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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 14 — Jul. 6, 2009
  • pp: 11415–11425
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Phase-sensitive image amplification with elliptical Gaussian pump

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


Optics Express, Vol. 17, Issue 14, pp. 11415-11425 (2009)
http://dx.doi.org/10.1364/OE.17.011415


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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

Phase-sensitive amplifiers (PSAs) have unique properties that allow them to break the 3-dB quantum limit of the optical-amplifier noise figure (NF) [1

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

], if the input signal phase is chosen for maximum gain, and to generate squeezed light, if the input phase is chosen for maximum attenuation. Such PSAs can be experimentally realized using optical parametric amplifiers (OPAs), operating either in a degenerate regime (where signal and idler modes are the same), or in a non-degenerate regime with both signal and idler modes simultaneously excited at the input. Over the last decade, PSAs based on 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

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]

], regenerators for timing and phase jitters [5

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

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]

], with capabilities for distributed [9

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]

] and multichannel [9

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

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]

] operation.

In this paper, we investigate the possibility of amplification of a multimode (multi-pixel) image with pump powers ≤10 kW typically available from the aforementioned compact and inexpensive DFB/EDFA combinations. For spatially inhomogeneous pumps, such as Gaussian beams, prior image amplification theory has considered transverse mode coupling in a short 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]

], where semi-analytical mode expansion over Laguerre-Gaussian polynomials was used. For travelling-wave PSAs, Refs. [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]

, 20

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]

] used a semi-analytical approach expanding the signal over radially symmetric Laguerre-Gaussian polynomials. While useful for finding best-squeezed modes, this approach in its present form cannot be applied to practical images because of its radial symmetry restrictions. In contrast to it, we permit any input signal and pump spatial profiles, and solve the PSA equations by direct numeric integration using split-step Fourier method in paraxial approximation with undepleted pump. This numerical method, commonly used in fiber optics, has already been proven helpful in the imaging PSA context for semi-classical modeling of noise [21

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

]. Here, we apply it to investigate the propagation of a realistic two-dimensional (2-D) multi-pixel image without radial symmetry through a PSA pumped by an elliptical Gaussian beam, which is a significant step beyond the approximations used in the prior imaging PSA studies. We show that, for typical nonlinear-crystal parameters, after optimization of the pump spot sizes in two dimensions, it is possible to achieve more than 6 dB of gain across an image with ~30 pixels, with a peak gain over 10 dB.

The rest of this paper is organized as follows: Section 2 describes the theoretical background of parametric amplification of images, Section 3 presents and Section 4 discusses the data from our modeling, and Section 5 summarizes the results.

2. Theory of parametric image amplification

A detailed theory of parametric amplification of multimode fields is summarized in a recently published book [22

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

]. Here, we will concentrate on OPA equations in undepleted-pump paraxial approximation with polarized (scalar) fields. We are looking for solutions in the form e(r⃗,t)=E(ρ⃗z)ei(kz-ωt)+ c.c., where E(ρ⃗,z) is a slowly-varying field envelope, ρ⃗ is a transverse vector with coordinates (x,y), and the intensity is given by I(ρ⃗,z)=2εnc|E(ρ⃗,z)|2. In the presence of a strong pump Ep(ρ⃗,z) at frequency ωp, the signal electric field Es(ρ⃗,z) at frequency ωs is coupled to the idler electric field Ei (ρ⃗,z) at frequency ωi=ωp-ωs through the following equation:

Esρzz=i2ksρ2Esρz+iωsdeffnscEpρzEi*ρzeikz,
(1)

where Δk=kp-ks-ki, and the equation for the idler beam is obtained by interchanging subscripts s and i in Eq. (1). Equation (1) describes the traveling-wave OPA in paraxial approximation with a pump of arbitrary spatial profile.

Let us define spatial-frequency (q⃗) domain via the direct and inverse Fourier transforms

E˜qz=Eρzeiq·ρdρ,Eρz=E˜qzeiq·ρdq(2π)2·
(2)

In the absence of the pump (Ep=0), Eq. (1) is reduced to the paraxial Helmholtz equation, whose solution in the Fourier domain is given by

E˜qz=E˜q0exp(iq22kz)·
(3)

Analytical solution 1: plane-wave pump

With the plane-wave pump Ep(ρ,z)=Epeiθp = const., Eq. (1) still retains the shift-invariance of the original paraxial Helmholtz equation and hence is easily solved in Fourier domain [23

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

]:

E˜sqz=μ(q)E˜s(q,0)+v(q)E˜i*(q,0),
(4)

where the coefficients of the input-output transformation are

μ(q)=(coshγziΔkeff2γsinhγz)×exp(iΔkeff2z)exp(iq22ksz),
v(q)=iκsγsinhγz×exp(iΔkeff2z)exp(iq22ksz),
(5)

the effective wavevector mismatch is

Δkeff=Δk+q22(1ks+1ki),
(6)

and the parametric gain coefficient γ is given by

γ=κ2Δkeff24,
(7)
κ2=κsκi*=ωsωideff2Iρ(2ε0nsninρc3),κs,i=ωs,ideffEρ(ns,ic).
(8)

The evolution of the quantum field operators also obeys Eq. (4), which leads to quantum correlations between q⃗+ and q⃗- 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]

]. Note that, in the degenerate case (signal and idler beam are the same), the product of the exponentials in Eqs. (5), containing the effective mismatch and diffraction phase terms, becomes simply exp(ikz/2). This is also approximately true for the non-degenerate case if kski. For maximum phase-sensitive gain, the magnitudes of Ẽs(q⃗,0) and s(-q⃗,0) should be the same. For a crystal of length L, the optimum input signal phase is given by

θsopt(q)=12{arg[E˜s(q˜,0)]+arg[E˜s(q,0)]}=θρ2+π4+12tan1[Δkeff2γtanhγL]+12arg[sinhγLγ].
(9)

The maximum PSA gain

GPSAopt=(μ+v)2
(10)

is achieved for optimum signal phase of Eq. (9) that is q-dependent and, therefore, may not be easily realizable. However, for small or moderate values of κL, optimum signal phase of Eq. (9) is

θsoptθp2+π4+ΔkL4+q24ksL,
(11)

i.e., the optimum (quadratic) spatial-frequency dependence can be straightforwardly obtained by placing the focus of the image at 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

For sufficiently short crystals, the diffraction term in Eq. (1) can be neglected, and the equation takes the following form:

Es(ρ,z)z=iκsEi*(ρ,z)eiΔkz.
(12)

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

μ(ρ,z)=(coshγzk2γsinhγz)×exp(iΔk2z),
v(ρ,z)=iκsγsihhγz×exp(iΔk2z),
(13)
γ(ρ)=κ2(ρ)Δk24,κ2(ρ)=κsκi*=ωsωideff2Ip(ρ)2ε0nsninpc3,
(14)

so that the solution takes the form of point-by-point (pixel-by-pixel) amplification:

Es(ρ,z)=μ(ρ,z)Es(ρ,0)+v(ρ,z)Ei*(ρ,0).
(15)

The optimum signal phase and the maximum PSA gain are still given by Eqs. (9) and (10), respectively, where Δk effk is assumed, and θp and γ may vary as functions of ρ⃗.

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

For the case of spatially-inhomogeneous pump (e.g., Gaussian beam) and non-negligible diffraction term, Eq. (1) has to be solved numerically. For circular-Gaussian pump and signal beams, the computation can be sped up by decomposing the signal over radially symmetric Laguerre-Gaussian modes [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]

, 20

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]

]. In the more general (non-radially-symmetric) case, 2-D Hermite-Gaussian modes need to be used, making the solution more computationally demanding. For our simulations, however, we do not use any mode decomposition, but instead employ numeric integration of Eq. (1) by split-step Fourier method (aka FFT-BPM), which enables us to use arbitrary pump and signal spatial profiles.

3. Modeling results

We concentrate on amplification of a 2-D image with unequal number of pixels in the two dimensions, which suggests using elliptical Gaussian spatial profile for the pump. First, we investigate the possibility of phase-sensitive amplification and de-amplification of several signal modes by a single elliptical Gaussian pump beam (Fig. 1a), where we observe very similar gains for the first three Hermite-Gaussian signal modes.

Next, we change the input signal phase by 90° and observe phase-sensitive de-amplification (PSD, Fig. 3). PSD gain is very sensitive to the input signal phase. In particular, -45° phase (optimum for the plane-wave-pump case with 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]

] for circular Gaussian pump. Figures 1(d) and 3 show that the de-amplified image is easily recognizable.

Fig. 1. (a) Phase-sensitive amplification and de-amplification gains for 3 signal modes (profiles shown on the left) with 1/e intensity radii a 0s=66 µm versus pump power; PSA crystal has d eff=6.5 pm/V and L=5.2 mm; signal wavelength is λs=1064 nm, elliptical pump waists are a0px=141 µm, a0py=47 µm. (b)–(d) Grayscale text images before (b) and after phase-sensitive amplification (c) and de-amplification (d) for PSA with d eff=8.7 pm/V, L=2.5 cm, λs=1560 nm, a0px=440 µm, a0py=25 µm, image size ~470 µm×90 µm, and 10 kW pump power. Linear grayscales of (b) and (d) are the same, whereas the grayscale in (c) is normalized by the maximum intensity of (c), which is ~10 dB higher than that in (b).

The observed classical PSA and PSD gain values can provide some indication of what to expect in the quantum case, where anti-squeezing and squeezing of noise quadratures will take place. In fact, Ref. [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]

] provides a reciprocity argument for OPA equations, which relates the observed classical total-power gains of a signal mode with arbitrary profile (the text “QUANTUM” in our case) to the degree of squeezing measurable in the mode of squeezed vacuum coming out of the same PSA and detected by a local oscillator with field profile conjugate to that of the signal. Thus, there exists such a conjugate mode whose anti-squeezing and squeezing factors at the output of our PSA are 11.2 dB and -2.2 dB, respectively.

Fig. 2. Intensity of the text image before (a, b) and after (c, d, e) phase-sensitive amplification. Vertical scale is linear and is normalized by the peak intensity of the input image. (b) and (d) are the top views of (a) and (c), respectively. (e) is the front-side view of (c).

4. Discussion of amplified image resolution

Let us estimate the resolution of our PSA from simple considerations based on plane-wave-pump theory. As we have shown in [25

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.

], the most accurate estimate of the spatial bandwidth of the PSA [-3-dB point of the (G PSA-1) function, to be precise] for Δk=0 is given by

Δq=(4ks2κ1nL)14,
(16)

and the -3-dB radius of the corresponding point-spread function (PSF) in space domain can be estimated as Δρ≈1/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): Δq=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 Δq=25 rad/mm (i.e., 4.0 lines/mm) and -3-dB PSF radius Δρ=44 µm.

For an inhomogeneous pump of a given spot size, the number of amplified pixels can be approximated by the ratio of the pump-beam area πa0pxa0py and the effective pixel area π(Δρ)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.

Fig. 3. (a) Dependence of total power gain and brightest pixel gain on input phase detuning from -45°. (c) De-amplified text image. (b) Top view of (c). (d) Front-side view of (c).

Let us illustrate this point by processing the same 470 µm×90 µm input image as that in Section 3 by two PSFs in plane-wave-pump approximation: one without the delta-function [i.e., corresponding to (GPSA(q)1) transfer function in spatial frequency domain], and the other with the delta-function [i.e., corresponding to Gpsa(q) transfer function in spatial frequency domain]. We assume that the input image has “almost optimum phase” [25

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.

], which differs from the optimum in Eq. (9) by lack of the 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|--1)2, while the second to G PSA=(|µ|-)2 (“almost optimum gain” [25

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.

]). 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.

In contrast, Figure 5, obtained using a complete PSF (including the delta-function), shows both the amplified image and its quality that are very similar to those obtained with elliptical Gaussian pump [Figs. 2 (c), (d), (e)]. The peak gain for the image is 10.1 dB and the total power gain is 12.4 dB, while the gain at zero spatial frequency is 17.5 dB. As we have mentioned above, to match the peak gain of 10.1 dB from Section 3 in this way, we use the plane-wave pump intensity equal to 25% of the Gaussian-pump peak intensity in Section 3. Thus, these semi-analytical plane-wave-pump results corroborate the numerical modeling results on multi-pixel amplification in the elliptical-Gaussian-pump case.

The key to the difference between Fig. 4 (which is supported by the low resolution estimates) and Fig. 5 (which is similar to elliptical-Gaussian-pump results showing good resolution) is the absence and presence of the delta-function in the PSFs used for Fig. 4 and Fig. 5, respectively. Thus, Fig. 5 can be interpreted as the constructive interference of the electric field of poorly resolved image of Fig. 4 with the electric field of the original well-resolved input image. In other words, the normalized intensity pattern [i.e., G Fig.5 PSA(ρ⃗) G] of Fig. 5 essentially equals to GPSAFig.5(ρ)=[GPSAFig.4(ρ)+Sin(ρ)]2,, where Sin(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 GFig. 4 PSA(ρ⃗). The resolved letters come from the interference of GFig. 4 PSA(ρ⃗) and S in(ρ⃗), rising above this background by

GPSAFig.5(ρ)GPSAFig.4(ρ)=2GPSAFig.4(ρ)Sin(ρ)+Sin(ρ)=2GPSAFig.5(ρ)Sin(ρ)Sin(ρ),
(17)

which for GFig. 5 PSA(ρ⃗)=10 yields G Fig. 5 PSA(ρ⃗)-GFig. 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.

Fig. 4. (a) Intensity of the text image processed by the modified PSF (without the delta-function), i.e. amplified by transfer function (|µ|--1) in the spatial frequency domain; (b) and (c) are the top and front-side views of (a). Vertical scale is linear and is normalized by the peak intensity of the input image.
Fig. 5. Same input image as that in Fig. 4, but processed by the complete PSF (with the delta-function), i.e. amplified by transfer function (|µ|-) in the spatial frequency domain; (b) and (c) are the top and front-side views of (a). Vertical scale is linear and is normalized by the peak intensity of the input image.

The signal-to-noise ratio (SNR) and the optimum detection of the resulting image promise to be interesting topics for future studies, as the image involves a multi-pixel signal mode and the noise with a correlation function spread over many pixels (such that the mode profile is different from the correlation-function profile). Even though the image contains poorly resolved background, it appears that from the reciprocity argument of Ref. [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]

], the SNR of the mode represented by the letters “QUANTUM” might be well preserved if the peak gain and the total power gain are sufficiently close, as is the case in Figs. 2 and 3. While the effect of the detector size on the observed SNR and noise-figure of the PSA was studied in Refs. [12

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

, 13

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

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

], the rigorous theory for the optimum detection of a multi-pixel image partially band-limited by the PSA has not been developed yet, as it involves decomposing the PSA output into independently squeezed modes (Karhunen-Loève expansion) and constructing a matched receiver for their detection. Even though the quantum noise properties of such an amplified image are not known yet, the 10-dB amplification shown in this paper can already improve the detected SNR if it is limited by the read noise and dark current of the conventional detector arrays.

5. Summary

We have numerically demonstrated that in an optical parametric amplifier with optimized pump, a text image with over 30 pixels can be phase-sensitively amplified by 6–10 dB with 10-kW peak pump power easily available from today’s compact sources. Under the same conditions, the image with 90°-shifted input phase can be attenuated by 2.2 dB. These results support the case for using phase-sensitive amplifiers for improvement of LADAR images.

This work was supported by the DARPA Quantum Sensors Program.

References and links

1.

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

2.

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

3.

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]

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]

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. 18, 802–804 (1993). [CrossRef] [PubMed]

7.

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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]

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]

11.

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]

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]

14.

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]

15.

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

16.

P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-Free Amplification: Towards Quantum Laser Radar,” the 14th Coherent Laser Radar Conference, Snowmass, CO, July 2007.

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 14, 1564–1575 (1997). [CrossRef]

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]

22.

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

23.

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

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]

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


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References

  1. C. M. Caves, "Quantum limits on noise in linear amplifiers," Phys. Rev. D. 26, 1817-1839 (1982). [CrossRef]
  2. D. Levandovsky, M. Vasilyev, and P. Kumar, "Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier," Opt. Lett. 24, 984-986 (1999). [CrossRef]
  3. 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]
  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]
  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. 18, 802-804 (1993). [CrossRef] [PubMed]
  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. 9, 1020-1022 (1997). [CrossRef]
  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]
  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]
  11. 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]
  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]
  14. 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]
  15. A. Mosset, F. Devaux, and E. Lantz, "Spatially noiseless optical amplification of images," Phys. Rev. Lett. 94, 223603-1-223603-4 (2005). [CrossRef]
  16. P. Kumar, V. Grigoryan, and M. Vasilyev, "Noise-Free Amplification: Towards Quantum Laser Radar," the 14th Coherent Laser Radar Conference, Snowmass, CO, July 2007.
  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 14, 1564-1575 (1997). [CrossRef]
  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]
  22. M. Kolobov, ed., Quantum Imaging, (Springer Verlag, New York, 2007). [CrossRef]
  23. A. Gavrielides, P. Peterson, D. Cardimona, "Diffractive imaging in three-wave interactions," J. Appl. Phys. 62, 2640-2645 (1987). [CrossRef]
  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]
  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.

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