## Abstract

We numerically analyze phase-sensitive parametric amplification and de-amplification of a multimode field representing a multi-pixel text image. We optimize pumping configuration and demonstrate that ~10-dB gain is achievable with relatively moderate ~10-kW total pump peak power available from compact pump sources.

©2009 Optical Society of America

## 1. Introduction

Phase-sensitive amplifiers (PSAs) have unique properties that allow them to break the 3-dB quantum limit of the optical-amplifier noise figure (NF) [1], if the input signal phase is chosen for maximum gain, and to generate squeezed light, if the input phase is chosen for maximum attenuation. Such PSAs can be experimentally realized using optical parametric amplifiers (OPAs), operating either in a degenerate regime (where signal and idler modes are the same), or in a non-degenerate regime with both signal and idler modes simultaneously excited at the input. Over the last decade, PSAs based on *third-order* nonlinear susceptibility *χ*
^{(3)} in single-mode fibers have been gaining popularity in optical communications as noiseless amplifiers [2–4], regenerators for timing and phase jitters [5–8], with capabilities for distributed [9] and multichannel [9, 10] operation.

On the other hand, travelling-wave OPAs based on *second-order* nonlinear susceptibility *χ*
^{(2)} in bulk crystals offer an additional advantage of being able to simultaneously amplify a number of spatial modes within their phase-matching bandwidth (acceptance angle) [11]. When operated as PSAs, these amplifiers enable noiseless optical amplification of images, discussed in [12, 13] and experimentally demonstrated in [14, 15]. We have recently proposed a scheme [16], in which a spatially broadband PSA overcomes the sensitivity and resolution impairments of a coherent laser radar (LADAR) from inefficient and noisy receiver as well as from soft input aperture by injecting squeezed vacuum into the spatial modes attenuated by the aperture and noiselessly amplifying the weak return image. The practical implementation of this approach, however, is rather challenging. Indeed, the number of amplified pixels in a spatially-broadband PSA is limited by the available pump power, which makes it important to optimize the pumping arrangement for the best efficiency. Although we have already experimentally demonstrated noiseless image amplification by a PSA [14], it required about 5 kW of pump peak power per pixel, and even with a Q-switched mode-locked (QSML) pump laser we could only process a very simple 2-slit image. While such pump lasers can produce very-high optical powers, their use is undesirable both from convenience (big size, weight, power consumption) and from PSA performance points of view. Indeed, in [17] it was shown that the two out of the three main causes of the PSA-noise-figure degradation are inherent to the use of a QSML because: a) the PSA gain is averaged over the Gaussian-pulse temporal profile of such laser’s output and b) the input-signal phase errors cause gain fluctuations (because low Q-switch repetition rate does not permit fast phase-lock loop). On the other hand, combination of a distributed feedback (DFB) diode laser and an erbium-doped-fiber-amplifier (EDFA) as a pump source permits flat-top pulse carving by direct or external modulators and the use of MHz-repetition-rate low-duty-cycle pulse trains allows achieving kilowatt-type peak powers from conventional 1–10W-class EDFAs, eliminating the two QSML-related problems above. The third cause of PSA-noise-figure degradation is the gain-induced diffraction [17], originating from the finite size of the pump beam and necessitating semi-analytical and numerical modeling of the PSA beyond the standard plane-wave-pump approximation.

In this paper, we investigate the possibility of amplification of a multimode (multi-pixel) image with pump powers ≤10 kW typically available from the aforementioned compact and inexpensive DFB/EDFA combinations. For spatially inhomogeneous pumps, such as Gaussian beams, prior image amplification theory has considered transverse mode coupling in a short *χ*
^{(2)} crystal inside a cavity [18], where semi-analytical mode expansion over Laguerre-Gaussian polynomials was used. For travelling-wave PSAs, Refs. [19, 20] used a semi-analytical approach expanding the signal over radially symmetric Laguerre-Gaussian polynomials. While useful for finding best-squeezed modes, this approach in its present form cannot be applied to practical images because of its radial symmetry restrictions. In contrast to it, we permit any input signal and pump spatial profiles, and solve the PSA equations by direct numeric integration using split-step Fourier method in paraxial approximation with undepleted pump. This numerical method, commonly used in fiber optics, has already been proven helpful in the imaging PSA context for semi-classical modeling of noise [21]. Here, we apply it to investigate the propagation of a realistic two-dimensional (2-D) multi-pixel image without radial symmetry through a PSA pumped by an elliptical Gaussian beam, which is a significant step beyond the approximations used in the prior imaging PSA studies. We show that, for typical nonlinear-crystal parameters, after optimization of the pump spot sizes in two dimensions, it is possible to achieve more than 6 dB of gain across an image with ~30 pixels, with a peak gain over 10 dB.

The rest of this paper is organized as follows: Section 2 describes the theoretical background of parametric amplification of images, Section 3 presents and Section 4 discusses the data from our modeling, and Section 5 summarizes the results.

## 2. Theory of parametric image amplification

A detailed theory of parametric amplification of multimode fields is summarized in a recently published book [22]. Here, we will concentrate on OPA equations in undepleted-pump paraxial approximation with polarized (scalar) fields. We are looking for solutions in the form *e*(*r*⃗,*t*)=*E*(*ρ⃗z*)*ei ^{(kz-ωt)}*+ c.c., where

*E*(

*ρ⃗,z*) is a slowly-varying field envelope,

*ρ*⃗ is a transverse vector with coordinates (

*x,y*), and the intensity is given by

*I*(

*ρ⃗,z*)=2

*εnc*|

*E*(

*ρ⃗,z*)|

^{2}. In the presence of a strong pump

*E*(

_{p}*ρ*⃗,

*z*) at frequency

*ω*, the signal electric field

_{p}*E*(

_{s}*ρ*⃗,

*z*) at frequency

*ω*is coupled to the idler electric field

_{s}*E*(

_{i}*ρ*⃗,

*z*) at frequency

*ω*=

_{i}*ω*-

_{p}*ω*through the following equation:

_{s}where Δ*k*=*k _{p}*-

*k*, and the equation for the idler beam is obtained by interchanging subscripts

_{s}-k_{i}*s*and

*i*in Eq. (1). Equation (1) describes the traveling-wave OPA in paraxial approximation with a pump of arbitrary spatial profile.

Let us define spatial-frequency (*q*⃗) domain via the direct and inverse Fourier transforms

In the absence of the pump (*E _{p}*=0), Eq. (1) is reduced to the paraxial Helmholtz equation, whose solution in the Fourier domain is given by

*Analytical solution 1: plane-wave pump*

With the plane-wave pump ${E}_{p}(\overrightarrow{\rho},z)=\mid {E}_{p}\mid {e}^{i{\theta}_{p}}$ = const., Eq. (1) still retains the shift-invariance of the original paraxial Helmholtz equation and hence is easily solved in Fourier domain [23]:

where the coefficients of the input-output transformation are

$$v\left(q\right)=i\genfrac{}{}{0.1ex}{}{{\kappa}_{s}}{\gamma}\mathrm{sinh}\mathrm{\gamma z}\times \mathrm{exp}\left(i\genfrac{}{}{0.1ex}{}{\Delta {k}_{\mathrm{eff}}}{2}z\right)\mathrm{exp}(-i\genfrac{}{}{0.1ex}{}{{q}^{2}}{2{k}_{s}}z),$$

the effective wavevector mismatch is

and the parametric gain coefficient *γ* is given by

The evolution of the quantum field operators also obeys Eq. (4), which leads to quantum correlations between *q*⃗+ and *q*⃗- spatial-frequency components of the image, demonstrated in [24]. Note that, in the degenerate case (signal and idler beam are the same), the product of the exponentials in Eqs. (5), containing the effective mismatch and diffraction phase terms, becomes simply exp(*ikz*/2). This is also approximately true for the non-degenerate case if *k _{s}*≈

*k*. For maximum phase-sensitive gain, the magnitudes of

_{i}*E*̃

*(*

_{s}*q*⃗,0) and

*Ẽ*(-

_{s}*q*⃗,0) should be the same. For a crystal of length

*L*, the optimum input signal phase is given by

The maximum PSA gain

is achieved for optimum signal phase of Eq. (9) that is *q*-dependent and, therefore, may not be easily realizable. However, for small or moderate values of *κL*, optimum signal phase of Eq. (9) is

i.e., the optimum (quadratic) spatial-frequency dependence can be straightforwardly obtained by placing the focus of the image at *z*
_{0}=*L*/2 (middle of the nonlinear crystal). The minimum PSA gain (de-amplification) is the inverse of the gain in Eq. (10) and is obtained by shifting the signal phase by *π*/2 from the optimum of Eq. (9).

*Analytical solution 2: short crystal with inhomogeneous pump*

For sufficiently short crystals, the diffraction term in Eq. (1) can be neglected, and the equation takes the following form:

Here κ* _{s}*, defined in Eq. (8), is, in general, a complex parameter that depends on the coordinate

*ρ*⃗ (if the pump is inhomogeneous); at the same time, due to the short crystal length, the

*z*-dependence of the pump is neglected. One can then introduce parameters µ and ν as

$$v(\overrightarrow{\rho},z)=i\genfrac{}{}{0.1ex}{}{{\kappa}_{s}}{\gamma}\mathrm{sihh}\mathrm{\gamma z}\times \mathrm{exp}\left(i\genfrac{}{}{0.1ex}{}{\Delta k}{2}z\right),$$

so that the solution takes the form of point-by-point (pixel-by-pixel) amplification:

The optimum signal phase and the maximum PSA gain are still given by Eqs. (9) and (10), respectively, where Δ*k*
_{eff}=Δ*k* is assumed, and θ* _{p}* and

*γ*may vary as functions of

*ρ*⃗.

*Case considered in this paper: finite-length crystal with inhomogeneous pump*

For the case of spatially-inhomogeneous pump (e.g., Gaussian beam) and non-negligible diffraction term, Eq. (1) has to be solved numerically. For circular-Gaussian pump and signal beams, the computation can be sped up by decomposing the signal over radially symmetric Laguerre-Gaussian modes [19, 20]. In the more general (non-radially-symmetric) case, 2-D Hermite-Gaussian modes need to be used, making the solution more computationally demanding. For our simulations, however, we do not use any mode decomposition, but instead employ numeric integration of Eq. (1) by split-step Fourier method (aka FFT-BPM), which enables us to use arbitrary pump and signal spatial profiles.

## 3. Modeling results

We concentrate on amplification of a 2-D image with unequal number of pixels in the two dimensions, which suggests using elliptical Gaussian spatial profile for the pump. First, we investigate the possibility of phase-sensitive amplification and de-amplification of several signal modes by a single elliptical Gaussian pump beam (Fig. 1a), where we observe very similar gains for the first three Hermite-Gaussian signal modes.

As the next step, we amplify a 2-D text image [the word “QUANTUM” in Fig. 1(b) and Figs. 2(a) and 2(b)] with 10 kW of pump power. For this 2-D image amplification, we assume a nonlinear crystal with *d*
_{eff}=8.7 pm/V and length *L*=2.5 cm (typical values for a periodically-poled KTP crystal), pumped by the second harmonic of 1560-nm light. The image to be amplified is obtained by illuminating a mask with transparent letters by a plane wave. In the absence of gain, the image plane is located at the center of the crystal. The image size is ~470 µm×90 µm, with line thickness of ~10 µm. We slightly smoothen the edges of the letters to avoid aliasing in split-step computation. 5.2:1 aspect ratio of the image makes a strong case for using elliptical rather than circular Gaussian pump. After tedious optimization, we have found that the highest total-power PSA gain takes place for pump with 1/e intensity radii 440 µm×25 µm, and amounts to 11.2 dB at 10 kW pump power. The amplified image is shown in Figs. 1(c), 2(c), 2(d), and 2(e). The maximum output intensity (center letters) is 10.1 dB above the input. The intensity gradually rolls off at the sides, while still gaining more than 6 dB over the input everywhere, including the peripheral letters. The fact that the total power gain is higher than the local gain for most of the letters can be explained by the presence of a broad pedestal under the amplified letters [particularly evident as a bright oval in the center of Fig. 2(d)]. As we discuss in the next Section, this appears to be a manifestation of low-pass spatial frequency filtering by the PSA, even though this concept in strict sense is only valid in plane-wave-pump (spatially invariant) case. Despite this filtering and gain non-uniformity across the image field, the amplified text is clearly recognizable.

Next, we change the input signal phase by 90° and observe phase-sensitive de-amplification (PSD, Fig. 3). PSD gain is very sensitive to the input signal phase. In particular, -45° phase (optimum for the plane-wave-pump case with *k*=0 and the image located at the crystal’s center) is not optimal here and fine phase tuning leads to -55° optimum [Fig. 3(a)], improving the total-power PSD gain from 0.05 dB to -2.2 dB. In contrast, the PSA gain varies less with the input phase: tuning it from +45° (optimum for the plane-wave pump) to +35° increases the total-power gain by only 0.1 dB and the peak gain by only 0.2 dB (the data in Figs. 1 and 2 correspond to the optimum +35° signal phase). The phenomenon of the optimal phase shifting away from its plane-wave-pump-case value is similar to that noticed in [19] for circular Gaussian pump. Figures 1(d) and 3 show that the de-amplified image is easily recognizable.

The observed classical PSA and PSD gain values can provide some indication of what to expect in the quantum case, where anti-squeezing and squeezing of noise quadratures will take place. In fact, Ref. [19] provides a reciprocity argument for OPA equations, which relates the observed classical total-power gains of a signal mode with arbitrary profile (the text “QUANTUM” in our case) to the degree of squeezing measurable in the mode of squeezed vacuum coming out of the same PSA and detected by a local oscillator with field profile conjugate to that of the signal. Thus, there exists such a conjugate mode whose anti-squeezing and squeezing factors at the output of our PSA are 11.2 dB and -2.2 dB, respectively.

## 4. Discussion of amplified image resolution

Let us estimate the resolution of our PSA from simple considerations based on plane-wave-pump theory. As we have shown in [25], the most accurate estimate of the spatial bandwidth of the PSA [-3-dB point of the (*G*
_{PSA}-1) function, to be precise] for Δ*k*=0 is given by

and the -3-dB radius of the corresponding point-spread function (PSF) in space domain can be estimated as Δρ≈1/*q*. Assuming plane-wave-pump intensity equal to 25% of the peak Gaussian-pump intensity used in Section 3 (this is needed to match the peak image gains in the two cases, as shown below), we obtain from Eq. (16): Δ*q*=26 rad/mm (i.e., 4.2 lines/mm) and Δρ≈1/*q*=38 µm. These numbers are not very different from the exact values: -3-dB bandwidth Δ*q*=25 rad/mm (i.e., 4.0 lines/mm) and -3-dB PSF radius Δρ=44 µm.

For an inhomogeneous pump of a given spot size, the number of amplified pixels can be approximated by the ratio of the pump-beam area π*a _{0px}a_{0py}* and the effective pixel area π(Δρ)

^{2}of the plane-wave-pump case, which yields between 5 and 8 effective pixels of resolution in our chosen example (depending on which of the two Δρ estimates above we use). Curiously enough, this order-of-magnitude-accurate value is 4–6 times lower than the observed number of amplified pixels in Fig. 1(c), estimated by counting the numbers of bright-dark line pairs in horizontal and vertical directions of the image, which yields at least 15×2=30 effective pixels of resolution.

The difference between the analytically estimated and numerically observed numbers of resolution pixels can be explained by the fact that the PSFs used in the analytical estimates are related to the (*G*
_{PSA}-1) rather than the *G*
_{PSA} (PSA gain) function in the spatial frequency domain. Because the PSA is a transparent amplifier (when pump is turned off, the signal is transmitted through it without degradation), *G*
_{PSA} equals unity at high spatial frequencies, i.e., the high-frequency content of the image is preserved, but not amplified. This leads to the presence of a delta-function δ(ρ⃗) in the complete PSF of the PSA, and this delta-function represents the original (unamplified) image. The analytical and semi-analytical estimates modify the PSF by subtracting this delta-function from it, which makes such modified PSF a smooth function given by the inverse 2-D Fourier transform of the $(\sqrt{{G}_{\mathrm{PSA}}\left(q\right)}-1)$ function in spatial-frequency domain. As a result, the pixel sizes above, based on the modified PSF, overestimate the radius of the effective pixel. Thus, even the exact -3-dB radius of the modified PSF is not accurate enough for * quantitative* arguments and cannot be straightforwardly applied to access the quality of the amplified image. The final verdict on resolution can only be given by observing the amplified image.

Let us illustrate this point by processing the same 470 µm×90 µm input image as that in Section 3 by two PSFs in plane-wave-pump approximation: one without the delta-function [i.e., corresponding to $(\sqrt{{G}_{\mathrm{PSA}}\left(q\right)}-1)$ transfer function in spatial frequency domain], and the other with the delta-function [i.e., corresponding to $\sqrt{{G}_{\mathrm{psa}}\left(q\right)}$ transfer function in spatial frequency domain]. We assume that the input image has “almost optimum phase” [25], which differs from the optimum in Eq. (9) by lack of the *π*/2 phase jumps at spatial frequencies beyond the -3-dB bandwidth, contributed by the last term in Eq. (9) (we prefer the smooth phase profile because it results in smoother PSF). We also assume that a similar phase profile is applied to the amplified image at the PSA output to bring it into focus. Then the first PSF corresponds to *G*
_{PSA}=(|µ|-*iν*-1)^{2}, while the second to *G*
_{PSA}=(|µ|-*iν*)^{2} (“almost optimum gain” [25]). The intensities of the images processed by the former and the latter functions are shown in Figs. 4 and 5, respectively, after normalization by the peak intensity of the input image. The text in Fig. 4 is virtually unrecognizable (too few effective pixels of resolution), with the peak gain of 6.9 dB and the total power gain of 10.8 dB.

In contrast, Figure 5, obtained using a complete PSF (including the delta-function), shows both the amplified image and its quality that are very similar to those obtained with elliptical Gaussian pump [Figs. 2 (c), (d), (e)]. The peak gain for the image is 10.1 dB and the total power gain is 12.4 dB, while the gain at zero spatial frequency is 17.5 dB. As we have mentioned above, to match the peak gain of 10.1 dB from Section 3 in this way, we use the plane-wave pump intensity equal to 25% of the Gaussian-pump peak intensity in Section 3. Thus, these semi-analytical plane-wave-pump results corroborate the numerical modeling results on multi-pixel amplification in the elliptical-Gaussian-pump case.

The key to the difference between Fig. 4 (which is supported by the low resolution estimates) and Fig. 5 (which is similar to elliptical-Gaussian-pump results showing good resolution) is the absence and presence of the delta-function in the PSFs used for Fig. 4 and Fig. 5, respectively. Thus, Fig. 5 can be interpreted as the constructive interference of the electric field of poorly resolved image of Fig. 4 with the electric field of the original well-resolved input image. In other words, the normalized intensity pattern [i.e., *G*
^{Fig.5}
_{PSA}(ρ⃗) *G*] of Fig. 5 essentially equals to ${G}_{\mathrm{PSA}}^{\mathrm{Fig}.5}\left(\overrightarrow{\rho}\right)={\left[\sqrt{{G}_{\mathrm{PSA}}^{\mathrm{Fig}.4}\left(\overrightarrow{\rho}\right)}+\sqrt{{S}_{\mathrm{in}}\left(\overrightarrow{\rho}\right)}\right]}^{2},$, where *S _{in}*(ρ⃗) is the normalized intensity of the input image, varying from 0 to 1. Thus, the raised background in Fig. 5 represents the poorly resolved image of Fig. 4, given by G

^{Fig. 4}

_{PSA}(ρ⃗). The resolved letters come from the interference of G

^{Fig. 4}

_{PSA}(ρ⃗) and

*S*

_{in}(ρ⃗), rising above this background by

which for G^{Fig. 5}
_{PSA}(ρ⃗)=10 yields *G*
^{Fig. 5}
_{PSA}(ρ⃗)-G^{Fig. 4}
_{PSA}(ρ⃗)=5.3. Hence, despite the small weight (max *S*
_{in}(ρ⃗)=1) of the input signal compared to the poorly-resolved background, their constructive interference produces a swing in the output intensity that is 5.3 times greater than the input signal, leading to better than 50% contrast of the resolved letters in the output image.

The signal-to-noise ratio (SNR) and the optimum detection of the resulting image promise to be interesting topics for future studies, as the image involves a multi-pixel signal mode and the noise with a correlation function spread over many pixels (such that the mode profile is different from the correlation-function profile). Even though the image contains poorly resolved background, it appears that from the reciprocity argument of Ref. [19], the SNR of the mode represented by the letters “QUANTUM” might be well preserved if the peak gain and the total power gain are sufficiently close, as is the case in Figs. 2 and 3. While the effect of the detector size on the observed SNR and noise-figure of the PSA was studied in Refs. [12, 13, 15], the rigorous theory for the optimum detection of a multi-pixel image partially band-limited by the PSA has not been developed yet, as it involves decomposing the PSA output into independently squeezed modes (Karhunen-Loève expansion) and constructing a matched receiver for their detection. Even though the quantum noise properties of such an amplified image are not known yet, the 10-dB amplification shown in this paper can already improve the detected SNR if it is limited by the read noise and dark current of the conventional detector arrays.

## 5. Summary

We have numerically demonstrated that in an optical parametric amplifier with optimized pump, a text image with over 30 pixels can be phase-sensitively amplified by 6–10 dB with 10-kW peak pump power easily available from today’s compact sources. Under the same conditions, the image with 90°-shifted input phase can be attenuated by 2.2 dB. These results support the case for using phase-sensitive amplifiers for improvement of LADAR images.

This work was supported by the DARPA Quantum Sensors Program.

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