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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 28134–28153
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Compact representation of the spatial modes of a phase-sensitive image amplifier

Muthiah Annamalai, Nikolai Stelmakh, Prem Kumar, and Michael Vasilyev  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 28134-28153 (2013)
http://dx.doi.org/10.1364/OE.21.028134


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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

1. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982). [CrossRef]

]. 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

2. D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett. 24(14), 984–986 (1999). [CrossRef] [PubMed]

7

7. Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18(14), 14820–14835 (2010). [CrossRef] [PubMed]

]. PSAs can also be used for noiseless image amplification, due to their broad spatial bandwidth, as was theoretically proposed in [8

8. M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52(6), 4930–4940 (1995). [CrossRef] [PubMed]

10

10. K. Wang, G. Yang, A. Gatti, and L. Lugiato, “Controlling the signal-to-noise ratio in optical parametric image amplification,” J. Opt. B Quantum Semiclassical Opt. 5(4), S535–S544 (2003). [CrossRef]

] and experimentally demonstrated in [11

11. S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).

15

15. L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett. 100(1), 013604 (2008). [CrossRef] [PubMed]

]. 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

16. P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” in 14th Coherent Laser Radar Conference, Snowmass, CO, July 2007.

18

18. O.-K. Lim, G. Alon, Z. Dutton, S. Guha, M. Vasilyev, and P. Kumar, “Optical resolution enhancement with phase-sensitive preamplification,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper CTuPP7.

]. The same devices can also generate spatially-broadband (multimode) squeezed vacuum for quantum information processing applications [19

19. M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98(8), 083602 (2007). [CrossRef] [PubMed]

, 20

20. S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun. 3, 1026 (2012). [CrossRef] [PubMed]

]. For example, simultaneous squeezing of frequency-degenerate families of Hermite- or Laguerre-Gaussian modes of an optical parametric oscillator has been predicted [21

21. C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A 79(4), 043820 (2009). [CrossRef]

, 22

22. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80(4), 043816 (2009). [CrossRef]

] and achieved experimentally in conventional [23

23. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A 81(6), 061804 (2010). [CrossRef]

] and self-imaging [24

24. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express 19(5), 4405–4410 (2011). [CrossRef] [PubMed]

] cavities; a spatially-multimode waveguide-based photon-pair source has been recently demonstrated [25

25. P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett. 103(23), 233901 (2009). [CrossRef] [PubMed]

].

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 ~(kp/L)1/2, where kp is the pump propagation constant in the nonlinear crystal of length L [8

8. M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52(6), 4930–4940 (1995). [CrossRef] [PubMed]

, 14

14. E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” J. Sel. Top. Quantum Electron. 14(3), 635–647 (2008). [CrossRef]

, 26

26. M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt. 56(18-19), 2029–2033 (2009). [CrossRef]

]). Indeed, the traveling-wave nature of gain in the PSA requires the use of a small-area pump beam (typically, the fundamental Gaussian TEM00 mode) of high power (~1 kW per pixel of resolution) [27

27. M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express 17(14), 11415–11425 (2009). [CrossRef] [PubMed]

]. 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

27. M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express 17(14), 11415–11425 (2009). [CrossRef] [PubMed]

, 28

28. V. Delaubert, M. Lassen, D. R. N. Pulford, H.-A. Bachor, and C. C. Harb, “Spatial mode discrimination using second harmonic generation,” Opt. Express 15(9), 5815–5826 (2007). [CrossRef] [PubMed]

]. These spatial mode-mixing effects are known as gain-induced diffraction [29

29. A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44(3), 2013–2022 (1991). [CrossRef] [PubMed]

31

31. R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51(5), R3429–R3432 (1995). [CrossRef] [PubMed]

], 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

31. R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51(5), R3429–R3432 (1995). [CrossRef] [PubMed]

, 32

32. C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett. 73(12), 1605–1608 (1994). [CrossRef] [PubMed]

], 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

33. E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A 69(2), 023802 (2004). [CrossRef]

, 34

34. E. Lantz, N. Treps, C. Fabre, and E. Brambilla, “Spatial distribution of quantum fluctuations in spontaneous down-conversion in realistic situations: comparison between the stochastic approach and the Green’s function method,” Eur. Phys. J. D 29(3), 437–444 (2004). [CrossRef]

], or on computing noise variance from pixel-to-pixel Green’s function [22

22. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80(4), 043816 (2009). [CrossRef]

, 34

34. E. Lantz, N. Treps, C. Fabre, and E. Brambilla, “Spatial distribution of quantum fluctuations in spontaneous down-conversion in realistic situations: comparison between the stochastic approach and the Green’s function method,” Eur. Phys. J. D 29(3), 437–444 (2004). [CrossRef]

] have been used.

Our paper [36

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

], 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

35. M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt. 57(19), 1908–1915 (2010). [CrossRef]

, 36

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

] 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.

This paper is organized as follows: Section 2 describes the PSA mode-coupling theory in the original basis (where the signal’s waist is 21/2 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

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

], 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 21/2 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

27. M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express 17(14), 11415–11425 (2009). [CrossRef] [PubMed]

, 36

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

]
Es(ρ,z)z=i2ksρ2Es(ρ,z)+i2ωsdeffnscEp(ρ,z)Es*(ρ,z)exp(iΔkz),
(1)
where Δk = kp – 2ks is the wavevector mismatch for propagation along the z direction. We are looking for a solution in the form ei(r,t)=Ei(ρ,z)ei(kizωit)+c.c., where Ei(ρ,z) is a slowly-varying field envelope satisfying Eq. (1), ρ is a transverse vector with coordinates (x, y), and the intensity is given by Ii(ρ,z)=2ε0nic|Ei(ρ,z)|2 with index i taking the value of either s or p, denoting the signal or the pump field, respectively, with ωp = 2ωs. 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

27. M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express 17(14), 11415–11425 (2009). [CrossRef] [PubMed]

], 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 HG00 mode with potentially unequal beam waists (1/e intensity radii) a0px and a0py along the x- and y-directions, respectively. Our “original” HG signal expansion basis uses the signal beam waists a0sx and a0sy that are 21/2 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., zRx = kpa0px2 = ksa0sx2, zRy = kpa0py2 = ksa0sy2. 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
Ep(ρ,z)=P02ε0npceiθpg0(x,z,a0px,kp)g0(y,z,a0py,kp),Es(ρ,z)=m,nAmn(z)2ε0nscgm(x,z,2a0px,ks)gn(y,z,2a0py,ks),
(2)
where the one-dimensional HG modes gm are defined as
gm(β,z,a0,k)=Hm[β/a(z)]2mm!π1/2a(z)eiθ(z)eβ22a2(z)eikβ22R(z)
(3)
with the orthonormality condition
gm(β,z,a0,k)gm*(β,z,a0,k)dβ=δmm,
(4)
and the Rayleigh range zR, the 1/e intensity radius a(z), the Gouy phase shift θ(z), and the beam’s radius of curvature R(z) are given by

zR=ka02,a(z)=a01+(z/zR)2,θ(z)=(m+1/2)tan1(z/zR),R(z)=z[1+(zR/z)2].
(5)

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 Amn(z) = Xmn(z) + iYmn(z):
dAmn(z)dz=ieiθpκeiΔkzm,nBmm(z/zRx)Bnn(z/zRy)Amn*(z),
(6)
where
κ=2ωs2deff2ε0ns2npc3P0πa0pxa0py,
(7)
P0 is the pump power, and θp is the initial pump phase. The overlap integral Bmm between the pump and the two signal modes with indices m and m′ has a closed-form expression given by
Bmm(ξ)={ei(m+m+1/2)tan1ξ1+ξ24(1)mm2(m+m1)!!2m+m+1m!m!foreven(m+m),0forodd(m+m).
(8)
As we have shown in [36

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

], Bmm exhibits fast Gaussian decay as a function of (mm′), 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 Amn(–L/2) are excited at the crystal input z = –L/2, one at a time, and the resulting output mode patterns Amn(L/2) at z = L/2 are recorded [36

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

]. The independently squeezed or amplified PSA eigenmodes are obtained by diagonalizing the quantum-noise correlator derived from the Green’s function [35

35. M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt. 57(19), 1908–1915 (2010). [CrossRef]

, 36

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

]. This diagonalization procedure is similar to the procedure we developed previously in the studies of squeezing in quantum solitons [44

44. D. Levandovsky, M. Vasilyev, and P. Kumar, “Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering,” Opt. Lett. 24(1), 43–45 (1999). [CrossRef] [PubMed]

46

46. D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror: errata,” Opt. Lett. 24, 423 (1999).

]. 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

43. G. Patera, N. Treps, C. Fabre, and G. J. de Valcárcel, “Quantum theory of synchronously pumped type I optical parametric oscillators: characterization of the squeezed supermodes,” Eur. Phys. J. D 56(1), 123–140 (2010). [CrossRef]

, 47

47. C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282(4), 583–593 (2009). [CrossRef]

, 48

48. A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys. 13(3), 033027 (2011). [CrossRef]

].

The symmetry of Eq. (6), arising from the pump-waist’s location at the center of the crystal, leads to two important consequences [38

38. K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A 60(5), 4122–4134 (1999). [CrossRef]

, 49

49. G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett. 38(8), 1268–1270 (2013). [CrossRef] [PubMed]

]. If we assume θp = –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 Amn 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 Amn* [38

38. K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A 60(5), 4122–4134 (1999). [CrossRef]

]. 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: Amn(L/2) = Amn*(–L/2) [49

49. G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett. 38(8), 1268–1270 (2013). [CrossRef] [PubMed]

]. We refer to these two properties of the PSA as reciprocity relations.

2.2. Procedure for identifying a compact basis HGc or LGc for the signal modes

f00(ρ,φ,z,a0,k)=g0(x,z,a0,k)×g0(y,z,a0,k).
(14)

2.3. PSA equation in the compact basis HGc

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 HGc. Since for the basis with hsx = a0sx / a0px ≠ 21/2 or hsy = a0sy / a0py ≠ 21/2 the wavefront curvatures of the pump and signal modes do not align with each other, the overlap integrals Bmm of Eq. (8) must be replaced in Eq. (6) by a more complex integral Dmm given by
Dmm(z,a0sx,a0px)={ei[(m+m+1)tan1(zzRsx)12tan1(zzRpx)]1+(z/zRpx)242m+mπm!m!k=0(m+m)/2c2kΓ(1+2k2)ξx(z)1+2k2foreven(m+m),0forodd(m+m),
(17)
where c2k are the even polynomial coefficients of a product of two Hermite polynomials Hm(x)Hm(x)=i=0m+mcixi (ci is also known as the discrete convolution of the sequences of coefficients of two Hermite polynomials),
ξx(z)=δx(z)iγx(z),δx(z)=asx2(z)2apx2(z)+1,γx(z)=kpasx2(z)2(1Rpx(z)1Rsx(z)),
(18)
and zRjx=ka0jx2,ajx(z)=a0jx1+(z/zRjx)2,Rjx(z)=z[1+(zRjx/z)2] for j = s, p. We note that for hsx = 21/2, Eq. (17) matches Eq. (8) with ξ = z/zRsx = z/zRpx, and at z = 0 we have D00(0, a0sx, a0px) = B00(0) = 2 –1/2.

Similarly, the overlap integral Bnn in Eq. (6) should be replaced by a new expression Dnn, 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 DmmDnn instead of BmmBnn 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 hsx = hsy = 21/2, the memory size scales quadratically with the pump beam area [36

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

]. 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

27. M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express 17(14), 11415–11425 (2009). [CrossRef] [PubMed]

, 36

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

], 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 a0px in that dimension and the spatial bandwidth of the crystal qc=πkp/L [26

26. M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt. 56(18-19), 2029–2033 (2009). [CrossRef]

]:
Nx=a0pxqc=πkpa0px2L=πzRpxL.
(19)
The space of the first Nx HG modes with waist size a0sx can be used to describe an image with maximum dimension ~a0sxNx and minimum feature size ~a0sx/Nx. When this mode space is used to approximate the PSA eigenmodes, the maximum dimension is determined by the pump waist size a0px (gain region) and the minimum feature size is determined by the inverse spatial bandwidth 1/qc:
a0sxNx=a0px,.
(20)
a0sxNx=1qc=Lπkp.
(21)
Combining Eqs. (20) and (21) yields an estimate of the beam waist size a0sx for the optimum HGc mode set, which turns out to be given by the geometric average of the pump waist and the inverse spatial bandwidth:
a0sx=a0pxqc=a0pxLπzRpx4,
(22)
as was pointed out in our paper [36

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

] and was also previously discussed in [42

42. W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006). [CrossRef]

] 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 Nx >> 1.

3. Results and discussion

In this Section, we present the modeling results for a PSA based on a PPKTP crystal (deff = 8.7 pm/V, length L = 2 cm) with signal wavelength of 1560 nm and zero wavevector mismatch Δk. First, we compute the eigenvalue spectra and eigenmode shapes in the xy, HG, and LG representations for several different waist sizes a0px × a0py of the pump beam (25 × 25 μm2 to 800 × 50 μm2), 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)
Fig. 1 (a) Eigenvalue (gain and squeezing factor) spectra for the first 16 PSA modes. The vertical scale is linear. The legend indicates the pump waist dimensions a0px × a0py in μm2. The horizontal dashed black line 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. (b) Dependence of the signal’s compact-basis waist size (obtained by best match to eigenmode #0) on the corresponding pump waist size plotted on a double logarithmic scale. “Circular” cases represent 25 × 25, 100 × 100, and 200 × 200 μm2 pump waist sizes (here, the horizontal scale corresponds to a0p), whereas “100×Y” cases represent 100 × 25, 100 × 50, 100 × 100, 100 × 200, and 100 × 400 μm2 pump waist sizes (here, the horizontal scale corresponds to a0py). Blue symbols correspond to pump powers that ensure a gain of ~15 for the eigenmode #0. Red symbols correspond to a pump power of ~4.1 kW (which produces a gain of ~15 for the eigenmode #0 in the case of 200 × 200 μm2 pump waist size). The three dotted lines indicate slopes proportional to a0p, a0p1/2, and a0p1/3.
. As shown, the number of the supported eigenmodes with significant gain increases with the pump waist size [36

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

].

3.1. Fundamental eigenmode #0 in the compact basis HGc0

For each of the pump sizes, we have found a compact basis HGc0 by adjusting the 1/e intensity half-widths a0sx and a0sy (or the 1/e intensity radius a0s for the circular pump cases) of the HGmn modes of the signal beam to produce maximum overlap |A00|2 of the mode HG00 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 μm2 pump case. The three dashed lines in Fig. 1(b) indicate slopes proportional to a0p,a0p, and a0p3. For tight focusing (e.g., 25 × 25 μm2 case, where Boyd-Kleinman parameter ξ = L / (2zRp) = 1.11), the pump waist size a0p 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 21/2 × a0p. Hence, such single mode operation is also inherent to the most power-efficient parametric interaction regime, corresponding to ξ = 2.84 [40

40. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968). [CrossRef]

]. 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 a0p 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 μm2 for pump waist sizes of 800 × 50, 400 × 100, 200 × 200, and 100 × 100 μm2, 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 μm2, respectively. Therefore, the estimates from Eq. (22) turn out to be within 15% of the actual optimum values.

Figure 2
Fig. 2 Overlap integral |A00|2 of the eigenmode #0 of the PSA (400 × 100 μm2 pump waist size and pump power P0 = 4.3 kW) with HG00 modes of various waist sizes a0sx and a0sy. The best overlap of 99.3% occurs for a HG00 signal beam of waist size 83 × 48 μm2, which is then taken to be the waist size for the compact basis HGc0.
illustrates dependence of the overlap integral |A00|2 on a0sx and a0sy for the 400 × 100 μm2 pump case and shows that a rather wide range of waist sizes (83 ± 10) × (48 ± 6) μm2 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
Fig. 3 xy and HG representations of the most amplified eigenmode #0 for the PSAs with pump waist sizes (left to right) of 100 × 100, 100 × 50, 100 × 25, and 25 × 25 μm2 and pump powers P0 corresponding to gains of ~15 for the eigenmode #0. Row (a): the xy intensity profiles on the same scale. Row (b): the HG representations |Amn|2 in our original basis (signal waist is 21/2 times larger than the pump waist). Overlap |A00|2 with the HG00 mode is (left to right): 35.4%, 49.9%, 60.9%, and 96.6%. Row (c): the HG representations |Amn|2 in the compact basis HGc0 with a0sx × a0sy sizes of (left to right) 49.6 × 49.6, 51.5 × 40.1, 51.5 × 31.8, and 32.2 × 32.2 μm2, which are optimized for the best overlap of mode HG00 with the eigenmode #0, yielding |A00|2 of 98.9%, 98.4%, 98.4%, and 97.7%, respectively.
, 4
Fig. 4 xy, HG, and LG representations of the eigenmodes #0 to #6 of the PSA with 100 × 100 μm2 pump waist size and pump power P0 = 1.25 kW (gain of mode #0 is 15.3). Note that the eigenmodes #2, #4, and #7 are degenerate with the eigenmodes #1, #3, and #6, respectively. Column 1: xy profiles. Column 2: HG representations |Amn|2 in our original basis (signal waist is 21/2 times larger than the pump waist). Also shown are HG representations |Amn|2 (column 3) and LG representations |Ap|l||2 (column 4) in the compact bases HGc0 and LGc0, respectively, with a0sx = a0sy = a0s = 49.6 μm, which are optimized for best overlap (98.9%) with the eigenmode #0.
, 5
Fig. 5 xy, HG, and LG representations of the eigenmodes #8 to #14 of the PSA with 100 × 100 μm2 pump waist size and pump power P0 = 1.25 kW (gain of mode #0 is 15.3). Note that the eigenmodes #9, #11, and #13 are degenerate with the eigenmodes #8, #10, and #12, respectively. Columns are the same as those in Fig. 4.
, 6
Fig. 6 xy, HG, and LG representations of the eigenmodes #0 to #6 of the PSA with 200 × 200 μm2 pump waist size and pump power P0 = 4.06 kW (gain of mode #0 is 15.1). Note that the eigenmodes #2, #4, and #7 are degenerate with the eigenmodes #1, #3, and #6, respectively. Column 1: xy profiles. Column 2: HG representations |Amn|2 in our original basis (signal waist is 21/2 times larger than the pump waist). Column 3: LG representations |Ap|l||2 in the compact basis LGc0 with a0s = 61.2 μm, optimized for best overlap (99.4%) with the eigenmode #0. Column 4: LG representations |Ap|l||2 in the compact basis LGc5 with a0s = 69.8 μm, optimized for best overlap (96.5%) with the eigenmode #5.
, 7
Fig. 7 xy, HG, and LG representations of the eigenmodes #8 to #14 of the PSA with 200 × 200 μm2 pump waist size and pump power P0 = 4.06 kW (gain of mode #0 is 15.1). Note that the eigenmodes #9, #11, and #13 are degenerate with the eigenmodes #8, #10, and #12, respectively. Columns are the same as those in Fig. 6.
, and 8
Fig. 8 xy and HG representations of some of the main eigenmodes of the PSA with 800 × 50 μm2 pump waist size and pump power P0 = 5.4 kW (gain of mode #0 is 15.7). Column 1: xy profiles. Column 2: HG representations |Amn|2 in our original basis (signal waist is 21/2 times larger than the pump waist). Column 3: HG representations |Amn|2 in the compact basis HGc0 with a0sx = 113.1 μm, a0sy = 38.4 μm, optimized for best overlap (99.0%) with the eigenmode #0. Column 4: HG representations |Amn|2 in the compact basis HGc4 with a0sx = 128.2 μm, a0sy = 41.1 μm, optimized for best overlap (98.3%) with the eigenmode #4.
show the eigenmode shapes in the xy, the original HG (with signal waist that is 21/2 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 μm2, the original basis with 1/e intensity radius a0s = 35.4 μm is already very close to the optimum (32.2 μm) and, hence, the exact compact basis HGc0 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 HGc0 (or LGc0) 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

49. G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett. 38(8), 1268–1270 (2013). [CrossRef] [PubMed]

]. 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 HG00 mode in the compact basis HGc0 with greater than 98% overlap.

Similarly to the small-to-medium pump-waist cases, we find even smaller overlaps |A00|2 of the HG00 mode of our original basis with the eigenmode #0 in the cases of larger pump waists: it is 16.2% for 800 × 50 μm2, 16.1% for 400 × 100 μm2, and 15.7% for 200 × 200 μm2 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 HGc0 with an optimum beam waist, we find 99% or better overlap of the HG00 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. 48. 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 HGc0 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 μm2 pump waist, where the low-order eigenmodes #n with n from 0 to 3 are largely represented by just one HGn0 mode in the compact basis HGc0 with waist size of 113.1 × 38.4 μm2, 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 HGc or the LGc compact basis. For 100 × 100 μm2 and 200 × 200 μm2 pump waists, an especially profound reduction in the number of required modes results from using the compact LGc0 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 LGpl mode, the LG expansion of the eigenmode also contains the LGp(–l) mode of the same magnitude (possibly with a different phase). Hence, Figs. 47 plot |Ap|l||2 = |Apl|2 + |Ap(–l)|2 = 2|Apl|2 for each non-zero value of |l| and plot |Ap|0||2 = |Ap0|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 LGc5 with 69.8 μm waist optimized for maximum (96.5%) overlap of eigenmode #5 with LG10 mode (in contrast to 88.1% overlap between these modes for compact basis LGc0 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 LG00 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 LGc14 with waist of 76.5 μm to maximize the overlap of the eigenmode #14 with the LG20 mode (this overlap is 94.7%, 83.6%, and 44.9% for LGc14, LGc5, and LGc0 bases, respectively) at the expense of further degrading the overlap of eigenmode #0 with the LG00 mode.

Similarly, an optimized 128.2 × 41.1 μm2 waist of the compact basis HGc4 can be chosen to maximize the overlap between the eigenmode #4 and mode HG40 (98.3%, to compare with 85.1% overlap for HGc0 basis with 113.1 × 38.4 μm2 waist) for the case of 800 × 50 μm2 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 HG00.

Figure 9
Fig. 9 Radial field profiles of the output eigenmodes #0 to #14 of the PSA with 200 × 200 μm2 pump waist size and pump power P0 = 4.06 kW (gain of mode #0 is 15.1), compared to the radial profiles fpl(ρ,0,0,a0s,ks) of the dominant LGpl modes of the corresponding compact representation in basis LGc5 with a0s = 69.8 μm, optimized for best overlap with eigenmode #5. The legend lists the eigenmode number, dominant LGpl mode, and their overlap |Ap|l||2, represented by the tallest bar on the corresponding graph in column 4 of Fig. 6 or Fig. 7. Solid red line – real part of the eigenmode profile, solid blue line – imaginary part of the eigenmode profile, dashed black line – dominant LGpl mode profile fpl(ρ,0,0,a0s,ks) at z = 0. To compare the eigenmode with fpl(ρ,0,0,a0s,ks) of Eq. (11), we have normalized the eigenmode by its most dominant mode coefficient Apl, and linearly back-propagated the eigenmode to the center of the crystal z = 0 (this situation corresponds to the output eigenmode profile observable by projecting the center of the crystal onto the image plane of a 1:1 telescope).
(Fig. 10
Fig. 10 Horizontal field profiles at y = 0 of the output eigenmodes #0 to #14 of the PSA with 800 × 50 μm2 pump waist size and pump power P0 = 5.4 kW (gain of mode #0 is 15.7), compared to the horizontal profiles fm(x,0,a0sx,ks) × f0(0,0,a0sy,ks) of the dominant HGm0 modes of the corresponding compact representation in basis HGc4 with a0sx = 128.2 μm and a0sy = 41.1 μm, optimized for best overlap with eigenmode #4. The legend lists the eigenmode number, dominant HGm0 mode, and their overlap |Am0|2, represented by the tallest bar on the corresponding graph in column 4 of Fig. 8. Solid red line – real part of the eigenmode profile, solid blue line – imaginary part of the eigenmode profile, dashed black line – dominant HGm0 mode profile at z = 0. To compare the eigenmode with fm(x,0,a0sx,ks) × f0(0,0,a0sy,ks) of Eq. (3), we have normalized the eigenmode by its most dominant mode coefficient Am0, and linearly back-propagated the eigenmode to the center of the crystal z = 0 (this situation corresponds to the output eigenmode profile observable by projecting the center of the crystal onto the image plane of a 1:1 telescope).
) shows the line graphs comparing the field profiles of the PSA eigenmodes for the 200 × 200 μm2 (800 × 50 μm2) pump waist size with the most dominant mode LGpl (HGm0) of their compact representations in basis LGc5 (HGc4), 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. 410 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. 68] 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 μm2 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

36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

]. On the other hand, in the compact LGc5 basis only ~10 × 15 LG modes are required for the same 200 × 200 μm2 pump waist size, which leads to computing and diagonalizing of a (2 × 10 × 15) × (2 × 10 × 15) = 300 × 300 real matrix, i.e., approximately 104-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
Fig. 11 (a) Overlaps of the various eigenmodes from Figs. 610 with the respective most dominant HG or LG mode in the eigenmode’s compact representation. (b) Magnified version of (a), but with added overlap curves (magenta) for the compact LGc bases with optimized z-offsets that are equal to 3.5 mm for basis LGc0 matched to the eigenmode #0 and to 3 mm for basis LGc5 matched to the eigenmode #5.
each eigenmode’s overlap with the most dominant HG or LG mode of its compact representation for the cases of 800 × 50 μm2 and 200 × 200 μm2 pump waist sizes. We see that by matching the compact basis to eigenmode #4 (basis HGc4 for the elliptical pump case) or eigenmode #5 (basis LGc5 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

49. G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett. 38(8), 1268–1270 (2013). [CrossRef] [PubMed]

]. Such optimization could be important for the detection of high degree of squeezing. For the case of the 200 × 200 μm2 pump waist size, a z-offset of the basis LGc0 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 LGc5 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

50. M. Annamalai, M. Vasilyev, and P. Kumar, “Impact of phase-sensitive-amplifier's mode structure on amplified image quality,” Conference on Lasers and Electro-Optics 2012, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CF1B.7.

]. To verify the phase response, we plot in Fig. 12(a)
Fig. 12 (a) Right vertical scale: Phases of the most dominant modes HGn0, HG(n+2)0, and HG(n+4)0 in the compact HG representation (basis HGc4 optimized for the eigenmode #4) of the nth output eigenmode of a PSA with 800 × 50 μm2 pump waist size and pump power P0 = 5.4 kW. (b) Right vertical scale: Phases of the most dominant modes LGp|l| in the compact LG representation (basis LGc5 optimized for the eigenmode #5 with the basis’ waist location offset from the pump waist location by + 3 mm in the z-direction) of the output eigenmodes of the PSA with 200 × 200 μm2 pump waist size and pump power P0 = 4.06 kW. For the eigenmodes with non-zero |l|, in (b) we plot the average phase of LGpl and LGp(–l) (their difference phase determines azimuthal rotation of the eigenmode). Left vertical scales on both (a) and (b): the corresponding eigenvalue (gain) spectrum [also shown in Fig. 1(a)], which crosses the –3-dB level from the gain of the eigenmode #0 between the eigenmodes #9 and #10 in (a) and between the eigenmodes #5 and #6 in (b). Pump phase θp = –π/2.
the phases of the most dominant modes in the compact HGc4 representation of the output eigenmodes of the PSA with a 800 × 50 μm2 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 LGc5 representation of the output eigenmodes of the PSA with a 200 × 200 μm2 pump waist size. For the plots in Fig. 12(b), we take the compact LGc5 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

49. G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett. 38(8), 1268–1270 (2013). [CrossRef] [PubMed]

].

Under the condition θp = –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

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

20. S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun. 3, 1026 (2012). [CrossRef] [PubMed]

, 23

23. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A 81(6), 061804 (2010). [CrossRef]

, 24

24. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express 19(5), 4405–4410 (2011). [CrossRef] [PubMed]

]) 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

52. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive image amplifier with higher-order Gaussian pump and phase mismatch,” in IEEE Photonics Conference, Arlington, VA, October 9–13, 2011, paper TuO4. [CrossRef]

]. 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

53. M. Annamalai and M. Vasilyev, “Phase-sensitive multimode parametric amplification in a parabolic-index waveguide,” IEEE Photonics Technol. Lett. 24(21), 1949–1952 (2012). [CrossRef]

] and quantum image converters [54

54. M. Vasilyev and P. Kumar, “Frequency up-conversion of quantum images,” Opt. Express 20(6), 6644–6656 (2012). [CrossRef] [PubMed]

].

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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P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B 23(4), 598–610 (2006). [CrossRef]

6.

Z. Tong, C. Lundström, M. Karlsson, M. Vasilyev, and P. A. Andrekson, “Noise performance of a frequency nondegenerate phase-sensitive amplifier with unequalized inputs,” Opt. Lett. 36(5), 722–724 (2011). [CrossRef] [PubMed]

7.

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18(14), 14820–14835 (2010). [CrossRef] [PubMed]

8.

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52(6), 4930–4940 (1995). [CrossRef] [PubMed]

9.

M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71(5), 1539–1589 (1999). [CrossRef]

10.

K. Wang, G. Yang, A. Gatti, and L. Lugiato, “Controlling the signal-to-noise ratio in optical parametric image amplification,” J. Opt. B Quantum Semiclassical Opt. 5(4), S535–S544 (2003). [CrossRef]

11.

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).

12.

S.-K. Choi, M. Vasilyev, and P. Kumar, “Erratum: Noiseless optical amplification of images,” Phys. Rev. Lett. 84, 1361 (2000).

13.

A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett. 94(22), 223603 (2005). [CrossRef] [PubMed]

14.

E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” J. Sel. Top. Quantum Electron. 14(3), 635–647 (2008). [CrossRef]

15.

L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett. 100(1), 013604 (2008). [CrossRef] [PubMed]

16.

P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” in 14th Coherent Laser Radar Conference, Snowmass, CO, July 2007.

17.

Z. Dutton, J. H. Shapiro, and S. Guha, “LADAR resolution improvement using receivers enhanced with squeezed-vacuum injection and phase-sensitive amplification,” J. Opt. Soc. Am. B 27(6), A63–A72 (2010). [CrossRef]

18.

O.-K. Lim, G. Alon, Z. Dutton, S. Guha, M. Vasilyev, and P. Kumar, “Optical resolution enhancement with phase-sensitive preamplification,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper CTuPP7.

19.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. 98(8), 083602 (2007). [CrossRef] [PubMed]

20.

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun. 3, 1026 (2012). [CrossRef] [PubMed]

21.

C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A 79(4), 043820 (2009). [CrossRef]

22.

L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80(4), 043816 (2009). [CrossRef]

23.

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A 81(6), 061804 (2010). [CrossRef]

24.

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express 19(5), 4405–4410 (2011). [CrossRef] [PubMed]

25.

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett. 103(23), 233901 (2009). [CrossRef] [PubMed]

26.

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt. 56(18-19), 2029–2033 (2009). [CrossRef]

27.

M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express 17(14), 11415–11425 (2009). [CrossRef] [PubMed]

28.

V. Delaubert, M. Lassen, D. R. N. Pulford, H.-A. Bachor, and C. C. Harb, “Spatial mode discrimination using second harmonic generation,” Opt. Express 15(9), 5815–5826 (2007). [CrossRef] [PubMed]

29.

A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44(3), 2013–2022 (1991). [CrossRef] [PubMed]

30.

S.-K. Choi, R.-D. Li, C. Kim, and P. Kumar, “Traveling-wave optical parametric amplifier: investigation of its phase-sensitive and phase-insensitive gain response,” J. Opt. Soc. Am. B 14(7), 1564–1575 (1997). [CrossRef]

31.

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51(5), R3429–R3432 (1995). [CrossRef] [PubMed]

32.

C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett. 73(12), 1605–1608 (1994). [CrossRef] [PubMed]

33.

E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A 69(2), 023802 (2004). [CrossRef]

34.

E. Lantz, N. Treps, C. Fabre, and E. Brambilla, “Spatial distribution of quantum fluctuations in spontaneous down-conversion in realistic situations: comparison between the stochastic approach and the Green’s function method,” Eur. Phys. J. D 29(3), 437–444 (2004). [CrossRef]

35.

M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt. 57(19), 1908–1915 (2010). [CrossRef]

36.

M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express 19(27), 26710–26724 (2011). [CrossRef] [PubMed]

37.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B 66(6), 685–699 (1998). [CrossRef]

38.

K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A 60(5), 4122–4134 (1999). [CrossRef]

39.

K. G. Köprülü and O. Aytür, “Analysis of the generation of amplitude-squeezed light with Gaussian-beam degenerate optical parametric amplifiers,” J. Opt. Soc. Am. B 18(6), 846–854 (2001). [CrossRef]

40.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968). [CrossRef]

41.

M. Annamalai, M. Vasilyev, N. Stelmakh, and P. Kumar, “Compact representation of spatial modes of phase-sensitive image amplifier,” in Conference on Lasers and Electro-Optics 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThB77.

42.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006). [CrossRef]

43.

G. Patera, N. Treps, C. Fabre, and G. J. de Valcárcel, “Quantum theory of synchronously pumped type I optical parametric oscillators: characterization of the squeezed supermodes,” Eur. Phys. J. D 56(1), 123–140 (2010). [CrossRef]

44.

D. Levandovsky, M. Vasilyev, and P. Kumar, “Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering,” Opt. Lett. 24(1), 43–45 (1999). [CrossRef] [PubMed]

45.

D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror,” Opt. Lett. 24, 89–91 (1999).

46.

D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror: errata,” Opt. Lett. 24, 423 (1999).

47.

C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282(4), 583–593 (2009). [CrossRef]

48.

A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys. 13(3), 033027 (2011). [CrossRef]

49.

G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett. 38(8), 1268–1270 (2013). [CrossRef] [PubMed]

50.

M. Annamalai, M. Vasilyev, and P. Kumar, “Impact of phase-sensitive-amplifier's mode structure on amplified image quality,” Conference on Lasers and Electro-Optics 2012, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CF1B.7.

51.

A. R. Bhagwat, G. Alon, O.-K. Lim, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Fundamental eigenmode of traveling-wave phase-sensitive optical parametric amplifier: experimental generation and verification,” Opt. Lett. 38(15), 2858–2860 (2013). [CrossRef] [PubMed]

52.

M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive image amplifier with higher-order Gaussian pump and phase mismatch,” in IEEE Photonics Conference, Arlington, VA, October 9–13, 2011, paper TuO4. [CrossRef]

53.

M. Annamalai and M. Vasilyev, “Phase-sensitive multimode parametric amplification in a parabolic-index waveguide,” IEEE Photonics Technol. Lett. 24(21), 1949–1952 (2012). [CrossRef]

54.

M. Vasilyev and P. Kumar, “Frequency up-conversion of quantum images,” Opt. Express 20(6), 6644–6656 (2012). [CrossRef] [PubMed]

OCIS Codes
(190.4970) Nonlinear optics : Parametric oscillators and amplifiers
(270.6570) Quantum optics : Squeezed states

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 23, 2013
Revised Manuscript: October 31, 2013
Manuscript Accepted: November 2, 2013
Published: November 8, 2013

Citation
Muthiah Annamalai, Nikolai Stelmakh, Prem Kumar, and Michael Vasilyev, "Compact representation of the spatial modes of a phase-sensitive image amplifier," Opt. Express 21, 28134-28153 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28134


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References

  1. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields26(8), 1817–1839 (1982). [CrossRef]
  2. D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett.24(14), 984–986 (1999). [CrossRef] [PubMed]
  3. D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana56, 281–285 (2001).
  4. W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett.35(22), 1954–1955 (1999). [CrossRef]
  5. P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B23(4), 598–610 (2006). [CrossRef]
  6. Z. Tong, C. Lundström, M. Karlsson, M. Vasilyev, and P. A. Andrekson, “Noise performance of a frequency nondegenerate phase-sensitive amplifier with unequalized inputs,” Opt. Lett.36(5), 722–724 (2011). [CrossRef] [PubMed]
  7. Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express18(14), 14820–14835 (2010). [CrossRef] [PubMed]
  8. M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A52(6), 4930–4940 (1995). [CrossRef] [PubMed]
  9. M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys.71(5), 1539–1589 (1999). [CrossRef]
  10. K. Wang, G. Yang, A. Gatti, and L. Lugiato, “Controlling the signal-to-noise ratio in optical parametric image amplification,” J. Opt. B Quantum Semiclassical Opt.5(4), S535–S544 (2003). [CrossRef]
  11. S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett.83, 1938–1941 (1999).
  12. S.-K. Choi, M. Vasilyev, and P. Kumar, “Erratum: Noiseless optical amplification of images,” Phys. Rev. Lett.84, 1361 (2000).
  13. A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett.94(22), 223603 (2005). [CrossRef] [PubMed]
  14. E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” J. Sel. Top. Quantum Electron.14(3), 635–647 (2008). [CrossRef]
  15. L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett.100(1), 013604 (2008). [CrossRef] [PubMed]
  16. P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” in 14th Coherent Laser Radar Conference, Snowmass, CO, July 2007.
  17. Z. Dutton, J. H. Shapiro, and S. Guha, “LADAR resolution improvement using receivers enhanced with squeezed-vacuum injection and phase-sensitive amplification,” J. Opt. Soc. Am. B27(6), A63–A72 (2010). [CrossRef]
  18. O.-K. Lim, G. Alon, Z. Dutton, S. Guha, M. Vasilyev, and P. Kumar, “Optical resolution enhancement with phase-sensitive preamplification,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper CTuPP7.
  19. M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007). [CrossRef] [PubMed]
  20. S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun.3, 1026 (2012). [CrossRef] [PubMed]
  21. C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A79(4), 043820 (2009). [CrossRef]
  22. L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A80(4), 043816 (2009). [CrossRef]
  23. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A81(6), 061804 (2010). [CrossRef]
  24. B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express19(5), 4405–4410 (2011). [CrossRef] [PubMed]
  25. P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett.103(23), 233901 (2009). [CrossRef] [PubMed]
  26. M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt.56(18-19), 2029–2033 (2009). [CrossRef]
  27. M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express17(14), 11415–11425 (2009). [CrossRef] [PubMed]
  28. V. Delaubert, M. Lassen, D. R. N. Pulford, H.-A. Bachor, and C. C. Harb, “Spatial mode discrimination using second harmonic generation,” Opt. Express15(9), 5815–5826 (2007). [CrossRef] [PubMed]
  29. A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A44(3), 2013–2022 (1991). [CrossRef] [PubMed]
  30. S.-K. Choi, R.-D. Li, C. Kim, and P. Kumar, “Traveling-wave optical parametric amplifier: investigation of its phase-sensitive and phase-insensitive gain response,” J. Opt. Soc. Am. B14(7), 1564–1575 (1997). [CrossRef]
  31. R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A51(5), R3429–R3432 (1995). [CrossRef] [PubMed]
  32. C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett.73(12), 1605–1608 (1994). [CrossRef] [PubMed]
  33. E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A69(2), 023802 (2004). [CrossRef]
  34. E. Lantz, N. Treps, C. Fabre, and E. Brambilla, “Spatial distribution of quantum fluctuations in spontaneous down-conversion in realistic situations: comparison between the stochastic approach and the Green’s function method,” Eur. Phys. J. D29(3), 437–444 (2004). [CrossRef]
  35. M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt.57(19), 1908–1915 (2010). [CrossRef]
  36. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express19(27), 26710–26724 (2011). [CrossRef] [PubMed]
  37. C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B66(6), 685–699 (1998). [CrossRef]
  38. K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A60(5), 4122–4134 (1999). [CrossRef]
  39. K. G. Köprülü and O. Aytür, “Analysis of the generation of amplitude-squeezed light with Gaussian-beam degenerate optical parametric amplifiers,” J. Opt. Soc. Am. B18(6), 846–854 (2001). [CrossRef]
  40. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39(8), 3597–3639 (1968). [CrossRef]
  41. M. Annamalai, M. Vasilyev, N. Stelmakh, and P. Kumar, “Compact representation of spatial modes of phase-sensitive image amplifier,” in Conference on Lasers and Electro-Optics 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThB77.
  42. W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A73(6), 063819 (2006). [CrossRef]
  43. G. Patera, N. Treps, C. Fabre, and G. J. de Valcárcel, “Quantum theory of synchronously pumped type I optical parametric oscillators: characterization of the squeezed supermodes,” Eur. Phys. J. D56(1), 123–140 (2010). [CrossRef]
  44. D. Levandovsky, M. Vasilyev, and P. Kumar, “Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering,” Opt. Lett.24(1), 43–45 (1999). [CrossRef] [PubMed]
  45. D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror,” Opt. Lett.24, 89–91 (1999).
  46. D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror: errata,” Opt. Lett.24, 423 (1999).
  47. C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun.282(4), 583–593 (2009). [CrossRef]
  48. A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys.13(3), 033027 (2011). [CrossRef]
  49. G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett.38(8), 1268–1270 (2013). [CrossRef] [PubMed]
  50. M. Annamalai, M. Vasilyev, and P. Kumar, “Impact of phase-sensitive-amplifier's mode structure on amplified image quality,” Conference on Lasers and Electro-Optics 2012, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CF1B.7.
  51. A. R. Bhagwat, G. Alon, O.-K. Lim, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Fundamental eigenmode of traveling-wave phase-sensitive optical parametric amplifier: experimental generation and verification,” Opt. Lett.38(15), 2858–2860 (2013). [CrossRef] [PubMed]
  52. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive image amplifier with higher-order Gaussian pump and phase mismatch,” in IEEE Photonics Conference, Arlington, VA, October 9–13, 2011, paper TuO4. [CrossRef]
  53. M. Annamalai and M. Vasilyev, “Phase-sensitive multimode parametric amplification in a parabolic-index waveguide,” IEEE Photonics Technol. Lett.24(21), 1949–1952 (2012). [CrossRef]
  54. M. Vasilyev and P. Kumar, “Frequency up-conversion of quantum images,” Opt. Express20(6), 6644–6656 (2012). [CrossRef] [PubMed]

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