## Phase-sensitive image amplification with elliptical Gaussian pump

Optics Express, Vol. 17, Issue 14, pp. 11415-11425 (2009)

http://dx.doi.org/10.1364/OE.17.011415

Acrobat PDF (2789 KB)

### Abstract

We numerically analyze phase-sensitive parametric amplification and de-amplification of a multimode field representing a multi-pixel text image. We optimize pumping configuration and demonstrate that ~10-dB gain is achievable with relatively moderate ~10-kW total pump peak power available from compact pump sources.

© 2009 Optical Society of America

## 1. Introduction

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

*third-order*nonlinear susceptibility

*χ*

^{(3)}in single-mode fibers have been gaining popularity in optical communications as noiseless amplifiers [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**, 984–986 (1999).
[CrossRef]

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**, 1954–1955 (1999).
[CrossRef]

5. H. P. Yuen, “Reduction Of Quantum Fluctuation And Suppression Of The Gordon-Haus Effect With Phase-Sensitive Linear-Amplifiers,” Opt. Lett. **17**, 73–75 (1992).
[CrossRef] [PubMed]

8. K. Croussore, I. Kim, Y. Han, C. Kim, G. Li, and S. Radic, “Demonstration of phase-regeneration of DPSK signals based on phase-sensitive amplification,” Opt. Express **13**, 3945–3950 (2005).
[CrossRef] [PubMed]

9. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express **13**, 7563–7571 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-19-7563.
[CrossRef] [PubMed]

9. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express **13**, 7563–7571 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-19-7563.
[CrossRef] [PubMed]

10. R. Tang, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “In-line phase-sensitive amplification of multi-channel CW signals based on frequency nondegenerate four-wave-mixing in fiber,” Opt. Express **16**, 9046–9053 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-9046.
[CrossRef] [PubMed]

*χ*

^{(2)}crystal inside a cavity [18

18. 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**, 685–699 (1998).
[CrossRef]

19. 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**, 4122–4134 (1999).
[CrossRef]

20. 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**, 846–854 (2001).
[CrossRef]

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

## 2. Theory of parametric image amplification

22. M. Kolobov, Ed., *Quantum Imaging*, Springer Verlag, New York, 2007.
[CrossRef]

*e*(

*r*⃗,

*t*)=

*E*(

*ρ⃗z*)

*ei*+ c.c., where

^{(kz-ωt)}*E*(

*ρ⃗,z*) is a slowly-varying field envelope,

*ρ*⃗ is a transverse vector with coordinates (

*x,y*), and the intensity is given by

*I*(

*ρ⃗,z*)=2

*εnc*|

*E*(

*ρ⃗,z*)|

^{2}. In the presence of a strong pump

*E*(

_{p}*ρ*⃗,

*z*) at frequency

*ω*, the signal electric field

_{p}*E*(

_{s}*ρ*⃗,

*z*) at frequency

*ω*is coupled to the idler electric field

_{s}*E*(

_{i}*ρ*⃗,

*z*) at frequency

*ω*=

_{i}*ω*-

_{p}*ω*through the following equation:

_{s}*k*=

*k*-

_{p}*k*, and the equation for the idler beam is obtained by interchanging subscripts

_{s}-k_{i}*s*and

*i*in Eq. (1). Equation (1) describes the traveling-wave OPA in paraxial approximation with a pump of arbitrary spatial profile.

*q*⃗) domain via the direct and inverse Fourier transforms

*E*=0), Eq. (1) is reduced to the paraxial Helmholtz equation, whose solution in the Fourier domain is given by

_{p}*Analytical solution 1: plane-wave pump*

23. A. Gavrielides, P. Peterson, and D. Cardimona, “Diffractive imaging in three-wave interactions,” J. Appl. Phys. **62**, 2640–2645 (1987).
[CrossRef]

*γ*is given by

*q*⃗+ and

*q*⃗- spatial-frequency components of the image, demonstrated in [24

24. M. L. Marable, S.-K. Choi, and P. Kumar, “Measurement of quantum-noise correlations in parametric image amplication,” Opt. Express **2**, 84–92 (1998), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-3-84.
[CrossRef] [PubMed]

*ikz*/2). This is also approximately true for the non-degenerate case if

*k*≈

_{s}*k*. For maximum phase-sensitive gain, the magnitudes of

_{i}*E*̃

*(*

_{s}*q*⃗,0) and

*Ẽ*(-

_{s}*q*⃗,0) should be the same. For a crystal of length

*L*, the optimum input signal phase is given by

*q*-dependent and, therefore, may not be easily realizable. However, for small or moderate values of

*κL*, optimum signal phase of Eq. (9) is

*z*

_{0}=

*L*/2 (middle of the nonlinear crystal). The minimum PSA gain (de-amplification) is the inverse of the gain in Eq. (10) and is obtained by shifting the signal phase by

*π*/2 from the optimum of Eq. (9).

*Analytical solution 2: short crystal with inhomogeneous pump*

*, defined in Eq. (8), is, in general, a complex parameter that depends on the coordinate*

_{s}*ρ*⃗ (if the pump is inhomogeneous); at the same time, due to the short crystal length, the

*z*-dependence of the pump is neglected. One can then introduce parameters µ and ν as

*k*

_{eff}=Δ

*k*is assumed, and θ

*and*

_{p}*γ*may vary as functions of

*ρ*⃗.

*Case considered in this paper: finite-length crystal with inhomogeneous pump*

19. 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**, 4122–4134 (1999).
[CrossRef]

20. 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**, 846–854 (2001).
[CrossRef]

## 3. Modeling results

*d*

_{eff}=8.7 pm/V and length

*L*=2.5 cm (typical values for a periodically-poled KTP crystal), pumped by the second harmonic of 1560-nm light. The image to be amplified is obtained by illuminating a mask with transparent letters by a plane wave. In the absence of gain, the image plane is located at the center of the crystal. The image size is ~470 µm×90 µm, with line thickness of ~10 µm. We slightly smoothen the edges of the letters to avoid aliasing in split-step computation. 5.2:1 aspect ratio of the image makes a strong case for using elliptical rather than circular Gaussian pump. After tedious optimization, we have found that the highest total-power PSA gain takes place for pump with 1/e intensity radii 440 µm×25 µm, and amounts to 11.2 dB at 10 kW pump power. The amplified image is shown in Figs. 1(c), 2(c), 2(d), and 2(e). The maximum output intensity (center letters) is 10.1 dB above the input. The intensity gradually rolls off at the sides, while still gaining more than 6 dB over the input everywhere, including the peripheral letters. The fact that the total power gain is higher than the local gain for most of the letters can be explained by the presence of a broad pedestal under the amplified letters [particularly evident as a bright oval in the center of Fig. 2(d)]. As we discuss in the next Section, this appears to be a manifestation of low-pass spatial frequency filtering by the PSA, even though this concept in strict sense is only valid in plane-wave-pump (spatially invariant) case. Despite this filtering and gain non-uniformity across the image field, the amplified text is clearly recognizable.

*k*=0 and the image located at the crystal’s center) is not optimal here and fine phase tuning leads to -55° optimum [Fig. 3(a)], improving the total-power PSD gain from 0.05 dB to -2.2 dB. In contrast, the PSA gain varies less with the input phase: tuning it from +45° (optimum for the plane-wave pump) to +35° increases the total-power gain by only 0.1 dB and the peak gain by only 0.2 dB (the data in Figs. 1 and 2 correspond to the optimum +35° signal phase). The phenomenon of the optimal phase shifting away from its plane-wave-pump-case value is similar to that noticed in [19

19. 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**, 4122–4134 (1999).
[CrossRef]

**60**, 4122–4134 (1999).
[CrossRef]

## 4. Discussion of amplified image resolution

*G*

_{PSA}-1) function, to be precise] for Δ

*k*=0 is given by

*q*. Assuming plane-wave-pump intensity equal to 25% of the peak Gaussian-pump intensity used in Section 3 (this is needed to match the peak image gains in the two cases, as shown below), we obtain from Eq. (16): Δ

*q*=26 rad/mm (i.e., 4.2 lines/mm) and Δρ≈1/

*q*=38 µm. These numbers are not very different from the exact values: -3-dB bandwidth Δ

*q*=25 rad/mm (i.e., 4.0 lines/mm) and -3-dB PSF radius Δρ=44 µm.

*a*and the effective pixel area π(Δρ)

_{0px}a_{0py}^{2}of the plane-wave-pump case, which yields between 5 and 8 effective pixels of resolution in our chosen example (depending on which of the two Δρ estimates above we use). Curiously enough, this order-of-magnitude-accurate value is 4–6 times lower than the observed number of amplified pixels in Fig. 1(c), estimated by counting the numbers of bright-dark line pairs in horizontal and vertical directions of the image, which yields at least 15×2=30 effective pixels of resolution.

*π*/2 phase jumps at spatial frequencies beyond the -3-dB bandwidth, contributed by the last term in Eq. (9) (we prefer the smooth phase profile because it results in smoother PSF). We also assume that a similar phase profile is applied to the amplified image at the PSA output to bring it into focus. Then the first PSF corresponds to

*G*

_{PSA}=(|µ|-

*iν*-1)

^{2}, while the second to

*G*

_{PSA}=(|µ|-

*iν*)

^{2}(“almost optimum gain” [25]). The intensities of the images processed by the former and the latter functions are shown in Figs. 4 and 5, respectively, after normalization by the peak intensity of the input image. The text in Fig. 4 is virtually unrecognizable (too few effective pixels of resolution), with the peak gain of 6.9 dB and the total power gain of 10.8 dB.

*G*

^{Fig.5}

_{PSA}(ρ⃗)

*G*] of Fig. 5 essentially equals to

*S*(ρ⃗) is the normalized intensity of the input image, varying from 0 to 1. Thus, the raised background in Fig. 5 represents the poorly resolved image of Fig. 4, given by G

_{in}^{Fig. 4}

_{PSA}(ρ⃗). The resolved letters come from the interference of G

^{Fig. 4}

_{PSA}(ρ⃗) and

*S*

_{in}(ρ⃗), rising above this background by

^{Fig. 5}

_{PSA}(ρ⃗)=10 yields

*G*

^{Fig. 5}

_{PSA}(ρ⃗)-G

^{Fig. 4}

_{PSA}(ρ⃗)=5.3. Hence, despite the small weight (max

*S*

_{in}(ρ⃗)=1) of the input signal compared to the poorly-resolved background, their constructive interference produces a swing in the output intensity that is 5.3 times greater than the input signal, leading to better than 50% contrast of the resolved letters in the output image.

**60**, 4122–4134 (1999).
[CrossRef]

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

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

15. A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett. **94**, 223603-1–223603-4 (2005).
[CrossRef]

## 5. Summary

## References and links

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

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. | H. P. Yuen, “Reduction Of Quantum Fluctuation And Suppression Of The Gordon-Haus Effect With Phase-Sensitive Linear-Amplifiers,” Opt. Lett. |

6. | J. N. Kutz, W. L. Kath, R.-D. Li, and P. Kumar, “Long-distance propagation in nonlinear optical fibers by using periodically spaced parametric amplifiers,” Opt. Lett. |

7. | G. D. Bartolini, D. K. Serkland, P. Kumar, and W. L. Kath, “All-optical storage of a picosecond-pulse packet using parametric amplification,” IEEE Photon. Technol. Lett. |

8. | K. Croussore, I. Kim, Y. Han, C. Kim, G. Li, and S. Radic, “Demonstration of phase-regeneration of DPSK signals based on phase-sensitive amplification,” Opt. Express |

9. | M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express |

10. | R. Tang, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “In-line phase-sensitive amplification of multi-channel CW signals based on frequency nondegenerate four-wave-mixing in fiber,” Opt. Express |

11. | E. Lantz and F. Devaux, “Parametric Amplification of Images: From Time Gating to Noiseless Amplification,” J. Sel. Top. Quantum Electron. |

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

13. | K. Wang, G. Yang, A. Gatti, and L. Lugiato, “Controlling the signal-to-noise ratio in optical parametric image amplification,” J. Opt. B: Quantum Semiclass. Opt. |

14. | S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless Optical Amplification of Images,” Phys. Rev. Lett.83, 1938–1941 (1999); erratum: |

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

16. | P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-Free Amplification: Towards Quantum Laser Radar,” |

17. | 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 |

18. | 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 |

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

20. | 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 |

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

22. | M. Kolobov, Ed., |

23. | A. Gavrielides, P. Peterson, and D. Cardimona, “Diffractive imaging in three-wave interactions,” J. Appl. Phys. |

24. | M. L. Marable, S.-K. Choi, and P. Kumar, “Measurement of quantum-noise correlations in parametric image amplication,” Opt. Express |

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

**OCIS Codes**

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

(280.3640) Remote sensing and sensors : Lidar

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 11, 2009

Revised Manuscript: May 30, 2009

Manuscript Accepted: June 18, 2009

Published: June 23, 2009

**Citation**

Michael Vasilyev, Nikolai Stelmakh, and Prem Kumar, "Phase-sensitive image amplification
with elliptical Gaussian pump," Opt. Express **17**, 11415-11425 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11415

Sort: Year | Journal | Reset

### References

- C. M. Caves, "Quantum limits on noise in linear amplifiers," Phys. Rev. D. 26, 1817-1839 (1982). [CrossRef]
- D. Levandovsky, M. Vasilyev, and P. Kumar, "Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier," Opt. Lett. 24, 984-986 (1999). [CrossRef]
- D. Levandovsky, M. Vasilyev, and P. Kumar, "Near-noiseless amplification of light by a phase-sensitive fibre amplifier," PRAMANA-J. Phys. 56, 281-285 (2001). [CrossRef]
- 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, 1954-1955 (1999). [CrossRef]
- H. P. Yuen, "Reduction Of Quantum Fluctuation And Suppression Of The Gordon-Haus Effect With Phase-Sensitive Linear-Amplifiers," Opt. Lett. 17, 73-75 (1992). [CrossRef] [PubMed]
- J. N. Kutz, W. L. Kath, R.-D. Li, and P. Kumar, "Long-distance propagation in nonlinear optical fibers by using periodically spaced parametric amplifiers," Opt. Lett. 18, 802-804 (1993). [CrossRef] [PubMed]
- G. D. Bartolini, D. K. Serkland, P. Kumar, and W. L. Kath, "All-optical storage of a picosecond-pulse packet using parametric amplification," IEEE Photon. Technol. Lett. 9, 1020-1022 (1997). [CrossRef]
- K. Croussore, I. Kim, Y. Han, C. Kim, G. Li, and S. Radic, "Demonstration of phase-regeneration of DPSK signals based on phase-sensitive amplification," Opt. Express 13, 3945-3950 (2005). [CrossRef] [PubMed]
- M. Vasilyev, "Distributed phase-sensitive amplification," Opt. Express 13, 7563-7571 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-19-7563. [CrossRef] [PubMed]
- R. Tang, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, "In-line phase-sensitive amplification of multi-channel CW signals based on frequency nondegenerate four-wave-mixing in fiber," Opt. Express 16, 9046-9053 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-9046. [CrossRef] [PubMed]
- E. Lantz and F. Devaux, "Parametric Amplification of Images: From Time Gating to Noiseless Amplification," J. Sel. Top. Quantum Electron. 14, 635-647 (2008). [CrossRef]
- M. Kolobov, "The spatial behavior of nonclassical light," Rev. Mod. Phys. 71, 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 Semiclass. Opt. 5, S535-S544 (2003). [CrossRef]
- S.-K. Choi, M. Vasilyev, and P. Kumar, "Noiseless Optical Amplification of Images," Phys. Rev. Lett. 83, 1938-1941 (1999); erratum: Ibid., 84, 1361-1361 (2000). [CrossRef]
- A. Mosset, F. Devaux, and E. Lantz, "Spatially noiseless optical amplification of images," Phys. Rev. Lett. 94, 223603-1-223603-4 (2005). [CrossRef]
- P. Kumar, V. Grigoryan, and M. Vasilyev, "Noise-Free Amplification: Towards Quantum Laser Radar," the 14th Coherent Laser Radar Conference, Snowmass, CO, July 2007.
- 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, 1564-1575 (1997). [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. B 66, 685-699 (1998). [CrossRef]
- 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, 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. B 18, 846-854 (2001). [CrossRef]
- E. Lantz and F. Devaux, "Numerical simulation of spatial fluctuations in parametric image amplification," Eur. Phys. J. D 17, 93-98 (2001). [CrossRef]
- M. Kolobov, ed., Quantum Imaging, (Springer Verlag, New York, 2007). [CrossRef]
- A. Gavrielides, P. Peterson, D. Cardimona, "Diffractive imaging in three-wave interactions," J. Appl. Phys. 62, 2640-2645 (1987). [CrossRef]
- M. L. Marable, S.-K. Choi, and P. Kumar, "Measurement of quantum-noise correlations in parametric image amplication," Opt. Express 2, 84-92 (1998), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-3-84. [CrossRef] [PubMed]
- M. Vasilyev, N. Stelmakh, and P. Kumar, "Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave-pump," to appear in J. Mod. Opt.

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