An effective holographic amplifier exploiting consistent periodic structures

Abstract

The energy transfer between interacting laser beams is known as the remarkable effect of self-action of coherent waves in nonlinear optics. We demonstrate both experimentally and theoretically that such an effect can be obtained in a holographic setup based on a permanent phase dual-grating. We have found a new mode of unidirectional energy transfer from a strong pump beam to a weak signal beam, achieved through their diffraction on a set of two matched thin gratings. A theoretical description of this effect is developed based on a system of coupled waves and the formalism of generation of a shifted (nonlocal) dynamic grating with a spatially localized amplitude due to wave-mixing inside a bulk medium. Under the condition of a decrease in the intensity of a weak signal beam, the theory predicts almost complete energy transfer with a diffraction efficiency of $\sim$ 100 % from a stronger pump beam. This unidirectional mode is realized in a thick dual-grating with high modulation depth and cannot be achieved during self-action process of wave-mixing in nonlinear media. The obtained results open the way to design a new type of optical amplifiers and optical phase conjugation systems with great efficiency for amplifying weak signals.

1 Introduction

Optical amplifiers, which exploit wave interaction in nonlinear media, are widely used photonic devices. The most famous of them, optical parametric amplifiers, are based on the interaction of waves of different frequencies and on the transfer of energy from the fundamental harmonic beam to the high harmonic beams [1, 2]. At the same time, the amplification of weak signals of the fundamental frequency is required in many specific tasks, for example, in lidar systems or for optical phase conjugation [3,4,5]. A successful approach to effective amplification of laser beams in modern technologies is the use of large distances for mixing laser beams in optical fibers [2, 6, 7].

Alternatively, the effect of energy transfer between two coherent waves resulting from their mixing in a nonlinear medium been seen as a valid method for this purpose [8, 9]. In this case, the energy exchange conditions are obtained in the Bragg regime, which is implemented in a bulk nonlinear optical media for two incident beams (pump and signal). Bragg diffraction by a regular grating became valid for many experimental techniques, such as X-ray diffraction [10, 11], acousto-optic modulators, the study of neutron or electron scattering [12]. The Pendellösung phenomenon, which exhibits as the exchange of energy between diffracted and direct beams due to constructive and destructive interference of waves, is highlighted in the dynamic theory of diffraction (see, e.g. [13,14,15]). The Pendellösung phenomenon manifests itself as a dependence of the intensity of a diffracted wave on parameters that affect its phase change, for example, changing the thickness of the medium or monitoring the direction of propagation of input waves. The formation of a dynamic grating in the case of two-beam coupling refers to the processes of self-action occurring in a nonlinear dynamic medium, therefore, it has some peculiarities. The first relates to self-diffraction, which is that when two input beams form a light interference pattern (light lattice), it generates a periodic structure of photoinduced refractive index (or/and photoinduced absorption), i.e. a dynamic grating, inside an optically nonlinear medium, for example, in a photorefractive crystal. During this process, the recording beams are diffracted by the dynamic grating created by them. Secondly, both gratings, the light lattice and the dynamic grating, have the same spatial period . The process of self-diffraction is described by the coupled-wave equations derived from the wave equation for the total field, which is a superposition of two coherent plane waves [16]. These equations indicate the phase shift that a wave gets due to diffraction from the dynamic grating. Thus, if the dynamic grating is local, i.e. the maxima of the dynamic grating coincide with the maxima of the interference pattern, there is no energy transfer between the interacting beams, but only a phase transfer as a result of the self-diffraction process. However, an effect similar to the Pendellösung phenomenon will be observed when two lattices are mutually displaced relative to each other [8, 9, 17]. Since the phase delay is acquired in the changed propagation path, as well as constructive and destructive interference between the diffracted and direct waves, an oscillatory behavior of the intensities in the output Bragg beams is observed. When a mutually shift is made within the spatial period $\varLambda$, the maximum output intensity will be observed at $\varLambda /4$ spatial shift, which corresponds to phase delay of $\pi /2$. Let’s denote the diffracted wave as a “signal” with input intensity $I_{s0}$ and the direct wave as a “pump” with the input intensity $I_{p0}$. The maximum energy transfer and amplification of the signal beam will be observed when the dynamic grating is shifted relative to the light lattice by $\varLambda /4$ in the direction of signal wave propagation due to the constructive interference between the diffracted and direct beams. This is the so-called “direct” energy transfer from the pump beam to the signal one. And vice versa, if the dynamic grating is shifted in the direction of the pump wave, $- \varLambda /4$, there is a “reverse” energy transfer from the signal beam to the pump. A similar energy transfer in Bragg orders was obtained using constant thick holographic gratings when illuminated by an interference light pattern and with its spatial displacement relative to the grating [18].

The exchange of energy occurs due to self-diffraction of two laser beams on a nonlocal dynamic phase grating formed by the same beams in the nonlinear medium is the subject of strong interest in dynamic holography. Theoretically, the complete (100%) energy transfer from the pump beam to the signal beam is predicted with wave-mixing in a bulk photorefractive medium [19, 20]. But the practical implementation of such an effect turned out to be very limited, associated with the significantly small amplitude of the registered dynamic grating ($\varDelta n$), which is usually $10^{-4} \div 10^{-6}$ in magnitude. In addition, with a large difference in the intensities of the pump and signal beams, the contrast of their interference pattern becomes almost invisible, which leads to an insignificant modulation depth of the recorded dynamic grating formed by this light contrast.

In our study, we propose to overcome this drawback by simulating a stationary dynamic grating in a bulk medium using a set of phase permanent gratings (a dual-grating of a special design having a layer of space), which is consistent with the light interference pattern formed by the pump and signal beams. We have shown that, owing to a layer of space, such a set of thin gratings supports the Bragg condition for diffraction of waves. Theoretically and experimentally, we have found that in the case of a small modulation depth for both gratings, an energy exchange traditional for nonlinear media occurs, namely, two cases: direct energy transfer (from the pump beam to the signal beam) and reverse energy transfer (from the signal beam to the pump beam), which are determined by the direction of the displacement of this thick dual-grating relative to the light interference pattern. But when the both gratings have a large modulation depth, we get a new unknown mode: only a weak beam is always amplified, regardless of the shear direction of the thick dual-grating. We provide a theoretical explanation of this phenomenon. We perform the first experimental observation of energy transfer in a holographic set-up based on a thick dual-grating we made. These studies confirm the existence of two different modes of energy transfer depending on the values of the modulation depth of the phase gratings.

2 Basic optical scheme of a holographic amplifier with a thick dual-grating

The efficiency of energy transfer between two interacting waves in a nonlinear medium is determined by the coupling strength $\varGamma = k \gamma _N d$, where $k = 2 \pi /\lambda$ is the wave vector, d is the thickness of the medium, and $\gamma _N = \varDelta n$ is the maximum amplitude of the nonlocal dynamic grating [16, 21]. Typical values for the observed energy exchange in photorefractive crystals should be $\varGamma \approx 5 \div 10$ (see, for example, [8, 9, 22]). That requires a sufficiently large crystal thickness: $d \approx 5 \div 10 \, \text {mm}$ for $\varDelta n = 10^{-4}$, or $d > 50 \, \text {mm}$ for $\varDelta n = 10^{-5}$ (for $\lambda = 633 \, \text {mm}$).

In this section, we prove that a thick stationary dynamic grating in such an extended medium can be successfully replaced by a set of thin matched gratings separated by layers of space. The basic scheme with a set of two gratings is shown in Fig. 1. It represents a traditional Fourier transform scheme, but the working area includes two separated holographic gratings. Transformation of the input interference pattern, which is formed by the pump $I_{p0}$ and a signal $I_{s0}$ beams, by two gratings is fixed in the far zone (in the Fourier plane of the lens). We study the intensity changes in the two main diffraction orders at the output.

Fig. 1

The principal optical scheme of coupled beams with two gratings. TDG shows a set of two matched gratings, ${G_1}$ and ${G_2}$ are thin phase gratings, L is the lens, F is the focal length of the lens, $Z_0$ is the distance between the gratings, $I_{s0}$ and $I_{p0}$ are the input intensities of the signal and pump beams, $I_{s}^\mathrm{out}$ and $I_{p}^\mathrm{out}$ are the output intensities of the signal and pump beams, respectively, DF is Fourier plane registering diffraction orders

Further, we prove that two thin phase gratings, $G_1$ and $G_2$, separated from each other by a distance $Z_0$, can, under certain conditions, simulate Bragg diffraction. The two matched gratings should have the same period ($\varLambda$) and they should not be offset from each other in the transverse direction. Therefore, we introduce the following notation for the thick dual-grating (TDG). To satisfy the Bragg diffraction, two conditions must be fulfilled. Firstly, the spatial period of the interference pattern ($\varLambda _\mathrm{int}$) should be a multiple of the period of the thin gratings. Secondly, the interference pattern of the input light must be shifted relative to the TDG in the transverse direction x. The features of beams’ transformation in such a system are demonstrated in Fig. 2. The figure shows the intensity distributions calculated in the output diffraction orders at different input configurations. We have chosen both gratings with a large modulation depth, which gives a phase delay for each grating $\varPhi _1 = \varPhi _2 = 1$, as well as a large distance between the gratings $Z_0 = 50 \, \text {mm}$. Calculations are based on scalar diffraction theory [23, 24]. (Note, that the definition of the phase delay of a thin holographic grating $\varPhi = k \varDelta n d$ formally coincides with the formula for the coupling constant of two-wave interaction in the steady state $\varGamma$.)

Figure 2a shows the intensity spectrum in diffraction orders when only single probe beam is at the input normal to the TDG. The spectrum is equivalent to the diffraction of a beam on one thin phase grating, the modulation depth of which is summed by two gratings in the TDG. It consists of the main strong order (denoted as $\{0\}$) and high diffraction orders, which have the same intensities in symmetric orders [25]. When two coherent beams are at the input (Fig. 2b), the intensity spectrum changes. It includes two main diffraction orders corresponding to two recording beams, which we define by analogy with the diffraction of a single input beam as $\{-0\}$ and $\{+0\}$, as well as high diffraction orders. The main diffraction orders $\{-0\}$ and $\{+0\}$ correspond to the Bragg orders in the case of diffraction on a thick grating. In our notation, we use the output “pump” beam for the order $\{-0\}$ and output “signal” beam for the orders $\{+0\}$.

Fig. 2

Calculations the distribution of the intensities in output diffraction orders for different input conditions in the optical scheme, which is shown in Fig. 1: a a single pump beam $I_{p0}$ enters normally to TDG; b both the signal beam and the pump beam are at the input with the same intensities, there is no mutual shift between the interference pattern and TDG ($\delta =0$); c “direct” energy transfer due to the spatial shift between the interference pattern and TDG, $\delta = 0.25 \,\varLambda$ ; d “reverse” energy transfer in the case of $\delta = 0.75 \,\varLambda$. The parameters of the scheme are as follows: $Z_0 = 50$ mm, $G_1$ and $G_2$ have the same modulation depth, which gives a phase delay $\varPhi _1 = \varPhi _2 = 1$ rad, $I_{s0}=I_{p0}$, $\varLambda _{int}=2 \varLambda$

As a rule, during diffraction on a thin phase grating, a rather large part of the total input intensity is distributed to high diffraction orders at the output. The Bragg-like condition can be satisfied when there exists a shift of the interference pattern with respect to the TDG. Just due to this shift, energy is mostly redistributed between the two main diffraction orders, $\{-0\}$ and $\{+0\}$. At the same time, high diffraction orders take off a small fraction of the total intensity. In the case of spatial shift ($\delta = + \varLambda /4$) of maximums of the interference pattern in the direction of gratings’ minimums, only one signal beam is greatly enhanced, $I^\mathrm{out}_s$. This case is called direct energy transfer. The others diffraction orders almost vanish (Fig. 2c). If the spatial shift occurs in the opposite direction ($\delta = - \varLambda /4$), the only pump beam is significantly amplified at the output, $I^\mathrm{out}_p$, and all other diffraction orders almost disappear, (Fig. 2d). This is a case of reverse energy transfer.

3 Modeling of energy exchange at two-wave coupling in an extended medium

We consider the analytical solutions of two-wave coupling in an extended nonlinear medium under the assumption that the medium has a purely nonlocal response. In this case, both the input interference light pattern, which is formed by two coupled waves, and the photoinduced dynamic grating have a sinusoidal profile along the transverse direction x, but they are mutually shifted by a quarter of the grating period. Then the energy transfer occurs between the two coupled waves that leads to a change in their mutual intensities over the thickness of the medium z. This process results in a change in the contrast of the light interference inside the medium. Modeling of such a process is based on the coupled-wave theory. The changes in the amplitudes of two coupled waves during their propagation in a dynamic nonlinear medium are described by the following system [8, 16, 26]:

$$\begin{aligned} \frac{\partial {\mathbf {E}}_s (z)}{\partial z} = k N(z) {\mathbf {E}}_p (z) \; ; \;\;\; \frac{\partial {\mathbf {E}}^*_p (z)}{\partial z} = - k N(z) {\mathbf {E}}^*_s (z) \end{aligned}$$

(1)

where N(z) denotes the amplitude of the nonlocal dynamic grating, $z/(\pm \cos \theta )$ is the direction of wave propagation, and here we take into account that the angle of convergence of the two waves, $\theta$, small, so $\cos (\theta) \simeq 1$.

The analytical solutions of the system (1), which relate the intensities of the waves at the input and output, have been reported in numerous publications for photorefractive materials that cover many important situations of experimental beam coupling (see, for example, [9, 19,20,21, 26, 27]). These solutions were obtained under the assumption that the light contrast remains constant over the thickness z. We will consider the case of varying the light contrast in the volume of the medium. This leads to a change in the amplitude of the photoinduced grating N(z) along the thickness z. The dependence of the grating amplitude on the light intensity is taken from the evolution equation (see [28,29,30,31,32]):

$$\begin{aligned} \frac{\partial N(z)}{\partial t / \tau } = \gamma _N J_m - N(z) \end{aligned}$$

(2)

where $\gamma _N$ is a dimensionless quantity determining the maximum possible amplitude of the shifted (nonlocal) grating, $\tau$ is the relaxation constant of the grating amplitude, and the term $J_m = ({\mathbf {E}}_s {\mathbf {E}}^*_p + {\mathbf {E}}^*_s {\mathbf {E}}_p)/I_0$ describes the light contrast inside the medium, where $I_0 = I_s + I_p = \mathrm{const}$ in the transmission geometry under condition to neglect of the absorption in the medium ($I_i = {\mathbf {E}}_i {\mathbf {E}}^*_i,\; {\mathbf {E}}_i = E_i e^{\varphi _i},\; i=s,p$). We use the approach in which we first obtain a solution for the distribution of the grating amplitude N(z), and then the output intensities will be expressed through this solution. For the steady state ($\partial N(z) /\partial t = 0$), the system (1)–(2) can be rewritten in new variables N(z), $J_m (z)$ and $J_d (z) = ({\mathbf {E}}_p {\mathbf {E}}^*_p - {\mathbf {E}}_s {\mathbf {E}}^*_s)/I_0$:

$$\begin{aligned}&N (z) = \gamma _N J_m (z) \; ; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \end{aligned}$$

(3)

$$\begin{aligned}&\frac{\text {d} J^2_m}{\text {d} z} = 4 k \gamma _N \cdot J_d J^2_m \; ; \;\;\;\;\;\; \frac{\text {d} J_d}{\text {d} z} = - 2 k \gamma _N \cdot J^2_m \end{aligned}$$

(4)

Dividing the first equation by the second in (4), we get the first integral of the system:

$$\begin{aligned} C^2 = J^2_m + J^2_d = I_s I_p / I^2_0 + (I_p - I_s)^2/I^2_0 = \mathrm{const} \end{aligned}$$

(5)

Using the definition of $C^2$, we can integrate the first equation in (4) and find a solution for $J_m (z)$. Then applying (3) the amplitude profile of the grating will take the following form:

$$\begin{aligned} N(z) = \frac{\gamma _N C}{\cosh (2 \gamma _N C \cdot k z - q)} \end{aligned}$$

(6)

where both the constants C and the integration constant q are determined by the input intensities at the boundary of the medium. The formula (6) shows that in the steady state the spatial profile of the grating amplitude takes on a localized soliton-like form. Both the degree of localisation of the soliton-like profile of the N(z) and the location of its maximum are determined by two parameters: the maximum amplitude $\gamma _N$ and the ratio of the input wave intensities.

Knowing the spatial distribution of the grating amplitude N(z), we can determine the output intensities, wich are expressed as an integral over N(z) within the boundaries of $z_1$ and $z_2$ of the medium ([31, 32]). To do this, we rewrite the system (1) for the amplitudes and phases of the waves:

$$\begin{aligned} \frac{\text {d} E_s}{\text {d} z}= & {} k N(z) E_p \; \cos (\varphi _{sp}) \; ; \;\;\; \frac{\text {d} E_p}{\text {d} z} = - k N(z) E_s \; \cos (\varphi _{sp}) \; ; \nonumber \\ \frac{\text {d} \varphi _{sp}}{\text {d} z}= & {} k N(z) \frac{E^2_p-E^2_s}{E_s E_p} \sin (\varphi _{sp}) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \end{aligned}$$

(7)

where $\varphi _{sp} = \varphi _s (z)- \varphi _p (z)$. This system has the first integral $D = E_s E_p \sin (\varphi _{sp}) = \mathrm{const}$, so we can find $\cos (\varphi _{sp}) = \sqrt{E^2_s (I_0 - E^2_s)- D^2}/(E_s E_p)$. Introducing this value of $\cos (\varphi _{sp})$, we can combine the first two equations in (7) in the following integral equation:

$$\begin{aligned}&s_{[s,p]} \int \limits ^{I_{[s,p]}(z_2)}_{I_{[s,p]}(z_1)} \frac{\text {d} I_{[s,p]}}{ \sqrt{-I^2_{[s,p]} + I_0 I_{[s,p]} - D^2}} \nonumber \\&\quad = 2 k \int \limits ^{z_2}_{z_1} N(z) \text {d}z \end{aligned}$$

(8)

where one need to choose either $I_s$ and the sign $s_s=1$, or $I_p$ with $s_p=-1$. We denote the right-hand side in (8) by $U_d=2 k \int _{z_1}^{z_2} N(z) \text {d}z$, which has the physical meaning of the area under the profile of the grating amplitude within the boundaries of the nonlinear medium. Substituting here the solutions (6), we find:

$$\begin{aligned} U_d= & {} 2 \arctan \frac{e^{\chi _2} -e^{\chi _1} }{1+e^{\chi _1+\chi _2}} \;,\;\;\;\;\;\;\; \nonumber \\ \chi _1= & {} 2 k \gamma _N C z_1 -q \;, \;\;\;\; \chi _2 = 2 k \gamma _N C z_2 -q \end{aligned}$$

(9)

Then the solutions for the output intensities are:

$$\begin{aligned} \frac{I^\mathrm{out}_{s}}{I_0}= & {} \frac{1}{2} \sin \Big [\pm U_d + \arcsin \Big (2 \frac{I_{s0}}{I_0}-1\Big ) \Big ] + \frac{1}{2} \end{aligned}$$

(10)

$$\begin{aligned} \frac{I^\mathrm{out}_{p}}{I_0}= & {} \frac{1}{2} \sin \Big [\mp U_d + \arcsin \Big (2 \frac{I_{p0}}{I_0}-1\Big ) \Big ] + \frac{1}{2} \end{aligned}$$

(11)

In formulas (10)–(11), the upper sign before $U_d$ is taken for the case of direct energy transfer, and the lower one is for the reverse energy transfer.

The calculations of the energy transfer according to the formulas (9)–(11) in a wide range of the input intensity ratio are shown in Fig. 3. The curves are plotted in a coordinate system in which the input intensity of the signal beam decreases and the input pump beam increases in the positive direction of the abscissa axis. In the indicated normalization, these curves correspond to diffraction efficiencies, which we define as follows: $\eta _s = I^\mathrm{out}_s / (I^\mathrm{out}_s + I^\mathrm{out}_p)$ for the signal wave; $\eta _p = I^\mathrm{out}_p / (I^\mathrm{out}_s + I^\mathrm{out}_p)$ for the pump wave, and $I^\mathrm{out}_s + I^\mathrm{out}_p = I_0 = 1$.

Fig. 3

Theoretical calculations of energy transfer between two coupled waves under Bragg conditions in a bulk medium for different maximum amplitudes of a stationary dynamic grating: a $\gamma _N = 5 \cdot 10 ^{-6}$ ; b $\gamma _N = 5 \cdot 10^{-7}$. Both cases for the sign before $U_d$ in the Eqs. (10)–(11) are plotted: solid lines are for the upper sign, and dash lines for the lower sign; dotted lines for the input intensities. Initial parameters of the optical system: $I_0 = I_{p0} + I_{s0} = 1$, $z_2 - z_1 = 50$ mm, $q=0$. Black lines for the signal beam, red lines for the pump beam

Figure 3 shows that we have obtained two different modes of energy transfer depending on whether the maximum amplitude $\gamma _N$ of the photoinduced grating is weak or strong. In the case of strong $\gamma _N$, the theory predicts the complete ($\sim 100 \%$) energy transfer from the pump beam to the signal one. Moreover, a monotonic increase of the signal intensity occurs for the entire range of the input intensity ratio (see Fig. 3a). The explanation of this result is related to the value of the total phase delay $U_d$ of the generated dynamic grating which depends on the value of $\gamma _N$, as follows from (6). For a weak value of $\gamma _N$, the value $U_d \le \pi /2$. Moreover, the degree of localization of the grating is small. The grating has a small amplitude and an almost uniform profile over the thickness of the medium. According to formulas (10)–(11), there is a traditional behavior of energy exchange: an increase of the signal intensity for the case of direct energy transfer, and a decrease of the signal intensity accompanied by an increase of the pump intensity during the reverse energy transfer (see Fig. 3b). When $\gamma _N$ is strong, the grating localization becomes more pronounced in an extended medium. The magnitude of $U_d$ is increased to $\pi$: $U_d \rightarrow \pi$, and then $\sin (U_d)$ changes its sign in the formulas (10)–(11). As a result, a new mode appears which represents an unidirectional energy transfer. In this case, the intensity of a weaker signal beam always increases regardless of the sign in front of $U_d$. In Fig. 4 shows the change in the absolute values of the output signal intensity with a decrease in $I_{s0}$ and a strong constant input pump intensity $I_{p0}$.

The appearance of asymmetric profiles of the output intensities at $I_{s0} \approx I_{po}$ requires additional research. We only note that a significant asymmetry is observed when the total phase delay is $U_d \rightarrow \pi /2$ due to diffraction on the grating (or, more generally, in the resonances when $U_d \rightarrow n \cdot \pi /2$, n-integer). The asymmetry of the profile, along with the interference many waves to form separate independent channels, indicates a similarity with the Fano effect. Similar asymmetric resonant profiles are observed for Fano effects resulting from the interference of (at least two) wave processes, in particular, in the case of wave interference in nanostructures [33, 34]. There are two inherent features of Fano resonance for TDG, such as destructive interference and asymmetric resonance profile. The same applies to the suppression of orders with different (multiple) spatial frequencies, shown in Fig. 2. Such suppression can be associated with destructive interference and relates to arising the Fano resonance. The similar was observed in the interaction of several chains of atoms, or guided resonances for transmitted light in photonic crystals [33, 35, 36]. Revealing Fano effects for our system deserves additional research.

Fig. 4

Normalized output intensity of the signal beam depending on the intensity of the input signal and the amplification coefficient for a strong input pump beam $I_{p0}/I_0 = 200$. The rest of the initial parameters are the same as in Fig. 3; case $+ U_d$

Note that our calculations were made under the assumption that the maximum of the grating amplitude is located in the middle of the extended medium ($q=0$) and does not change its position with varying the input intensity ratio. That is, in this calculation $z_1 = -25$ mm, $z_2 = 25$ mm for the thickness of the medium $Z_0=50$ mm. In real dynamic photorefractive media, when the mutual intensities of the input waves change, not only the degree of localization of the grating changes, but also its maximum changes its location along z-axis [31, 32, 37]. An exception is four-wave mixing with four equal input intensities $I_{1} (z_1) = I_{2} (z_1) = I_{3} (z_2) = I_{4} (z_2)$ or symmetric intensity ratio at the input boundaries: $I_{1} (z_1) / I_{2} (z_1) = I_{4} (z_2) / I_{4} (z_2)$. Moreover, it can be shown that the value of $U_d$ can not exceed $\pi$ in the case of a dynamic grating recording during self-diffraction in a bulk nonlinear medium. Thus, our calculations are made formally and correspond to some kind of artificial case. But in the following sections, we will show that such a case can be realized in the system of a thick dual-grating presented in Fig. 1.

4 Experimental study of energy transfer in the holographic amplifier

We experimentally studied the energy transfer between two laser beams at a holographic system with a thick dual-grating. An experimental layout is shown in Fig. 5. Laser beam with $\lambda =632.8$ nm divided by beam splitter Bs on two beams, each of them paths through of the digital grids $DG_1$ and $DG_2$, which form two incident beams with the intensities $I_{p0}$ and $I_{s0}$ and the convergence angle $2\theta$ between them. Digital phase gratings were produced by the SLM. Fresnel rhomb RFr and analyzer A providing the phase operating mode of the SLM. Changing of the phase modulation of $DG_2$ in the range of [0-$\pi$] allow us to produce the beam $I_{s0}$ with a given change in intensity. The another beam $I_{p0}$ which is generated by $DG_1$, possesses a stable intensity. CCD camera with a lens L register the output intensities of both beams, $I_s^\mathrm{out}$ and $I_p^\mathrm{out}$. In these experiments on changing the signal intensity using SLM, we have revealed that the both outgoing beams, $I_{s0}$ and $I_{p0}$, do not change their phase under any modulations of $DG_1$ and $DG_2$.

Fig. 5

Optical holographic set-up containing TDG for experimental study of energy transfer between two coherent waves: laser—He–Ne laser (632.8 nm); BEx—beam splitter; $Mr_{1}$, $Mr_{2}$, $Mr_{3}$—mirrors; RFr—Fresnel rhomb; BS—beam splitter; SLM—spatial light modulator HoloEye HEO-0017; $DG_1$, $DG_2$—digital phase grids; D—diaphragm; A—analyzer; $G_{1}$, $G_{2}$—phase gratings with distance $Z_{0}$=50 mm between them, L—Fourier-lens; $\textit{CCD}$—a camera; $PC_1$ and $PC_2$—computers; $I_{s0}$, $I_{p0}$ with 2$\theta$ angle between them and $I_{0}$ are the input intensities; $I_{s}^\mathrm{out}$ and $I_{p}^\mathrm{out}$ are the output intensities

The experimental set-up shown in the Fig. 5 was also used to record holographic gratings, which formed TDG. Holographic plates (VRL-2) were placed sequentially in the $P_r$ plane for exposure by a light interference pattern formed by beams with the same intensities $I_{p0} = I_{s0}$. After light exposure, the plates were developed and bleached. Changing the exposure time, the plates recorded different bright contrast, which after processing gave different modulation depths of the holographic gratings. All gratings have the same period $\varLambda =22\,\mu$m and thickness of $d = 18 \pm 3\,\mu$m.

The above scheme was used to register the intensity spectrum of diffraction orders to define the modulation depth of fabricated gratings. To register the intensity spectrum formed by an individual grating, we use a mirror $Mr_3$ and a beam $I_0$, which propagated through the phase grating, as well as the lens L and CCD connected with PC. An experimental intensity spectrum generated by a grating was compared to the theoretically calculated spectrum (see our works [23, 24]). In this manner, we finally selected four gratings, $G_1, G_2, G_3, G_4$, with following experimentally defined phase delays: $\varPhi _1 = 1.5$, $\varPhi _2=1.465$, $\varPhi _3=0.355$ and $\varPhi _4=0.25$ rad. In the TDG design, one can change the modulation depths of the two gratings and the distance between them. We have developed two TDG that have similar modulation depths for both gratings and whose characteristics are given in Table 1.

Table 1 The elements of two different TDGs used in the experiment

Fig. 6

Pictures of energy transfer recorded on a CCD for the case of equal input intensities. a Direct energy transfer, that is, the signal beam is amplified (right spot). b Reverse energy transfer, that is, the pump beam is amplified (left spot)

We have studied the change in intensities in two main diffraction orders due to diffraction of two input coherent beams on the TDG depending on the input intensity ratio $I_{p0}/I_{s0}$. At the beginning of each experimental measurement, equal input intensities $I_{p0}=I_{s0}$ were set, and we adjusted the shift $\delta$ of the TDG relative to the interference pattern to achieve the maximum value of the energy transfer. In the Fig. 6 we present the typical output intensity spectrum detected at the CCD for equal input intensities, where we show the direct energy transfer from the pump beam to the signal beam (a), and the reverse energy transfer from the signal beam to the pump (b). After adjusting the maximum value of the energy transfer for the case of $I_{p0}=I_{s0}$, we started to control a phase modulation of the signal beam in the SLM, which causes a decrease in $I_{s0}$.

Fig. 7

Experimental energy transfer in $TDG_1$ (consisting of gratings with strong modulation depths)—I, and $TDG_2$ (consisting of gratings with weak modulation depths)—II, depending on the input intensity ratio: a direct energy transfer, b reverse energy transfer, c input intensities. Black marks for the signal beam, red marks for the pump beam. (Lines are only for visualisation)

The experimental results of the energy transfer in the dependency of the input intensity ratio are presented in Fig. 7 for the $TDG_1$ (Fig. 7-I), and for the $TDG_2$ (Fig. 7-II). There we show the normalized values of the output intensities, $I_{p}^\mathrm{out}/I_0$ and $I_{s}^\mathrm{out}/I_0$ for the direct energy transfer (graphs (I-a), (II-a)), for the reverse energy transfer (graphs (I-b), (II-b)), and normalized input intensities, $I_{p0}/I_0$ and $I_{s0}/I_0$ (graphs (I-c), (II-c)), where $I_0 = I_{s0} + I_{p0}$.

It can be seen that for $TDG_1$ (Fig. 7-I), which contains gratings with strong modulation depths, the signal beam is amplified by a pump beam even under a strong decrease in the input signal. In case of reverse energy transfer at the beginning, Fig. 7(I-b), the direct energy transfer is restored with decreasing input signal intensity, and the signal beam again becomes amplified.

But for $TDG_2$, which consists of gratings with weak modulation depths, the situation is vice versa. In this case, the amplifying beam is the pump beam. With decreasing the input signal intensity, the output signal is reduced (Fig. 7-II).

The real experimental values of the intensities obtained in our set-up were as follows. The input intensity of the pump beam did not change, and its measured value was $I_{p0} \approx 240 \; \mathrm{a.u.}$ for $TDG_1$ and $I_{p0} \approx 250 \; \mathrm{a.u.}$ for $TDG_2$. At the same time, the input intensity of the signal beam was decreased, the measured values of $I_{s0}$ were in the range $I_{s0} \approx 240 ... 7 \; \mathrm{a.u.}$. For the output signal beam in the scheme with $TDG_1$, we obtained an increase in the intensity of the smallest input signal by more than 20 times for both cases of direct $+ U_d$ and reverse $- U_d$ energy transfer; the absolute value reached $\approx 160 \; \mathrm{a.u.}$ Simultaneously, the output pump beam dropped to a value of $\approx 30 \; \mathrm{a.u.}$ But for $TDG_2$, the smallest signal increased only by 10–20 a.u. in the case of a direct energy transfer; and for a reverse energy transfer, it slightly decreased or did not change.

In addition, in our experiments, we observed that under the condition of maximum energy transfer, the total energy is almost completely concentrated in two main diffraction orders with the smallest part scattered in high diffraction orders. Thus, in our case, the TDG system works similarly to one thick Bragg grating, despite the fact that the TDG includes thin gratings. We confirm this important finding in our further theoretical modeling.

5 Theoretical modeling of energy transfer in TDG

In this section we show that the Bragg diffraction of two coupled waves on the TDG can be described analytically on the base of coupled-wave theory. To apply the coupled wave model, we can approximate TDG as a bulk medium $Z_0$ thick, on the boundaries of which there are thin phase gratings $G_1$ and $G_2$, and they determine the boundary conditions for the grating amplitude. We assume the bulk medium contains one thick phase grating $G_{SMG}$. The modulation depth of the thick grating $G_{TDG}$ is denoted by $\gamma _{\mathtt {N}}$. We can determine the value of $\gamma _{\mathtt {N}}$ from the assumption that the total phase delay, $\varPhi _{TDG}$, which the wave would receive due to diffraction by one thick grating $G_{TDG}$, should be equal to the sum of the phase delays, $\varPhi _1$ and $\varPhi _2$, that the wave would receive by diffraction on each of thin gratings $G_1$ and $G_2$, forming the TDG. Thus, $\varPhi _{TDG}$ satisfies the following relation:

$$\begin{aligned} \varPhi _{TDG}= & {} \gamma _N k Z_0 = \varPhi _1 + \varPhi _2 \end{aligned}$$

(12)

$$\begin{aligned} \varPhi _{1,2}= & {} \varDelta n_{1,2} \cdot k d \end{aligned}$$

(13)

where $\varDelta n_1$ and $\varDelta n_2$ are the modulation depths of the corresponding thin gratings (that is, the maximum deviation of the refractive index of the grating from its average value $n_0$ ($n= n_0 + \varDelta n$)); d is the thickness of a thin grating. The phase delay is an integral of the grating amplitude along the thickness within the boundaries of the medium multiplied by k.

Table 2 The parameters used to calculate the energy transfer in the TDG in accordance with the formulas (9)–(11)

Fig. 8

Calculation of energy transfer between two main diffraction orders using the formulas (9)–(11) for experimental input intensity ratios (according to Fig. 7) and TDGs (Table 1); solid lines for $+U_d$—direct energy transfer, dash-and-dot lines for $-U_d$—reverse energy transfer, dotted lines for the input intensities; a $TDG_1$; b $TDG_2$. Black lines for the signal beam, red lines for the pump beam

We performed theoretical calculations of energy transfer using the formulas (9)–(11) for the schemes containing $TDG_1$, $TDG_2$, and for such input intensity ratio as in experiment. To do this, we needed to determine the parameters $\gamma _N$ and q, which correspond to the characteristics of the TDGs. The value of $\gamma _N$ is calculated by the ratio (12). The value of C is obtained from (5) using the input intensities: $C=\sqrt{I^2_{s0} - I_{s0} I_{p0} + I^2_{p0}} /I_0$. The parameter q determines the offset of the maximum of the amplitude profile of the grating N(z) relative to the origin. Since we assume that thin gratings are located at the boundaries $z_1$ and $z_2$, this parameter can be found by constructing the function N(z) by the formula (6) with the boundary conditions: $[N(z_1) ; N(z_2)] = [\varPhi _1 ; \varPhi _2 ]$. In Table 2 we have collected all the parameters, $\gamma _N$ and q, which correspondent to real TDGs, to carry out theoretical calculations.

Theoretical calculations for two different TDGs using experimental intensities for input beams are shown in the Fig. 8. The obtained theoretical graphs are in very good agreement with the experimental data (Fig. 7). Thus, these calculations, based on a model developed for a thick Bragg grating, confirm that TDG fulfills the coupling of two beams with the Bragg diffraction conditions.

6 Conclusion

We have demonstrated both experimentally and theoretically that two matched phase gratings separated by a space layer (the so-called TDG is a thick dual-grating) can work as one volume Bragg grating. When the TDG is illuminated by two coherent laser beams that form a light interference pattern, their diffraction by the TDG results in predominant energy transfer between two Bragg diffraction orders provided there is a shift $\delta$ of the interference pattern relative the TDG in the transverse direction. To calculate the maximum energy transfer in the TDG we have developed analytical model that is based on coupled wave theory (see Sects. 3 and 5).

When both grating in TDG have a small modulation depth, the classical effect of wave coupling takes place, the same as in a bulk medium with nonlocal optical nonlinearity. That is: direct energy transfer from the pump beam to the signal beam when the spatial shift $\delta = \varLambda / 4$; and reverse energy transfer from the signal beam to the pump beam for the spatial shift $\delta = - \varLambda / 4$. This spatial shift can be achieved by adjusting the phase difference between the two input laser beams.

We have discovered a new mode of unidirectional energy transfer for the case when both gratings in the TDG have a large modulation depth. Then there is only direct energy transfer to amplify the weaker beam by the stronger pump beam, regardless of the direction of the spatial shift $\delta$. This mode was obtained for TDG both experimentally and theoretically, and it was not known for nonlinear photorefractive media.

Thus, our studies open up prospects for using a set of periodic gratings spaced at some distance, for a significant increase in intensity in an individual diffraction order, using all-optical methods: by changing the ratio of the input intensities of beams or/and modulation of the phase of one of the beams. The proposed holographic amplifier, in a simple and natural way, allows it to be adapted to a four-wave mixing scheme to creation optical phase conjugation (OPC) with unprecedented amplification of the OPC-beam. The advantages of this scheme are a high gain for weak laser beams in combination with the ability of image amplification when using a nonlinear medium instead of one of the grating, as well the simplicity of manufacturing TDG by laser engraving in a single monoblock.