Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps

Muthiah Annamalai; Nikolai Stelmakh; Michael Vasilyev; Prem Kumar

doi:10.1364/OE.19.026710

1. Introduction

Phase-sensitive optical parametric amplifiers (PSAs), unlike any other optical amplifiers, can increase the magnitude of the signal without adding any noise [1]. This property, coupled with the wide temporal bandwidth of fiber-based parametric amplifiers, has lead to their use as nearly noiseless inline amplifiers for optical communication systems [2–6]. Similarly, the broad spatial bandwidth of the parametric amplifiers enables their use as noiseless amplifiers of faint optical images, as was theoretically proposed in [7–9] and experimentally demonstrated in [10–13]. The signal-to-noise ratio improvement provided by such image amplifiers can in turn lead to resolution improvement in the detection of faint images [14–16]. The same devices can also be used for the generation of spatially-broadband squeezed vacuum for quantum information processing applications. The design of practical parametric image amplifiers, however, represents a big challenge. Indeed, the traveling-wave nature of gain in these devices requires the use of a tightly focused pump beam (typically, the fundamental Gaussian TEM₀₀ mode) of very high power (~1 kW per pixel of resolution) [17]. The resulting spatially-varying PSA gain, together with the limited spatial bandwidth of the PSA [determined from phase-matching conditions to be ~(k_p/L)^1/2, where k_p is the pump propagation constant and L is the nonlinear crystal’s length] [7, 12, 18], couples and mixes up in both space and spatial-frequency domains the modes representing the information content of the image [17]. These spatial mode-mixing effects, known as gain-induced diffraction [19, 20], also make it difficult to detect the lowest-noise mode of a traveling-wave squeezer [21, 22], because the properly mode-matched homodyne detector requires exact knowledge of this mode’s spatial profile. The coupling between the various spatial modes of the signal was previously studied either in a travelling-wave PSA in which only the radially symmetric signal modes were considered [23, 24], or in a cavity-based sub-threshold optical-parametric oscillator [25], all pumped by a circular Gaussian beam. In this paper, we extend the mode-coupling analysis to the travelling-wave PSA with a more general elliptical Gaussian TEM₀₀ pump and then develop a procedure to find, for the first time to our knowledge, the orthogonal set of independently squeezed (or amplified) eigenmodes of such a PSA. We presented the preliminary results of this approach in a recent conference paper [26].

The rest of this paper is organized as follows: Section 2 describes the theoretical model of mode coupling in the PSA and the procedure for finding the PSA’s eigenmodes, Section 3 describes the details of our computational approach, Section 4 reports and discusses the results, and Section 5 summarizes our work.

2. Theory of 2-D spatially-multimode PSA

2.1 Hermite-Gaussian mode representation

Our work builds upon the Laguerre-Gaussian (LG) expansion of the parametric amplifier equations, originally developed for cavities [25] and later extended to 1-D (radially symmetrical) traveling-wave PSAs [23]. We generalize this approach by replacing the LG expansion with a Hermite-Gaussian (HG) expansion to describe arbitrary 2-D images and a general elliptical Gaussian pump beam with potentially unequal 1/e intensity radii a₀_px and a₀_py in x- and y-dimensions, respectively. By expanding the PSA input over signal TEM_mn HG modes having the same Rayleigh ranges z_Rx = k_pa₀_px² and z_Ry = k_pa₀_py² as the pump, we reduce the PSA propagation to a system of coupled ordinary differential equations for the HG-mode amplitudes A_mn = X_mn + iY_mn. After integrating this system of equations, we can obtain the PSA’s Green’s function and all the quantum correlators needed for finding the independently squeezed or amplified modes [27].

We start by considering the nonlinear paraxial wave equation of a degenerate optical parametric amplifier in the undepleted pump approximation [17, 27, 28]

\frac{\partial E_{s} (\vec{ρ}, z)}{\partial z} = \frac{i}{2 k_{s}} \nabla_{ρ}^{2} E_{s} (\vec{ρ}, z) + \frac{i 2 ω_{s} d_{e f f}}{n_{s} c} E_{p} (\vec{ρ}, z) E_{s}^{*} (\vec{ρ}, z) \exp (i Δ k z) .

We are looking for the solutions in the form $e_{i} (\vec{r}, t) = E_{i} (\vec{ρ}, z) e^{i (k_{i} z - ω_{i} t)} + c .c .$ , where $E_{i} (\vec{ρ}, z)$ is a slowly-varying field envelope, $\vec{ρ}$ is a transverse vector with coordinates (x,y), the intensity is given by $I_{i} (\vec{ρ}, z) = 2 ε_{0} n_{i} c | E_{i} (\vec{ρ}, z) |^{2}$ with index i taking value of either s or p, denoting the signal or pump field, respectively, and ω_p = 2ω_s. We expand the signal and pump beams over the HG basis with potentially unequal 1/e intensity radii in x- and y-dimensions:

\begin{array}{l} E_{p} (\vec{ρ}, z) = \sqrt{\frac{P_{0}}{2 ε_{0} n_{p} c}} e^{i θ_{p}} g_{0} (x, z, a_{0 x}^{p}, k_{p}) g_{0} (y, z, a_{0 y}^{p}, k_{p}), \\ E_{s} (\vec{ρ}, z) = \sum_{m, n} \frac{A_{m n} (z)}{\sqrt{2 ε_{0} n_{s} c}} g_{m} (x, z, \sqrt{2} a_{0 x}^{p}, k_{s}) g_{n} (y, z, \sqrt{2} a_{0 y}^{p}, k_{s}), \end{array}

where the one-dimensional HG modes g_m are defined as (β = x or y)

g_{m} (β, z, a_{0}, k) = \frac{H_{m} [β / a (z)]}{\sqrt{2^{m} m! π^{1 / 2} a (z)}} e^{i θ (z)} e^{- \frac{β^{2}}{2 a^{2} (z)}} e^{i k \frac{β^{2}}{2 R (z)}}

with the orthogonality condition

\int_{- \infty}^{\infty} g_{m} (β, z, a_{0}, k) g_{m^{'}}^{*} (β, z, a_{0}, k) d β = δ_{m m^{'}},

and the Rayleigh range z_R, 1/e intensity radius a(z), Gouy phase shift θ(z), and beam’s radius of curvature R(z), respectively, given by

\begin{array}{l} z_{R} = k a_{0}^{2}, \\ a (z) = a_{0} \sqrt{1 + {(z / z_{R})}^{2}}, \\ θ (z) = - (m + \frac{1}{2}) \tan^{- 1} \frac{z}{z_{R}}, \\ R (z) = z [1 + (^{z_{R} / z) 2}] . \end{array}

The x- and y-radii for the signal expansion basis are chosen to be 2^1/2 times greater than those of the pump, which ensures that the signal and the pump have the same wavefront curvature and the same Rayleigh ranges z_Rx = k_pa₀_px² and z_Ry = k_pa₀_py². The beam waists for both the pump and the signal HG basis occur at z = 0.

After substituting the HG expansions of the signal and the pump [Eq. (2)] into Eq. (1) and projecting the results onto the signal’s HG basis, we arrive at the following coupled-mode equations for the signal’s HG mode amplitudes A_mn(z):

\frac{\partial A_{m n} (z)}{\partial z} = i e^{i θ_{p}} κ e^{i Δ k z} \sum_{m^{'}, n^{'}} B_{m m^{'}} (z / z_{R x}) B_{n n^{'}} (z / z_{R y}) A_{m^{'} n^{'}}^{*} (z),

where

κ = \sqrt{\frac{2 ω_{s}^{2} d_{eff}^{2}}{ε_{0} n_{s}^{2} n_{p} c^{3}}} \sqrt{\frac{P_{0}}{π a_{0 x}^{p} a_{0 y}^{p}}},

Δk is the wavevector mismatch, P₀ is the pump power and θ_p is the initial pump phase. The overlap integral B_mm_′ of the pump and two signal modes with indices m and m′ has a closed-form expression

B_{m m^{'}} (ξ) = {\begin{matrix} \begin{array}{l} \frac{e^{i (m + m^{'} + 1 / 2) \tan^{- 1} ξ}}{\sqrt[4]{1 + ξ^{2}}} \frac{{(- 1)}^{\frac{m - m^{'}}{2}} (m + m^{'} - 1)!!}{\sqrt{2^{m + m^{'} + 1} m! m^{'}!}} \end{array} & f o r e v e n (m + m^{'}), \\ 0 & f o r o d d (m + m^{'}), \end{matrix}

where the double factorial (m + m′ – 1)!! = 1 for m + m′ = 0. Let us list several important properties of the coupling matrix B_mm_′ comprising coefficients B_mm_′. First of all,

B_{m m^{'}} (ξ) = \frac{e^{i (m + m^{'} + 1 / 2) \tan^{- 1} ξ}}{\sqrt[4]{1 + ξ^{2}}} B_{m m^{'}} (0),

i.e., propagation away from the center of the crystal reduces the magnitude of B_mm_′ and introduces a phase shift due to the Gouy phase mismatch between the pump and the two signal modes. Second, the magnitude of B_mm_′(0) for large indices m and m′ can be approximated as

| B_{m m^{'}} (0) | \approx \frac{e^{- \frac{{(m - m^{'})}^{2}}{4 (m + m^{'})}}}{\sqrt{π (m + m^{'})}},

i.e., for a fixed (m + m′), it exhibits a fast Gaussian decay as a function of (m – m′), whereas the decay along the main diagonal m = m′ is very slow and proportional to (m + m′)^–1/2, as illustrated in Fig. 1 . Hence, the fast decay of B_mm_′ versus (m – m′) serves as the selection rule favoring coupling between the signal modes with close indices. On the other hand, the slow decay versus (m + m′) means that the maximum range of the amplified signal modes is determined not by the magnitude of the coupling coefficient, but by its Gouy phase mismatch [numerator in Eq. (9)] that leads to fast oscillations at large (m + m′), which limits the PSA gain. The signal evolution in the PSA in Eq. (6) is governed by the outer product of matrices B_mm_′ and B_nn_′, yielding a 4th-rank tensor in which only 25% of the elements are not zero.

Fig. 1 Magnitude (absolute value) of the coefficients of the coupling matrix B_mm_′(0) either as a function of (m + m′) in the direction of main diagonal m = m′ (red) or as a function of (m – m′) in the direction orthogonal to the main diagonal for m + m ′ = 20 (blue), 50 (green), 80 (purple), and 100 (black). Filled circles correspond to the exact Eq. (8) and solid curves to the approximate Eq. (10).

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2.2 Laguerre-Gaussian mode representation

In the case of a circular Gaussian pump beam with a₀_px = a₀_py = a₀^p, it is more convenient to use the LG expansion [25]:

\begin{array}{l} E_{p} (\vec{ρ}, z) = \sqrt{\frac{P_{0}}{2 ε_{0} n_{p} c}} e^{i θ_{p}} f_{00} (r, φ, z, a_{0}^{p}, k_{p}), \\ E_{s} (\vec{ρ}, z) = \sum_{p, l} \frac{A_{p l} (z)}{\sqrt{2 ε_{0} n_{s} c}} f_{p l} (r, φ, z, \sqrt{2} a_{0}^{p}, k_{s}), \end{array}

where the LG modes f_pl of radial index p and azimuthal index l have the form

f_{p l} (r, φ, z, a_{0}, k) = \sqrt{\frac{p!}{(p + | l |)! π a^{2} (z)}} {(\frac{r}{a (z)})}^{| l |} L_{p}^{| l |} (\frac{r^{2}}{a^{2} (z)}) e^{i l φ} e^{i θ (z)} e^{- \frac{r^{2}}{2 a^{2} (z)}} e^{i k \frac{r^{2}}{2 R (z)}},

r, φ are the polar coordinates,

\begin{array}{l} z_{R} = k a_{0}^{2}, \\ a (z) = a_{0} \sqrt{1 + {(z / z_{R})}^{2}}, \\ θ (z) = - (2 p + | l | + 1) \tan^{- 1} \frac{z}{z_{R}}, \\ R (z) = z [1 + (^{z_{R} / z) 2}], \end{array}

and the orthogonality condition is

\int_{0}^{2 π} \int_{0}^{\infty} f_{p l} (r, φ, z, a_{0}, k) f_{p^{'} l^{'}}^{*} (r, φ, z, a_{0}, k) r d r d φ = δ_{p p^{'}} δ_{l l^{'}} .

The fundamental LG mode LG₀₀ coincides with the fundamental circular HG mode TEM₀₀:

f_{00} (r, φ, z, a_{0}, k) = g_{0} (x, z, a_{0}, k) \times g_{0} (y, z, a_{0}, k) .

The coupled-mode equations in the LG representation take the form of

\frac{\partial A_{p l} (z)}{\partial z} = i e^{i θ_{p}} κ e^{i Δ k z} \sum_{p^{'}} Λ_{p p^{'}}^{| l |} (z / z_{R}) A_{p^{'}, - l}^{*} (z),

where

κ = \sqrt{\frac{2 ω_{s}^{2} d_{eff}^{2}}{ε_{0} n_{s}^{2} n_{p} c^{3}}} \sqrt{\frac{P_{0}}{π {(a_{0}^{p})}^{2}}},

the overlap integral of the pump mode LG₀₀, with the signal modes LG_pl and LG_p_′_l_′ is given by

Λ_{p p^{'}}^{| l |} (ξ) = \frac{e^{2 i (p + p^{'} + | l | + 1 / 2) \tan^{- 1} ξ}}{\sqrt{1 + ξ^{2}}} \frac{(p + p^{'} + | l |)!}{2^{p + p^{'} + | l | + 1} \sqrt{p! p^{'}! (p + | l |)! (p^{'} + | l |)!}},

and the coupling coefficients between the modes LG_pl and LG_p_′_l_′ are zero for l ≠ –l′. The latter selection rule means that the circular symmetry of the pump reduces the rank of the coupling tensor in the PSA propagation equation from 4 to 3. Apart from this simplification, other properties of the coupling coefficients in Eq. (16) are similar to those in the HG case. Namely, these coefficients can be expressed as

Λ_{p p^{'}}^{| l |} (ξ) = \frac{e^{2 i (p + p^{'} + | l | + 1 / 2) \tan^{- 1} ξ}}{\sqrt{1 + ξ^{2}}} Λ_{p p^{'}}^{| l |} (0),

and for large value of (p + p′ + |l|) they have an asymptotic dependence

Λ_{p p^{'}}^{| l |} (0) \approx \frac{e^{- \frac{{(p - p^{'})}^{2} + l^{2}}{2 (p + p^{'} + | l |)}}}{\sqrt{2 π (p + p^{'} + | l |)}},

which imposes fast Gaussian decay when indices deviate from p = p′ and l = 0 condition, as well as a slow p ^–1/2 decay when they satisfy this condition. We also note that, as a consequence of Eq. (15), the first diagonal element of the coupling tensor is the same in both LG and HG representations:

Λ_{00}^{0} (0) = B_{00}^{2} (0) = 1/2 .

2.3 PSA Green’s functions and eigenmodes

In what follows, we will use index notation corresponding to the HG representation; results for the LG representation can be obtained in a similar manner. We will assume that the pump beam waist is located exactly at the center of a nonlinear crystal of length L, i.e., the crystal input is at z = –L/2 and crystal output is at z = L/2. Following our approach to solving the PSA equation and finding its eigenmodes, presented in [27], we re-write the complex signal mode amplitudes A_mn = X_mn + iY_mn in a matrix-vector form

A_{m n}^{} (z) = [\begin{matrix} X_{m n}^{} (z) \\ Y_{m n}^{} (z) \end{matrix}],

where

X_{m n}

and

Y_{m n}

are real-valued matrices. Then, we can express the solution as

A_{m n}^{} (L / 2) = G_{m n m^{'} n^{'}} (L / 2, - L / 2) A_{m^{'} n^{'}}^{} (- L / 2),

where the tensor

G_{m n m^{'} n^{'}}

is the PSA Green’s function in HG representation, relating the mode amplitudes at the input of the crystal to those at the output. Unlike the numerically challenging evaluation of the PSA Green’s function in xy-representation [29],

G_{m n m^{'} n^{'}}

can be easily computed by exciting a finite number of input HG modes

A_{m^{'} n^{'}}^{} (- L / 2)

, one at a time, and recording the output mode pattern

A_{m n}^{} (L / 2)

in each case. Once the Green’s function is known, the amount of squeezing can be found using the Rayleigh quotient

λ (L / 2) = \frac{A_{m n}^{T} G_{m n m^{″} n^{″}} G_{m^{'} n^{'} m^{″} n^{″}}^{T} A_{m^{'} n^{'}}}{A_{m n}^{T} A_{m n}} .

From the symplectic property of the tensor $G_{m n m^{'} n^{'}}$ [27, 30–32] it follows that the output can be decomposed into a set of independently squeezed modes (a Karhunen-Loève expansion) determined from the eigenvalue equation

G_{m n m^{″} n^{″}} G_{m^{'} n^{'} m^{″} n^{″}}^{T} A_{m^{'} n^{'}} = λ (L / 2) A_{m n},

where λ is the gain (same as squeezing factor) of the eigenmode

A_{m n}^{λ}

. Symplecticity of the tensor

G_{m n m^{'} n^{'}}

ensures that for each eigenmode

A_{m n}^{λ}

with eigenvalue λ, there is an eigenmode

A_{m n}^{1 / λ}

with eigenvalue 1/λ that is obtained by –90° phase shift of

A_{m n}^{λ}

(i.e., by substituting

Y_{m n}^{}

for

X_{m n}^{}

, and

- X_{m n}^{}

for

Y_{m n}^{}

):

A_{m n}^{λ} = [\begin{matrix} X_{m n} \\ Y_{m n} \end{matrix}], A_{m n}^{1 / λ} = [\begin{matrix} Y_{m n} \\ - X_{m n} \end{matrix}] .

Equation (24) allows one to find the independently squeezed / amplified modes at the PSA output (at z = L/2). In order to obtain their shapes at the PSA input (at z = –L/2), one can invoke the reciprocity argument [23], which stems from the equivalence between z-reversal and conjugation of the coupling coefficients in Eq. (6) (for this, it is helpful to assume θ_p = –π/2 as the initial phase reference). The reciprocity principle states that the squeezing factor detected by a particular spatial profile of the local oscillator is the same as the classical PSA gain seen by the input signal whose profile is conjugate to that of the local oscillator. When applied to the local-oscillator shape that matches one of the PSA eigenmodes (and for the PSA eigenmode the gain and squeezing factors are the same), this principle means that the eigenmode profile at the PSA input is the conjugate of the eigenmode profile at the PSA output.

3. Computational considerations

We numerically integrate Eq. (6) with 4^th-rank coupling tensor B_mm_′B_nn_′ by the 4^th-order Runge-Kutta method (RK-4). The two critical parameters of the computation are the maximum HG order m_max, at which the tensor is truncated, and the step size Δz over the propagation distance z.

To estimate the truncation order m_max for each dimension, we recall that it is determined by the Gouy phase mismatch [the numerator in Eq. (9)]. When the mismatch reaches π, the coupling coefficient changes sign, i.e., switches from amplification to de-amplification. If this reversal occurs within the crystal, it means that the mode is outside of the crystal’s spatial bandwidth and can be ignored. Thus, at m = m′ = m_max the mismatch should reach π at the end of the crystal:

(2 m_{\max} + 1 / 2) \tan^{- 1} (L / 2 z_{R}) \approx \frac{m_{\max} L}{z_{R}} = π,

which results in the truncation-order condition

m_{\max} = π \frac{z_{R}}{L} = \frac{π k_{p} {(a_{0}^{p})}^{2}}{L} .

The same condition can be obtained by noting that the spatial-frequency bandwidth of the m_max-order HG function, which is approximately $\sqrt{m_{\max}} / a_{0}^{p}$ , should be equal to the spatial bandwidth of the crystal (πk_p/L)^1/2 [17]. For a PPKTP crystal with L = 2 cm, n_s ≈n_p ≈1.78, free-space λ_p = 780 nm, and a₀^p = 100 μm, Eq. (27) yields m_max = 22.5, which we have confirmed by observing the sufficient decay of the Green’s function versus (m + m′) at this index value. To provide additional accuracy margin, we use m_max = 32 for these PSA parameters. For a circular pump spot size, the x- and y-dimensions require 32 modes each.

As the pump waist increases, Eq. (27) demands significant increase in the number of considered modes in both the x- and y-dimensions, proportional to the square of the corresponding waist radii:

m_{\max}^{x} n_{\max}^{y} = π^{2} \frac{z_{R x} z_{R y}}{L^{2}} = {(\frac{π k_{p}}{L})}^{2} {(a_{0 x}^{p} a_{0 y}^{p})}^{2} .

Thus, for a₀^p = 200 μm we need to use m_max = 128, i.e., m_max that is 4 times greater than that for a₀^p = 100 μm. Thus, for a 200-μm circular pump waist, Eq. (24) requires diagonalization of a tensor consisting of 2×2×(128×128)×(128×128) = 2³⁰ elements (only quarter of which are non-zero), leading to multi-gigabyte memory requirements for the computer. This unfortunately fast scaling with pump waist size takes place because our choice of the HG expansion basis for the signal is not optimal. More specifically, in order to eliminate the beam curvature from the overlap integral, the signal waist of the expansion basis has been chosen to be 2^1/2 times greater than the pump’s, whereas all the interesting dynamics takes place on a smaller transverse scale that varies between the inverse spatial bandwidth of the crystal on the low end and the pump waist size on the high end. The number of independently amplified (or squeezed) PSA modes can be estimated by the product of the pump beam area and the square of the crystal’s spatial bandwidth [17, 33]:

N u m b e r o f P S A \mod e s = \frac{π k_{p}}{L} a_{0 x}^{p} a_{0 y}^{p} = π \frac{\sqrt{z_{R x} z_{R y}}}{L},

which is square root of the number of modes required for computation by Eq. (28). Significant potential computational memory savings [reduction of m^x_maxn^y_max down to the value given by Eq. (29)] can be achieved by using a different HG expansion basis for the signal, that with a beam waist in the vicinity of the geometric average of the pump waist and the inverse spatial bandwidth of the crystal [34], as was previously discussed in the context of temporal modes [35]. These savings, however, are achieved at the expense of a considerably more demanding computation of the coupling coefficients at every z-step, and have not been attempted in the present work.

The estimation of the step size Δz can be done by demanding both the coefficients B_mm_′ and the solution to not change much over one step. The fastest variation in the coupling coefficients comes from the Gouy phase mismatch for the maximum order m_max, and we require this phase to change over one step by no more than π/150 ≈0.02 rad:

(2 m_{\max} + 1 / 2) \frac{Δ z}{z_{R}} \approx 2 m_{\max} \frac{Δ z}{z_{R}} \leq \frac{π}{150},

which leads to the step size that is independent of the pump waist radius:

Δ z \leq \frac{π z_{R}}{300 m_{\max}} = \frac{L}{300} .

The second requirement on variation of the solution over one step can be met by requiring the magnitude of the most amplified eigenmode to change by no more than 2% (0.086 dB) over one step, which means that the step size given by Eq. (31) meets this requirement up to the gain of ≈300 × 0.086 dB = 25.8 dB for the most amplified mode.

4. Results and discussion

Using the approach discussed in Sections 2.1 and 2.3, we have numerically generated the Green’s functions of Eq. (6) in the HG representation and obtained the PSA eigenmodes for various pump powers P₀ and spot sizes a₀_px × a₀_py for a PPKTP crystal of length L = 2 cm, d_eff = 8.7 pm/V, n_s ≈n_p ≈1.78, and signal wavelength of 1560 nm. For pump spot sizes of 100×100 μm² and smaller (100×50, 100×25, and 25×25 μm²) we have used 32×32 HG functions (i.e., m^x_max = n^y_max = 32) as the signal expansion basis. For pump waists larger than 100 μm, we have increased the truncation order according to Eq. (28); e.g., for 400×100 μm² pump spot size we have used 512×32 expansion basis. For 100×100 μm² pump spot size, the Rayleigh range equals z_R = k_p(a₀^p)² ≈143 mm, i.e., it is ~7 times longer than the crystal.

In the first set of results, whose eigenvalue spectra are shown in Fig. 2 , we have chosen the pump powers such that the gains (which are also the same as the squeezing factors) of the most amplified (fundamental) eigenmode for each spot size are approximately the same and equal to 15 (or 11.8 dB). These powers are shown in Fig. 3(b) and also listed in the legends of Figs. 4 , 5 . From Fig. 2 one can see that the number of modes amplified by the PSA grows with the pump spot size. For circular pump waists, some eigenvalues are degenerate, whereas for elliptical waists the degeneracy is lifted. The smallest pump spot size (25×25 μm²) seems to support the amplification of only the fundamental mode.

Fig. 2 Eigenvalue (gain and squeezing) spectra for the first 16 (a) and the first 101 (b) PSA modes. The vertical scale is linear in (a) and logarithmic in (b). The legend indicates the pump spot sizes. The horizontal dashed black line in (a) marks the –3-dB level from the gain of the fundamental eigenmode #0. The pump powers are chosen in each case to achieve approximately the same gain of ~15 for mode #0.

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Fig. 3 (a) Eigenvalue spectra for the second data set, where the pump powers are chosen to provide the same peak intensity as that in the 100 × 100 μm² case. (b) Blue circles: pump power needed for the PSA gain of ~15 for mode #0 in Fig. 2(a), as a function of the pump spot size a₀_px × a₀_py. Red triangles: PSA gain of mode #0 for the four spot sizes in (a). Blue dashed line: linear power dependence (guide for the reader’s eyes).

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Fig. 4 xy-profiles of the few most prominent eigenmodes of the PSA for five different pump spot sizes. The distance scale is shown below the profile of mode #0 for the 200 × 200 μm² case. The inset graph shows the number of well-amplified PSA modes [those with gains above the –3-dB line in Fig. 2(a)] versus the pump power.

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Fig. 5 xy-profiles of the few most prominent eigenmodes of the PSA for the two largest elliptic pump spot sizes considered (800 × 50 and 400 × 100 μm²). The distance scale is shown above the profile of mode #0 for the 800 × 50 μm² case and is the same as that in Fig. 4.

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In the second set of results, whose eigenvalue spectra are shown in Fig. 3(a), we have adjusted the pump powers so that the peak intensity of the pump for all spot sizes is the same as that in the 100×100 μm² case of the first set of results, i.e., the pump power is scaled proportionally to the beam area. The spectra shown in Fig. 3(a) indicate that, under the constant intensity condition, the gain of the fundamental eigenmode decreases with decreasing spot size. This point is further illustrated in Fig. 3(b) (red), which shows the gain, at constant pump intensity, as a function of the spot size a₀_px × a₀_py. In the same Fig. 3(b), we also plot the pump powers (blue) needed to achieve the fundamental-mode gain of ~15, as in the first set of data. The fact that this dependence is not linear indicates that higher intensity is required to achieve the same gain for the fundamental eigenmode when the tightly focused pump size starts to approach the inverse spatial bandwidth of the crystal (L/πk_p)^1/2 ≈21 μm.

All the remaining Figs. 4–9 present results from the first set of data (i.e., where the pump powers are adjusted to produce the same gain of ~15 for the fundamental eigenmode). Figures 4 and 5 show the (x,y) representation of the first few computed eigenmodes (i.e., the spatial profiles of their intensity), whereas Figs. 6 –9 show the HG representation of these eigenmodes (i.e., |A_mn|²). The inset in Fig. 4 also shows the number of modes with gains within 3 dB of the gain of the fundamental eigenmode [i.e., the number of modes above the dashed black line in Fig. 2(a)] as a function of the pump power, exhibiting roughly linear dependence with the inverse slope of ~690 W / mode, which can be considered as the power efficiency of the PSA under consideration. The large required pump power justifies the use of the undepleted pump approximation in Eq. (1) for input signal powers up to ~1 W (i.e., as long as the amplified signal is much weaker than the pump).

Fig. 9 HG representation (i.e., |A_mn|²) of mode #14 for the four largest pump spot sizes studied. Pump power for each spot size is chosen such that the gain for mode #0 is ~15. The corresponding xy-profiles are shown in Fig. 4 (for the left two graphs) and in Fig. 5 (for the right two graphs).

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Fig. 6 Mode #0 in the HG representation (i.e., |A_mn|² for mode #0) for various pump spot sizes. The gain for the PSA mode #0 is ~15 in all cases. The corresponding xy-profiles are shown in Fig. 4 (for all graphs in the top row and the first graph in the bottom row of the present Figure) and in Fig. 5 (for the last two graphs in the bottom row of the present Figure).

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From Figs. 4–9, one can make two main observations: 1) the fundamental eigenmode always exhibits a TEM₀₀-like single-lobe (x,y) shape, even though its HG representation consists of a superposition of many modes (except in the 25×25 μm² case, for which almost no HG modes other than the TEM₀₀ are present); 2) although at small pump spot sizes the higher-order eigenmodes have complicated (x,y) shapes, at spot sizes of 100×100 μm² and larger the eigenmodes’ amplitude and phase profiles, at least qualitatively, resemble those of the HG modes TEM_mn (for elliptical pump waists) or the LG modes (for circular pump waists). We discuss these two observations separately below.

The first observation is further illustrated in Fig. 6, which shows the PSA’s fundamental eigenmode #0 in the HG representation for various pump spot sizes. The overlap of this mode with the TEM₀₀ mode of our HG basis is 97% for the 25×25 μm², 61% for the 100×25 μm², 50% for the 100×50 μm², 35% for the 100×100 μm², 16.2% for the 800×50 μm², 16.1% for the 400×100 μm², and 15.7% for the 200×200 μm² pump spot sizes. Thus, even though the eigenmode #0 is a single-lobe in all 7 cases, it requires a significant number of HG modes for its representation in all cases except the 25×25 μm² pump spot size. Higher-order eigenmodes involve superpositions of even greater numbers of HG modes, as shown in Figs. 7 –9.

Fig. 7 HG representation (i.e., |A_mn|²) of the few most prominent eigenmodes of the PSA for 100 × 100 μm² pump spot size, corresponding to the xy-profiles shown in Fig. 4. Pump power P₀ = 1.25 kW and the gain of mode #0 is 15.3. Note that mode #1 is degenerate with mode #2, mode #3 with #4, mode #6 with #7, mode #8 with #9, mode #10 with #11, and mode #12 with #13.

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The second observation leads us to the conclusion that, for PSAs with pump spot sizes large enough to support amplification of many eigenmodes, it should be possible to represent each eigenmode by a superposition of just a few HG modes. The fact that our HG basis requires a great number of modes for representing the PSA eigenmodes simply indicates that it is not the optimal basis, as we have discussed in Section 3 after Eq. (28). We have recently obtained some preliminary results on finding the optimum (compact) basis for representation of the PSA modes [34], and the complete study of that will be the subject of a separate publication.

Let us also point out that for the 25×25 μm² pump spot size, only the fundamental PSA eigenmode #0 sees any significant gain, and this mode has 97% overlap with the TEM₀₀ HG mode. This indicates that, in spite of gain-induced diffraction [20], for tightly focused pumps or long crystals (z_R / L ≈0.45 in this case) the crystal’s limited spatial bandwidth forces the PSA to produce squeezed vacuum in a single, well-defined fundamental Gaussian mode. This fact greatly simplifies mode matching of the local oscillator for subsequent homodyne detection and opens the possibilities for generation and observation of significant squeezing factors in such a regime of the PSA operation.

Fig. 8 HG representation (i.e., |A_mn|²) of the few most prominent eigenmodes of the PSA for 200 × 200 μm² pump spot size, corresponding to the xy-profiles shown in Fig. 4. Pump power P₀ = 4.06 kW and the gain of mode #0 is 15.1. Note that mode #1 is degenerate with mode #2, mode #3 with #4, mode #6 with #7, mode #8 with #9, mode #10 with #11, and mode #12 with #13.

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5. Conclusion

To summarize, we have developed and demonstrated an approach to rigorous calculation of the independently squeezed (or amplified) modes of the spatially-broadband PSA. While an order of magnitude estimate of the number of PSA modes is (pump waist × spatial bandwidth)², our rigorous method determines the exact number and the shapes of the PSA modes supported by the pump beam of a given spot size. This method is the spatial version of the quantum Karhunen-Loève expansion [36] previously used in the temporal domain to study squeezing during quantum soliton propagation [32] and in four-wave-mixing processes [37] in optical fiber. The generated eigenmodes are the spatial analogs of the temporal “supermodes” found in optical parametric oscillators [38]. The PSA eigenmodes are also closely related to the Schmidt modes of spontaneous parametric down conversion [39], previously discussed in both temporal [40] and spatial [41] contexts: at very low PSA gains, the PSA modes correspond to frequency-degenerate transverse Schmidt modes.

The obtained results indicate that the PSA has two important regimes: a) when the pump is tightly focused (z_R / L < 1), the limited spatial bandwidth forces the PSA to produce squeezing in a single, well-defined TEM₀₀ Gaussian eigenmode; b) for larger pump spot sizes that support many PSA eigenmodes, the shapes of the most-amplified PSA modes are close to the first few Hermite-Gaussian (for elliptical pump waists) or Laguerre-Gaussian (for circular pump waists) modes. The exact amount of this resemblance (mode overlap) will be the subject of a separate study.

The mode-calculation procedure and the obtained eigenmode shapes are important for both squeezing (finding the maximally squeezed modes as well as maximizing the number of highly-squeezed modes) and noiseless-amplifier (boosting a faint image before detection) applications of the PSA. For the squeezing application, the mode shape gives the profile of the optimum matched local oscillator needed to measure the high-degree of quadrature noise suppression in a travelling-wave PSA. For the image-amplifier application, the optimum performance will be reached when the image modes are mapped onto the PSA eigenmodes, so that they can be amplified without cross-coupling to each other.

This material is based upon work funded by DARPA’s Quantum Sensor Program under AFRL Contract No. FA8750-09-C-0194. 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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Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps

Abstract

1. Introduction

2. Theory of 2-D spatially-multimode PSA

2.1 Hermite-Gaussian mode representation

2.2 Laguerre-Gaussian mode representation

2.3 PSA Green’s functions and eigenmodes

3. Computational considerations

4. Results and discussion

5. Conclusion

References and links

Cited By

Figures (9)

Equations (31)

Optics Express