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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 6 — Mar. 12, 2012
  • pp: 6644–6656
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Frequency up-conversion of quantum images

Michael Vasilyev and Prem Kumar  »View Author Affiliations


Optics Express, Vol. 20, Issue 6, pp. 6644-6656 (2012)
http://dx.doi.org/10.1364/OE.20.006644


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Abstract

We develop the theory of frequency up-conversion of an arbitrary spatially-broadband quantum state (quantum image), deriving the analytical solutions for the cases of plane-wave pump and short crystal with arbitrary pump. By using an example of the quantum image imposed by the orbital angular momentum of the pump beam in spontaneous parametric down-conversion, we show that 99%-fidelity up-conversion of quantum images from an infrared wavelength to the visible wavelength can be obtained in periodically-poled lithium-niobate crystals at reasonable pump intensities well below the crystal damage threshold.

© 2012 OSA

1. Introduction

Light with quantum correlations among various spatial modes (a quantum image) can be used in quantum communications, computing, and measurements. From a practical point of view, it is convenient to produce such quantum images via optical parametric processes that are pumped by easily available high-power laser sources; e.g., those with wavelengths at 532 nm, 780 nm, 1064 nm, or 1550 nm. The resulting quantum images, however, fall into the infrared region of the electromagnetic spectrum, where arrayed detectors have very poor quantum efficiencies, making it difficult to observe and utilize the spatial quantum features of light. One potential way to overcome these low detection efficiencies is to use phase-sensitive parametric amplifiers, which are uniquely capable of amplifying input signals without degrading their signal-to-noise ratios (noiseless amplification) [1

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

]. This property, in combination with the wide temporal bandwidth of the parametric processes, has already been used to make noiseless amplifiers for fiber-optic communications [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

6. Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011). [CrossRef]

]. The wide spatial bandwidth of these processes also enables their use for noiseless amplification of classical images [7

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

11

11. 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]

]. It was also recently proposed to use a phase-sensitive parametric amplifier to noiselessly amplify some quantum images, in which multimode squeezed vacuum is used to suppress the loss of information at the periphery of a soft aperture of a spatially-broadband optical receiver [12

12. 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

]. While this approach allows one to extract the quantum features of the original input light even with low-efficiency detectors (e.g., those available in the infrared region of the electromagnetic spectrum), it can amplify only one quadrature of the input light, making this method unsuitable for quantum images in which the information is encoded in photon-number correlations.

An alternative method for overcoming the low infrared detection efficiency is the sum-frequency-generation (SFG) based up-conversion of images into the visible spectrum [13

13. J. E. Midwinter, “Image conversion from 1.6 μm to the visible in lithium niobate,” Appl. Phys. Lett. 12(3), 68–70 (1968). [CrossRef]

], where efficient arrayed detectors are readily available. This approach has already been demonstrated to provide high-fidelity up-conversion for the parametric twin-beam state [14

14. J. M. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. 68(14), 2153–2156 (1992). [CrossRef] [PubMed]

], single-photon states [15

15. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105(9), 093604 (2010). [CrossRef] [PubMed]

, 16

16. M. T. Rakher, L. Ma, O. T. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nat. Photonics 4(11), 786–791 (2010). [CrossRef]

], and quantum memory [17

17. A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6(11), 894–899 (2010). [CrossRef]

], all of which were spatially single-mode states. A theory of frequency-degenerate image up-conversion via second-harmonic generation was recently introduced in [18

18. P. Scotto, P. Colet, A. Jacobo, and M. S. Miguel, “Optical image processing in second-harmonic generation,” chapter 8 in Quantum Imaging, ed. by M. Kolobov, Springer Verlag, New York, 2007.

], but its practical implementation requires high input signal powers that restrict the types of quantum states that can be up-converted. Another mechanism of frequency conversion relying on quantum teleportation protocol has also been proposed [19

19. L. V. Magdenko, I. V. Sokolov, and M. I. Kolobov, “Quantum teleportation of optical images with frequency conversion,” Opt. Spectrosc. 103(1), 62–66 (2007). [CrossRef]

]. A quantum theory for simultaneous parametric amplification and up-conversion in the same crystal has recently been developed [20

20. A. S. Chirkin and E. V. Makeev, “Simultaneous phase-sensitive parametric amplification and up-conversion of an optical image,” J. Opt. B Quantum Semiclassical Opt. 7(12), S500–S506 (2005). [CrossRef]

, 21

21. E. V. Makeev and A. S. Chirkin, “Quantum fluctuations of parametrically amplified and up-converted optical images in consecutive wave interactions,” J. Russ. Laser Res. 27(5), 466–474 (2006). [CrossRef]

], but it discussed only the signal-to-noise-ratio evolution of an input coherent-state (i.e., classical) image. In this paper, we discuss the up-conversion of a spatially-multimode quantum state (i.e., a quantum image). In Section 2 we derive the analytical expressions for the up-conversion efficiencies in the case of a plane-wave pump or a short crystal length. In Section 3 we calculate the fidelities of the up-converted images and the pump powers required for up-conversion. We do so by using the example of a quantum image produced via spontaneous parametric down-conversion in a χ(2) crystal pumped by a beam carrying orbital angular momentum [22

22. G. A. Barbosa and H. H. Arnaut, “Twin photons with angular-momentum entanglement: phase matching,” Phys. Rev. A 65(5), 053801 (2002). [CrossRef]

, 23

23. A. R. Altman, K. G. Köprülü, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum imaging of nonlocal spatial correlations induced by orbital angular momentum,” Phys. Rev. Lett. 94(12), 123601 (2005). [CrossRef] [PubMed]

]. Our proposed up-conversion method is not limited only to that particular type of quantum image and can be equally well applied to other types, e.g., to the quantum images with sub-shot-noise spatial correlations [24

24. M. L. Marable, S.-K. Choi, and P. Kumar, “Measurement of quantum-noise correlations in parametric image amplification,” Opt. Express 2(3), 84–92 (1998). [CrossRef] [PubMed]

26

26. J. L. Blanchet, F. Devaux, L. Furfaro, and E. Lantz, “Measurement of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime,” Phys. Rev. Lett. 101(23), 233604 (2008). [CrossRef] [PubMed]

] and continuous-variable entanglement of many spatial modes [27

27. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008). [CrossRef] [PubMed]

] that have been recently generated. We conclude in Section 4 by summarizing the results.

2. Theory of quantum image conversion

κ(ρ,z)=4ω1ω3deff2n1n3c2E2(ρ,z).
(3)

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

a˜1,3(q,z)=a1,3(ρ,z)eiqρdρ,a1,3(ρ,z)=a˜1,3(q,z)eiqρdq(2π)2.
(4)

In the absence of the pump (E2 = 0), Eq. (1) is reduced to the paraxial Helmholtz equation, whose solution for the normalized fields in the Fourier domain is given by
a˜1,3(q,z)=a˜1,3(q,0)exp(iq22k1,3z),
(5)
where q=|q|. In general, similarly to the case of optical parametric amplification, Eq. (2) has to be solved numerically in order to account for coupling among the various spatial modes due to the effects akin to gain-induced diffraction [31

31. 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 (1997). [CrossRef]

]. Some semi-analytical methods mirroring those used in solving spatially-broadband parametric-amplifier equations [32

32. 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]

36

36. M. Annamalai, M. Vasilyev, N. Stelmakh, and P. Kumar, “Compact representation of spatial modes of phase-sensitive image amplifier,” the Conference on Lasers and Electro-Optics, Baltimore, MD, May 1–6, 2011, paper JThB77.

] can be applied here as well. In this paper, however, we will restrict ourselves to the two particular cases that admit analytical solutions: the case of a plane-wave pump and the case of a short crystal with arbitrary pump.

Analytical solution 1: plane-wave pump

With the plane-wave pump E2(ρ,z)=|E2|eiθ2=const., Eq. (1) still retains the shift-invariance of the original paraxial Helmholtz equation and hence it is easily solved in the Fourier domain:
a˜1(q,z)=eiq22k1z[t(q)a˜1(q,0)+r(q)a˜3(q,0)],a˜3(q,z)=eiq22k3z[r*(q)a˜1(q,0)+t*(q)a˜3(q,0)],
(6)
where the coefficients of the above unitary input-output transformation are
t(q)=(cosγziΔkeff2γsinγz)×exp(iΔkeff2z),r(q)=iκ*γsinγz×exp(iΔkeff2z),
(7)
the effective wavevector mismatch is
Δkeff=ΔkFC+q22(1k11k3),
(8)
and the parametric gain coefficient γ is given by

γ=|κ|2+Δkeff2/4.
(9)

Equation (6) couples the input and up-converted fields of the same spatial frequency q. This transformation is equivalent to that of a beam-splitter with reflectance (up-conversion efficiency)
R=|r|2=|κ|2γ2sin2γz
(10)
and transmittance

T=|t|2=1|κ|2γ2sin2γz=1R.
(11)

The evolution of the quantum field operators also obeys Eq. (6), which leads to quantum correlations between the same spatial-frequency components of the input and the up-converted images. The correlations in the ρ-domain, on the other hand, are not ideal, because the limited spatial bandwidth of the up-conversion process, determined by the wavevector mismatch of Eq. (8), results in image distortion. The spatial frequencies, for which the mismatch in (8) is nearly zero, can be up-converted with near-unity efficiency R, whereas those outside the up-conversion spatial bandwidth experience very low conversion efficiency. For input images whose quantum features are contained within the spatial bandwidth of the up-conversion process, the output quantum image will reproduce all the information from the input with high fidelity. And vice versa, for images with broader spatial frequency content, the non-unity conversion efficiency R is equivalent to loss and results in reduction of the fidelity of the up-converted state. We also note that the up-conversion takes place equally for both quadratures of the image, and no underlying symmetry of the image is required (this is in contrast to the phase-sensitive amplification process, which requires symmetry between the signal’s positive- and negative-spatial-frequency components, leading to a flat phase front of the image [7

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

,30

30. 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]

,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]

]).

Analytical solution 2: short crystal with inhomogeneous pump

For sufficiently short crystals, the diffraction terms in Eqs. (1) and (2) can be neglected, and the coupled-wave equations take the following form:

{a1(ρ,z)z=iκ*(ρ)a3(ρ,z)eiΔkFCz,a3(ρ,z)z=iκ(ρ)a1(ρ,z)eiΔkFCz.
(12)

Here κ, defined in (3), is, in general, a complex parameter that depends on the coordinate ρ (if the pump is inhomogeneous) and, at the same time, due to the short crystal length, the z-dependence of the pump is neglected. One can then introduce parameters t and r as
t(ρ,z)=(cosγziΔkFC2γsinγz)×exp(iΔkFC2z),r(ρ,z)=iκ*γsinγz×exp(iΔkFC2z),
(13)
where
γ(ρ)=|κ|2(ρ)ΔkFC2/4,|κ|2(ρ)=2ω1ω3deff2I2(ρ)ε0n1n3n2c3,
(14)
so that the solution takes the form of point-by-point (pixel-by-pixel) up-conversion with efficiency R = |r|2:
a1(ρ,z)=t(ρ,z)a1(ρ,0)+r(ρ,z)a3(ρ,0),a3(ρ,z)=r*(ρ,z)a1(ρ,0)+t*(ρ,z)a3(ρ,0),
(15)
which is equivalent to the beam-splitter transformation.

3. Example of quantum image conversion

In this section, we will provide an example of frequency conversion of the quantum image [22

22. G. A. Barbosa and H. H. Arnaut, “Twin photons with angular-momentum entanglement: phase matching,” Phys. Rev. A 65(5), 053801 (2002). [CrossRef]

, 23

23. A. R. Altman, K. G. Köprülü, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum imaging of nonlocal spatial correlations induced by orbital angular momentum,” Phys. Rev. Lett. 94(12), 123601 (2005). [CrossRef] [PubMed]

, 37

37. K. G. Köprülü, Y.-P. Huang, G. A. Barbosa, and P. Kumar, “Lossless single-photon shaping via heralding,” Opt. Lett. 36(9), 1674–1676 (2011). [CrossRef] [PubMed]

] produced via the frequency-degenerate spontaneous parametric down-conversion (SPDC) process of type-I with spatially-modulated z-propagating pump Ep(x, y) of frequency ωp = 2ω1 (here we use subscript “p” to distinguish the SPDC pump from the up-conversion pump denoted by subscript “2”). Assuming that the SPDC crystal length lSPDC is much smaller than the Rayleigh distance of the pump, the SPDC state can be approximated for low pump intensities (i.e., neglecting multiple-pair generation) by an unnormalized wavefunction
|ψ=|0+dqsdqiF(qs,qi)|1qs|1qi,
(16)
where qs and qi are the transverse components (spatial frequencies) of the signal and idler wavevectors ks and ki, respectively, and |1qs|1qi denotes a state having one photon with spatial frequency qs and one photon with spatial frequency qi, with all other modes in the vacuum state. F is the joint two-photon wavefunction in the momentum space given by
F(qs,qi)drEp(r)eiΔkr=dzeiΔkzzdxdyEp(x,y)ei(Δkxx+Δkyy)=lSPDCeiΔkzlSPDC/2sinc(ΔkzlSPDC2)E˜p(qs+qi),
(17)
where the z-integral gives the phase-mismatch factor, and the (x,y)-integral gives the spatial Fourier transform E˜p(q) of the transverse spatial profile of the pump, with
Δk=kpkski=z^Δkzqsqiz^(kpkski+qs22ks+qi22ki)qsqiz^qs2+qi22q022ksqsqi,
(18)
qs,i2=(kx)s,i2+(ky)s,i2 and q02=ks(kp2ks). In the last line of Eq. (18) we assumed kiks.

The peculiar properties of the state of Eq. (16), obtained with orbital-momentum-carrying pump, are illustrated in the far-field (transverse momentum space) diagram shown in Fig. 1(a)
Fig. 1 (a) Illustration of the phase matching, as well as the quantum image generation and up-conversion in the far field (transverse momentum space) when both the SPDC pump and the up-conversion pump are propagating in the same direction. (b) A schematic of the corresponding experimental setup. Inserts 2 and 3 in (a) show the quantum image function |F(qs,qi0)|2 for the cases of SPDC pumping by LG00 and LG02 modes, respectively, and insert 1 is the image obtained by scanning a single photon-counting detector across the output plane in the far field. All three inserts are equally applicable to the configuration shown in Fig. 2. SPDC–spontaneous parametric down-conversion, FC–frequency up-conversion.
, where pump profile of Laguerre-Gaussian mode LG02 with orbital momentum l = 2 is shown in purple as an example. For non-collinear phase matching (i.e., for kp ≠ 2ks), the SPDC photons are emitted in the pattern of a phase-matched cone [red ring of radius q0 in Fig. 1(a)]. The image obtained by direct detecting the SPDC simply reflects this ring shape [see insert 1 in Fig. 1(a)]. The coincidence count measurement, on the other hand, reveals the underlying quantum structure of this image. In particular, if an idler photon at spatial frequency qi0 is detected (heralded) by a small-area detector [yellow dot in Fig. 1(a)], then the probability amplitude of finding a signal photon with spatial frequency qs (or, in other words, the spatial profile of the resulting heralded single-photon mode) will be proportional to the function F(qs,qi0) from Eq. (17). The magnitude of this function is a product of the spatial profile of the phase-matching condition [red ring in Fig. 1(a)] and the far-field pattern of the SPDC pump centered at qi0 [checkered purple shape in Fig. 1(a), which has the profile of a ring if the pump field is given by LG02]. The resulting product is sketched by the two green dots in Fig. 1(a), while its actual calculated shape is shown by the insert 3 in the same Figure. If the pump is a simple fundamental Gaussian beam LG00, this product takes the form shown by the insert 2. Thus, the spatial mode of the heralded single-photon state of the signal depends on the pump profile. This property is only observable in coincidence counting and represents a quantum image.

Next, we study the up-conversion of the SPDC state of Eq. (16). Since the underlying quantum image is revealed in coincidence counting, it is convenient to use the same heralding procedure as that described above to check the fidelity of the up-converted quantum image (we use this procedure for convenience of calculation only; heralding is not required for up-conversion scheme itself to work). In order to up-convert the quantum image given by Eq. (16), we will assume that the output of the SPDC crystal is imaged into the input of the up-conversion crystal of length lFC by a 1:1 telescope and that the up-conversion pump is a plane wave (i.e., the conversion obeys the formulas for Analytical Solution 1 in Section 2), propagating in the same direction as the SPDC pump, as depicted in Fig. 1(b). The mode at frequency ω3 is in the vacuum state at the input of the up-converter. For a single-photon input at frequency ω1 and spatial frequency qs or qi, the input-output transformation (6) performs a beam-splitting procedure on signal and idler modes independently, which takes the following form in Schrödinger picture:
|Ψqs,i=t*(qs,i)|10qs,ir*(qs,i)|01qs,i,
(19)
wherein the transformation coefficients t(qs,i),r(qs,i) are given by Eq. (7) with z = lFC, and |10q (or |01q) refers to a state having one photon with spatial frequency q and frequency ω1 (or ω3, respectively), with all other modes in a vacuum state. Thus, the SPDC state (16) after up-conversion becomes:

|ψFC=|0+dqsdqiF(qs,qi)t*(qs)t*(qi)|10qs|10qi+dqsdqiF(qs,qi)r*(qs)r*(qi)|01qs|01qidqsdqiF(qs,qi)r*(qs)t*(qi)|01qs|10qidqsdqiF(qs,qi)t*(qs)r*(qi)|10qs|01qi.
(20)

In Eqs. (19) and (20) we have omitted the non-essential diffractive phase q2z/(2k3), which is equivalent to observing the far-field pattern in the Fourier plane of a lens.

Upon detection of the idler photon with transverse wavevector qi0, the up-converted state can be reduced to a heralded up-converted state

|ψHeralded,FC=dqsF(qs,qi0)[t*(qs)|10qsr*(qs)|01qs].
(21)

The fidelity η of the frequency up-converted quantum image can be quantified by the overlap of the heralded states obtained with ideal and non-ideal frequency conversion (the former corresponding to Eq. (21) with r = 1 and t = 0 and denoted simply “Heralded” below; it is also the same as the ideal up-conversion of the state heralded prior to frequency conversion):
η=|Heraldedψ|ψHeralded,FC|2Heraldedψ|ψHeraldedHeralded,FCψ|ψHeralded,FC=|dqs|F(qs,qi0)|2r*(qs)|2[dqs|F(qs,qi0)|2]2,
(22)
which is, in turn, determined by the overlap of the two-photon wavefunction F and the up-conversion coefficient r* in the transverse momentum space. Thus, to maximize the fidelity of the up-converted quantum image, one needs to ensure that r* is close to unity over the range of spatial frequencies where F has non-zero values. We note that the fidelity is the same regardless of whether the idler photon is detected before or after up-conversion (however, the probability of detection of the idler photon will be lower after imperfect up-conversion).

To achieve appreciable count rates, it makes sense to place the idler detector in the vicinity of qi0 = q0 (which corresponds to Δkz = 0 in the plane-wave pump case), so we will assume qi0=x^q0. Then, the up-conversion should be set up so that Δkeff = 0 takes place at q = qs = q0, i.e.,
ΔkFC=q022(1k11k3)<0,Δkeff=qs2q022(1k11k3),
(23)
and k1=ks. Thus, the phase-matched bands of both the up-conversion and the SPDC are centered at |qs|=q0, and to achieve the high fidelity we just need to ensure that the spatial bandwidth of the up-conversion [represented by the half-width of the blue ring in Fig. 1(a)] is greater than that of the SPDC [represented by the half-width of the red ring in Fig. 1(a)]. To estimate the former, we note that at |qs|=q0, r* = 1 for γlFC = |κ|lFC = π/2, i.e. lFC = π/(2|κ|). The first zero of r* in Eq. (7) takes place at |qs|=q0+ΔqcFC so that γlFC = π (i.e., Δkeff=π3/lFC), where the detuning from the center to the first zero quantifies the spatial bandwidth of the up-conversion:

ΔqcFC=π3q0lFCk1k3k3k1.
(24)

Similarly, the spatial bandwidth of the SPDC ΔqcSPDC can be estimated from the first zero of the sinc function in Eq. (17) by using Δkz from Eq. (18) with qi = q0 and |qs|=q0+ΔqcSPDC:

ΔqcSPDC=2πksq0lSPDC=ΔqcFC23lFCnslSPDCn1(1k1k3)=ΔqcFC2ω2lFCns3ω3lSPDCn1.
(25)

High fidelity is realized when ΔqcFC>ΔqcSPDC, i.e.,
lSPDClFC>2ω2ns3ω3n1,
(26)
where ns / n1 is the ratio of the signal-wave refractive indices in the SPDC and up-conversion crystals.

It is worth noting that one can also try an alternative configuration of the up-conversion pumping, where the up-conversion pump wavevector is collinear with the ki02qi0 direction, which corresponds to the direction of the signal SPDC photon in the plane-wave-pump approximation. The corresponding phase-matching diagram in the far field and the up-conversion setup are shown in Figs. 2(a)
Fig. 2 (a) Illustration of the phase matching, as well as the quantum image generation and up-conversion in the far field (transverse momentum space) when the up-conversion pump is propagating along the ki02qi0 direction. (b) A schematic of the corresponding experimental setup. For SPDC pumping by LG02, the quantum image function |F(qs,qi0)|2 is proportional to the product of the red and checkered-purple rings in (a), which is sketched as two green dots. The single- and coincidence-count images for SPDC pumping by LG00 and LG02 modes are still represented by inserts 1–3 in Fig. 1. SPDC–spontaneous parametric down-conversion, FC–frequency up-conversion.
and 2(b), respectively. The SPDC state is still the same as that in Fig. 1, but at the up-converted stage r*(qs) in Eq. (22) should be replaced by r*(q), where q=qs+qi0. Equation (23) should be replaced by

ΔkFC=0,Δkeff=q22(1k11k3).
(27)

In the calculations of the spatial bandwidth and the fidelity for the configurations of both Fig. 1 (up-conversion pump co-propagating with the SPDC pump) and Fig. 2 (up-conversion pump propagating in the direction of the SPDC signal), two dimensionless parameters are of great importance. The first is the ratio of the center spatial frequency q0 to the SPDC spatial bandwidth ΔqcSPDC from Eq. (25):

N=q0ΔqcSPDC=q02lSPDC2πks.
(29)

M=2πzRlSPDC.
(30)

The SPDC spatial bandwidth given by Eq. (25) is the spatial bandwidth in the so-called “bandpass” configuration of the optical parametric process, where the signal and idler beams are centered at non-zero spatial frequency q0. It is also helpful to calculate the spatial bandwidth (at the first zero of the sinc function) of the signal for the “lowpass” configuration (i.e., assuming qi = q0 = 0):
ΔqcLP-SPDC=4πkslSPDC,
(31)
which allows us to express the dimensionless parameters M and N as
M=a0pΔqcLP-SPDC,N=2(q0ΔqcLP-SPDC)2,
(32)
and

a0pq0=MN2.
(33)

The dependence of the up-conversion fidelity upon the pump’s orbital angular momentum l, the parameter M, and the ratio of the up-conversion to the SPDC crystal lengths lFC / lSPDC is shown in Fig. 3
Fig. 3 The fidelity of the up-converted quantum image as a function of the ratio between the lengths of the up-conversion and SPDC crystals. Black dash-dotted traces correspond to the configuration of Fig. 1, whereas colored solid and dashed traces correspond to the configuration of Fig. 2. The cases of orbital momenta l = 0, l = 2, and l = 4 correspond to the SPDC pumping by LG00, LG02, and LG04 modes, respectively. We do not show the black traces with M = 3, because they are almost indistinguishable from the black traces with M = 4.
for ω2 / ω3 = 2/3 and N = 10. The results indicate that high fidelity can be achieved in all cases by having a sufficiently short lFC. However, because shortening lFC requires increasing the up-conversion pump power, the plots show that the configuration of Fig. 2 is clearly preferable as it achieves higher fidelity for a given up-conversion crystal length.

Let us consider a specific case of using λ2 = λp = 780 nm, λ1 = 2λ2 = 1560 nm, and the up-converted wavelength λ3 = (1/λ1 + 1/λ2)–1 = 2λ2/3 = λ1/3 = 520 nm. Both SPDC and wavelength conversion can be implemented, for example, in PPLN crystals (ns = n1n2 = npn3 ≈2.14). The (one-sided) angle at which the signal beam emerges in the free space is θs = q0ns / ks. Assuming the parameters θs = 0.04 rad, M = 4, and N = 10 — which are similar to those in [23

23. A. R. Altman, K. G. Köprülü, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum imaging of nonlocal spatial correlations induced by orbital angular momentum,” Phys. Rev. Lett. 94(12), 123601 (2005). [CrossRef] [PubMed]

] (θs = 0.07 rad, M = 4, and N = 8.3), but at a different signal wavelength — we obtain lSPDC = N λs ns / θs2 = 20.9 mm and a0p = (λp lSPDC / np)1/2M/(2π) = 55.5 μm. Then, in order to achieve the fidelity η = 99% with the configuration of Fig. 2, one needs to employ the up-conversion crystals with lengths lFC = 16.7 mm, 6.26 mm, and 4.17 mm for the cases of SPDC pumping by the LG00, LG02, and LG04 modes, respectively. This corresponds to the peak up-conversion pump intensities of 2.4, 17.2, and 38.7 MW/cm2, respectively, where all three intensity numbers are well below the ~250-MW/cm2 optical damage threshold of the lithium-niobate crystal. In contrast to the case of Fig. 2, the configuration of Fig. 1 requires significantly shorter lengths lFC of 2.5 mm, 1.67 mm, and 1.46 mm to achieve the same fidelity, taking the up-conversion pump requirements to the intensity levels of 108, 241, and 318 MW/cm2, respectively, where only the first number is safely below the crystal damage threshold. While the configuration of Fig. 2 requires lower pump intensities, it also makes it more difficult to spatially separate the up-converted image from the up-conversion pump. For our chosen example, where the up-conversion pump’s near-infrared wavelength lies within the spectral response of silicon detectors, a strong dichroic filtering might be needed to prevent the pump beam from affecting the signal detection.

4. Summary

We have developed the theory of frequency up-conversion of an arbitrary spatially-broadband quantum state (quantum image), with analytical solutions derived for the cases of a) plane-wave pump and b) short crystal with arbitrary pump. Using an example of the quantum image imposed by the orbital angular momentum of the pump beam in spontaneous parametric down-conversion, we have obtained the up-conversion fidelities in two up-converter configurations: one where the SPDC and up-conversion pumps co-propagate in the respective crystals and the other where the up-conversion pump co-propagates with the signal beam in the up-conversion crystal. We have shown that the former configuration requires unreasonably high levels of pump intensity, whereas in the latter configuration, 99%-fidelity quantum image up-conversion from 1560 nm to 520 nm wavelength can be obtained in PPLN crystals at moderate up-conversion pump intensities well below the PPLN damage threshold. This indicates the practicality of the quantum image up-conversion with readily available solid-state or fiber-based pump sources.

Acknowledgments

This work was supported in part by DARPA’s Quantum Sensor Program under AFRL Contract No. FA8750-09-C-0194. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of DARPA or the U.S. Air Force.

References and links

1.

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

2.

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

3.

D. Levandovsky, M. Vasilyev, and P. Kumar, ““Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” PRAMANA–,” J. Phys. 56, 281 (2001).

4.

W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35(22), 1954 (1999). [CrossRef]

5.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundstrom, M. Sjodin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Gruner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010). [CrossRef]

6.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011). [CrossRef]

7.

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

8.

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

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]

10.

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

11.

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]

12.

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

13.

J. E. Midwinter, “Image conversion from 1.6 μm to the visible in lithium niobate,” Appl. Phys. Lett. 12(3), 68–70 (1968). [CrossRef]

14.

J. M. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett. 68(14), 2153–2156 (1992). [CrossRef] [PubMed]

15.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105(9), 093604 (2010). [CrossRef] [PubMed]

16.

M. T. Rakher, L. Ma, O. T. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nat. Photonics 4(11), 786–791 (2010). [CrossRef]

17.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6(11), 894–899 (2010). [CrossRef]

18.

P. Scotto, P. Colet, A. Jacobo, and M. S. Miguel, “Optical image processing in second-harmonic generation,” chapter 8 in Quantum Imaging, ed. by M. Kolobov, Springer Verlag, New York, 2007.

19.

L. V. Magdenko, I. V. Sokolov, and M. I. Kolobov, “Quantum teleportation of optical images with frequency conversion,” Opt. Spectrosc. 103(1), 62–66 (2007). [CrossRef]

20.

A. S. Chirkin and E. V. Makeev, “Simultaneous phase-sensitive parametric amplification and up-conversion of an optical image,” J. Opt. B Quantum Semiclassical Opt. 7(12), S500–S506 (2005). [CrossRef]

21.

E. V. Makeev and A. S. Chirkin, “Quantum fluctuations of parametrically amplified and up-converted optical images in consecutive wave interactions,” J. Russ. Laser Res. 27(5), 466–474 (2006). [CrossRef]

22.

G. A. Barbosa and H. H. Arnaut, “Twin photons with angular-momentum entanglement: phase matching,” Phys. Rev. A 65(5), 053801 (2002). [CrossRef]

23.

A. R. Altman, K. G. Köprülü, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum imaging of nonlocal spatial correlations induced by orbital angular momentum,” Phys. Rev. Lett. 94(12), 123601 (2005). [CrossRef] [PubMed]

24.

M. L. Marable, S.-K. Choi, and P. Kumar, “Measurement of quantum-noise correlations in parametric image amplification,” Opt. Express 2(3), 84–92 (1998). [CrossRef] [PubMed]

25.

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93(24), 243601 (2004). [CrossRef] [PubMed]

26.

J. L. Blanchet, F. Devaux, L. Furfaro, and E. Lantz, “Measurement of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime,” Phys. Rev. Lett. 101(23), 233604 (2008). [CrossRef] [PubMed]

27.

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008). [CrossRef] [PubMed]

28.

A. Gavrielides, P. Peterson, and D. Cardimona, “Diffractive imaging in three-wave interactions,” J. Appl. Phys. 62(7), 2640–2645 (1987). [CrossRef]

29.

M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express 17(14), 11415–11425 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11415. [CrossRef] [PubMed]

30.

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]

31.

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 (1997). [CrossRef]

32.

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]

33.

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]

34.

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 (2001). [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, M. Vasilyev, N. Stelmakh, and P. Kumar, “Compact representation of spatial modes of phase-sensitive image amplifier,” the Conference on Lasers and Electro-Optics, Baltimore, MD, May 1–6, 2011, paper JThB77.

37.

K. G. Köprülü, Y.-P. Huang, G. A. Barbosa, and P. Kumar, “Lossless single-photon shaping via heralding,” Opt. Lett. 36(9), 1674–1676 (2011). [CrossRef] [PubMed]

OCIS Codes
(190.7220) Nonlinear optics : Upconversion
(190.4975) Nonlinear optics : Parametric processes

ToC Category:
Nonlinear Optics

History
Original Manuscript: October 26, 2011
Revised Manuscript: January 19, 2012
Manuscript Accepted: February 14, 2012
Published: March 7, 2012

Citation
Michael Vasilyev and Prem Kumar, "Frequency up-conversion of quantum images," Opt. Express 20, 6644-6656 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6644


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References

  1. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields26(8), 1817–1839 (1982). [CrossRef]
  2. D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett.24(14), 984–986 (1999). [CrossRef] [PubMed]
  3. D. Levandovsky, M. Vasilyev, and P. Kumar, ““Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” PRAMANA–,” J. Phys.56, 281 (2001).
  4. W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett.35(22), 1954 (1999). [CrossRef]
  5. R. Slavík, F. Parmigiani, J. Kakande, C. Lundstrom, M. Sjodin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Gruner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010). [CrossRef]
  6. Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011). [CrossRef]
  7. M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys.71(5), 1539–1589 (1999). [CrossRef]
  8. S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett.83(10), 1938–1941 (1999). [CrossRef]
  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]
  10. A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett.94(22), 223603 (2005). [CrossRef] [PubMed]
  11. 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]
  12. 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
  13. J. E. Midwinter, “Image conversion from 1.6 μm to the visible in lithium niobate,” Appl. Phys. Lett.12(3), 68–70 (1968). [CrossRef]
  14. J. M. Huang and P. Kumar, “Observation of quantum frequency conversion,” Phys. Rev. Lett.68(14), 2153–2156 (1992). [CrossRef] [PubMed]
  15. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett.105(9), 093604 (2010). [CrossRef] [PubMed]
  16. M. T. Rakher, L. Ma, O. T. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nat. Photonics4(11), 786–791 (2010). [CrossRef]
  17. A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys.6(11), 894–899 (2010). [CrossRef]
  18. P. Scotto, P. Colet, A. Jacobo, and M. S. Miguel, “Optical image processing in second-harmonic generation,” chapter 8 in Quantum Imaging, ed. by M. Kolobov, Springer Verlag, New York, 2007.
  19. L. V. Magdenko, I. V. Sokolov, and M. I. Kolobov, “Quantum teleportation of optical images with frequency conversion,” Opt. Spectrosc.103(1), 62–66 (2007). [CrossRef]
  20. A. S. Chirkin and E. V. Makeev, “Simultaneous phase-sensitive parametric amplification and up-conversion of an optical image,” J. Opt. B Quantum Semiclassical Opt.7(12), S500–S506 (2005). [CrossRef]
  21. E. V. Makeev and A. S. Chirkin, “Quantum fluctuations of parametrically amplified and up-converted optical images in consecutive wave interactions,” J. Russ. Laser Res.27(5), 466–474 (2006). [CrossRef]
  22. G. A. Barbosa and H. H. Arnaut, “Twin photons with angular-momentum entanglement: phase matching,” Phys. Rev. A65(5), 053801 (2002). [CrossRef]
  23. A. R. Altman, K. G. Köprülü, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum imaging of nonlocal spatial correlations induced by orbital angular momentum,” Phys. Rev. Lett.94(12), 123601 (2005). [CrossRef] [PubMed]
  24. M. L. Marable, S.-K. Choi, and P. Kumar, “Measurement of quantum-noise correlations in parametric image amplification,” Opt. Express2(3), 84–92 (1998). [CrossRef] [PubMed]
  25. O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett.93(24), 243601 (2004). [CrossRef] [PubMed]
  26. J. L. Blanchet, F. Devaux, L. Furfaro, and E. Lantz, “Measurement of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime,” Phys. Rev. Lett.101(23), 233604 (2008). [CrossRef] [PubMed]
  27. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science321(5888), 544–547 (2008). [CrossRef] [PubMed]
  28. A. Gavrielides, P. Peterson, and D. Cardimona, “Diffractive imaging in three-wave interactions,” J. Appl. Phys.62(7), 2640–2645 (1987). [CrossRef]
  29. M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express17(14), 11415–11425 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11415 . [CrossRef] [PubMed]
  30. 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]
  31. 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 (1997). [CrossRef]
  32. 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]
  33. 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]
  34. 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 (2001). [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, M. Vasilyev, N. Stelmakh, and P. Kumar, “Compact representation of spatial modes of phase-sensitive image amplifier,” the Conference on Lasers and Electro-Optics, Baltimore, MD, May 1–6, 2011, paper JThB77.
  37. K. G. Köprülü, Y.-P. Huang, G. A. Barbosa, and P. Kumar, “Lossless single-photon shaping via heralding,” Opt. Lett.36(9), 1674–1676 (2011). [CrossRef] [PubMed]

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