## Abstract

We compute the eigenmodes of a spatially-broadband optical parametric amplifier with elliptical Gaussian pump and show that the well-amplified eigenmodes can be compactly represented by a low-dimensional subspace of the first few Laguerre- or Hermite-Gaussian (LG or HG, respectively) modes of an appropriate waist size. We also show that the first few eigenmodes are well matched to single LG or HG modes. For sufficiently large pump waists, the optimum waist size of the compact basis is in the vicinity of the geometric average of the pump waist size and the inverse spatial bandwidth of the nonlinear crystal in the parametric amplifier. The use of such compact representation can greatly simplify numerical computation of the spatial eigenmodes of the amplifier and thus lead to improving the experiments on traveling-wave image amplification and spatially-broadband vacuum squeezing.

© 2013 Optical Society of America

## 1. Introduction

Phase-sensitive optical parametric amplifiers (PSAs) are unique in their ability to amplify a signal without adding any noise [1]. This property, in addition to the wide *temporal* bandwidth of fiber-based parametric amplifiers, has led to their use as nearly noiseless inline amplifiers for optical communication systems [2–7]. PSAs can also be used for noiseless image amplification, due to their broad *spatial* bandwidth, as was theoretically proposed in [8–10] and experimentally demonstrated in [11–15]. The signal-to-noise ratio improvement by such image pre-amplifiers before lossy or noisy detectors can enhance the resolution in the detection of faint images [16–18]. The same devices can also generate spatially-broadband (multimode) squeezed vacuum for quantum information processing applications [19, 20]. For example, simultaneous squeezing of frequency-degenerate families of Hermite- or Laguerre-Gaussian modes of an optical parametric oscillator has been predicted [21, 22] and achieved experimentally in conventional [23] and self-imaging [24] cavities; a spatially-multimode waveguide-based photon-pair source has been recently demonstrated [25].

For proper design of a parametric image amplifier, one needs to understand image propagation through the PSA under practical constraints of finite pump power and finite spatial bandwidth (the latter is determined from phase-matching conditions to be ~(*k _{p}*/

*L*)

^{1/2}, where

*k*is the pump propagation constant in the nonlinear crystal of length

_{p}*L*[8, 14, 26]). Indeed, the traveling-wave nature of gain in the PSA requires the use of a small-area pump beam (typically, the fundamental Gaussian TEM

_{00}mode) of high power (~1 kW per pixel of resolution) [27]. The resulting spatially-varying PSA gain, together with the limited spatial bandwidth of the PSA, couples and mixes up in both the space and spatial-frequency domains the modes that represent the information content of the image, and, in general, numerical methods are required to process the image propagation [27, 28]. These spatial mode-mixing effects are known as gain-induced diffraction [29–31], because the spatially-varying gain accelerates the diffraction of the amplified central region of the signal, which scatters light from the signal’s original mode into other modes and makes portions of the signal wavefront switch from amplification to deamplification. The gain-induced diffraction also makes it difficult to detect the lowest-noise mode of a traveling-wave squeezer [31, 32], because the design of a properly mode-matched homodyne detector requires exact knowledge of this mode’s spatial profile. To calculate the PSA noise properties, numerical methods based on either propagation of stochastic (Wigner-function-based) input [33, 34], or on computing noise variance from pixel-to-pixel Green’s function [22, 34] have been used.

To obtain a comprehensive description of the traveling-wave PSA, we have recently developed a procedure for finding the orthogonal set of independently squeezed (or amplified) PSA modes (eigenmodes) [35, 36] that is based on solving the parametric amplifier equation via a 2D Hermite-Gaussian (HG) expansion over the TEM* _{mn}* modes, which was originally developed for cavities [37] and for traveling-wave PSAs with circular Gaussian pump [38, 39]. Unlike the previous mode-expansion work, our method [36] can a) handle amplification by an elliptical TEM

_{00}pump with potentially different 1/

*e*intensity radii

*a*

_{0}

*and*

_{px}*a*

_{0}

*in*

_{py}*x*- and

*y*-dimensions, respectively, and b) find the eigenmodes of the spatially-broadband PSA once its Green’s function has been computed [35, 36]. For convenience, we expanded the signal over the TEM

*HG modes (denoted as HG*

_{mn}*modes below) with the same Rayleigh ranges*

_{mn}*z*=

_{Rx}*k*

_{p}a_{0}

_{px}^{2}and

*z*=

_{Ry}*k*

_{p}a_{0}

_{py}^{2}as the pump (i.e., with the signal having 2

^{1/2}times larger waist than the pump) and found the PSA Green’s function by solving the resulting system of coupled-HG-mode equations. For circularly-symmetric pumps, a similar procedure was also implemented by Laguerre-Gaussian (LG) expansion. With the knowledge of the eigenmodes and their gains, any multimode signal, quantum or classical, can be decomposed over the input eigenmode profiles, propagated through the PSA by multiplying each eigenmode by its gain, and recombined from the output eigenmode profiles. The eigenmodes remain independent and uncoupled from one another even in the presence of gain-induced diffraction. The results obtained in [36] have shown that for tightly focused pump (e.g., near Boyd-Kleinman parameter ξ =

*L*/(2

*z*) = 2.84, corresponding to the most power-efficient parametric interaction [40]) only one mode is amplified (or squeezed), and this mode has a Gaussian TEM

_{R}_{00}shape with waist 2

^{1/2}times larger than the pump’s waist. For larger pump spot sizes that support multiple modes, the shapes of all well-amplified modes at least qualitatively resemble Hermite- or Laguerre-Gaussian profiles. Similar eigenmode profiles have previously been found in a one-dimensional case by pixel-to-pixel Green’s function method [22] for a self-imaging optical parametric oscillator with a moderate-sized pump beam.

Our paper [36], however, has left two important questions unanswered. First, are the eigenmodes not only qualitatively, but also quantitatively close to HG or LG modes? If yes, how closely can an eigenmode be matched by an HG or LG mode? If not, the eigenmodes could be difficult to match experimentally. Second, even though our original approach [35, 36] provided a simple way of finding the PSA Green’s function, it also had a serious drawback: the choice of the signal’s HG expansion basis required the use of a large number (proportional to the square of pump-beam area) of HG modes, which demanded large computational resources and complicated the interpretation of the results.

In this paper, we resolve the two questions stated above. We report better (i.e., smaller) HG (for the elliptical pump case) and LG (for the circular pump case) bases, which provide compact representations for the corresponding PSA eigenmodes. These compact bases, referred to as HG* ^{c}* (or LG

*) below, differ from the original HG (or LG) basis by the choice of the signal beam waist: instead of the signal waist being 2*

^{c}^{1/2}times larger than the pump waist, the compact bases have the beam waists that are roughly equal to the geometric average of the pump waist and the inverse spatial bandwidth of the PSA. The resulting requirement on the number of HG or LG modes needed for solving the PSA propagation equations grows linearly, rather than quadratically, with the pump-beam area, which drastically reduces the memory needs and the computation time for numerical determination of the Green’s function (by integration) and the corresponding eigenmodes (by diagonalization). Since in the new basis the PSA eigenmodes can be approximated by just a handful of HG or LG modes, this also makes it easy to interpret the eigenmodes and generate them experimentally using spatial light modulators. In particular, we show that the most amplified (most squeezed) eigenmodes can be very closely matched by single HG or LG modes of the compact basis with an appropriate waist, which drastically simplifies the detection of a multimode squeezed vacuum. We have presented the initial results of this work at a recent conference [41]. A related idea of a computationally optimum basis has been previously discussed in the study of temporal eigenmodes in pulsed squeezers [42] and in cavity-based optical parametric oscillators [43]. An observation that the spatial eigenmodes of a self-imaging optical parametric oscillator are close to HG profiles has been also made in the context of a three-mode squeezing experiment [24].

This paper is organized as follows: Section 2 describes the PSA mode-coupling theory in the original basis (where the signal’s waist is 2^{1/2} times larger than the pump’s) and in the new compact basis, Section 3 presents the results, and Section 4 summarizes our work.

## 2. Theory of the PSA in compact basis

In this Section, we first present the coupled-mode theory in the original basis [36], then generalize it to arbitrary waist size of the signal-mode basis, and finally discuss the optimum basis that requires the fewest number of modes to represent the PSA eigenmodes.

#### 2.1. PSA equations in the original HG basis (signal’s waist is 2^{1/2} times the pump’s)

We start by considering the nonlinear paraxial wave equation of a degenerate optical parametric amplifier in the undepleted pump approximation [27, 36]

*k*=

*k*– 2

_{p}*k*is the wavevector mismatch for propagation along the

_{s}*z*direction. We are looking for a solution in the form ${e}_{i}(\overrightarrow{r},t)={E}_{i}(\overrightarrow{\rho},z){e}^{i({k}_{i}z-{\omega}_{i}t)}+\text{c}\text{.c}.$, where ${E}_{i}(\overrightarrow{\rho},z)$ is a slowly-varying field envelope satisfying Eq. (1), $\overrightarrow{\rho}$ is a transverse vector with coordinates (

*x*,

*y*), and the intensity is given by ${I}_{i}(\overrightarrow{\rho},z)=2{\epsilon}_{0}{n}_{i}c|{E}_{i}(\overrightarrow{\rho},z){|}^{2}$ with index

*i*taking the value of either

*s*or

*p*, denoting the signal or the pump field, respectively, with ω

*= 2ω*

_{p}*. In the traveling-wave PSAs, the pump powers required to obtain any noticeable gains are on the order of hundreds of Watts per pixel [27], which makes negligible the chances of pump depletion either by quantum signals or by classical pilot beams used for bright squeezing or local-oscillator generation. We assume the pump to be a fundamental HG*

_{s}_{00}mode with potentially unequal beam waists (1/

*e*intensity radii)

*a*

_{0}

*and*

_{px}*a*

_{0}

*along the*

_{py}*x*- and

*y*-directions, respectively. Our “original” HG signal expansion basis uses the signal beam waists

*a*

_{0}

*and*

_{sx}*a*

_{0}

*that are 2*

_{sy}^{1/2}times larger than those of the pump, which makes the pump and the signal HG modes to have the same wavefront curvatures and Rayleigh ranges, i.e.,

*z*=

_{Rx}*k*

_{p}a_{0}

_{px}^{2}=

*k*

_{s}a_{0}

_{sx}^{2},

*z*=

_{Ry}*k*

_{p}a_{0}

_{py}^{2}=

*k*

_{s}a_{0}

_{sy}^{2}. We assume both the pump and the signal beam waists to be co-located at the same axial position,

*z*= 0. The pump and signal basis expansions can therefore be written as

*g*are defined as

_{m}*z*, the 1/

_{R}*e*intensity radius

*a*(

*z*), the Gouy phase shift θ(

*z*), and the beam’s radius of curvature

*R*(

*z*) are given by

We use the expansion of Eq. (2) and project Eq. (1) onto the modes of Eq. (3) to arrive at the coupled-mode equations for the signal’s HG mode amplitudes *A _{mn}*(

*z*) =

*X*(

_{mn}*z*) +

*iY*(

_{mn}*z*):

*P*

_{0}is the pump power, and θ

*is the initial pump phase. The overlap integral*

_{p}*B*

_{mm}_{′}between the pump and the two signal modes with indices

*m*and

*m*′ has a closed-form expression given by

*B*

_{mm}_{′}exhibits fast Gaussian decay as a function of (

*m*–

*m*′), which serves as the selection rule favoring coupling between the signal modes with close indices. On the other hand, its slow decay versus (

*m*+

*m*′) means that the maximum range of the amplified signal modes is determined not by the magnitude of the overlap integral, but by its Gouy phase mismatch [numerator of the first fraction in Eq. (8)].

The Green’s function of Eq. (6) is a 4-dimensional tensor that can be straightforwardly obtained by numerical integration of Eq. (6), where the various mode amplitudes *A _{m}*

_{′}

_{n}_{′}(–

*L*/2) are excited at the crystal input

*z*= –

*L*/2, one at a time, and the resulting output mode patterns

*A*(

_{mn}*L*/2) at

*z*=

*L*/2 are recorded [36]. The independently squeezed or amplified PSA eigenmodes are obtained by diagonalizing the quantum-noise correlator derived from the Green’s function [35, 36]. This diagonalization procedure is similar to the procedure we developed previously in the studies of squeezing in quantum solitons [44–46]. The obtained eigenvalues represent the eigenmodes’ gains, which are also the same as their anti-squeezing factors. Each eigenvalue λ comes in a pair with an eigenvalue 1/λ representing the magnitude of the de-gain and squeezing experienced by the same mode shifted in phase by π/2. Similar diagonalizations or singular-value decompositions of the Green’s functions have been previously discussed in the context of temporal modes [43, 47, 48].

The symmetry of Eq. (6), arising from the pump-waist’s location at the center of the crystal, leads to two important consequences [38, 49]. If we assume θ* _{p}* = –π/2, then the first consequence is the fact that the amount of vacuum squeezing (anti-squeezing) observed via homodyne detection of the PSA output with an arbitrary local oscillator

*A*is exactly the same as the classical power de-gain (gain) seen in the same PSA by an input signal having a complex-conjugate field profile

_{mn}*A*[38]. The second consequence, under the same assumption, is the fact that the shape of each amplified eigenmode at the output of the crystal is the complex conjugate of its shape at the input of the crystal:

_{mn}^{*}*A*(

_{mn}*L*/2) =

*A*(–

_{mn}^{*}*L*/2) [49]. We refer to these two properties of the PSA as

*reciprocity relations*.

#### 2.2. Procedure for identifying a compact basis HG^{c} or LG^{c} for the signal modes

In order to find a basis that is more closely related to the PSA eigenmodes, we re-write the signal expansion from Eq. (2) as

*a*

_{0}

_{sx}= h_{sx}a_{0}

*,*

_{px}*a*

_{0}

_{sy}= h_{sy}a_{0}

*, and the signal-to-pump waist ratios*

_{py}*h*,

_{sx}*h*are no longer restricted to a value of 2

_{sy}^{1/2}(typically,

*h*,

_{sx}*h*≤ 2

_{sy}^{1/2}). In the case of a circular Gaussian pump beam with

*a*

_{0}

*=*

_{px}*a*

_{0}

*=*

_{py}*a*

_{0}

*, it is more convenient to use the LG expansion [36]:*

_{p}*f*of radial index

_{pl}*p*and azimuthal index

*l*have the form

*ρ*, φ are the polar coordinates, and the orthonormality condition is

_{00}coincides with the fundamental circular HG mode HG

_{00}:

Once the *j*-th PSA eigenmode ${E}_{s}^{j}(\overrightarrow{\rho},L/2)$ has been obtained at the PSA output (*z* = *L*/2) by the procedure outlined in Section 2.1 for our original basis, we can find a more compact basis HG* ^{cj}* or LG

*[41] by adjusting its waist to maximize the eigenmode’s overlap with the most closely-related HG or LG mode, respectively:*

^{cj}*i*≠

*j*, the two bases HG

*and HG*

^{ci}*(or LG*

^{cj}*and LG*

^{ci}*) that optimize overlaps for eigenmodes*

^{cj}*i*and

*j*, respectively, will, in general, be different (i.e., the basis HG

*that maximizes overlap for mode*

^{ci}*i*may not necessarily maximize overlap for mode

*j*). Since the lower-order eigenmodes are most important, we are primarily interested in the basis HG

^{c}^{0}(or LG

^{c}^{0}) that maximizes overlap of the elliptical (or circular) eigenmode #0 with the fundamental TEM

_{00}HG (or LG) mode, respectively.

#### 2.3. PSA equation in the compact basis HG^{c}

Instead of obtaining the PSA eigenmodes in the original basis first and subsequently re-expressing them in the compact basis, one can re-write and solve Eq. (6) directly in the compact basis HG* ^{c}*. Since for the basis with

*h*=

_{sx}*a*

_{0}

*/*

_{sx}*a*

_{0}

*≠ 2*

_{px}^{1/2}or

*h*=

_{sy}*a*

_{0}

*/*

_{sy}*a*

_{0}

*≠ 2*

_{py}^{1/2}the wavefront curvatures of the pump and signal modes do not align with each other, the overlap integrals

*B*

_{mm}_{′}of Eq. (8) must be replaced in Eq. (6) by a more complex integral

*D*

_{mm}_{′}given by

*c*

_{2}

*are the even polynomial coefficients of a product of two Hermite polynomials ${H}_{m}(x){H}_{{m}^{\prime}}(x)={\displaystyle {\sum}_{i=0}^{m+{m}^{\prime}}{c}_{i}{x}^{i}}$ (*

_{k}*c*is also known as the discrete convolution of the sequences of coefficients of two Hermite polynomials),

_{i}*j*=

*s*,

*p*. We note that for

*h*2

_{sx}=^{1/2}, Eq. (17) matches Eq. (8) with ξ =

*z*/

*z*=

_{Rsx}*z*/

*z*, and at

_{Rpx}*z*= 0 we have

*D*

_{00}(0,

*a*

_{0}

*,*

_{sx}*a*

_{0}

*) =*

_{px}*B*

_{00}(0) = 2

^{–1/2}.

Similarly, the overlap integral *B _{nn}*

_{′}in Eq. (6) should be replaced by a new expression

*D*

_{nn}_{′}, which can be obtained from Eqs. (17) and (18) after substituting

*y*for

*x*,

*n*for

*m*, and

*n*′ for

*m*′.

The modified Eq. (6) with *D _{mm}*

_{′}

*D*

_{nn}_{′}instead of

*B*

_{mm}_{′}

*B*

_{nn}_{′}can be used to compute the PSA eigenmodes for very large pump waists, because in the compact basis the required memory size (proportional to the number of needed HG modes) scales linearly with the pump beam area, whereas in the original basis with

*h*2

_{sx}= h_{sy}=^{1/2}, the memory size scales quadratically with the pump beam area [36]. The memory reduction in the compact basis, however, comes at the expense of the need to compute the more complicated expression for the overlap integral in Eq. (17) at each propagation step along the

*z*axis.

#### 2.4. Scaling of the optimum beam waist of the compact basis with the pump waist

As was discussed previously in our papers [27, 36], the number of modes supported by the PSA in one dimension (e.g., the *x*-dimension) can be estimated by the product of the pump waist size *a*_{0}* _{px}* in that dimension and the spatial bandwidth of the crystal ${q}_{c}=\sqrt{\pi {k}_{p}/L}$ [26]:

*N*HG modes with waist size

_{x}*a*

_{0}

*can be used to describe an image with maximum dimension $~{a}_{0sx}\sqrt{{N}_{x}}$ and minimum feature size $~{a}_{0sx}/\sqrt{{N}_{x}}$. When this mode space is used to approximate the PSA eigenmodes, the maximum dimension is determined by the pump waist size*

_{sx}*a*

_{0}

*(gain region) and the minimum feature size is determined by the inverse spatial bandwidth 1/*

_{px}*q*:

_{c}*a*

_{0}

*for the optimum HG*

_{sx}*mode set, which turns out to be given by the geometric average of the pump waist and the inverse spatial bandwidth:as was pointed out in our paper [36] and was also previously discussed in [42] in the context of temporal modes. Thus, the optimum beam waist size for the compact basis scales as the square root of the pump beam waist size. This estimate is valid for the cases where*

^{c}*N*>> 1.

_{x}## 3. Results and discussion

In this Section, we present the modeling results for a PSA based on a PPKTP crystal (*d*_{eff} = 8.7 pm/V, length *L* = 2 cm) with signal wavelength of 1560 nm and zero wavevector mismatch Δ*k*. First, we compute the eigenvalue spectra and eigenmode shapes in the *xy*, HG, and LG representations for several different waist sizes *a*_{0}* _{px}* ×

*a*

_{0}

*of the pump beam (25 × 25 μm*

_{py}^{2}to 800 × 50 μm

^{2}), and in each case the pump power is adjusted to produce similar gains of ~15 for the eigenmode #0. The eigenvalue spectra, representing the gains (which for eigenmodes are the same as squeezing factors) for the 16 most prominent eigenmodes, are shown in Fig. 1(a). As shown, the number of the supported eigenmodes with significant gain increases with the pump waist size [36].

#### 3.1. Fundamental eigenmode #0 in the compact basis HG^{c0}

For each of the pump sizes, we have found a compact basis HG^{c}^{0} by adjusting the 1/*e* intensity half-widths *a*_{0}* _{sx}* and

*a*

_{0}

*(or the 1/*

_{sy}*e*intensity radius

*a*

_{0}

*for the circular pump cases) of the HG*

_{s}*modes of the signal beam to produce maximum overlap |*

_{mn}*A*

_{00}|

^{2}of the mode HG

_{00}with the eigenmode #0. The scaling of this optimum signal waist size versus the pump waist size is shown in Fig. 1(b), which also provides a comparison with the results obtained for a fixed pump power of ~4.1 kW corresponding to a gain of ~15 in the 200 × 200 μm

^{2}pump case. The three dashed lines in Fig. 1(b) indicate slopes proportional to ${a}_{0p},\sqrt{{a}_{0p}},$ and $\sqrt[3]{{a}_{0p}}$. For tight focusing (e.g., 25 × 25 μm

^{2}case, where Boyd-Kleinman parameter ξ =

*L*/ (2

*z*) = 1.11), the pump waist size

_{Rp}*a*

_{0}

*is comparable to the inverse spatial bandwidth of the crystal and the PSA operates in a single-mode regime with the optimum signal waist close to 2*

_{p}^{1/2}×

*a*

_{0}

*. Hence, such single mode operation is also inherent to the most power-efficient parametric interaction regime, corresponding to ξ = 2.84 [40]. For large pump waists, where the (pump waist size) × (spatial bandwidth) product and the resulting number of the amplified PSA modes [see Eq. (19)] are large, the optimum signal waist sizes (i.e., the compact basis waist sizes) in Fig. 1(b) start aligning along the $\propto \sqrt{{a}_{0p}}$ asymptote, as discussed in Section 2.4. Under conditions where the gain is ~15 for the eigenmode #0, the compact basis waists are 113.1 × 38.4, 83.0 × 48.0, 61.2 × 61.2, and 49.6 × 49.6 μm*

_{p}^{2}for pump waist sizes of 800 × 50, 400 × 100, 200 × 200, and 100 × 100 μm

^{2}, respectively, whereas the geometric average of the pump waist size and the inverse spatial bandwidth [see Eq. (22)] yields 129.8 × 32.5, 91.8 × 45.9, 64.9 × 64.9, and 45.9 × 45.9 μm

^{2}, respectively. Therefore, the estimates from Eq. (22) turn out to be within 15% of the actual optimum values.

Figure 2 illustrates dependence of the overlap integral |*A*_{00}|^{2} on *a*_{0}* _{sx}* and

*a*

_{0}

*for the 400 × 100 μm*

_{sy}^{2}pump case and shows that a rather wide range of waist sizes (83 ± 10) × (48 ± 6) μm

^{2}yields overlaps within ± 1% of the 99.3% maximum. Because of this wide range, the compact basis does not have to use the exact optimum beam waist, but could simply use the estimate found from Eq. (22).

Figures 3, 4, 5, 6, 7, and 8 show the eigenmode shapes in the *xy*, the original HG (with signal waist that is 2^{1/2} times larger than the pump waist), and the compact representations (degenerate modesare not repeated). These shapes for the eigenmode #0 in small-to-medium pump-waist cases are shown in Fig. 3. In the smallest pump-waist case of 25 × 25 μm^{2}, the original basis with 1/*e* intensity radius *a*_{0}* _{s}* = 35.4 μm is already very close to the optimum (32.2 μm) and, hence, the exact compact basis HG

^{c}^{0}provides only a minor improvement in the overlap from 96.6% to 97.7%. This tight-focusing case produces only one significantly amplified eigenmode, and a further improvement in the mode matching to 99.0% can be obtained by using an HG

^{c}^{0}(or LG

^{c}^{0}) basis whose waist location is offset from the pump waist location by 1.9 mm in the

*z*-direction. The benefit of such a longitudinal waist offset in the tight-focusing case has been predicted, experimentally verified, and studied in detail in [49]. For the other pump cases shown in Fig. 3 we see a significant improvement by switching to the compact basis: the fundamental eigenmode #0, requiring representation by many HG modes in the original basis, can be represented by just one HG

_{00}mode in the compact basis HG

^{c}^{0}with greater than 98% overlap.

Similarly to the small-to-medium pump-waist cases, we find even smaller overlaps |*A*_{00}|^{2} of the HG_{00} mode of our original basis with the eigenmode #0 in the cases of larger pump waists: it is 16.2% for 800 × 50 μm^{2}, 16.1% for 400 × 100 μm^{2}, and 15.7% for 200 × 200 μm^{2} pump waist sizes. A significant number of HG modes are thus required to represent the eigenmode #0 in our original basis, and even more modes are needed to represent higher-order eigenmodes. However, after switching to the corresponding compact basis HG^{c}^{0} with an optimum beam waist, we find 99% or better overlap of the HG_{00} with the eigenmode #0 in all of these large pump-waist cases.

#### 3.2. Higher-order eigenmodes in the compact basis for medium-to-large pump-waist sizes

We now turn to the discussion of higher-order eigenmodes in Figs. 4–8. While all these eigenmodes require a large number of HG modes for representation in our original basis (column 2 of these figures), switching to the compact basis HG^{c}^{0} reduces the number of the required HG modes [column 3 in Figs. 4, 5, and 8]. The reduction is particularly profound in the highly-elliptical case of 800 × 50 μm^{2} pump waist, where the low-order eigenmodes #*n* with *n* from 0 to 3 are largely represented by just one HG_{n}_{0} mode in the compact basis HG^{c}^{0} with waist size of 113.1 × 38.4 μm^{2}, whereas each of the higher-order eigenmodes is represented by a mere handful of the HG modes. The eigenmodes for the circular-pump cases can be decomposed in either the HG* ^{c}* or the LG

*compact basis. For 100 × 100 μm*

^{c}^{2}and 200 × 200 μm

^{2}pump waists, an especially profound reduction in the number of required modes results from using the compact LG

^{c}^{0}bases with the optimized waists of 49.6 μm and 61.2 μm, respectively, where the first few eigenmodes are largely represented by single LG modes and the higher-order eigenmodes are represented by superpositions of 4–5 LG modes with the same |

*l*| but with various

*p*indices [column 4 of Figs. 4 and 5, column 3 of Figs. 6 and 7]. One should note that for every LG

*mode, the LG expansion of the eigenmode also contains the LG*

_{pl}

_{p}_{(–}

_{l}_{)}mode of the same magnitude (possibly with a different phase). Hence, Figs. 4–7 plot |

*A*|

_{p|l|}^{2}= |

*A*|

_{pl}^{2}+

*|A*

_{p}_{(–}

_{l}_{)}|

^{2}= 2|

*A*|

_{pl}^{2}for each non-zero value of |

*l*| and plot |

*A*

_{p|}_{0}

*|*

_{|}^{2}= |

*A*

_{p}_{0}|

^{2}for

*l*= 0.

In order to further reduce the number of modes needed to represent the higher-order eigenmodes, the compact basis’ waist can be chosen so that there is optimum match between one of the higher-order eigenmodes and the corresponding HG or LG mode. For example, column 4 of Figs. 6 and 7 shows results in the compact basis LG^{c}^{5} with 69.8 μm waist optimized for maximum (96.5%) overlap of eigenmode #5 with LG_{10} mode (in contrast to 88.1% overlap between these modes for compact basis LG^{c}^{0} with 61.2 μm waist). As a result, even higher-order eigenmodes up to #14 require no more than two LG modes for their representation. This simplification comes at the expense of a slight reduction in the overlap of the eigenmode #0 with the LG_{00} mode (from 99.4% to 97.9%). The compact basis finding procedure can also be used to better represent modes that are far away from the eigenmode #0. For example, it is possible [although not shown in Figs. 6 and 7] to choose a compact basis LG^{c}^{14} with waist of 76.5 μm to maximize the overlap of the eigenmode #14 with the LG_{20} mode (this overlap is 94.7%, 83.6%, and 44.9% for LG^{c}^{14}, LG^{c}^{5}, and LG^{c}^{0} bases, respectively) at the expense of further degrading the overlap of eigenmode #0 with the LG_{00} mode.

Similarly, an optimized 128.2 × 41.1 μm^{2} waist of the compact basis HG^{c}^{4} can be chosen to maximize the overlap between the eigenmode #4 and mode HG_{40} (98.3%, to compare with 85.1% overlap for HG^{c}^{0} basis with 113.1 × 38.4 μm^{2} waist) for the case of 800 × 50 μm^{2} pump waist, with the results shown in Fig. 8, column 4. In this basis, the eigenmodes #0 to #6 are represented by a single HG mode, whereas the higher-order eigenmodes up to #14 require no more than five HG modes for their representation. This simplification comes at the expense of a slight reduction (from 99.0% to 98.4%) in the overlap of the eigenmode #0 and mode HG_{00}.

Figure 9 (Fig. 10) shows the line graphs comparing the field profiles of the PSA eigenmodes for the 200 × 200 μm^{2} (800 × 50 μm^{2}) pump waist size with the most dominant mode LG* _{pl}* (HG

_{m}_{0}) of their compact representations in basis LG

^{c}^{5}(HG

^{c}^{4}), corresponding to the tallest bar of various graphs in column 4 of Figs. 6 and 7 (column 4 of Fig. 8).

Another way to look at the results in Figs. 4–10 is to note that in the compact basis, whether optimized for the eigenmode #0 or for a higher-order eigenmode, all the eigenmodes with gains within –3 dB from the maximum gain (gain of mode #0) are well represented by either one or, at most, two HG or LG modes, whereas the eigenmodes outside of the –3 dB gain range require a few more HG or LG modes for their compact representation.

Thus, all of the first 15 prominent (i.e., well-amplified) eigenmodes in all the pump cases considered above can be represented by a low-dimensional space of ~20 HG modes (for the elliptical pump cases) or ~25 (counting both + *l* and –*l* modes) LG modes (for the circular pump cases) with appropriate waist. This is in sharp contrast to our original basis, which required hundreds of HG or LG modes to represent the same PSA eigenmodes.

While there is a big difference between the numbers of modes required by the original and compact bases, we observe [e.g., in columns 3 and 4 of Figs. 6–8] that moderate changes in the signal basis waist size around the optimum do not affect the low-dimensional nature (compact support size) of the space representing the PSA eigenmodes. Hence the approximate optimal signal waist can be computed from Eq. (22) (geometric average of the pump waist size and the inverse spatial bandwidth of the crystal) *prior* to solving the PSA Eq. (1). Equation (1) can then be expanded over the compact basis with this optimum waist size (see Sec. 2.3), and the coupled-mode system of Eq. (6) with coupling coefficients in Eq. (17) can be efficiently solved with a drastically reduced number of modes needed for signal expansion. For example, 200 × 200 μm^{2} pump waist size requires 128 × 128 modes for computation in the original HG basis, which leads to computing and diagonalizing of a (2 × 128 × 128) × (2 × 128 × 128) = 32768 × 32768 real matrix (Green’s tensor). At the same time, the number of modes with gain noticeably differing from unity is less than 100 [36]. On the other hand, in the compact LG^{c}^{5} basis only ~10 × 15 LG modes are required for the same 200 × 200 μm^{2} pump waist size, which leads to computing and diagonalizing of a (2 × 10 × 15) × (2 × 10 × 15) = 300 × 300 real matrix, i.e., approximately 10^{4}-fold memory savings.

#### 3.3. Fine tuning of the eigenmode overlaps with the HG or LG modes

To further illustrate how well the PSA eigenmodes can be approximated by single HG or LG modes, we plot in Fig. 11 each eigenmode’s overlap with the most dominant HG or LG mode of its compact representation for the cases of 800 × 50 μm^{2} and 200 × 200 μm^{2} pump waist sizes. We see that by matching the compact basis to eigenmode #4 (basis HG^{c}^{4} for the elliptical pump case) or eigenmode #5 (basis LG^{c}^{5} for the circular pump case) produces better overlaps for all eigenmodes other than #0 at the expense of a slightly worse overlap for the eigenmode #0. If an even better overlap is desired, then additional improvement can be obtained by optimizing the offset of the *z*-position of the waist of the compact basis from the pump waist location (*z* = 0) [49]. Such optimization could be important for the detection of high degree of squeezing. For the case of the 200 × 200 μm^{2} pump waist size, a *z*-offset of the basis LG^{c}^{0} matched to the eigenmode #0 improves the best overlap of the eigenmode #0 from 99.4% to 99.7% and those of the eigenmodes #1–#5 by 1–2%. The *z*-offset of the basis LG^{c}^{5} matched to the eigenmode #5 improves the overlaps of the eigenmodes #0–#5 by 0.3–0.8%, compared to the zero-offset case.

#### 3.4. Phase response of the PSA (input and output eigenmode phases)

The capability of the compact basis to approximate each of the most prominent PSA eigenmodes by just one HG or LG mode comes from the fact that as the pump waist size increases the PSA behavior approaches that of a PSA with a plane-wave pump. In the latter case, the spatial frequencies that are well within the spatial bandwidth of the PSA are nearly equally amplified, and thus any image contained well within the spatial bandwidth is amplified without distortion. Thus, this flat-gain region of the plane-wave-pump PSA is equivalent to free space and, instead of the standard spatial-frequency eigenmodes, can be equivalently described by free-space HG or LG eigenmodes. Thus, a PSA with a large pump waist size will have its low-order eigenmodes very similar to the free-space HG or LG eigenmodes, but the eigenmodes beyond the –3-dB gain point will progressively deviate from the free-space HG or LG modes, because the propagation of these eigenmodes will be constrained simultaneously by the limited pump size and by the limited spatial bandwidth of the PSA. As the pump size increases, the eigenvalue (gain) spectrum flattens (i.e., more and more eigenmodes have gains within –3-dB from the maximum). This flat gain ensures undistorted image amplification if there is no additional phase distortion, i.e., significant deviation of the eigenmode’s phase from the free-space HG- or LG-mode phase [50]. To verify the phase response, we plot in Fig. 12(a) the phases of the most dominant modes in the compact HG^{c}^{4} representation of the output eigenmodes of the PSA with a 800 × 50 μm^{2} pump waist size [corresponding to the representations in column 4 of Fig. 8] and in Fig. 12(b) the phases of the most dominant modes in the compact LG^{c}^{5} representation of the output eigenmodes of the PSA with a 200 × 200 μm^{2} pump waist size. For the plots in Fig. 12(b), we take the compact LG^{c}^{5} basis matched to the eigenmode #5 [shown in column 4 of Figs. 6 and 7] and further optimize it by moving the *z*-position of its waist by 3 mm in the propagation direction from the position of the pump waist at *z* = 0 [the corresponding overlaps are shown by the filled magenta diamonds in Fig. 11(b)]. The resulting maximum phase excursion across the eigenmodes within the –3-dB gain range is 3.5° for the plot in Fig. 12(a) and is 0.7° for the plot in Fig. 12(b) (the latter becomes 3.6° if we do not optimize the *z*-offset of the basis waist location), which means that the amplified image will have practically no phase distortion. According to the reciprocity relations discussed at the end of Section 2.1, the input and output eigenmode shapes are complex conjugates of each other and, therefore, the eigenmode phases at the PSA input are negatives of their phases at the PSA output. With the *z*-offset basis, this also means that, in order to get the minimum distortion of the image, the input image must be focused in the plane located 3 mm *before* the pump waist, whereas the output image will appear to be emerging from a plane located 3 mm *after* the pump waist [49].

Under the condition θ* _{p}* = –π/2, used in obtaining the plots in Fig. 12, the optimum (for maximum gain) input signal phase of a plane-wave-pump PSA is 0°. We observe in Fig. 12(a) that the optimum input phase for the eigenmode #0 would be –4.6°. This absolute phase deviation from the plane-wave-pump case is owing to the rather tight pump focusing in one of the dimensions (i.e.,

*y*).

## 4. Conclusions

To summarize, we have found the eigenmodes of a spatially-broadband PSA and expressed them as superpositions of a very small number of HG or LG modes. This low-dimensional HG or LG subspace forms the compact basis for representing the PSA eigenmodes. The compact basis can be found by optimizing its waist size to maximize the overlap with one of the eigenmodes, in contrast to our previous work [36] that used a waist 2^{1/2} times larger than the pump waist. We have found that for all pump waist sizes considered, the fundamental PSA eigenmode #0 has very good (99% or better) overlap with the fundamental Gaussian mode HG_{00} of the compact basis. This holds promise for the detection of a large degree of vacuum squeezing from a traveling-wave PSA with use of a simple Gaussian-shaped local oscillator of a proper waist. Other low-order eigenmodes of the PSA (with gains well within the –3-dB range from the gain of the eigenmode #0) also have high overlaps with the corresponding HG or LG modes of the compact basis, whereas the higher-order eigenmodes may require several modes for their representation. The compact representation greatly simplifies the generation of the eigenmodes in the laboratory and has recently enabled experimental verification of the fundamental eigenmode for several pump waist sizes [49, 51]. While small adjustments in the beam waist allow for the compact basis to be customized for best overlap with a specific (fundamental or higher-order) eigenmode, the optimum waist size, however, remains in the general vicinity of the geometric average of the pump waist size and the inverse spatial bandwidth of the PSA crystal, except for the case of a tightly-focused pump.

The great simplification enabled by the eigenmode’s representation in the compact basis is important for both classical (e.g., boosting the power of an image before lossy or noisy detection) and quantum (e.g., generating massively multimode squeezed light beyond the current several-mode state of the art [20, 23, 24]) applications of the spatially-broadband PSA. It makes it easier to match the input or output spatial patterns to the PSA eigenmodes in order to minimize image distortions in the classical case and to observe the maximum degree of squeezing in homodyne detection in the quantum case. Moreover, using the compact basis to solve the PSA propagation equation drastically reduces the memory requirements for numerical computation of the eigenmodes. The insights gained from the simplified eigenmode representation have recently enabled us to extend the studies of imaging PSAs by including higher-order or multimode Gaussian pumps [by merely modifying overlap integrals in Eq. (6)] and negative wavevector mismatch [52]. Both modifications have been shown to shift the maximum PSA gain to higher-order HG or LG modes. Further recent extensions of this model have been applied to self-imaging waveguides [53] and quantum image converters [54].

## Acknowledgments

This work was supported in part by the DARPA Quantum Sensors Program under AFRL Contract # FA8750-09-C-0195 and by the DARPA Quiness Program under Grant # W31P4Q-13-1-0004. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of DARPA or the U.S. Air Force.

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