BD\left(-\frac{\vec{\alpha}}{m_2}+\vec{\xi}_0\right) - B_3D^*\left(\frac{\vec{\alpha}}{m_2}-\vec{\xi}_0\right) \Bigg\} \exp\left\{\frac{i\cdot k\cdot |\vec{\alpha}|^2}{2z}\right\}, \tag{15} \end{aligned} \]

where \(B, B_1, B_2, B_3\) are constants depending on the recording and reconstruction conditions;

\[ \Phi_0(\vec{\alpha}) \]

is the component due to the background \(I_0\);

\[ m_1=\frac{\lambda_1+\lambda_2}{\lambda_2-\lambda_1}-1;\qquad m_2=\frac{\lambda_1+\lambda_2}{\lambda_2-\lambda_1}+1 \]

are the magnification coefficients.

Analysis

From relation (13) one can establish the connection between the recording and reconstruction parameters. We obtain

\[ \Lambda z = 2 z_1 \cdot \frac{\lambda_1 \cdot \lambda_2}{\lambda_1 - \lambda_2}, \tag{16} \]

where \(\Lambda = \dfrac{2\pi}{K}\) is the reconstruction wavelength,

\(z\) is the distance from the hologram to the image.

For a sufficiently small difference \(\lambda_1 - \lambda_2\), it is possible to make the value \(\Lambda z\) correspond to reconstruction in visible light. In accordance with the value of \(\Lambda\), the image will appear at different distances \(z\). From (15) it is seen that an image undistorted by the term

\[ \exp \left\{ \frac{i \cdot k \cdot |\vec{\alpha}|^2}{2z} \right\} \]

is obtained at \(z = \infty\). By using a focusing lens, it can be brought from infinity to a distance more convenient for observation.

According to (15), we obtain two undistorted images, symmetric with respect to the point \(+\vec{\xi}_0\), and two distorted ones, symmetric with respect to \(\vec{\xi}_0\). In both cases one image is brighter than the background, the other darker; one is erect, the other inverted.

Conclusions

The calculations performed show that the formation of holograms by radiation lying outside the visible part of the spectrum can be carried out in such a way that there is no need to change the scale of the holograms during image reconstruction. The images obtained are magnified, and the magnification factor \(m\) depends on the ratio of the wavelengths used at the recording stage, in contrast to the usual holography schemes, where \(m\) depends on the ratio of the wavelengths at the recording and reconstruction stages.

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