Measurement of quantum-noise correlations

in parametric image amplification

Michael L. Marable, Sang-Kyung Choi, and Prem Kumar

Department of Electrical and Computer Engineering,Northwestern University, Evanston, Illinois 60208-3118

Phone: (847)491-4128; Fax: (847)491-4455

kumarp@nwu.edu

Abstract: We demonstrate quantum-noise correlations between thespatial frequencies of a parametrically amplified signal image and thegenerated conjugate (idler) image. Test images were amplified by anoptical parametric amplifier that can be operated either as a low-pass ora band-pass amplifier for spatial frequencies. Direct difference detectionof the signal and idler spatial frequencies at ±16 mm−1 resulted innoise that fell below the shot-noise level by '5 dB. Parametric-gain andphase-mismatch dependence of the observed quantum-noise reductionis in good agreement with the theory of a spatially-broadband opticalparametric amplifier.c©1998 Optical Society of AmericaOCIS codes: (270.6570) Squeezed states; (190.4970) Parametric oscillators andamplifiers; (190.4420) Nonlinear optics, transverse effects in; (110.4280) Noise inimaging systems.

References

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(C) 1998 OSA 2 February 1998 / Vol. 2, No. 3 / OPTICS EXPRESS 84#4000 - $10.00 US Received November 19, 1997; Revised January 29, 1998

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1. Introduction

Research involving squeezed states of light has evolved from demonstrations of novelgeneration schemes to applications where performance can be enhanced beyond theshot-noise limit. While applications in the temporal domain1−8 have been reported overthe years, there has been little attention devoted to the applications of squeezed light tophenomena in the spatial domain, such as optical imaging, diffraction, and holography.However, it has been proposed that spatially-broadband squeezed light can be used toimage faint objects with sensitivity exceeding the shot-noise limit.9,10 In this scheme,the phase object to be imaged is placed in one arm of a Mach-Zehnder interferometer,whose normally-unused input port is illuminated with spatially-broadband squeezedlight generated by a traveling-wave optical-parametric amplifier (OPA).11

We have recently begun to experimentally investigate the noise properties ofparametrically-amplified images.12 Since an OPA can generate correlated photons overa broad band of spatial frequencies, it is an ideal device for image amplification. Earlywork in this field concentrated on parametric up-conversion of infrared images to visi-ble wavelengths.13−15 More recently, parametric amplification of images16−19 has beenshown to have practical applications in time-gated image recovery.20 An example is theamplification of ballistic photons through a turbid medium for biomedical imaging.21−23

In this paper, we expand on our recently-reported first observation24 of quantum-noise correlations in parametric image amplification. An object is imaged into the OPA,where the real image is amplified in a twin-beams configuration. Quantum noise is meas-ured by means of direct difference detection of the twin beams.25 In particular, we showthat there are strong quantum correlations between the appropriate spatial frequencies

(C) 1998 OSA 2 February 1998 / Vol. 2, No. 3 / OPTICS EXPRESS 85#4000 - $10.00 US Received November 19, 1997; Revised January 29, 1998

of a parametrically-amplified image and its generated conjugate (idler) image. Portionsof the amplified signal and idler images that are at different spatial frequencies, however,are not correlated.

Previous experiments have produced squeezed light with uniform features in thespatial domain. Quantum-noise reduction of more than 6 dB has been achieved usingtwin beams generated by means of a traveling-wave OPA.25 Subtraction of signal andidler photocurrents in this twin-beams configuration shows quantum-noise reductionbelow the shot-noise limit as measured when an equivalent coherent-state photon fluxis incident on each of the two detectors. In the preceding experiment, the correlatedsignal and idler photons were created at a spatial frequency of zero.

In the case of an image, the signal input is composed of a spectrum of spatialfrequencies, which are amplified by the OPA within the limits of its spatial bandwidth.Both horizontal and vertical components of the spatial frequency must be included for atrue two-dimensional image. For simplicity, however, we consider only one-dimensionalimages. Since the spatial frequencies are the transverse components of the wavevectors,for a plane-wave pump beam with no transverse components, the phase-matching condi-tion in parametric down-conversion dictates that a signal photon at a spatial frequencyof q be co-generated with a conjugate idler photon at −q. Therefore, to observe quantumcorrelations in a parametrically-amplified image, we need to sample both the signal andthe idler photons at the same magnitude of the spatial frequency.

2. Theory

A simple theoretical description for quantum-noise reduction as a function of spatialfrequency can be constructed from the standard equations of optical-parametric ampli-fication. The equations governing a traveling-wave OPA are

bs = µas + νa†i , (1)

bi = µai + νa†s, (2)

where as and ai are the input and bs and bi are the output annihilation operators forthe signal and idler fields, respectively. In our experiment, there is no idler input fieldso that 〈ai〉 = 〈a†i 〉 = 0. After taking into account the effect of non-ideal detectionefficiency η, the expressions for the parametric gain and the quantum-noise reductionin the case of a collinear twin-beams experiment25 can be written as

g = |µ|2, (3)

R = 1− η +η

|µ|2 + |ν |2. (4)

In the non-collinear case, g and R have the same forms as in the collinear case,but the coupling coefficients µ and ν are given by26

µ(q) =

{cosh (h`) + i

∆keff`

2

[sinh (h`)

(h`)

]}exp

(−i

∆keff`

2

), (5)

ν(q) =

{iκ`

2

[sinh (h`)

(h`)

]}exp

(i∆keff`

2

), (6)

where h ≡ 12

√κ2 − (∆keff)

2, ∆keff ≡

(~kp − ~ks − ~ki

)·~iz is the effective phase mismatch

within the crystal along the nominal propagation direction z, κ is the parametric-gaincoefficient which is proportional to the intensity of the pump beam, and ` is the length

(C) 1998 OSA 2 February 1998 / Vol. 2, No. 3 / OPTICS EXPRESS 86#4000 - $10.00 US Received November 19, 1997; Revised January 29, 1998

of the χ(2)-nonlinear medium—a 5.21-mm long KTP crystal in our experiment. Opti-mum amplification occurs when ∆keff = 0. At the spatial frequency of q = 0, this phasematching condition is fulfilled for ∆k = kp−ks−ki = 0. However, at higher spatial fre-quencies, phase matching occurs only when ∆k 6= 0. Using the paraxial approximation,26

it can be shown that the effective phase mismatch for a spatial frequency q is given by

∆keff(q) = ∆k+q2

2

(1

ks+

1

ki

). (7)

Therefore, by making ∆k become progressively more negative, it is possible to bringincreasingly higher spatial frequencies into the phase match condition.

-30 -20 -10 0 10 20 30Spatial Frequency (mm-1)

-10

-8

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-4

-2

0

2

4

Noi

se R

educ

tion

(dB

) ∆k=0

Bare SignalAmp SignalIdlerNoise Reduction

Low-Pass OPA

-30 -20 -10 0 10 20 30Spatial Frequency (mm-1)

-10

-8

-6

-4

-2

0

2

4

Noi

se R

educ

tion

(dB

)

∆k= -0.95 rad/mm

Band-Pass OPA

Inte

nsity

(B

are=

1);

Inte

nsity

(B

are=

1);

(a)

(b)

Figure 1. Theoretical plots of the quantum-noise reduction as a function of spatialfrequency for the case of a low-pass amplifier with ∆k = 0 (a), band-pass amplifierwith ∆k = −0.95 rad/mm (b). Also shown are the amplified signal and generatedidler for a spatially narrowband bare (input) signal centered at ξ = 0 in (a), andξ = 16 mm−1 in (b).

Since µ and ν depend on q through ∆keff , we can evaluate the signal and idleroutputs, and the quantum-noise reduction, as a function of the spatial frequency for anygiven signal input. The simplest case is for an input signal with a small spread centered

(C) 1998 OSA 2 February 1998 / Vol. 2, No. 3 / OPTICS EXPRESS 87#4000 - $10.00 US Received November 19, 1997; Revised January 29, 1998

at q = 0, as shown in Fig. 1(a). We define ξ = q/2π so that the spatial frequency ξ isin units of mm−1. Here the phase matching condition is satisfied for ∆k = 0 and wehave chosen the OPA gain g ≡ |µ(0)|2 = 4. As expected, the signal and idler outputs aswell as the noise reduction are maximized for ξ = 0. From the noise-reduction curve, weestimate that the spatial bandwidth of our OPA is approximately 15 mm−1 (HWHM).In this configuration, the OPA functions as a low-pass amplifier for spatial frequencies.

Noise Measurements

Pulsed Nd:YAG Laser

KTP

CCD

CCD

HWPPs

Pump

(532 nm)

SignalInput

(1064 nm)Object

Image PlaneFourier Plane

DBS

Filter

(OPA)

Pump

KTP

ObjectIrisSignal Output(+16 mm-1)

-

Ph

oto

de

tect

ors

Idler(-16 mm-1)

Image Amplification

Figure 2. Experimental layout for parametric image amplification (top) andmeasurement of quantum-noise correlations (bottom).

In the experiments, we are interested in observing the quantum correlationsat a spatial frequency of 16 mm−1. From Fig. 1(a), it is evident that a signal input at+16 mm−1 (or −16 mm−1) will be amplified very little when ∆k = 0. For this spatialfrequency, the phase-matching condition is fulfilled for ∆k = −0.95 rad/mm for the pa-rameters of our KTP crystal. In practice, ∆k can be adjusted by rotating the azimuthalangle of the KTP crystal in the OPA so that the incidence angle of the signal changes

(C) 1998 OSA 2 February 1998 / Vol. 2, No. 3 / OPTICS EXPRESS 88#4000 - $10.00 US Received November 19, 1997; Revised January 29, 1998

while that of the pump remains fixed. Note that q remains unchanged in this processsince the polar angle is kept fixed. As shown in Fig. 1(b), when ∆k = −0.95 rad/mm,the signal at +16 mm−1 is amplified with an OPA gain of g = 4, and the conjugate idleris generated at −16 mm−1. Since the maximum gain and the noise reduction depend onthe ability to achieve optimum phase matching for the appropriately chosen ∆k, we canobtain a gain of 4 for the same value of κ as in the low-pass configuration (∆k = 0).Therefore, the noise reduction at ξ = ±16 mm−1 is not diminished from that at ξ = 0in the low-pass case, although the spatial-frequency bandwidth is somewhat reduced.

For values of ∆k < 0, the OPA acts like a band-pass amplifier, allowing us toamplify higher spatial frequencies more effectively. This feature of optical parametricamplification makes it possible to optimize phase matching, and hence quantum-noisecorrelations, for a single incident spatial frequency of our choosing. To observe cor-relations at many different spatial frequencies simultaneously, an OPA with a spatialbandwidth covering all the desired frequencies would be required. Also, a band-pass OPAcan be utilized for edge and contrast enhancement in classical parametric amplificationof images.16,23

baresignal

amplifiedsignal

idler

(a)

Bare Signal

0-16 +16 mm-1

(b)

Amplified Signal(Low-Pass OPA)

(c)

Amplified Signal(Band-Pass OPA)

(d)

Figure 3. (a) Bare signal, amplified signal, and idler patterns in the output imageplane. (b) Bare signal pattern in the output Fourier plane. (c) Amplified signalpattern in the output Fourier plane with the OPA aligned for low-pass amplification.(d) Same as in (c) but with the OPA aligned for band-pass amplification.

3. Experiments

The layout of our parametric image amplification experiment is depicted in Fig. 2. A5.21-mm long KTP crystal (the OPA) is pumped by a Q-switched, mode-locked, andfrequency-doubled Nd:YAG laser. The IR (1064 nm) signal input and the green (532 nm)pump are each p polarized (parallel to the crystal z-axis) for type II phase-matchingin the crystal. The object is placed in the signal-beam path in front of the OPA. Areal image of this object is formed in the center of the KTP crystal by a ×1 telescopeconsisting of two 10-cm focal-length lenses. The spatial frequencies of this image areamplified by the pump beam, which is made coincident with the signal beam usinga dichroic beamsplitter. The green pump is blocked after the crystal by using a filter

(C) 1998 OSA 2 February 1998 / Vol. 2, No. 3 / OPTICS EXPRESS 89#4000 - $10.00 US Received November 19, 1997; Revised January 29, 1998

which passes only the IR. CCD cameras are placed in the output image as well as theFourier planes of a 20-cm focal-length lens that is placed after the filter. The generatedidler is orthogonally polarized relative to the amplified signal because of type II phasematching. Therefore, a half-wave plate followed by a polarizing beamsplitter placed afterthe 20-cm lens allows us to observe either the signal or the idler output in the image aswell as the Fourier planes by simply rotating the half-wave plate.

1 2 3 4 5 6 7 8

OPA Gain

-6

-5

-4

-3

-2

-1

0

1

Noi

se R

educ

tion

(dB

)∆k= -0.95 rad/mm

TheoryData

(a)

-4 -3 -2 -1 0 1 2

∆k (rad/mm)

-6

-5

-4

-3

-2

-1

0

1

Noi

se R

educ

tion

(dB

)

TheoryData

(b)

Figure 4. Noise reduction as a function of (a) OPA gain and (b) phase mismatch∆k.

For parametric image amplification, we used a negative test pattern of threevertical lines with a uniform spacing of 62.5µm (16 lines/mm). The horizontal Fouriertransform of this object consists of three main peaks at ξ = 0,±16 mm−1 with twosmaller peaks in between at ξ = ±8 mm−1. As recorded in the output image plane, realimages of the bare signal (i.e., with the pump turned off), the amplified signal, and thegenerated idler are shown in Fig. 3(a) for an OPA gain of'1.2. The transverse pattern ofthe bare signal as recorded in the output Fourier plane is shown in Fig. 3(b). Figure 3(c)shows the transverse pattern of the amplified signal in the output Fourier plane when theOPA was aligned in the low-pass configuration (∆k = 0) and the pump power adjusted

(C) 1998 OSA 2 February 1998 / Vol. 2, No. 3 / OPTICS EXPRESS 90#4000 - $10.00 US Received November 19, 1997; Revised January 29, 1998

for a gain of '4. As shown, the central peak (ξ = 0) was strongly amplified with littleamplification occurring at the side peaks (ξ = ±16 mm−1). Transverse pattern in theoutput Fourier plane for band-pass amplification with ∆k = −0.95 rad/mm is shown inFig. 3(d). Here, the pump power was the same as in Fig. 3(c) and the OPA was alignedfor maximum amplification of the two side peaks at ±16 mm−1. These results comparefavorably with the theoretical predictions presented in Fig. 1 above.

For the measurement of quantum correlations, our goal was to observe noisereduction at a single spatial frequency. First, the OPA was optimized for maximum gainby aligning the signal and idler patterns to be simultaneously coincident in both thereal-image and the Fourier planes. By placing an iris in the Fourier plane that is in frontof the OPA (halfway in between the two lenses of the ×1 telescope), we blocked allspatial-frequency components of the input signal pattern except the peak centered atξ = +16 mm−1. Thus the input signal had a well defined spatial frequency. The OPA wasadjusted for maximum gain at ±16 mm−1, corresponding to an azimuthal rotation ofthe KTP crystal of about 0.85◦ from the angle for phase matching at ξ = 0. In this waythe input signal Fourier component was band-pass amplified, and a conjugate Fouriercomponent at ξ = −16 mm−1 of the idler beam was generated. Since the amplifiedsignal Fourier component at +16 mm−1 exits the OPA at an angle of 2× 17 = 34 mradwith respect to the idler Fourier component at −16 mm−1, it was easy to separate thetwo with use of plane mirrors. The mirror that sent the beams to the CCD cameraswas removed (cf. Fig. 2), and each beam was focused onto a separate photodetectorlocated in the far-field (i.e., in the Fourier plane). Therefore, one detector saw theamplified signal Fourier component at +16 mm−1, while the other detected the idlercomponent at −16 mm−1. Twin-beams type of noise measurements were made usingdirect difference detection, similar to those described in Ref. 25. Noise power of thedifference photocurrent was measured at 27 MHz with a 3 MHz resolution bandwidth.

Figure 4(a) shows the quantum-noise reduction below the shot-noise level forthe detected signal and idler Fourier components as a function of the signal gain. ForOPA gains below ' 4, the data is in good agreement with the theory (solid curve), oncea detection quantum efficiency of 0.76 is taken into account. We are in the process ofinvestigating the reasons for the discrepancy between the theory and the experiment forOPA gains above' 4. We have also measured the quantum-noise reduction as a functionof the phase mismatch ∆k for various values of the OPA gain. Results from one set ofdata for an OPA gain of ' 3.9 are shown in Fig. 4(b). As pointed out previously, ∆k wasvaried by rotating the azimuthal angle of the KTP crystal in the OPA. The micrometerreadings on the KTP rotation stage were calibrated and converted into units of ∆k forcomparison with the theory. As shown, the experimental data are once again in goodagreement with the theory (solid curve), once the detection quantum efficiency is takeninto account.

4. Conclusion

In conclusion, we have demonstrated quantum-noise reduction of almost 5 dB in directdifference detection of the correlated spatial frequencies of parametrically-amplified sig-nal and idler images. Test images were amplified by a spatially-broadband OPA that canbe operated either as a low-pass or a band-pass amplifier. Parametric-gain and phase-mismatch dependence of the observed quantum-noise reduction is in good agreementwith the theory of a spatially-broadband optical parametric amplifier. The observationof such a large quantum-noise reduction suggests that this amplifier can be configuredto produce spatially-broadband squeezed light with a bandwidth that is on the orderof 10 mm−1, thus making it a potential squeezer for enhancing the imaging of faintobjects.9,10 In order to image two-dimensional objects, it would be necessary to place a

(C) 1998 OSA 2 February 1998 / Vol. 2, No. 3 / OPTICS EXPRESS 91#4000 - $10.00 US Received November 19, 1997; Revised January 29, 1998

pair of two-dimensional detector arrays in the Fourier plane, match the pair of detectorsat each spatial frequency to subtract the corresponding photocurrents, and process theresulting Fourier image to recover the real image. The spatial resolution of the finalimage will depend on the resolution of the detector arrays placed in the Fourier plane.Future experiments will explore this approach, using a pair of one-dimensional detectorarrays to study the imaging of one-dimensional objects.

This work was supported in part by the U. S. Office of Naval Research. We aregrateful to M. Vasilyev for useful discussions.

(C) 1998 OSA 2 February 1998 / Vol. 2, No. 3 / OPTICS EXPRESS 92#4000 - $10.00 US Received November 19, 1997; Revised January 29, 1998