## Compact representation of the spatial modes of a phase-sensitive image amplifier |

Optics Express, Vol. 21, Issue 23, pp. 28134-28153 (2013)

http://dx.doi.org/10.1364/OE.21.028134

Acrobat PDF (3876 KB)

### 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

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*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. 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]

*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. 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]

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]

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]

*k*/

_{p}*L*)

^{1/2}, where

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

_{p}*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. 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. 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]

_{00}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]

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]

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]

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]

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. 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]

*modes, which was originally developed for cavities [37*

_{mn}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]

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]

_{00}pump with potentially different 1/

*e*intensity radii

*a*

_{0}

*and*

_{px}*a*

_{0}

*in*

_{py}*x*- and

*y*-dimensions, respectively, and b) find the eigenmodes of the spatially-broadband PSA once its Green’s function has been computed [35

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]

*HG modes (denoted as HG*

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

_{mn}*z*=

_{Rx}*k*

_{p}a_{0}

_{px}^{2}and

*z*=

_{Ry}*k*

_{p}a_{0}

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

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

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

*L*/(2

*z*) = 2.84, corresponding to the most power-efficient parametric interaction [40

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

_{00}shape with waist 2

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

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]

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

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]

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

^{1/2}times larger than the pump’s) and in the new compact basis, Section 3 presents the results, and Section 4 summarizes our work.

## 2. Theory of the PSA in compact basis

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

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

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]

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

*k*=

*k*– 2

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

_{s}*z*direction. We are looking for a solution in the form

*x*,

*y*), and the intensity is given by

*i*taking the value of either

*s*or

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

*= 2ω*

_{p}*. In the traveling-wave PSAs, the pump powers required to obtain any noticeable gains are on the order of hundreds of Watts per pixel [27*

_{s}**17**(14), 11415–11425 (2009). [CrossRef] [PubMed]

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

*e*intensity radii)

*a*

_{0}

*and*

_{px}*a*

_{0}

*along the*

_{py}*x*- and

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

*a*

_{0}

*and*

_{sx}*a*

_{0}

*that are 2*

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

*z*=

_{Rx}*k*

_{p}a_{0}

_{px}^{2}=

*k*

_{s}a_{0}

_{sx}^{2},

*z*=

_{Ry}*k*

_{p}a_{0}

_{py}^{2}=

*k*

_{s}a_{0}

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

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

*g*are defined aswith the orthonormality conditionand the Rayleigh range

_{m}*z*, the 1/

_{R}*e*intensity radius

*a*(

*z*), the Gouy phase shift θ(

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

*R*(

*z*) are given by

*A*(

_{mn}*z*) =

*X*(

_{mn}*z*) +

*iY*(

_{mn}*z*):where

*P*

_{0}is the pump power, and θ

*is the initial pump phase. The overlap integral*

_{p}*B*

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

*m*and

*m*′ has a closed-form expression given byAs we have shown in [36

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

*B*

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

*m*–

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

*m*+

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

*A*

_{m}_{′}

_{n}_{′}(–

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

*z*= –

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

*A*(

_{mn}*L*/2) at

*z*=

*L*/2 are recorded [36

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

**57**(19), 1908–1915 (2010). [CrossRef]

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

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]

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. 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]

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. 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]

*= –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*

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

_{mn}*A*[38

_{mn}^{*}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]

*A*(

_{mn}*L*/2) =

*A*(–

_{mn}^{*}*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]

*reciprocity relations*.

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

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

*. Since for the basis with*

^{c}*h*=

_{sx}*a*

_{0}

*/*

_{sx}*a*

_{0}

*≠ 2*

_{px}^{1/2}or

*h*=

_{sy}*a*

_{0}

*/*

_{sy}*a*

_{0}

*≠ 2*

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

*B*

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

*D*

_{mm}_{′}given by

*c*

_{2}

*are the even polynomial coefficients of a product of two Hermite polynomials*

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

_{i}*j*=

*s*,

*p*. We note that for

*h*2

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

*z*/

*z*=

_{Rsx}*z*/

*z*, and at

_{Rpx}*z*= 0 we have

*D*

_{00}(0,

*a*

_{0}

*,*

_{sx}*a*

_{0}

*) =*

_{px}*B*

_{00}(0) = 2

^{–1/2}.

*B*

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

*D*

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

*y*for

*x*,

*n*for

*m*, and

*n*′ for

*m*′.

*D*

_{mm}_{′}

*D*

_{nn}_{′}instead of

*B*

_{mm}_{′}

*B*

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

*h*2

_{sx}= h_{sy}=^{1/2}, the memory size scales quadratically with the pump beam area [36

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

*z*axis.

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

**17**(14), 11415–11425 (2009). [CrossRef] [PubMed]

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

*x*-dimension) can be estimated by the product of the pump waist size

*a*

_{0}

*in that dimension and the spatial bandwidth of the crystal*

_{px}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]

*N*HG modes with waist size

_{x}*a*

_{0}

*can be used to describe an image with maximum dimension*

_{sx}*a*

_{0}

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

_{px}*q*: Combining Eqs. (20) and (21) yields an estimate of the beam waist size

_{c}*a*

_{0}

*for the optimum HG*

_{sx}*mode set, which turns out to be given by the geometric average of the pump waist and the inverse spatial bandwidth:as was pointed out in our paper [36*

^{c}**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

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]

*N*>> 1.

_{x}## 3. Results and discussion

*d*

_{eff}= 8.7 pm/V, length

*L*= 2 cm) with signal wavelength of 1560 nm and zero wavevector mismatch Δ

*k*. First, we compute the eigenvalue spectra and eigenmode shapes in the

*xy*, HG, and LG representations for several different waist sizes

*a*

_{0}

*×*

_{px}*a*

_{0}

*of the pump beam (25 × 25 μm*

_{py}^{2}to 800 × 50 μm

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

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

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

^{c}^{0}by adjusting the 1/

*e*intensity half-widths

*a*

_{0}

*and*

_{sx}*a*

_{0}

*(or the 1/*

_{sy}*e*intensity radius

*a*

_{0}

*for the circular pump cases) of the HG*

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

_{mn}*A*

_{00}|

^{2}of the mode HG

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

^{2}pump case. The three dashed lines in Fig. 1(b) indicate slopes proportional to

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

*L*/ (2

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

_{Rp}*a*

_{0}

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

_{p}^{1/2}×

*a*

_{0}

*. Hence, such single mode operation is also inherent to the most power-efficient parametric interaction regime, corresponding to ξ = 2.84 [40*

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

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

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

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

*A*

_{00}|

^{2}on

*a*

_{0}

*and*

_{sx}*a*

_{0}

*for the 400 × 100 μm*

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

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

*xy*, the original HG (with signal waist that is 2

^{1/2}times larger than the pump waist), and the compact representations (degenerate modesare not repeated). These shapes for the eigenmode #0 in small-to-medium pump-waist cases are shown in Fig. 3. In the smallest pump-waist case of 25 × 25 μm

^{2}, the original basis with 1/

*e*intensity radius

*a*

_{0}

*= 35.4 μm is already very close to the optimum (32.2 μm) and, hence, the exact compact basis HG*

_{s}

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

^{c}^{0}(or LG

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

*z*-direction. The benefit of such a longitudinal waist offset in the tight-focusing case has been predicted, experimentally verified, and studied in detail in [49

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]

_{00}mode in the compact basis HG

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

*A*

_{00}|

^{2}of the HG

_{00}mode of our original basis with the eigenmode #0 in the cases of larger pump waists: it is 16.2% for 800 × 50 μm

^{2}, 16.1% for 400 × 100 μm

^{2}, and 15.7% for 200 × 200 μm

^{2}pump waist sizes. A significant number of HG modes are thus required to represent the eigenmode #0 in our original basis, and even more modes are needed to represent higher-order eigenmodes. However, after switching to the corresponding compact basis HG

^{c}^{0}with an optimum beam waist, we find 99% or better overlap of the HG

_{00}with the eigenmode #0 in all of these large pump-waist cases.

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

^{c}^{0}reduces the number of the required HG modes [column 3 in Figs. 4, 5, and 8]. The reduction is particularly profound in the highly-elliptical case of 800 × 50 μm

^{2}pump waist, where the low-order eigenmodes #

*n*with

*n*from 0 to 3 are largely represented by just one HG

_{n}_{0}mode in the compact basis HG

^{c}^{0}with waist size of 113.1 × 38.4 μm

^{2}, whereas each of the higher-order eigenmodes is represented by a mere handful of the HG modes. The eigenmodes for the circular-pump cases can be decomposed in either the HG

*or the LG*

^{c}*compact basis. For 100 × 100 μm*

^{c}^{2}and 200 × 200 μm

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

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

*l*| but with various

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

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

_{pl}

_{p}_{(–}

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

*A*|

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

*A*|

_{pl}^{2}+

*|A*

_{p}_{(–}

_{l}_{)}|

^{2}= 2|

*A*|

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

*l*| and plot |

*A*

_{p|}_{0}

*|*

_{|}^{2}= |

*A*

_{p}_{0}|

^{2}for

*l*= 0.

^{c}^{5}with 69.8 μm waist optimized for maximum (96.5%) overlap of eigenmode #5 with LG

_{10}mode (in contrast to 88.1% overlap between these modes for compact basis LG

^{c}^{0}with 61.2 μm waist). As a result, even higher-order eigenmodes up to #14 require no more than two LG modes for their representation. This simplification comes at the expense of a slight reduction in the overlap of the eigenmode #0 with the LG

_{00}mode (from 99.4% to 97.9%). The compact basis finding procedure can also be used to better represent modes that are far away from the eigenmode #0. For example, it is possible [although not shown in Figs. 6 and 7] to choose a compact basis LG

^{c}^{14}with waist of 76.5 μm to maximize the overlap of the eigenmode #14 with the LG

_{20}mode (this overlap is 94.7%, 83.6%, and 44.9% for LG

^{c}^{14}, LG

^{c}^{5}, and LG

^{c}^{0}bases, respectively) at the expense of further degrading the overlap of eigenmode #0 with the LG

_{00}mode.

^{2}waist of the compact basis HG

^{c}^{4}can be chosen to maximize the overlap between the eigenmode #4 and mode HG

_{40}(98.3%, to compare with 85.1% overlap for HG

^{c}^{0}basis with 113.1 × 38.4 μm

^{2}waist) for the case of 800 × 50 μm

^{2}pump waist, with the results shown in Fig. 8, column 4. In this basis, the eigenmodes #0 to #6 are represented by a single HG mode, whereas the higher-order eigenmodes up to #14 require no more than five HG modes for their representation. This simplification comes at the expense of a slight reduction (from 99.0% to 98.4%) in the overlap of the eigenmode #0 and mode HG

_{00}.

^{2}(800 × 50 μm

^{2}) pump waist size with the most dominant mode LG

*(HG*

_{pl}

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

^{c}^{5}(HG

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

*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.

*prior*to solving the PSA Eq. (1). Equation (1) can then be expanded over the compact basis with this optimum waist size (see Sec. 2.3), and the coupled-mode system of Eq. (6) with coupling coefficients in Eq. (17) can be efficiently solved with a drastically reduced number of modes needed for signal expansion. For example, 200 × 200 μm

^{2}pump waist size requires 128 × 128 modes for computation in the original HG basis, which leads to computing and diagonalizing of a (2 × 128 × 128) × (2 × 128 × 128) = 32768 × 32768 real matrix (Green’s tensor). At the same time, the number of modes with gain noticeably differing from unity is less than 100 [36

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

^{c}^{5}basis only ~10 × 15 LG modes are required for the same 200 × 200 μm

^{2}pump waist size, which leads to computing and diagonalizing of a (2 × 10 × 15) × (2 × 10 × 15) = 300 × 300 real matrix, i.e., approximately 10

^{4}-fold memory savings.

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

^{2}and 200 × 200 μm

^{2}pump waist sizes. We see that by matching the compact basis to eigenmode #4 (basis HG

^{c}^{4}for the elliptical pump case) or eigenmode #5 (basis LG

^{c}^{5}for the circular pump case) produces better overlaps for all eigenmodes other than #0 at the expense of a slightly worse overlap for the eigenmode #0. If an even better overlap is desired, then additional improvement can be obtained by optimizing the offset of the

*z*-position of the waist of the compact basis from the pump waist location (

*z*= 0) [49

**38**(8), 1268–1270 (2013). [CrossRef] [PubMed]

^{2}pump waist size, a

*z*-offset of the basis LG

^{c}^{0}matched to the eigenmode #0 improves the best overlap of the eigenmode #0 from 99.4% to 99.7% and those of the eigenmodes #1–#5 by 1–2%. The

*z*-offset of the basis LG

^{c}^{5}matched to the eigenmode #5 improves the overlaps of the eigenmodes #0–#5 by 0.3–0.8%, compared to the zero-offset case.

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

^{c}^{4}representation of the output eigenmodes of the PSA with a 800 × 50 μm

^{2}pump waist size [corresponding to the representations in column 4 of Fig. 8] and in Fig. 12(b) the phases of the most dominant modes in the compact LG

^{c}^{5}representation of the output eigenmodes of the PSA with a 200 × 200 μm

^{2}pump waist size. For the plots in Fig. 12(b), we take the compact LG

^{c}^{5}basis matched to the eigenmode #5 [shown in column 4 of Figs. 6 and 7] and further optimize it by moving the

*z*-position of its waist by 3 mm in the propagation direction from the position of the pump waist at

*z*= 0 [the corresponding overlaps are shown by the filled magenta diamonds in Fig. 11(b)]. The resulting maximum phase excursion across the eigenmodes within the –3-dB gain range is 3.5° for the plot in Fig. 12(a) and is 0.7° for the plot in Fig. 12(b) (the latter becomes 3.6° if we do not optimize the

*z*-offset of the basis waist location), which means that the amplified image will have practically no phase distortion. According to the reciprocity relations discussed at the end of Section 2.1, the input and output eigenmode shapes are complex conjugates of each other and, therefore, the eigenmode phases at the PSA input are negatives of their phases at the PSA output. With the

*z*-offset basis, this also means that, in order to get the minimum distortion of the image, the input image must be focused in the plane located 3 mm

*before*the pump waist, whereas the output image will appear to be emerging from a plane located 3 mm

*after*the pump waist [49

**38**(8), 1268–1270 (2013). [CrossRef] [PubMed]

*= –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.,*

_{p}*y*).

## 4. Conclusions

**19**(27), 26710–26724 (2011). [CrossRef] [PubMed]

^{1/2}times larger than the pump waist. We have found that for all pump waist sizes considered, the fundamental PSA eigenmode #0 has very good (99% or better) overlap with the fundamental Gaussian mode HG

_{00}of the compact basis. This holds promise for the detection of a large degree of vacuum squeezing from a traveling-wave PSA with use of a simple Gaussian-shaped local oscillator of a proper waist. Other low-order eigenmodes of the PSA (with gains well within the –3-dB range from the gain of the eigenmode #0) also have high overlaps with the corresponding HG or LG modes of the compact basis, whereas the higher-order eigenmodes may require several modes for their representation. The compact representation greatly simplifies the generation of the eigenmodes in the laboratory and has recently enabled experimental verification of the fundamental eigenmode for several pump waist sizes [49

**38**(8), 1268–1270 (2013). [CrossRef] [PubMed]

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]

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. 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]

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]

## Acknowledgments

## References and links

1. | C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields |

2. | D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett. |

3. | D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana |

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. |

5. | P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ |

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. |

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 |

8. | M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A |

9. | M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. |

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. |

11. | S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. |

12. | S.-K. Choi, M. Vasilyev, and P. Kumar, “Erratum: Noiseless optical amplification of images,” Phys. Rev. Lett. |

13. | A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett. |

14. | E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” J. Sel. Top. Quantum Electron. |

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. |

16. | P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” in |

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 |

18. | O.-K. Lim, G. Alon, Z. Dutton, S. Guha, M. Vasilyev, and P. Kumar, “Optical resolution enhancement with phase-sensitive preamplification,” in |

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. |

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. |

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 |

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 |

23. | B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A |

24. | B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express |

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. |

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. |

27. | M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express |

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 |

29. | A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A |

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 |

31. | R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A |

32. | C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett. |

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 |

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 |

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. |

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 |

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 |

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 |

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 |

40. | G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. |

41. | M. Annamalai, M. Vasilyev, N. Stelmakh, and P. Kumar, “Compact representation of spatial modes of phase-sensitive image amplifier,” in |

42. | W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A |

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 |

44. | D. Levandovsky, M. Vasilyev, and P. Kumar, “Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering,” Opt. Lett. |

45. | D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror,” Opt. Lett. |

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

47. | C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. |

48. | A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys. |

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. |

50. | M. Annamalai, M. Vasilyev, and P. Kumar, “Impact of phase-sensitive-amplifier's mode structure on amplified image quality,” |

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. |

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 |

53. | M. Annamalai and M. Vasilyev, “Phase-sensitive multimode parametric amplification in a parabolic-index waveguide,” IEEE Photonics Technol. Lett. |

54. | M. Vasilyev and P. Kumar, “Frequency up-conversion of quantum images,” Opt. Express |

**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

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- 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]
- D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana56, 281–285 (2001).
- 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]
- 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]
- 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]
- 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]
- M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A52(6), 4930–4940 (1995). [CrossRef] [PubMed]
- M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys.71(5), 1539–1589 (1999). [CrossRef]
- 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]
- S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett.83, 1938–1941 (1999).
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- 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]
- 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]
- P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” in 14th Coherent Laser Radar Conference, Snowmass, CO, July 2007.
- 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]
- 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.
- 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]
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- 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]
- 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]
- 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]
- 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]
- 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]
- M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express17(14), 11415–11425 (2009). [CrossRef] [PubMed]
- 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]
- A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A44(3), 2013–2022 (1991). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
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