IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 14, NO. 3, MAY/JUNE 2008
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Parametric Amplification of Images: From Time Gating to Noiseless Amplification Eric Lantz and Fabrice Devaux (Invited Paper)
Abstract—Image parametric amplification and its applications are reviewed in this paper. Phase matching conditions allow the resolution to be determined and spatial frequency filtering to be performed. The phase conjugate idler image can be used to cancel aberrations. Then, time gating properties are applied to ultrafast imaging, imaging through diffusing media, and lifetime fluorescence imaging. Degenerate schemes permit polychromatic amplification as well as phase-sensitive amplification, which is proved to be noiseless with respect to spatial fluctuations at the quantum level. Finally, it is shown that the spatial fluctuations of the signal and the idler are subshot noise correlated. Index Terms—Image processing, noise measurement, nonlinear optics, parametric amplifiers, phase matching, quantum optics.
Fig. 1.
Schematic setup. (After [12].)
II. PHASE MATCHING AND RESOLUTION A. Amplification of a Monochromatic Image
I. INTRODUCTION INCE its early stages, nonlinear optics has been applied to image processing to obtain infrared-to-visible parametric up-conversion [1], amplification in photorefractive materials [2], or visualization of ultrafast phenomena with Kerr gates [3]. More recently, power increase of current picosecond lasers enabled high-gain parametric amplification in only one pass through a birefringent crystal. Hence, parametric amplification of images has become possible, provided that phase matching could be fulfilled. This paper aims to summarize the different schemes that have been proposed, the underlying theory and the applications. Section II describes specific phase matching conditions that must be fulfilled for the amplification of an image and applications to image restoration by phase conjugation. The amplification of a polychromatic image is also addressed. Section III describes the use of parametric image amplification as an ultrafast optical gate. Such a scheme is applied to imaging through diffusing media. Sections IV and V deal with quantum-noise-limited images. Section IV is devoted to noiseless image amplification while Section V reviews properties of images formed by parametric fluorescence, i.e., without input image.
S
Manuscript received November 6, 2007; revised November 26, 2007. The authors are with the Institut Femto-ST Unit´e Mixtes de Recherche (UMR) 6174 Centre National de la Recherche Scientifique (CNRS) Laboratoire d’Optique P. M. Duffieux, Universit´e de Franche-Comt´e, 25030 Besanc¸on, France (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTQE.2008.918650
Consider a partially transparent object illuminated by a monochromatic plane wave. The scattered light forming the signal can be considered as a superposition of plane waves, each plane wave being associated by the laws of Fourier optics to a spatial frequency of the object. The pump beam, as well collimated and monochromatic as possible, is superimposed with the signal in a nonlinear birefringent crystal. The formation of an amplified image can be obtained in (at least) three different ways. 1) No imaging system is used and the image is formed with the idler wave, that is phase-conjugate of the signal wave [4]–[7]. 2) The crystal is placed in the Fourier plane of the imaging system [8]–[10]. The range of amplified spatial frequencies is then determined in this plane by the transverse cross section of either the crystal or the pump beam (the latter situation occurs if the diameter of the pump beam is not large enough to ensure uniform illumination of the crystal). On the other hand, the field-of-view is determined by the angular acceptance that can be derived from the phase matching conditions. 3) The crystal is placed in a first image plane and a second image is formed on the detector [11]–[14]. The field-ofview is determined by the lateral size of the crystal (or of the pump beam), while the amplified spatial frequencies are determined by the phase matching conditions. We consider in the following this latter scheme that corresponds to the setup used in [11] and [12] and shown in Fig. 1. Infrared, 50 ps pulses at 1.064 µm are delivered by a Q-switched mode-locked Nd:YAG laser. The radiation is partially frequency-doubled in a potassium dihydrogen phosphate (KDP) crystal. The remaining infrared light is separated from the green light by a dichro¨ıc mirror and is then attenuated and horizontally polarized. This beam illuminates a transparency
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(the object), which is imaged at the input of the KTP crystal. The frequency-doubled pulse is superimposed in the crystal by means of a delay line. A second polarizer, placed before the camera, selects the idler, cross-polarized with the signal in a type II interaction. The charge-coupled device (CCD) camera is placed either in the image or the Fourier plane. The basic equations of parametric amplification [15] give the amplitude of the signal wave As after the traversal of a crystal of thickness L: j ∆K j (∆ K L /2) sinh (bL) = As (0) cosh (bL) + As (L)e 2b g − j A∗i (0) sinh (bL) 2b with b=
1 2
g 2 − (∆K)2 ,
K p −K s − K i ∆K=
(1)
where g is a gain term proportional to the pump amplitude Ap propagating along z. The idler wave Ai obeys the same equation, with i and s inverted. To amplify a signal forming directed along z an image, the phase mismatch vector ∆K, [16], must conserve a small modulus for the widest range of spatial frequencies, each one being associated by the laws of s directed along the angle ϕs . The Fourier optics to a vector K best resolution is obtained if phase matching is noncritical with respect to a rotation of the signal wave vector, compensated at first order by an opposite rotation of the idler of the signal wave [17], [18]: s i ∂K ∂K + = 0. ∂ϕs ∂ϕi
with the idler wave in order to obtain a true bandpass filter with zero transmission at the zero spatial frequency (Fig. 2).
(2) B. Image Restoration by Phase Conjugation
Hence, the phase mismatch vector includes only second-order terms: 1 ∂ 2 ns ns λi 1 ∂ 2 ni πns 2 ∆ϕs 1 − + 1− , |∆K| = λs ns ∂ϕ2s ni λs ni ∂ϕ2i ∆ϕs = νs λs
Fig. 2. (a) Domain of quasi-phase-matching versus the pump and the signal angles: a rotation of the crystal corresponds to a translation from a low-pass amplifier (point A) to a bandpass amplifier (point B). (b) Corresponding phase matching domains versus both signal angles. (c) Experimental spatial frequency spectrum of a random plate in the B configuration. (d) Edge enhancement of the number 4. (After [12].)
(3)
where νs is the considered spatial frequency. If we define a spatial cutoff frequency as corresponding to a total gain equal to half the gain for perfect phase matching, (1) and (3) mean that this spatial frequency, proportional to the inverse of the size of the resolution cell (or coherence area), evolves as L−1/4 for a gain g much larger than the phase mismatch. For a 5-mm-long type II KTP crystal, the phase matching range of 20 mrad corresponds to a resolution of about 60 µm in the crystal, for the 1.064 µm wavelength of a Nd:YAG laser. While the field-of-view and the resolution on the object depend on the magnification of the optical system, the resolution on the crystal and the number of resolved points in the image remain constant. A simple rotation of the crystal yields a transition from amplification of the low spatial frequencies to amplification of higher spatial frequencies [19]. In this configuration, phase matching acts as a coherent spatial filter, permitting basic image processing operations, like edges enhancement, by forming the image
With no imaging system, the idler is phase conjugate with the signal, but propagates in the forward direction. Hence, it forms an image of the object plane in a conjugate plane, symmetrically located with respect to the crystal, while the amplified signal still gives a far-field representation of the diffracted wave. The resolution is determined by the smallest value between the phase matching aperture and the solid angle occupied by the crystal (or by the pump beam) seen from the object [6]. The most promising application of the optical phase conjugation is to correct deterministic aberrations. χ(2) phase conjugation can be used for this purpose [20], like usual optical phase conjugation in χ(3) media. In Fig. 3, the signal diffracted by a transparency crosses an aberrant plate before to be amplified in a KTP crystal. The output face is used as a mirror (R ∼ 7%) and is exactly turned toward the normal direction of propagation of the pump plane wave. The reflected waves propagate back through the aberrant plate. While the signal wave (Σs ) is again degraded by the backward travel through the aberrant medium, distortions of the idler wave (Σi ) are compensated and the initial diffracted wave is restored. A beam splitter directs the counterpropagating waves on a single-shot CCD camera placed in the object conjugate plane at the same distance D of the crystal. A polarizer (P) placed in front of the camera rejects the signal and permits the detection of the idler wave. The resolution is
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Fig. 3. (a) Experimental setup. (b) Direct images. (c) Direct images through the aberrant plate. (d) Restored idler images. (After [20].)
in good agreement with the prediction from the phase matching conditions and is not degraded by the aberrant medium. C. Amplification of a Polychromatic Image To obtain such an amplification, phase matching must be simultaneously satisfied over the widest area of both spatial and temporal frequencies. We have theoretically shown [17] that, in a type I crystal (same polarization for the signal and the idler wave), parametric amplification is one beam noncritical for the signal wavelength as well as for the signal direction around a collinear degenerate configuration. It means that a variation of the refractive index for the signal wave, due to a shift of either its wavelength or its direction of propagation, is compensated by an opposite variation with the same amplitude of the index for the idler wave, provided that these variations are small enough to locally describe the index as a linear function of both the wavelength and the propagation angle. Hence, it is possible to amplify the signal for a large cone of wave vectors centered on the pump direction, with a large wavelength bandwidth centered on twice the pump wavelength [Fig. 4(a)]. The resulting wave forms a polychromatic image whose mean propagation direction is collinear with the pump direction and corresponds to the zero spatial frequency. Such a polychromatic image amplification was experimentally demonstrated, first by using as polychromatic image a Raman cell [21], then quantum noise [22] [see Fig. 4(b)]. Note that a wave with such a spatiotemporal spectrum has been recently studied as an “X wave,” and similar experiment has been performed [23]; see also Section V-A. III. TIME GATING APPLICATIONS If not present at the input, the idler wave is generated only during the interaction with the pump pulse. Hence, parametric image amplification acts for the idler image as an ultrafast shut-
Fig. 4. (a) Effective gain on the signal G (in decibels) in a 15-mm-long LBO crystal, versus the signal wavelength and the signal direction. The angular width (FWHM) of the plateau is 8 mrad, and the wavelength bandwidth is 140 nm. The direction of the pump beam with respect to the crystal axes is fixed and corresponds to collinear degenerate phase matching. (After [21].) (b) Experimental amplification of the quantum noise (parametric fluorescence) in these conditions, recorded by placing the entrance slit of a spectroscope in the Fourier plane. (After [22].)
ter that permits imaging with a picosecond exposure time. We describe three applications using this property. A. Ultrafast Imaging Fig. 5 presents picosecond pulses illuminating obliquely a diffusing screen [13]. With no image amplification, the exposure time of the CCD camera is much longer than the propagation time of the pulse on the screen. Hence, all the beam is visible [Fig. 5(b)]. With amplification and selection of the idler by polarization (see Fig. 1), the pulse is imaged at a position depending on the delay line [Fig. 5(c)]. B. Time-Resolved Imaging Through Thick Biological Tissues In the recent years, optical imaging through biological media has attracted a considerable interest, owing to the small absorption in the therapeutic window (600–1300 nm) and the nonionizing character of light. However, the light is strongly scattered, and various methods have been proposed to select the less scattered photons. In relatively thin media (typically a thickness of 1 mm or less except in eye), a part of the incident light is not scattered and images are now routinely obtained by using optical coherence tomography, which detects this ballistic light by interferometry [24]. In thicker media, like human breast, imaging tumors remains challenging, because the optical thickness is so important that ballistic light does not exist any more and images must be formed with scattered photons. Since
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Fig. 7. (a) Amplified image of the cross without biological tissue. (b) and (c) Amplified images through 4-cm-thick chicken breast tissue. The delays are, respectively, 0 and 158 ps. In (b), the strips are resolved and the best SNR ≈ 2 is obtained. (d) Resolved image of a 1 cm3 piece of liver embedded in the same chicken breast sample. (After [25].)
Fig. 5. (a) Light beam cut in part by a mirror, propagating from right to left on a diffusing screen. (b) Image without amplification. (c) Amplified image of the pulse in flight at four successive times. Graduation in time is given by the length of the delay line. (After ( [13].)
Fig. 7 shows experimental results. Fig. 7(a) presents an amplified image of the object, a cross formed by two 9-mm-wide metal strips, without biological tissue. Fig. 7(b) and (c) shows images of this cross embedded in 4-cm-thick chicken breast tissues. The delays are, respectively, 0 and 158 ps, where the time origin corresponds to the pump and signal pulses synchronized in the crystal in the absence of biological tissue. In Fig. 7(b), the 9-mm-wide metal strips are resolved and the best signal-tonoise ratio is obtained. When the delay is greater than 33 ps, the strips are no longer resolved, and only diffused light is amplified with a maximum intensity of the stretched pulse for a delay of 158 ps, in good agreement with a simple model developed in [26] [Fig. 7(c)]. Fig. 7(d) shows a resolved image of a 1 cm3 piece of liver embedded in the same biological sample. These results are important because the thickness of the biological tissues corresponds to that of human breast in mammography. Though the resolution is in the range of several millimeters, the nonionizing and noninvasive character of the method could be useful for routine preventive exams. Using parametric image amplification, images in a reflection configuration were also obtained with femtosecond pulses [27]. This pulse duration allows precise tomography, but for thinner samples. C. Fluorescence Life-Time Imaging
Fig. 6. Principle of selection of the less scattered photons at the forefront of the scattered pulse.
diffusion in a biological medium is strongly anisotropic, the photons are preferentially scattered in the forward direction and the front part of a short pulse, largely stretched by the medium, is formed with the least scattered photons that have traveled on the shortest paths. Hence, the front part still carries some amount of spatial information, with a resolution that depends on the thickness of the diffusing medium. We have used parametric image amplification to separate this front part from the multidiffused light (Fig. 6) [25]. The image is formed with the idler in order to eliminate the multiple scattered light. The most efficient scheme combines time gating and forward phase conjugation for a better rejection of the diffused light.
As the fluorescence lifetimes depend on the local fluorophore environment, they inform about their chemical and structural situations. The measurement of fluorescence lifetime is achieved by either frequency-domain (FD) or time-domain (TD) techniques. Most FD instruments are known to be limited to an upper modulation frequency near 200 MHz that limits this technique to the measurement of lifetimes greater than few hundred picoseconds. TD measurement consists in illuminating the sample with a short pulse of light and then measuring the fluorescence decay with a rapid detection system. The most frequently used technique of fluorescence lifetime determination is time-correlated single-photon counting (TSCPC). In [28], picosecond fluorescent lifetimes were measured with a subpicosecond precision. Nevertheless, this process requires scanning to obtain images. 2-D fluorescence lifetime imaging has been achieved using intensified cameras for time gating [29]. The temporal resolution is then limited by the time-gate width (about 50 ps). To improve this resolution, we have performed parametric image amplification of a fluorescence signal in order to spatially resolve different picosecond lifetimes in the same image [30].
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be possible to record a fluorescence lifetime map with a single excitation, opening the way to detect a rapid evolution of the lifetime map. Note however that the fluorescence signal is very weak and was acquired during several shots in our experiments. IV. NOISELESS AMPLIFICATION OF OPTICAL IMAGES We begin in this section the study of quantum optics effects in image amplification. This is part of a new area named “quantum imaging” that has been recently the focus of an important research effort, in great part in the framework of the European project “QUANTIM.” Obtained results are reviewed in [31]. In Section IV, we consider an image at the amplifier input that is perfect from a classical point of view, i.e., whose noise properties are entirely connected to the quantum nature of light. In Section V, we review the case of no input image, i.e., spatial properties of PF. A. Theoretical and Experimental Noise Figures Fig. 8. (a) Relative position of the pump pulses with respect to the signal, for different positions of the delay line. (b) Corresponding fluorescence signal pulses after amplification (only one of the three peaks exists for one delay). (c) Idler pulses. (d) Position of fluorophores. (e) Fluorescence lifetime map.
Fluorescence images are recorded at different delays. Then, the fluorescence lifetime map is reconstructed by fitting pixel per pixel the exponential fluorescence decay by means of a nonlinear least square algorithm. Experimentally, pulses at 532 and 355 nm, with a duration of, respectively, 38 and 20 ps, are delivered by a frequency doubled and tripled Nd:YAG laser. The green light illuminates the sample in order to generate a fluorescence signal around 710 nm. This signal, emitted in all directions and over a wide wavelength range, is spatially and spectrally filtered with, respectively, a pinhole and a narrow-band interferential filter centered on 700 nm (∆λFW HM = 3 nm). The sample is imaged on the input face of a lithium triborate (LBO) crystal designed for type 1 phase matching around degeneracy. Because of noncritical phase matching (see Section II-C), images can be amplified over a wide 650–770 nm wavelength range. The idler wave generated during the interaction is separated from the amplified signal by a second narrow-band interferential filter centered on 720 nm (∆λFW HM = 5 nm) and is imaged on a cooled CCD camera. Because of the weakness of the signal, an image of the parametric fluorescence (PF) must be subtracted before applying the fitting algorithm (see Section V-B for details). Several samples containing different kinds of dye embedded in polymethyl methacrylate (PMMA) were prepared [Fig. 8(d)]. The corresponding lifetime map is given in Fig. 8(e). Four different lifetimes are measured in this map: 85 ± 8 ps for styryl 8, 135 ± 18 ps for styryl 7, 111 ± 9 ps for pyridine 2, and 126 ± 12 ps for LDS 720. These values are in good agreement with results obtained with a streak camera. To conclude this section, the temporal resolution of the method depends on the pump pulse duration. In this experiment, it was 20 ps but it can go down 1 ps while preserving phase matching conditions (for shorter pulses, the crystal thickness must be small [27]). By using several crystals, it could
The quantum noise properties of an optical parametric amplifier are described by the noise figure NF = SNRin /SNRout . However, if SNRout is the output signal-to-noise ratio after detection with a quantum efficiency η < 1, the input signal-tonoise ratio SNRin is not a measurable quantity. What can be effectively measured is the SNR after detection without amplification, resulting in multiplying both the input SNR and the noise figure by the global quantum efficiency of the system ηtot . Unlike the “theoretical” noise figure NF, the practical ratio R = ηtot NF can be smaller than 1, meaning that the amplified image is less degraded by a poor detector quantum efficiency than a nonamplified image, because amplification introduces redundancy (i.e., bunching) between photons. Photons in the input image with no classical noise are described by a Poissonian statistics and R is given by, depending if amplification is phase sensitive [PSA, Ai (0) = 0 in (1)] or phase insensitive [PIA, Ai (0) = 0] [32], [33]: SNRwithout am pl SNRin =η SNRwith am pl SNRout 1 1 =η 1− + GPSA ηGPSA 2 1 =η 2− + . GPIA ηGPIA
RPSA =
RPIA
(4)
Physically speaking, if the three interacting waves are present at the amplifier input, amplification is sensitive to the phase relationship between these waves and is noiseless (R = 1 for unit quantum efficiency). On the other hand, if the idler is not injected, amplification becomes phase-insensitive and vacuum fluctuations enter the idler unused port [34], resulting in a 3 dB degradation of R. Amplification improves the detected SNR whatever the quantum efficiency for a PSA scheme, and only for η < 0.5 in a PIA scheme. Note that this improvement is effective whatever the gain, though increasing with it. Sokolov et al. [35] have shown that using a detector with a pixel area Sd smaller than the coherence area Sdiff in the amplified image is equivalent to multiplying the quantum efficiency
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by the ratio Sd /Sdiff in (4). It means physically that, even for a strongly multimode beam, detecting only a part of one of these spatial modes is equivalent to introducing losses, as for a single-mode beam. We have verified (see Section IV-D) that the measured R indeed decreases with the pixel size when this size is smaller than the coherence area. We discuss in the following if this measurement corresponds to a real improvement of the SNR. In order to be real, this improvement must occur over the whole bandpass of the amplifier, either temporal or spatial. This bandpass is very wide in the temporal domain, and in practice, the bandpass of the system is not limited by the optical amplifier but rather by the detector (typically a photodiode). The signal and noise power are measured at frequencies well within this bandpass, respectively, 0 and 27 MHz in [36]. Hence, a too small size of this detector can be actually compensated by amplification: among the redundant photons that carry the same temporal information, some arrive in the coherence area outside the photodetector but some are detected. The situation is completely different if we consider spatial fluctuations in one image because the amplifier rejects the spatial quantum noise above its cutoff frequency 1/Ldiff , where Ldiff is the lateral size of the coherence area. Because what is measured is the variance of the pixels, i.e., the integral of the noise power 1/2 until the sampling frequency 1/Sd = 1/Ld , the rejection of the high-frequency noise results in multiplying the variance by Sd /Sdiff . On the other hand, the signal is measured as the mean of a continuous background, i.e., at the zero spatial frequency: it is not affected by the reduction of the bandpass, and an apparent improvement of the SNR is measured. This improvement is an artifact: the bandpass is reduced for the signal just like for the noise and no spatial signal information can be transmitted above the cutoff frequency 1/Ldiff . Kolobov pioneered theoretically the spatial behavior of squeezed states at the output of a parametric amplifier [37] and noise properties of an image. To summarize, a phase-sensitive amplifier is noiseless for a relative phase corresponding either to perfect amplification or deamplification, and noisy for intermediate phases. If the crystal is placed in the Fourier plane, as mostly considered in [37], phase matching limits the field-ofview of the object that can be noiseless amplified. However, in the experiments on noiseless amplification [32], [36], the input face of the crystal was placed in the image plane, as theoretically assessed in [38]. In this case, noiseless amplification is obtained for low spatial frequencies, with a relative phase corresponding in practice to maximum amplification, i.e., in the range of maximum gain centered on the null frequency in Fig. 9(d).
Fig. 9. (a), (b) Images of gain obtained, for two different laser shots, by dividing pixel per pixel the (de)amplified image by the reference image. (c), (d) Corresponding gain profiles, fitted by theoretical curves, with a relative phase of (c) π/4 or (d) −π/4.
ever, a small phase mismatch, corresponding to middle spatial frequencies, leads to a small amplification, because this phase mismatch compensates in part the input phase [21]. To assess experimentally the phase-sensitive gain versus the image spatial frequencies, the far-field image in the Fourier plane was compared to a nonamplified reference beam [39]. Theoretically, a phase-sensitive amplifier leads to amplification or deamplification whatever the pump power. Nevertheless, in practice, the presence of small added noise, whatever its origin, precludes strong deamplification when strong parametric gain is experienced. Hence, we used in this experiment a weak pump pulse, leading to gains in the tenths of decibels range. Fig. 9 presents, for two different laser shots, the gain distribution in the Fourier plane, obtained by dividing the amplified images of a pinhole by the nonamplified images. For both shots, we observed circular fringes centered on the zero spatial frequency that corresponds to perfect phase matching. However, since the relative phase between the signal and the pump pulses fluctuated from one laser shot to another, strong variations of intensity modulation in the fringes pattern were observed, and Fig. 9 presents two shots corresponding to maximum amplification or deamplification, with fitting of the experimental curves by theoretical curves for a relative phase of, respectively, π/4 or −π/4. The agreement is correct, with a fitted parametric gain that corresponds to the measured mean energy of pump pulses. C. Noiseless Image Amplification for Temporal Fluctuations
B. Gain in Phase-Sensitive Amplification In practice, PSA is obtained either at total degeneracy in a type I crystal, where the idler and signal waves are indistinguishable, or in a type II crystal with a 45◦ polarization of the input wave [32]. If the relative phase φs of the input wave with respect to the pump wave is equal to −π/4, the amplification is maximum for perfect phase matching and presents a plateau around the null spatial frequency (see Section II-C.) For φs = π/4, the input wave is deamplified for perfect phase matching. How-
The first experiment that demonstrated noiseless image amplification, performed by Choi et al., considered temporal fluctuations that affect a spatial pattern [36]. The signal and pump pulses are provided, respectively, by the fundamental (1064 nm) and the second harmonic (532 nm) of a Q-switched modelocked Nd:YAG laser at a repetition rate of 1 kHz. The resulting Q-switch envelopes of the pump and signal pulses are 145 and 200 ns in duration, respectively. The mode-locked pump and signal pulses underneath these Q-switch envelopes are estimated
LANTZ AND DEVAUX: PARAMETRIC AMPLIFICATION OF IMAGES: FROM TIME GATING TO NOISELESS AMPLIFICATION
to be 85 and 120 ps, respectively. The amplification is performed in a KTP crystal, with a length of either 3.25 or 5.21 mm. The signal is polarized at 45◦ of the crystal neutral axes, in order to inject both the signal and the idler waves into the amplifier and obtain PSA. The phase of the pump beam relative to that of the signal beam is locked to maximize the parametric gain, by means of a feedback loop that drives the piezoelectric transducer (PZT). The object illuminated by the signal beam consists of two vertical lines of a U.S. Air Force test pattern. These lines are imaged with a unity magnification into the center of the crystal by a telescope. The spatial frequency corresponding to these lines, 10.1 lines/mm, is chosen sufficiently low to lie well within the spatial bandwidth of the amplifier, and sufficiently high to ensure uniform illumination by the pump beam. The amplified image is magnified 24 times after the crystal in order to be spatially resolved when scanned by an InGaAs detector with a diameter of 300 µm. In the temporal domain, the bandwidth of the parametric amplifier is much larger than that of the detector, ensuring an identical gain for dc and for 27 MHz photocurrents. Hence, the experimental noise figure can be simply determined as NF =
noise power gain R = η η (mean intensity gain)2
(5)
where the photocurrent noise power is recorded at 27 MHz. To minimize the effect of spatial averaging caused by the finite size of the photodetector, the noise figure was characterized using experimental values of gains measured at the peaks of the spatial profile. Intensity gains G ∼ = 2.5 were obtained at the peaks, with a noise power gain lower than the intensity gain to the square, because the amplified image is less degraded by the detection than the shot-noise-limited input. By taking into account the measured overall detection efficiency η = 0.82 and using (5), the following values of the total NF of the optical amplifier were obtained: (0.2 ± 0.6) dB at G = 2.5 for the 3.25 mm KTP crystal and (0.4 ± 0.6) dB at G = 2.6 for the 5.21 mm KTP crystal. These values agree with the theoretical values predicted for a PSA amplifier, and clearly show the improvement due to preamplification when compared with the NF = 0.86 dB of the detector. D. Noiseless Image Amplification in the Continuous-Wave (CW) Regime Gigan et al. used a transverse degenerate optical parametric oscillator (OPO) below threshold (i.e., in the amplification regime) to obtain the first experimental observation of PSA of an image in the continuous-wave (CW) regime [40]. The relative phase between the pump and the input image was controlled by a piezoelectric transducer. The hemiconfocal triply resonant cavity was multimode but only partially imaging: the output image corresponds to the even part of the input image, plus its spatial Fourier transform. On the other hand, a confocal cavity is able to amplify symmetrical images, with a resolution assessed in [41] and [42]. Both hemiconfocal and confocal cavities have recently been proved to be able to noiselessly amplify a multimode beam [43], where noise refers to temporal fluctuations measured on the entire beam. Moreover, the hemiconfocal
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cavity operating with a type II crystal produced signal and idler images with subshot noise level either on the difference between the total intensities for PSA or on the sum of these intensities for deamplification. In parallel, Lopez showed experimentally [44] that a second harmonic image can be generated at the output of a self-imaging cavity. Such a cavity, obtained by inserting a lens inside the cavity, transmits entire images and appears as the ideal candidate for noiseless image amplification in the CW regime, though amplifying images in stable conditions of triple resonance seems highly challenging. E. Noiseless Image Amplification for Spatial Fluctuations The experiment described in Section IV-C shows that a phasesensitive scheme allows the signal-to-noise ratio to be unmodified over an entire image, where the noise is recorded at a frequency of 27 MHz by a photodiode. As the photodiode scanned the image, this result proves that PSA improves the regularity in time of the distribution of photons for each point of the image but, because only fluctuations in the TD were recorded, it does not show any regularity in space. However, patterns in an image are pure spatial information, without any time aspect, that are ultimately degraded by spatial fluctuations of quantum origin for very weak images. The experiment described in this section [33] was designed to assess these spatial fluctuations. The experimental setup is similar to that shown in Fig. 1, with differences however in light sources. The signal and pump pulses are provided, respectively, by the second harmonic [1.2 ps duration (full-width at half-maximum, FWHM) at 527.5 nm] and the fourth harmonic (0.93 ps duration at 263.7 nm) of a Q-switched mode-locked Nd:glass laser at a repetition rate of 33 Hz. Indeed, no quantitative statistics can be performed after detection with a silicon camera for a wavelength beyond 1 µm, because of diffusion of light in the silicon. The amplification is performed in a β-barium borate (BBO) crystal whose transverse area, 7 mm × 7 mm, is chosen in order to obtain a sufficient number of resolution cells in the amplified image to perform valid statistics. The crystal length, 4 mm, is limited by the group-velocity difference between the UV pump and the green signal. Because of the high dispersion of the crystal in the UV, only type 1 amplification is possible for this couple of wavelengths. Hence, collinear interaction is phase sensitive while PIA is obtained by a slight angular shift between the pump and signal beams. A filtering hole, placed in the Fourier plane and centered around the zero spatial frequency of the signal image, limits the detected intensity of spontaneous down-conversion (SPDC) and ensures the elimination of the idler in the PIA scheme. However, the spatial spectral bandwidth in the detected image is reduced by this hole, which determines the size of the coherence area Sdiff , equal to 7 pixels × 7 pixels on the camera. In the following, the results will be presented for different binning of the pixels (achieved by software) in order to consider effective detector areas Sd smaller or greater than Sdiff (see Section IV-A). The NF depends on the resulting total quantum efficiency ηtot = 0.6, product of the quantum efficiency of the CCD camera by the transmission of the optical elements after the crystal.
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The experimental procedure to measure R is achieved in three main steps that are identical in PIA and PSA. The first step consists in measuring the SNR without amplification (i.e., ηtot SNRin ) as well as a statistical verification of the Poissonian hypothesis. In the second step, the intensity of the SPDC is measured, and its level is subtracted from the amplified images. Finally, SNRout is measured. As for the first step, we have verified that a nongrouped image is well described by a Poissonian distribution, while binning degrades this distribution because of residual deterministic defects. Nevertheless, the experimental statistics on the difference between two images remains Poissonian because the subtraction of images eliminates deterministic structures. To estimate the SPDC level, 20 images were recorded by injecting only the pump. The measurement of SNRout is performed by using single-shot images, in order to take into account the strong variations of the gain from one shot to another in PSA because of the noncontrolled variations of the relative phase between the signal and the pump. Classical noise is predominant in the amplified images, because of deterministic imperfections of the system (pump beam, lenses, etc.) leading to phase distortions. Good results have been obtained only by performing differences of images, in order to eliminate the spatial defects that are reproducible from one shot to another. SNRout is measured as follows. First, an area where the intensity (i.e., the gain) is as constant as possible is selected in the amplified image. Second, the measurement of SNR on all amplified images without subtraction allows the selection of images with the highest SNRs, which correspond to φs = π/4. Third, pairs of images are defined by all permutations between the selected images. Fourth, the mean of the images and the variance of the difference of the two images are calculated for each pair and each binning, and a mean value of SNRout is determined. Experiments were conducted in both PIA and PSA configurations with results in good agreement with theory. In the PSA scheme, about 500 images were recorded, and the gain was measured for each image with a range from 1 to 6. Because the relative phase is not controlled, it is more difficult than in the PIA case to find pairs of images that correspond both to the same (maximum) gain and the same phase. Nevertheless, the criteria based on the highest SNR allow the selection of five images that were amplified in the same conditions. Fig. 10(a) shows an example of a selected image. The gain is clearly nonhomogeneous along the line because of residual variations of the relative phase. Fig. 10(b) shows the evolution of R, as defined in Section IV-A, versus the detector size. The agreement between the experimental data and the theoretical curve is good and proves the noiseless character of the PSA scheme. Note that the measured R increases with the detector size until this size exceeds the size of the coherence area. Let us recall that the small value of R for small pixels does not correspond to any improvement, as analyzed in Section IV-A. F. Improvement of Detection by Parametric Preamplification In the preceding section, the demonstration of noiseless amplification used difference of images to eliminate the classical
Fig. 10. (a) Example of amplified image in the PSA scheme. Statistics are performed in the rectangular area of 3266 pixels. (b) Noise figure after detection (R) in the PSA scheme versus the lateral detector size. Squares: experimental data (full line error bars). Line: theoretical curve (heavy dotted error bars).
noise reproducible from an image to another. Clearly, this elimination, useful to verify that spatial quantum fluctuations obey theory like temporal fluctuations, is not possible for practical use. In [45], we showed, in a PIA scheme without difference of images, that the experimental noise figure is improved in close agreement with theory. Moreover, image amplification improves the detected SNR without any correction if the quantum efficiency of the detector is low. This improvement can also be considered as an improvement of an equivalent quantum efficiency [46]. The experimental setup is similar to that of the previous section in the PIA scheme. The major difference is the addition of a neutral density filter that ensures a low overall quantum efficiency η = 1.34%. In the Fourier plane, a pinhole of variable diameter first selects the signal and rejects the idler in order to ensure PIA amplification in a frequency degenerate but noncollinear scheme, second filters spatial frequencies (low-pass filter) in order to precisely adjust the dimensions of the resolution cell (coherence area) to the size of the pixel on the CCD camera. Two sets of either nonamplified or amplified images were recorded and an area of 240 pixels in a line of the chart
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was selected, where no deterministic patterns appeared in the amplified image. In this area, the mean value of a series of 20 nonamplified image was 3.0 photoelectrons (pe) per pixel, with a variance of 14.6 pe2 without any correction, giving an SNR of 3.02 /14.6 = 0.62. The sum of the Poissonian photon noise and detector readout noise fully accounts for this value. The mean value in the selected area of the amplified images was 9.1 pe/pixel, for a noncorrected variance of 22.4 pe2 . To determinate the SNR, the mean level of PF must be subtracted from the measured amplified signal to determine the mean signal level that carries information. Hence, 20 images of SPDC were recorded, with a mean value of 2.0 pe/pixel. After subtraction of the mean PF, the SNR in the amplified image was calculated as (9.1–2.0)2 /22.4 = 2.25, i.e., about four times the SNR in the detected image without amplification (R = 0.28): the amplified image is less sensitive than the nonamplified one to losses due to a poor quantum efficiency of the detector, but also in a similar way to detector noises. This conclusion still holds though the amplified image is slightly degraded by the remaining PF after filtering. We have also shown in [45] that experimental results are in good agreement with theory if theory takes into account all noises. V. PARAMETRIC FLUORESCENCE PF is called SPDC when the gain is so weak that light issued from spontaneous splitting of pump photons experiences negligible parametric amplification further in the crystal. Unlike parametric image amplification, SPDC has been extensively studied for more than 40 years [47], [48], and we will restrict this review to works devoted to characterize images of the amplified quantum noise, mostly in the high-gain regime and its spatial fluctuations. A. Spatiotemporal Shape of Parametric Fluorescence Because quantum noise includes all spatial and temporal frequencies, images of PF in the far field are direct representations of the gain, nonnegligible only near phase matching. Far-field images of PF were obtained already in the 1960s by positioning the entrance slit of a spectrometer in the Fourier plane, giving wavelength-angle phase matching curves [49]. At degeneracy, these curves are X-shaped [17], [21] (see Section II-C). Berzanskis et al. [50] gave the whole phase matching curves for an LBO crystal, numerically as well as experimentally. These studies have been recently pursued by Jedrkiewicz et al. [23]: Fig. 11 shows a spatiotemporal image for a BBO crystal, which shows the spreading of PF near degeneracy due to noncritical phase matching. To assess the PF power, waves amplitudes in (1) are replaced by operators with unity commutator, resulting in an output per spatiotemporal mode, expressed in photons, of g2 [sinh(bL)/2b)]2 [47]. In the pure SPDC regime, g is so weak that the term in the square brackets can be replaced by sin(∆KL/2)/∆K [15]. Hence, the gain is proportional to the pump intensity (because g is proportional to the pump amplitude), leading to a total SPDC power proportional to the pump power, whatever its intensity repartition [48]. Actually, it is
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Fig. 11 Single-shot spectrum of parametric down-conversion radiation generated by a type I LBO crystal pumped at 527 nm (left part obtained by duplication of the experimental right part) and comparison with the theoretical phase matching curve (white curve). (After [23].)
stated in [48] that this property also holds for a detector intercepting a small solid angle centered on the phase matching curve. It is not always true for a focused pump beam and quasimonochromatic detection. Indeed, for detection at signal frequency, a hot spot of fluorescence occurs on the phase matching curve when the Poynting vectors of the idler and signal waves are collinear [51]. Because these vectors are orthogonal to the normal surface of the crystal, and consequently to ∂ K/∂ϕ, such a situation occurs when [18] i) i s p − K p ∂K ∂K ∂(K ∂K = ⇒ = =0 ∂ϕp ∂ϕp ∂ϕp ∂ϕp
(6)
p s for the range of wave vectors K leading to almost the same K characterizing the focused pump beam. In the high-gain regime, the PF power is no more proportional to the pump power and the actual size of a spatiotemporal mode experiencing gain must be determined. It turns out that this size is also the size of a resolution cell [52] discussed in Section II-A. The number of spatial modes in a ring configuration [top of Fig. 2(b)] is twice the number of modes around a collinear configuration [bottom of Fig. 2(b)]. By multiplying the number of phase-matched modes by the gain, the total power of PF can be determined [52]. B. Absolute Radiance Imaging In the presence of a signal image, the number of idler photons exiting from the crystal in one mode can be written as in nout idler = (G − 1)nsignal + (G − 1)
(7)
where G is the total gain on the signal. The last term comes from PF. Equation (7) means that the number of photons per mode nin signal in the input image can be retrieved by comparing pixel per pixel the level of the output idler and the level of PF at the same frequency: nin signal =
nout idler − nPF nPF
(8)
Remarkably, the gain is not involved in (8), provided it remains constant when recording both images, with and without input
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Fig. 12. Radiance imaging of dye fluorescence. (a) Images at 700 nm of SPDC only. (b) Images at 700 nm of idler without correction. (c) Images at 700 nm of idler minus SPDC. (d) 2D radiance of the input signal.
image. On the other hand, this gain may vary between pixels, because of a nonhomogeneous pump beam [see Fig. 12(a)]. The method works only if amplified modes are the same for the signal and PF, i.e., if the signal light is emitted in a solid angle larger than the phase matching range. In practice, the signal image comes from an incoherent source, dye fluorescence in [53], and a blackbody in [54]. In this latter experiment, no imaging aspect was involved, and the radiance of an infrared signal (λ = 4 µm) was measured by recording the idler beam in the visible, where detectors are much more efficient. Incoherent images radiate in a wide solid angle but their level is often smaller than one photon per mode, as shown in Fig. 12(d). Though the amplified image [obtained by subtraction: Fig. 12(c)] is less intense than PF, radiance imaging is still possible provided that the rms fluctuations of PF are smaller than the level of this amplified image [55].
Fig. 13. Far-field SPDC pattern recorded on one laser shot. The statistics in Fig. 14 is performed in the area surrounded by the white circle.
C. Bose–Einstein Statistics of Spatial Fluctuations The last two sections are devoted to the study of spatial fluctuations of quantum origin in PF. Their distribution is random from one temporal mode to another, and they are visible only if the image is formed by the addition on each pixel of intensities of a very small number of temporal modes, ideally one, to avoid averaging. Therefore, the pump pulse must contain a sufficient energy per temporal mode to amplify an image. This fact explains that such fluctuations were observed only relatively recently, with short pulses [50], [52]. The formation of patterns in parametric oscillators was observed some time before [56] but is outside the scope of this paper. For one temporal mode, spatial fluctuations of amplified spontaneous emission obey a Bose–Einstein statistics [57], as it was experimentally demonstrated 30 years ago for a laser amplifier [58]. We observed [59] the same statistics for PF. The pump laser, described in Section IV-E, delivers 0.93 ps pulses in the UV. In order to select an unique temporal mode, a narrow interferential filter centered at degeneracy (∆λ = 0.4 nm at 527.5 nm) was placed before the camera, and the crystal length was sufficiently small to avoid temporal walk-off that would stretch the PF. Pure Bose–Einstein statistics is theoretically obtained for a single temporal mode and infinitely small pixels. In [59], the pixels were much smaller than the coherence cell
Fig. 14. Histogram of the experimental pixels from one image. Solid line: Bose–Einstein distribution, with the degeneracy factor M = 1.73. Dashed line: Poissonian distribution with the same mean.
in the far field, the number of temporal modes was close to one and the experimental variance was in good agreement with a theoretical curve calculated with a “degeneracy factor” that takes into account the finite size of the pixels and the nonperfect monomode character of the fluorescence. This degeneracy factor is fully explained by the actual size of the pixel and the actual spectrotemporal repartition of the fluorescence pulse, corresponding to 1.3 temporal modes, as assessed by numerical simulation. Fig. 13 shows one experimental image and Fig. 14 gives the corresponding histogram of the pixels. D. Twin Images Parametric amplification occurs when a pump photon splits in a signal and an idler photon. This relation is exact in the sense that signal and idler photons are always produced by pairs, even
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Fig. 16.
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Correlated signal–idler areas. (After [69].)
Fig. 15. Covariance between intensity fluctuations measured on pixel (x = 22, y = 18) and all the other pixels (photons2 ). Left: Green’s function method; right: average of 104 stochastic simulations.
if the number of this pairs fluctuates. Actually, beams are entangled, i.e., the phase and the amplitude are simultaneously strongly quantum correlated. Despite its basic interest in quantum optics, we will not discuss further entanglement, but one of its aspects, i.e., sub-Poissonian correlations of photon numbers. Indeed, in this paper, our interest lies in spatial fluctuations recorded by a CCD camera and not by homodyne detection. We also refer to [60] and references therein to discuss ghost imaging, i.e., imaging by temporal coincidences. At the output of a two-mode nondegenerate OPO, the spectrum of temporal fluctuations in the signal and intensity difference is reduced below shot noise, as first demonstrated experimentally by Heidmann et al. [61]. If the two detectors do not intercept the whole beams, the correlation is reduced because, for some pairs, one photon is detected while the other is not intercepted. Because “intercepted” can be replaced by “detected” in the previous sentence, such insufficient size of a detector is exactly equivalent to a reduction of the quantum efficiency. The situation is different for a strongly multimode beam issued either from a traveling wave amplifier or a degenerate oscillator. Brambilla et al. showed [62] that, for unity quantum efficiency, the variance of the signal–idler difference goes to zero (in shot noise units) if the pixel size is much greater than the coherence area (see also [63] for the case where an input image is present). 2 + 1D numerical simulations with a realistic pump beam support this assertion. We first showed [64], at the classical level, that the random shape of spatial fluctuations observed at degeneracy in [52] was reproduced by a stochastic simulation that takes into account the experimental intensity repartition of the pump beam. Because of degeneracy, fluctuations are symmetrical (see Fig. 14), experimentally [52], [65] and numerically [64]. To obtain results at the quantum level, the groups of Como [66] and Besanc¸on [67] performed averages of a great number of stochastic simulations, corresponding to Wigner formalism. Quantum expectations are then obtained by applying corrections to pass from symmetrized operators to normal order corresponding to detection. It is also possible to obtain directly quantum expectations [67] by applying the Green’s function method, originally designed by Treps and Fabre to study squeezing in spatial solitons [68]. Fig. 15 confirms that results are identical with both methods: the covariance of a pixel with its neighbors is much smaller than with its opposites, and the shape of this covariance in far field is reciprocal of the pump shape, which is wider than
Fig. 17. Intensity difference variance normalized to the shot noise. Each point corresponds to a single-shot measurement. Inset: correlation function between symmetrical areas. (After [69].)
high in the experimental map [52] used for this simulation. Another interesting result is that the variance of the difference signal–idler increases linearly with the lateral size of the pixel. Hence, this difference is more and more sub-Poissonian because the mean increases as the pixel area. An intuitive explanation resides in that only pairs corresponding to a coherence area at the frontier between pixels can arrive on nonopposite pixels. The area of this frontier zone evolves as the pixel lateral size. Jedrkiewicz et al. performed the first experimental demonstration of subshot noise behavior of spatial fluctuations on the difference signal–idler [69]. After generation in a type II BBO crystal, the idler and signal PF were separated by a polarizing beam splitter and imaged on a deep-depletion CCD camera. Fig. 16 shows evidence of strong correlation of the spatial fluctuations. After centering, signal pixels are associated to their idler counterpart and the variance of the difference is calculated on an area of 40 pixels × 100 pixels as 2 2 = σs+b−(i+b) − 2σb2 σs−i
(9)
where b represents the background noise, having a standard deviation of seven photoelectrons measured in presence of pulse illumination over an area of the same size of the acquisition area and suitably displaced from the directly illuminated region. Fig. 17 presents the obtained results. The values lie well below the shot noise limits if the intensity is sufficiently small. It would be interesting to obtain results for higher intensities to avoid a background correction greater
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than the result after correction [70]. We are trying at Besanc¸on another way, i.e., using very low-intensity images recorded by an electron multiplying CCD (EMCCD) camera, which allows real single-photon sensitivity but is unable to distinguish between a single and several photons on a pixel. Preliminary results are encouraging.
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Eric Lantz was born in 1957. He received the Ph.D. degree in superresolution in microscopy by inverse problem methods from the Universit´e de Franche Comt´e, Besanc¸on Cedex, France, in 1989. From 1979 to 1985, he worked in the field of solar energy and heat pumps. In 1986, he joined the Laboratoire d’Optique P. M. Duffieux, Universit´e de Franche Comt´e, where he is currently a Professor, and since 1989, has been engaged in the nonlinear optics group of the laboratory, and initiated studies about travelingwave parametric image amplification in crystals. His current research interests include parametric amplification in fibers, spatial solitons, and optimal photon counting strategies with electron multiplying charge-coupled device (EMCCD) cameras.
Fabrice Devaux was born in Caen, France, on December 27, 1967. He received the Graduate degree from the Ecole Sup´erieure d’Optique, Orsay, France, in 1991, and the Ph.D. degree in nonlinear optics from the Universit´e de Franche Comt´e, Besancon, France, in 1996. He is currently a Professor at the Universit´e de Franche-Comt´e, where he is engaged in research on χ (2 ) imaging systems, quantum optics, and optical spatial vortex solitons using photorefractive effects.
