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

Optics Express, Vol. 19, Issue 27, pp. 26710-26724 (2011)

http://dx.doi.org/10.1364/OE.19.026710

Acrobat PDF (3160 KB)

### Abstract

We develop a method for finding the number and shapes of the independently squeezed or amplified modes of a spatially-broadband, travelling-wave, frequency- and polarization-degenerate optical parametric amplifier in the general case of an elliptical Gaussian pump. The obtained results show that for tightly focused pump only one mode is squeezed, and this mode has a Gaussian TEM_{00} shape. For larger pump spot sizes that support multiple modes, the shapes of the most-amplified modes are close to Hermite- or Laguerre-Gaussian profiles. These results can be used to generate matched local oscillators for detecting high amounts of squeezing and to design parametric image amplifiers that introduce minimal distortion.

© 2011 OSA

## 1. Introduction

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

*temporal*bandwidth of fiber-based parametric amplifiers, has lead 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]

6. 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 of the parametric amplifiers enables their use as noiseless amplifiers of faint optical images, as was theoretically proposed in [7

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

9. 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]

13. 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]

14. P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” *the 14th Coherent Laser Radar Conference*, Snowmass, CO, July 2007. http://space.hsv.usra.edu/CLRC/presentations/Kumar.ppt

_{00}mode) of very high power (~1 kW per pixel of resolution) [17

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

*k*/

_{p}*L*)

^{1/2}, where

*k*is the pump propagation constant and

_{p}*L*is the nonlinear crystal’s length] [7

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

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

18. 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]

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

20. 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]

21. 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]

22. 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]

23. 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]

24. 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]

25. 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]

_{00}pump and then develop a procedure to find, for the first time to our knowledge, the orthogonal set of independently squeezed (or amplified) eigenmodes of such a PSA. We presented the preliminary results of this approach in a recent conference paper [26].

## 2. Theory of 2-D spatially-multimode PSA

### 2.1 Hermite-Gaussian mode representation

25. 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]

23. 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]

*e*intensity radii

*a*

_{0}

*and*

_{px}*a*

_{0}

*in*

_{py}*x*- and

*y*-dimensions, respectively. By expanding the PSA input over signal TEM

*HG modes having 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, we reduce the PSA propagation to a system of coupled ordinary differential equations for the HG-mode amplitudes

*A*=

_{mn}*X*+

_{mn}*iY*. After integrating this system of equations, we can obtain the PSA’s Green’s function and all the quantum correlators needed for finding the independently squeezed or amplified modes [27

_{mn}27. 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]

17. 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, 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]

28. Please note that the definition of *d*_{eff} in our prior work (Refs. 17, 18, and 27) is different from that in the present paper. The prior-work *d*_{eff} denotes the quantity that is more commonly known as the effective χ^{(2)} and equals 2*d*_{eff} in the present paper’s notations. As a result, the nonlinear paraxial wave equation in Refs. 17, 18, and 27 does not have the factor of 2 in front of *d*_{eff}. One fallout of this unfortunate choice of notation in our prior work is that Ref. 17 assumes effective χ^{(2)} = 8.7 pm/V for PPKTP crystal, which is about half of the actual value of that crystal’s nonlinearity, and the resulting pump powers listed in Refs. 17 and 26 are four times larger than those required for the same gain in a real PPKTP crystal. The present paper’s definitions rectify the previous inconsistencies.

*x*,

*y*), the intensity is given by

*i*taking value of either

*s*or

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

*= 2ω*

_{p}*. We expand the signal and pump beams over the HG basis with potentially unequal 1/*

_{s}*e*intensity radii in

*x*- and

*y*-dimensions:

*g*are defined as (β =

_{m}*x*or

*y*)with the orthogonality conditionand the Rayleigh range

*z*, 1/

_{R}*e*intensity radius

*a*(

*z*), Gouy phase shift θ(

*z*), and beam’s radius of curvature

*R*(

*z*), respectively, given by

*x*- and

*y*-radii for the signal expansion basis are chosen to be 2

^{1/2}times greater than those of the pump, which ensures that the signal and the pump have the same wavefront curvature and the same Rayleigh ranges

*z*=

_{Rx}*k*

_{p}a_{0}

_{px}^{2}and

*z*=

_{Ry}*k*

_{p}a_{0}

_{py}^{2}. The beam waists for both the pump and the signal HG basis occur at

*z*= 0.

*A*(

_{mn}*z*):whereΔ

*k*is the wavevector mismatch,

*P*

_{0}is the pump power and θ

*is the initial pump phase. The overlap integral*

_{p}*B*

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

*m*and

*m*′ has a closed-form expressionwhere the double factorial (

*m*+

*m*′ – 1)!! = 1 for

*m*+

*m*′ = 0. Let us list several important properties of the coupling matrix

**B**comprising coefficients

_{mm}_{′}*B*

_{mm}_{′}. First of all,i.e., propagation away from the center of the crystal reduces the magnitude of

*B*

_{mm}_{′}and introduces a phase shift due to the Gouy phase mismatch between the pump and the two signal modes. Second, the magnitude of

*B*

_{mm}_{′}(0) for large indices

*m*and

*m*′ can be approximated asi.e., for a fixed (

*m*+

*m*′), it exhibits a fast Gaussian decay as a function of (

*m*–

*m*′), whereas the decay along the main diagonal

*m*=

*m*′ is very slow and proportional to (

*m*+

*m*′)

^{–1/2}, as illustrated in Fig. 1 . Hence, the fast decay of

*B*

_{mm}_{′}versus (

*m*–

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

*m*+

*m*′) means that the maximum range of the amplified signal modes is determined not by the magnitude of the coupling coefficient, but by its Gouy phase mismatch [numerator in Eq. (9)] that leads to fast oscillations at large (

*m*+

*m*′), which limits the PSA gain. The signal evolution in the PSA in Eq. (6) is governed by the outer product of matrices

**B**and

_{mm}_{′}**B**, yielding a 4th-rank tensor in which only 25% of the elements are not zero.

_{nn}_{′}### 2.2 Laguerre-Gaussian mode representation

*a*

_{0}

*=*

_{px}*a*

_{0}

*=*

_{py}*a*

_{0}

*, it is more convenient to use the LG expansion [25*

^{p}25. 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]

*f*of radial index

_{pl}*p*and azimuthal index

*l*have the form

*r*, φ are the polar coordinates,and the orthogonality condition is

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

_{00}:

_{00}, with the signal modes LG

*and LG*

_{pl}

_{p}_{′}

_{l}_{′}is given byand the coupling coefficients between the modes LG

*and LG*

_{pl}

_{p}_{′}

_{l}_{′}are zero for

*l*≠ –

*l*′. The latter selection rule means that the circular symmetry of the pump reduces the rank of the coupling tensor in the PSA propagation equation from 4 to 3. Apart from this simplification, other properties of the coupling coefficients in Eq. (16) are similar to those in the HG case. Namely, these coefficients can be expressed asand for large value of (

*p*+

*p*′ + |

*l*|) they have an asymptotic dependencewhich imposes fast Gaussian decay when indices deviate from

*p*=

*p*′ and

*l*= 0 condition, as well as a slow

*p*

^{–1/2}decay when they satisfy this condition. We also note that, as a consequence of Eq. (15), the first diagonal element of the coupling tensor is the same in both LG and HG representations:

### 2.3 PSA Green’s functions and eigenmodes

*L*, i.e., the crystal input is at

*z*= –

*L*/2 and crystal output is at

*z*=

*L*/2. Following our approach to solving the PSA equation and finding its eigenmodes, presented in [27

27. 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]

*A*=

_{mn}*X*+

_{mn}*iY*in a matrix-vector formwhere

_{mn}*xy*-representation [29

29. E. Lantz and F. Devaux, “Numerical simulation of spatial fluctuations in parametric image amplification,” Eur. Phys. J. D **17**(1), 93–98 (2001). [CrossRef]

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

30. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A **13**(6), 2226–2243 (1976). [CrossRef]

*z*=

*L*/2). In order to obtain their shapes at the PSA input (at

*z*= –

*L*/2), one can invoke the reciprocity argument [23

23. 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]

*z*-reversal and conjugation of the coupling coefficients in Eq. (6) (for this, it is helpful to assume θ

*= –π/2 as the initial phase reference). The reciprocity principle states that the squeezing factor detected by a particular spatial profile of the local oscillator is the same as the classical PSA gain seen by the input signal whose profile is conjugate to that of the local oscillator. When applied to the local-oscillator shape that matches one of the PSA eigenmodes (and for the PSA eigenmode the gain and squeezing factors are the same), this principle means that the eigenmode profile at the PSA input is the conjugate of the eigenmode profile at the PSA output.*

_{p}## 3. Computational considerations

^{th}-rank coupling tensor

**B**by the 4

_{mm}_{′}B_{nn}_{′}^{th}-order Runge-Kutta method (RK-4). The two critical parameters of the computation are the maximum HG order

*m*

_{max}, at which the tensor is truncated, and the step size Δ

*z*over the propagation distance

*z*.

*m*

_{max}for each dimension, we recall that it is determined by the Gouy phase mismatch [the numerator in Eq. (9)]. When the mismatch reaches π, the coupling coefficient changes sign, i.e., switches from amplification to de-amplification. If this reversal occurs within the crystal, it means that the mode is outside of the crystal’s spatial bandwidth and can be ignored. Thus, at

*m*=

*m*′ =

*m*

_{max}the mismatch should reach π at the end of the crystal:which results in the truncation-order condition

*m*

_{max}-order HG function, which is approximately

*k*/

_{p}*L*)

^{1/2}[17

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

*L*= 2 cm,

*n*≈

_{s}*n*≈1.78, free-space λ

_{p}*= 780 nm, and*

_{p}*a*

_{0}

*= 100 μm, Eq. (27) yields*

^{p}*m*

_{max}= 22.5, which we have confirmed by observing the sufficient decay of the Green’s function versus (

*m*+

*m*′) at this index value. To provide additional accuracy margin, we use

*m*

_{max}= 32 for these PSA parameters. For a circular pump spot size, the

*x-*and

*y-*dimensions require 32 modes each.

*x-*and

*y-*dimensions, proportional to the square of the corresponding waist radii:

*a*

_{0}

*= 200 μm we need to use*

^{p}*m*

_{max}= 128, i.e.,

*m*

_{max}that is 4 times greater than that for

*a*

_{0}

*= 100 μm. Thus, for a 200-μm circular pump waist, Eq. (24) requires diagonalization of a tensor consisting of 2×2×(128×128)×(128×128) = 2*

^{p}^{30}elements (only quarter of which are non-zero), leading to multi-gigabyte memory requirements for the computer. This unfortunately fast scaling with pump waist size takes place because our choice of the HG expansion basis for the signal is not optimal. More specifically, in order to eliminate the beam curvature from the overlap integral, the signal waist of the expansion basis has been chosen to be 2

^{1/2}times greater than the pump’s, whereas all the interesting dynamics takes place on a smaller transverse scale that varies between the inverse spatial bandwidth of the crystal on the low end and the pump waist size on the high end. The number of independently amplified (or squeezed) PSA modes can be estimated by the product of the pump beam area and the square of the crystal’s spatial bandwidth [17

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

33. L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A **72**(1), 013806 (2005). [CrossRef]

*m*

^{x}_{max}

*n*

^{y}_{max}down to the value given by Eq. (29)] can be achieved by using a different HG expansion basis for the signal, that with a beam waist in the vicinity of the geometric average of the pump waist and the inverse spatial bandwidth of the crystal [34], as was previously discussed in the context of temporal modes [35

35. 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]

*z*-step, and have not been attempted in the present work.

*z*can be done by demanding both the coefficients

*B*

_{mm}_{′}and the solution to not change much over one step. The fastest variation in the coupling coefficients comes from the Gouy phase mismatch for the maximum order

*m*

_{max}, and we require this phase to change over one step by no more than π/150 ≈0.02 rad:which leads to the step size that is independent of the pump waist radius:

## 4. Results and discussion

*P*

_{0}and spot sizes

*a*

_{0}

*×*

_{px}*a*

_{0}

*for a PPKTP crystal of length*

_{py}*L*= 2 cm,

*d*

_{eff}= 8.7 pm/V,

*n*≈

_{s}*n*≈1.78, and signal wavelength of 1560 nm. For pump spot sizes of 100×100 μm

_{p}^{2}and smaller (100×50, 100×25, and 25×25 μm

^{2}) we have used 32×32 HG functions (i.e.,

*m*

^{x}_{max}=

*n*

^{y}_{max}= 32) as the signal expansion basis. For pump waists larger than 100 μm, we have increased the truncation order according to Eq. (28); e.g., for 400×100 μm

^{2}pump spot size we have used 512×32 expansion basis. For 100×100 μm

^{2}pump spot size, the Rayleigh range equals

*z*=

_{R}*k*(

_{p}*a*

_{0}

*)*

^{p}^{2}≈143 mm, i.e., it is ~7 times longer than the crystal.

^{2}) seems to support the amplification of only the fundamental mode.

^{2}case of the first set of results, i.e., the pump power is scaled proportionally to the beam area. The spectra shown in Fig. 3(a) indicate that, under the constant intensity condition, the gain of the fundamental eigenmode decreases with decreasing spot size. This point is further illustrated in Fig. 3(b) (red), which shows the gain, at constant pump intensity, as a function of the spot size

*a*

_{0}

*×*

_{px}*a*

_{0}

*. In the same Fig. 3(b), we also plot the pump powers (blue) needed to achieve the fundamental-mode gain of ~15, as in the first set of data. The fact that this dependence is not linear indicates that higher intensity is required to achieve the same gain for the fundamental eigenmode when the tightly focused pump size starts to approach the inverse spatial bandwidth of the crystal (*

_{py}*L*/π

*k*)

_{p}^{1/2}≈21 μm.

*x*,

*y*) representation of the first few computed eigenmodes (i.e., the spatial profiles of their intensity), whereas Figs. 6 –9 show the HG representation of these eigenmodes (i.e., |

*A*|

_{mn}^{2}). The inset in Fig. 4 also shows the number of modes with gains within 3 dB of the gain of the fundamental eigenmode [i.e., the number of modes above the dashed black line in Fig. 2(a)] as a function of the pump power, exhibiting roughly linear dependence with the inverse slope of ~690 W / mode, which can be considered as the

*power efficiency*of the PSA under consideration. The large required pump power justifies the use of the undepleted pump approximation in Eq. (1) for input signal powers up to ~1 W (i.e., as long as the amplified signal is much weaker than the pump).

_{00}-like single-lobe (

*x*,

*y*) shape, even though its HG representation consists of a superposition of many modes (except in the 25×25 μm

^{2}case, for which almost no HG modes other than the TEM

_{00}are present); 2) although at small pump spot sizes the higher-order eigenmodes have complicated (

*x*,

*y*) shapes, at spot sizes of 100×100 μm

^{2}and larger the eigenmodes’ amplitude and phase profiles, at least qualitatively, resemble those of the HG modes TEM

*(for elliptical pump waists) or the LG modes (for circular pump waists). We discuss these two observations separately below.*

_{mn}_{00}mode of our HG basis is 97% for the 25×25 μm

^{2}, 61% for the 100×25 μm

^{2}, 50% for the 100×50 μm

^{2}, 35% for the 100×100 μm

^{2}, 16.2% for the 800×50 μm

^{2}, 16.1% for the 400×100 μm

^{2}, and 15.7% for the 200×200 μm

^{2}pump spot sizes. Thus, even though the eigenmode #0 is a single-lobe in all 7 cases, it requires a significant number of HG modes for its representation in all cases except the 25×25 μm

^{2}pump spot size. Higher-order eigenmodes involve superpositions of even greater numbers of HG modes, as shown in Figs. 7 –9.

^{2}pump spot size, only the fundamental PSA eigenmode #0 sees any significant gain, and this mode has 97% overlap with the TEM

_{00}HG mode. This indicates that, in spite of gain-induced diffraction [20

20. 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]

*z*/

_{R}*L*≈0.45 in this case) the crystal’s limited spatial bandwidth forces the PSA to produce squeezed vacuum in a single, well-defined fundamental Gaussian mode. This fact greatly simplifies mode matching of the local oscillator for subsequent homodyne detection and opens the possibilities for generation and observation of significant squeezing factors in such a regime of the PSA operation.

## 5. Conclusion

*z*/

_{R}*L*< 1), the limited spatial bandwidth forces the PSA to produce squeezing in a single, well-defined TEM

_{00}Gaussian eigenmode; b) for larger pump spot sizes that support many PSA eigenmodes, the shapes of the most-amplified PSA modes are close to the first few Hermite-Gaussian (for elliptical pump waists) or Laguerre-Gaussian (for circular pump waists) modes. The exact amount of this resemblance (mode overlap) will be the subject of a separate study.

## 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–J. Phys. |

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, 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 |

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

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

9. | 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. |

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

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

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

13. | L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett. |

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

15. | 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 |

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

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

18. | M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt. |

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

20. | 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 |

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

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

23. | K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A |

24. | 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 |

25. | 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 |

26. | M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive image amplifier with elliptical Gaussian pump,” in |

27. | 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. |

28. | Please note that the definition of |

29. | E. Lantz and F. Devaux, “Numerical simulation of spatial fluctuations in parametric image amplification,” Eur. Phys. J. D |

30. | H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A |

31. | H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B |

32. | D. Levandovsky, “Quantum noise suppression using optical fibers,” Ph.D. thesis, Northwestern University, 1999. |

33. | L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A |

34. | M. Annamalai, M. Vasilyev, N. Stelmakh, and P. Kumar, “Compact Representation of Spatial Modes of Phase-Sensitive Image Amplifier,” in |

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

36. | J. H. Shapiro and A. Shakeel, “Optimizing homodyne detection of quadrature-noise squeezing by local-oscillator selection,” J. Opt. Soc. Am. B |

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

38. | 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 |

39. | A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. |

40. | C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett. |

41. | C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. |

**OCIS Codes**

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 22, 2011

Revised Manuscript: November 23, 2011

Manuscript Accepted: November 28, 2011

Published: December 14, 2011

**Citation**

Muthiah Annamalai, Nikolai Stelmakh, Michael Vasilyev, and Prem Kumar, "Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps," Opt. Express **19**, 26710-26724 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26710

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### References

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- 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]
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- 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]
- 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]
- 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]
- 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]
- M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive image amplifier with elliptical Gaussian pump,” in Laser Science, OSA Technical Digest (CD) (Optical Society of America, 2010), paper LTuB5.
- 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]
- Please note that the definition of deff in our prior work (Refs. 17, 18, and 27) is different from that in the present paper. The prior-work deff denotes the quantity that is more commonly known as the effective χ(2) and equals 2deff in the present paper’s notations. As a result, the nonlinear paraxial wave equation in Refs. 17, 18, and 27 does not have the factor of 2 in front of deff. One fallout of this unfortunate choice of notation in our prior work is that Ref. 17 assumes effective χ(2) = 8.7 pm/V for PPKTP crystal, which is about half of the actual value of that crystal’s nonlinearity, and the resulting pump powers listed in Refs. 17 and 26 are four times larger than those required for the same gain in a real PPKTP crystal. The present paper’s definitions rectify the previous inconsistencies.
- E. Lantz and F. Devaux, “Numerical simulation of spatial fluctuations in parametric image amplification,” Eur. Phys. J. D17(1), 93–98 (2001). [CrossRef]
- H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A13(6), 2226–2243 (1976). [CrossRef]
- H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B6, 309–313 (1989). [CrossRef]
- D. Levandovsky, “Quantum noise suppression using optical fibers,” Ph.D. thesis, Northwestern University, 1999.
- L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A72(1), 013806 (2005). [CrossRef]
- 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, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThB77.
- 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]
- J. H. Shapiro and A. Shakeel, “Optimizing homodyne detection of quadrature-noise squeezing by local-oscillator selection,” J. Opt. Soc. Am. B14(2), 232–249 (1997). [CrossRef]
- C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun.282(4), 583–593 (2009). [CrossRef]
- 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]
- A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys.63(5), 415–423 (1995). [CrossRef]
- C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett.84(23), 5304–5307 (2000). [CrossRef] [PubMed]
- C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett.92(12), 127903 (2004). [CrossRef] [PubMed]

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