- Email: [email protected]
Fluctuations in the phase-conjugate signal generated via degenerate four-wave mixing
Volume 50, number 3
OPTICS COMMUNICATIONS
1 June 1984
FLUCTUATIONS IN THE PHASE-CONJUGATE SIGNAL GENERATED VIA DEGENERATE FOUR-WAVE MIXING Prem KUMAR, Jeffrey H. SHAPIRO Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Roy S. BONDURANT Massachusetts Institute of Technology Lincoln Laboratory, P.O. Box 73, Lexington, MA 02173, USA Received 23 January 1984
The excess fluctuations of the phase-conjugate (PC) signal generated via pulsed degenerate four-wave mixing (DFWM) in sodium vapor are investigated. Various dye-laser oscillator and amplifier combinations are employed to trace the sources of these fluctuations. For DFWM driven by an externally-stabilized cw dye-laser oscillator amplified with a dye system pumped by the smoothed pulses of a Nd :YAG laser, the PC excess fluctuations are very highly correlated with the energy fluctuations of the input amplified dye-laser pulses. Thus, input-pulse selection on the basis of energy alone permits observation of the quantum-mechanical fluctuations of the PC signal.
1. Introduction ,_ B$ BS
This paper reports the results of an investigation of the fluctuations present on the phase conjugate (PC) reflection produced in a pulsed degenerate fourwave mixing (DFWM) experiment. Our interest in understanding these fluctuations stems from our desire to measure the quantum statistics (via photon counting) of light produced by DFWM, as it has been predicted that this light will possess novel q u a n t u m statistical properties [ 1]. Fluctuations of a classical nature will tend to mask those of quantum mechanical origin, hence it is necessary to eliminate, to a large degree, the effects of the former.
2. Experimental setup Our DFWM setup, shown in fig. 1, is similar to that which was used by Bloom et al. [2], except that our setup utilizes true counter-propagating pumps rather than a retro-reflection arrangement. Sodium vapor is generated in a heat-pipe oven maintained at 0 030-4018/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
r-~ ~
" a
ER
1
l
TO PULSE MONITORING ELECTRONICS
,-
,o '
o.d
°/d
N°
Fig. 1. Schematic of the experimental setup. YAG = frequency doubled Nd :YAG laser, DL = dye laser (Littman type pulse or cw ring), M = total reflector, BS = beam splitter, DET = detector, a/d = analog-to-digital converter, HWP = half-wave plate, PBS = polarization beam splitter.
290°C, implying a sodium vapor density of 2 X 1014 atom/cm 3. One to two torr of helium is used as a buffer. In all instances reported herein, the D 2 ( 2 S 1 / 2 2P3/2) resonance at 589.0 nm was utilized. The reso183
Volume 50, number 3
OPTICS COMMUNICATIONS
nantly enhanced electronic Kerr effect acts as the DFWM nonlinearity. For near-resonant excitation at frequency 6o with short pulses (4 ns in our experiments), the adiabatic following model [3] gives the following expression for the third-order susceptibility X(3) X(3) = 7rNapa/h3 ( ~ - 6o)3 ,
(1)
where N a is the atomic number density, ~2 is the transition frequency, and p is the dipole matrix element of the transition. The strong frequency dependence of X(3) is shown clearly by (1), and it is evident that fluctuations in 6o will lead to fluctuations in X(3) and, consequently, fluctuations in the PC beam. Furthermore, eq. (1) must be modified [4] when saturation occurs, and fluctuations in X(3) can then arise from pump-strength fluctuations. The output of the dye laser system employed is attenuated and split into two parts by a 50/50 beamsplitter for use as counter-propagating pump beams of nominal intensity 1 - 1 0 kW/cm 2. A small portion of the amplifier output is further split into two parts and directed to a pair of photodetectors. The first detector (called the "tagging" detector) has a slow response, so its peak output measures the energy of the dye laser pulses. The second detector has a fast response (50 ps risetime), and is used in conjunction with a 1 GHz bandwidth oscilloscope to monitor the
o
0
2
I
4
6
DYE LASER FREQUENCY (GHz)
8
I0
="
Fig. 2. a) PC signal as a function of the dye laser frequency; b) simultaneously recorded fluorescence signal.
184
1 June 1984
temporal quality of the pulses generated by the dye laser system. A weak probe beam, derived from one of the pumps by reflection off a piece of glass, intersects the pump beams inside the oven at a small angle of about 1°. The mirror spacings are adjusted so that all the beams arrive in time coincidence at the center of the oven. Polarization selection is employed to separate all of the PC beam from the counter-propagating probe beam. The PC signal is directed onto another slow-response detector to measure its energy. The outputs of the two slowresponse detectors are fed into a computer through two independent analog-to-digital (a/d) converters, which digitize the peak outputs of the detectors. Care is taken not to saturate any of the detectors, so that the a/d outputs are proportional to the pulse energies falling on the detectors. The PC signal shows two peaks, one on either side of the D 2 line, as the dye laser frequency is varied. Fig. 2 shows the observed signal and a simultaneously taken trace of the fluorescence emitted in a direction perpendicular to the plane of the four-wave mixing beams. The relative height and width of the two peaks is a function of the intensity of the pump beams. For the results presented in this paper, the dye laser is tuned to the peak of the higher frequency signal. In an earlier experiment, Jabr et al. [5] did not observe the lower frequency peak with the same choice of the pump and probe beam polarizations that we employ. Their pump intensities were at least an order of magnitude larger than ours, and so self-focusing (higher frequency side) and defocussing (lower frequency side) played an important role. Due to strong self-defocussing of the pump beams, no PC signal is observed on the lower frequency side of the resonance. We obtain similar results when the pump intensities are varied; for the data presented in this paper self-defocussing played little role.
3. Experimental results The main goal of our experiment is to measure the photon-counting statistics of the light generated by DFWM. Classical excess fluctuations will mask the quantum effects that we desire to observe. One major source of such classical noise is the shot-to-shot dye laser pulse energy fluctuations. Our photon counting
Volume 50, number 3
OPTICS COMMUNICATIONS
tuations, then as the window is narrowed, the excess noise will be reduced. Of course, depending upon the amplitude stability of the laser, narrowing the tagging window may require that a much larger total number of pulses be examined. In order to verify that pulse tagging can indeed eliminate the effects of amplitude fluctuations, let us examine histograms of the dye laser pulse energy and the PC pulse energy. These histograms exhibit the behavior of 10 240 events. Each event consists of a dye laser pulse energy and the corresponding PC pulse energy for that particular laser shot. Each event is stored individually in a computer data base. Thus, the average due laser energy can be determined, and then a window established (say +10% around the mean dye laser pulse energy, etc.) and histograms of PC energies corresponding to dye laser energies within the window can then be computed from the data base. When the experiments were started, a Littman type pulsed dye laser [7] was used in conjunction with a three stage dye laser amplifier pumped by the second harmonic of a Nd:YAG laser. The histograms obtained employing this source are shown in fig. 3(A). Histogram (a) shows the PC pulse energy distribution whereas (d) shows the dye laser pulse energy distri-
scheme, described elsewhere [6], attempts to circumvent this problem by the use of pulse tagging: only PC pulses originating from dye laser pulses falling within a specified energy window are employed in the photon counting. If input pulse energy fluctuations are the dominant cause of PC pulse excess fluc-
i.ll
(A)
PULSE ENERGY ~
PULSE ENERGY ~
(B)
C
"9
==
/ PULSEENERGY--~
PULSEENERGY~
(c)
I d
PULSEENERGY
PULSEENERGY
1 June 1984
Fig. 3. Normalized pulse energy fluctuation histograms. Horizontal scale, which is linearly proportional to pulse energy, is the same for all the plots. The vertical scales, which are linearly proportional to the number of pulses, have been chosen to achieve common peak values. A) Resuits obtained using a Littman type pulse dye laser, a) For the PC beam, x (standard deviation/mean) = 0.63; d) for the output of the dye laser oscillator/amplifier chain, x = 0.40; b) for the PC beam when only those pulses whose input probe beam pulse energy falls within ±10% of the mean value are selected,x = 0.52; c) same as b when ±2% selection is employed, x achieves a minimum value of 0.51. B) Results obtained using a cw ring dye laser but without the Nd:YAG laser pulse smoothing scheme, a) For the PC beam, x = 0.22; d) for the output of the dye laser oscillator/ amplifier chain, x = 0.11 ; b) for the PC beam when -*5% pulse selection is employed, x = 0.21; c) when ±1% pulse selection is employed, x = 0.22. C) Results obtained when the Nd :YAG laser pulse smoothing scheme is introduced. a) For the PC beam, x = 0.26; e) for the output of the dye laser oscillator/amplifier chain, x = 0.19; b) for the PC beam when ±10% pulse selection is employed, x = 0.06; d) x achieves a minimum value of 0.04 when ± 1% pulse selection is employed. 185
Volume 50, number 3
OPTICS COMMUNICATIONS
bution. Since the two pump beams are also derived from the same laser source, the width (measured by the standard deviation) of the PC histogram is ex• pected to be three times that of the dye laser pulse energy histogram at low conjugate reflectivity. This is not the case because of the saturation of the X(3) nonlinearity [4] ; three times as wide histograms are observed when the pump pulse energy is reduced to avoid saturation of the nonlinearity at the expense of reduced conjugate reflectivity. In all cases reported herein, the conjugate reflectivity was much less than one. New PC energy histograms are computed by selecting pulses falling within a specified window around the mean dye laser pulse energy. Histogram (b) in fig. 3(A) is obtained by defining the window to be 10% around the mean value, and histogram (c) in fig. 3 (A) uses a window of 2% around the mean value. The width of the histograms is reduced slightly by this energy tagging procedure, but it is clear that the majority of the PC pulse energy fluctuations is independent of the input dye laser pulse energy. From eq. (1) we see that the nonlinear response of the medium has a cubic resonant denominator. Resonant enhancement of the nonlinearity is necessary to achieve significant PC reflectivities. So operation near a resonance makes the PC signal very sensitive to the frequency content of the input laser pulses. For input pulses of a given energy, any randomness in their frequency leads to random fluctuations of the PC signal through a random change in nonlinearity seen by each pulse. For this reason a thorough investigation was made of the frequency content of the pulses generated with the pulsed oscillator/amplifier system, by employing a fast detector (50 ps risetime) in conjunction with a 1 GHz bandwidth oscilloscope and a streak camera with 200 ps resolution. Similar studies of pulsed dye lasers employing a streak camera have recently been published [8]. During this investigation we found that it is possible, by employing two gratings, to make a Littman type pulse dye laser run single mode for a short time. However, during the course of a long ( 1 7 60 min) photon counting run the system mode hops, and the center frequency of any particular mode drifts from pulse to pulse due to transient heating of the dye solution and other environmental perturbations. A study of thermal effects in similar dye lasers has recently been published [9]. Because of the pulsed na186
1 June 1984
ture of such a source, it is very difficult to lock it to an external reference frequency. Slow temperature drifts also become important for long data collection intervals. In light of these difficulties, the pulsed oscillator was abandoned and a cw dye laser locked to a pair of external cavities producing a linewidth of 250 kHz and less than 50 MHz/hour drift was employed. The output of this dye laser was sent through a similar chain of dye laser amplifiers as used with the pulsed dye laser. The only difference being the first amplifier, which now was used in a double pass mode to extract more gain from the first stage. The second harmonic of the same Nd:YAG laser was used as a pump source. Similar histograms were collected for the PC reflected pulses and the input dye laser pulses to the four-wave mixing medium. The results are shown in fig. 3(B). Histogram (d) of fig. 3(B) is the dye laser pulse energy distribution, and histogram (a) of fig. 3(B) gives the PC pulse energy distribution. The dye laser pulse energy histogram is narrower than that shown in (d) of fig. 3(A) because of higher saturation of the dye laser amplifier chain. The total PC pulse energy fluctuations are only about one third of those observed when using a pulsed dye laser oscillator. To see if these fluctuations track with the input laser energy fluctuations, new PC histograms similar to those in (b) and (c) of fig. 3(A) were calculated by defining windows of 5% and 1% around the mean input pulse energy. The results are shown in (b) and (c) of fig. 3(B) respectively. Once again we see that the PC pulse energy fluctuations are independent of the input dye laser pulse energy fluctuations. Thus, we were led to investigate the frequency characteristics of the output of the cw dye laser-pulsed amplifier system. It was found that, although the center frequency stability is established by the cw dye laser, there is random modulation of the amplified output. Fourier analysis of this modulation showed it to be related to similar modulation of the output of the Nd :YAG laser, which gets transferred to the amplified dye laser pulses during the amplification process. The output of the Nd:YAG laser is deeply modulated because of the multi-axial mode nature of the Q-switched laser. A new smoothing scheme, described in an earher paper [10], was developed and installed at the output of the Nd:YAG laser. The output of the pulse
Volume 50, n u m b e r 3
OPTICS COMMUNICATIONS
smoothing network was used to pump the amplifier chain. The output dye laser pulses were used to perform four wave mixing and the results are shown in fig. 3 (C). Histogram (d) of fig. 3(C) gives the output dye laser pulse energy distribution and histogram (a) of fig. 3 (C) is the corresponding PC pulse energy distribution. PC histograms were calculated after selecting pulses as before. Fig. 3 (C) histograms (b), (c) and (e) employ input pulses lying within a 10% window, a 5% window, and a 1% window about the mean value, respectively. The width of these histograms is clearly correlated with the input dye laser pulse energy spread, and there is a residual PC fluctuation of 4%. Most of this residual fluctuation is electrical noise on the two detection - a/d channels, as was confirmed by directing parts of the same pulses onto the two detectors. The histograms so generated correlated with each other to within 3%.
4. Discussion In the preceding sections we have shown that by sufficiently stabilizing the frequency of the dye laser pulses, we were able to track out all other PC excess fluctuations via pulse-energy tagging. The extensive effort required to achieve this situation, while not justifiable in many experiments, was, in our case, absolutely essential. In our experiments, photon counting was performed on the PC beam. For that purpose the PC output is directed onto a special purpose photomultiplier tube (PMT). From the photon counting distributions thus obtained, the value of the normalized second factorial moment of the photon number operator, g2, is evaluated. For coherent-state light generated by an ideal laser,g 2 = 1 and the photon counts are Poisson distributed. For a single-mode chaotic state, g2 = 2 and the photon counts are Bose-Einstein distributed. For nonclassical states such as two-photon coherent states [11 ] or squeezed states [12] exhibiting sub-poissonian photon statistics, g2 could be less than 1. Weak classically-random fluctuations on a single-mode light beam tend to increase its g2 value according to
--g2(1 +x2),
I0 °
1 June 1984
I
I
I
}
I 0
I 1 I I 2 3 NUMBER OF PHOTONS n
I
,o-'
10 -2
I 4
Fig. 4. P h o t o n counting distribution o f the PC beam. n is the n u m b e r o f primary photo-electrons counted per pulse with probability P(n). n !P(n) is plotted versus n on a semilog scale. g2 = 1.00 -+ 0.03 for the measured distribution.
where gF is the normalized second factorial moment with fluctuations and x = oE/E gives the relative pulse energy fluctuation in terms of the mean energy ff and the standard deviation oE of the fluctuation. Thus the value of x 2 directly determines the accuracy to which g2 values can be estimated. On the basis of this analysis, therefore, our system is capable of determining g2 to better than 0.2% accuracy. That is, departures o f g 2 value from 1 by as little as 0.002 are measurable. The ultimate sensitivity is actually limited by the photon counting system as described elsewhere [6]. Fig. 4 shows the photon counting distribution obtained for the PC pulses employing the above setup; the measured g2 valus is 1.00 consistent with theoretical predictions. Detailed analysis of the photon counting statistics of the PC signal have already been published [13,14]. To the best of the author's knowledge, this is the first measurement of the quantum behavior of light that has been generated through the third-order nonlinear interaction in an isotropic medium. Our understanding of the classically-random fluctuations has played a key role in these measurement. This research was supported by Office of Naval Research Contract N00014-81 -K-0662.
(2) 187
Volume 50, number 3
OPTICS COMMUNICATIONS
References [1] H.P. Yuen and J.H. Shapiro, Optics Lett. 4 (1979) 334; M.D. Reid and D.F. Wails, Optics Comm., to be pubfished. [2] D.M. Bloom, P.F. Liao and N~P. Economu, Optics Lett. 2 (1978) 58. [3] D. Grischkowsky, N.S. Shixen and R.J. Bennett, Appl. Phys. Lett. 33 (1978) 805. [4] D. Grischkowsky, in: Physics of quantum electronics, Vol. II, eds. S.F. Jacobs, M. Sargent III, J.F. Scott and M.O. Scully (Addison-Wesley, Reading, 1975) p. 437452. [5] S.N. Jabr, L.K. Lam and R.W. Hellwarth, Phys. Rev. A24 (1981) 3264.
188
1 June 1984
[6] R.S. Bondurant, P. Kumar, J.H. Shapiro and M.M. Salour, Optics Lett. 7 (1982) 529. [7] M.G. Littman, Optics Lett. 3 (1978) 138. [8] D.S. King and R.R. Cavanagh, Optics Lett. 8 (1983) 18. [9] F.J. Duarte, IEEE J. Quant. Electron. QE-19 (1983) 1345. [ 10] P. Kumar and R.S. Bondurant, Appl. Optics 22 (1983) 1284. [11] H.P. Yuen, Phys. Rev. AI3 (1976) 2226. [12] D.F. Walls, Nature 306 (1983) 141. [13] P. Kumar, R.S. Bondurant, J.H. Shapiro and M.M. Salour, in: Coherence and quantum optics, Vol. V, eds. L. Mandel and E. Wolf (Plenum Press, New York, 1983) p. 43-49. [14] R.S. Bondurant, P. Kumar, J.H. Shapiro and M. Maeda, submitted to Phys. Rev. A.
OPTICS COMMUNICATIONS
1 June 1984
FLUCTUATIONS IN THE PHASE-CONJUGATE SIGNAL GENERATED VIA DEGENERATE FOUR-WAVE MIXING Prem KUMAR, Jeffrey H. SHAPIRO Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Roy S. BONDURANT Massachusetts Institute of Technology Lincoln Laboratory, P.O. Box 73, Lexington, MA 02173, USA Received 23 January 1984
The excess fluctuations of the phase-conjugate (PC) signal generated via pulsed degenerate four-wave mixing (DFWM) in sodium vapor are investigated. Various dye-laser oscillator and amplifier combinations are employed to trace the sources of these fluctuations. For DFWM driven by an externally-stabilized cw dye-laser oscillator amplified with a dye system pumped by the smoothed pulses of a Nd :YAG laser, the PC excess fluctuations are very highly correlated with the energy fluctuations of the input amplified dye-laser pulses. Thus, input-pulse selection on the basis of energy alone permits observation of the quantum-mechanical fluctuations of the PC signal.
1. Introduction ,_ B$ BS
This paper reports the results of an investigation of the fluctuations present on the phase conjugate (PC) reflection produced in a pulsed degenerate fourwave mixing (DFWM) experiment. Our interest in understanding these fluctuations stems from our desire to measure the quantum statistics (via photon counting) of light produced by DFWM, as it has been predicted that this light will possess novel q u a n t u m statistical properties [ 1]. Fluctuations of a classical nature will tend to mask those of quantum mechanical origin, hence it is necessary to eliminate, to a large degree, the effects of the former.
2. Experimental setup Our DFWM setup, shown in fig. 1, is similar to that which was used by Bloom et al. [2], except that our setup utilizes true counter-propagating pumps rather than a retro-reflection arrangement. Sodium vapor is generated in a heat-pipe oven maintained at 0 030-4018/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
r-~ ~
" a
ER
1
l
TO PULSE MONITORING ELECTRONICS
,-
,o '
o.d
°/d
N°
Fig. 1. Schematic of the experimental setup. YAG = frequency doubled Nd :YAG laser, DL = dye laser (Littman type pulse or cw ring), M = total reflector, BS = beam splitter, DET = detector, a/d = analog-to-digital converter, HWP = half-wave plate, PBS = polarization beam splitter.
290°C, implying a sodium vapor density of 2 X 1014 atom/cm 3. One to two torr of helium is used as a buffer. In all instances reported herein, the D 2 ( 2 S 1 / 2 2P3/2) resonance at 589.0 nm was utilized. The reso183
Volume 50, number 3
OPTICS COMMUNICATIONS
nantly enhanced electronic Kerr effect acts as the DFWM nonlinearity. For near-resonant excitation at frequency 6o with short pulses (4 ns in our experiments), the adiabatic following model [3] gives the following expression for the third-order susceptibility X(3) X(3) = 7rNapa/h3 ( ~ - 6o)3 ,
(1)
where N a is the atomic number density, ~2 is the transition frequency, and p is the dipole matrix element of the transition. The strong frequency dependence of X(3) is shown clearly by (1), and it is evident that fluctuations in 6o will lead to fluctuations in X(3) and, consequently, fluctuations in the PC beam. Furthermore, eq. (1) must be modified [4] when saturation occurs, and fluctuations in X(3) can then arise from pump-strength fluctuations. The output of the dye laser system employed is attenuated and split into two parts by a 50/50 beamsplitter for use as counter-propagating pump beams of nominal intensity 1 - 1 0 kW/cm 2. A small portion of the amplifier output is further split into two parts and directed to a pair of photodetectors. The first detector (called the "tagging" detector) has a slow response, so its peak output measures the energy of the dye laser pulses. The second detector has a fast response (50 ps risetime), and is used in conjunction with a 1 GHz bandwidth oscilloscope to monitor the
o
0
2
I
4
6
DYE LASER FREQUENCY (GHz)
8
I0
="
Fig. 2. a) PC signal as a function of the dye laser frequency; b) simultaneously recorded fluorescence signal.
184
1 June 1984
temporal quality of the pulses generated by the dye laser system. A weak probe beam, derived from one of the pumps by reflection off a piece of glass, intersects the pump beams inside the oven at a small angle of about 1°. The mirror spacings are adjusted so that all the beams arrive in time coincidence at the center of the oven. Polarization selection is employed to separate all of the PC beam from the counter-propagating probe beam. The PC signal is directed onto another slow-response detector to measure its energy. The outputs of the two slowresponse detectors are fed into a computer through two independent analog-to-digital (a/d) converters, which digitize the peak outputs of the detectors. Care is taken not to saturate any of the detectors, so that the a/d outputs are proportional to the pulse energies falling on the detectors. The PC signal shows two peaks, one on either side of the D 2 line, as the dye laser frequency is varied. Fig. 2 shows the observed signal and a simultaneously taken trace of the fluorescence emitted in a direction perpendicular to the plane of the four-wave mixing beams. The relative height and width of the two peaks is a function of the intensity of the pump beams. For the results presented in this paper, the dye laser is tuned to the peak of the higher frequency signal. In an earlier experiment, Jabr et al. [5] did not observe the lower frequency peak with the same choice of the pump and probe beam polarizations that we employ. Their pump intensities were at least an order of magnitude larger than ours, and so self-focusing (higher frequency side) and defocussing (lower frequency side) played an important role. Due to strong self-defocussing of the pump beams, no PC signal is observed on the lower frequency side of the resonance. We obtain similar results when the pump intensities are varied; for the data presented in this paper self-defocussing played little role.
3. Experimental results The main goal of our experiment is to measure the photon-counting statistics of the light generated by DFWM. Classical excess fluctuations will mask the quantum effects that we desire to observe. One major source of such classical noise is the shot-to-shot dye laser pulse energy fluctuations. Our photon counting
Volume 50, number 3
OPTICS COMMUNICATIONS
tuations, then as the window is narrowed, the excess noise will be reduced. Of course, depending upon the amplitude stability of the laser, narrowing the tagging window may require that a much larger total number of pulses be examined. In order to verify that pulse tagging can indeed eliminate the effects of amplitude fluctuations, let us examine histograms of the dye laser pulse energy and the PC pulse energy. These histograms exhibit the behavior of 10 240 events. Each event consists of a dye laser pulse energy and the corresponding PC pulse energy for that particular laser shot. Each event is stored individually in a computer data base. Thus, the average due laser energy can be determined, and then a window established (say +10% around the mean dye laser pulse energy, etc.) and histograms of PC energies corresponding to dye laser energies within the window can then be computed from the data base. When the experiments were started, a Littman type pulsed dye laser [7] was used in conjunction with a three stage dye laser amplifier pumped by the second harmonic of a Nd:YAG laser. The histograms obtained employing this source are shown in fig. 3(A). Histogram (a) shows the PC pulse energy distribution whereas (d) shows the dye laser pulse energy distri-
scheme, described elsewhere [6], attempts to circumvent this problem by the use of pulse tagging: only PC pulses originating from dye laser pulses falling within a specified energy window are employed in the photon counting. If input pulse energy fluctuations are the dominant cause of PC pulse excess fluc-
i.ll
(A)
PULSE ENERGY ~
PULSE ENERGY ~
(B)
C
"9
==
/ PULSEENERGY--~
PULSEENERGY~
(c)
I d
PULSEENERGY
PULSEENERGY
1 June 1984
Fig. 3. Normalized pulse energy fluctuation histograms. Horizontal scale, which is linearly proportional to pulse energy, is the same for all the plots. The vertical scales, which are linearly proportional to the number of pulses, have been chosen to achieve common peak values. A) Resuits obtained using a Littman type pulse dye laser, a) For the PC beam, x (standard deviation/mean) = 0.63; d) for the output of the dye laser oscillator/amplifier chain, x = 0.40; b) for the PC beam when only those pulses whose input probe beam pulse energy falls within ±10% of the mean value are selected,x = 0.52; c) same as b when ±2% selection is employed, x achieves a minimum value of 0.51. B) Results obtained using a cw ring dye laser but without the Nd:YAG laser pulse smoothing scheme, a) For the PC beam, x = 0.22; d) for the output of the dye laser oscillator/ amplifier chain, x = 0.11 ; b) for the PC beam when -*5% pulse selection is employed, x = 0.21; c) when ±1% pulse selection is employed, x = 0.22. C) Results obtained when the Nd :YAG laser pulse smoothing scheme is introduced. a) For the PC beam, x = 0.26; e) for the output of the dye laser oscillator/amplifier chain, x = 0.19; b) for the PC beam when ±10% pulse selection is employed, x = 0.06; d) x achieves a minimum value of 0.04 when ± 1% pulse selection is employed. 185
Volume 50, number 3
OPTICS COMMUNICATIONS
bution. Since the two pump beams are also derived from the same laser source, the width (measured by the standard deviation) of the PC histogram is ex• pected to be three times that of the dye laser pulse energy histogram at low conjugate reflectivity. This is not the case because of the saturation of the X(3) nonlinearity [4] ; three times as wide histograms are observed when the pump pulse energy is reduced to avoid saturation of the nonlinearity at the expense of reduced conjugate reflectivity. In all cases reported herein, the conjugate reflectivity was much less than one. New PC energy histograms are computed by selecting pulses falling within a specified window around the mean dye laser pulse energy. Histogram (b) in fig. 3(A) is obtained by defining the window to be 10% around the mean value, and histogram (c) in fig. 3 (A) uses a window of 2% around the mean value. The width of the histograms is reduced slightly by this energy tagging procedure, but it is clear that the majority of the PC pulse energy fluctuations is independent of the input dye laser pulse energy. From eq. (1) we see that the nonlinear response of the medium has a cubic resonant denominator. Resonant enhancement of the nonlinearity is necessary to achieve significant PC reflectivities. So operation near a resonance makes the PC signal very sensitive to the frequency content of the input laser pulses. For input pulses of a given energy, any randomness in their frequency leads to random fluctuations of the PC signal through a random change in nonlinearity seen by each pulse. For this reason a thorough investigation was made of the frequency content of the pulses generated with the pulsed oscillator/amplifier system, by employing a fast detector (50 ps risetime) in conjunction with a 1 GHz bandwidth oscilloscope and a streak camera with 200 ps resolution. Similar studies of pulsed dye lasers employing a streak camera have recently been published [8]. During this investigation we found that it is possible, by employing two gratings, to make a Littman type pulse dye laser run single mode for a short time. However, during the course of a long ( 1 7 60 min) photon counting run the system mode hops, and the center frequency of any particular mode drifts from pulse to pulse due to transient heating of the dye solution and other environmental perturbations. A study of thermal effects in similar dye lasers has recently been published [9]. Because of the pulsed na186
1 June 1984
ture of such a source, it is very difficult to lock it to an external reference frequency. Slow temperature drifts also become important for long data collection intervals. In light of these difficulties, the pulsed oscillator was abandoned and a cw dye laser locked to a pair of external cavities producing a linewidth of 250 kHz and less than 50 MHz/hour drift was employed. The output of this dye laser was sent through a similar chain of dye laser amplifiers as used with the pulsed dye laser. The only difference being the first amplifier, which now was used in a double pass mode to extract more gain from the first stage. The second harmonic of the same Nd:YAG laser was used as a pump source. Similar histograms were collected for the PC reflected pulses and the input dye laser pulses to the four-wave mixing medium. The results are shown in fig. 3(B). Histogram (d) of fig. 3(B) is the dye laser pulse energy distribution, and histogram (a) of fig. 3(B) gives the PC pulse energy distribution. The dye laser pulse energy histogram is narrower than that shown in (d) of fig. 3(A) because of higher saturation of the dye laser amplifier chain. The total PC pulse energy fluctuations are only about one third of those observed when using a pulsed dye laser oscillator. To see if these fluctuations track with the input laser energy fluctuations, new PC histograms similar to those in (b) and (c) of fig. 3(A) were calculated by defining windows of 5% and 1% around the mean input pulse energy. The results are shown in (b) and (c) of fig. 3(B) respectively. Once again we see that the PC pulse energy fluctuations are independent of the input dye laser pulse energy fluctuations. Thus, we were led to investigate the frequency characteristics of the output of the cw dye laser-pulsed amplifier system. It was found that, although the center frequency stability is established by the cw dye laser, there is random modulation of the amplified output. Fourier analysis of this modulation showed it to be related to similar modulation of the output of the Nd :YAG laser, which gets transferred to the amplified dye laser pulses during the amplification process. The output of the Nd:YAG laser is deeply modulated because of the multi-axial mode nature of the Q-switched laser. A new smoothing scheme, described in an earher paper [10], was developed and installed at the output of the Nd:YAG laser. The output of the pulse
Volume 50, n u m b e r 3
OPTICS COMMUNICATIONS
smoothing network was used to pump the amplifier chain. The output dye laser pulses were used to perform four wave mixing and the results are shown in fig. 3 (C). Histogram (d) of fig. 3(C) gives the output dye laser pulse energy distribution and histogram (a) of fig. 3 (C) is the corresponding PC pulse energy distribution. PC histograms were calculated after selecting pulses as before. Fig. 3 (C) histograms (b), (c) and (e) employ input pulses lying within a 10% window, a 5% window, and a 1% window about the mean value, respectively. The width of these histograms is clearly correlated with the input dye laser pulse energy spread, and there is a residual PC fluctuation of 4%. Most of this residual fluctuation is electrical noise on the two detection - a/d channels, as was confirmed by directing parts of the same pulses onto the two detectors. The histograms so generated correlated with each other to within 3%.
4. Discussion In the preceding sections we have shown that by sufficiently stabilizing the frequency of the dye laser pulses, we were able to track out all other PC excess fluctuations via pulse-energy tagging. The extensive effort required to achieve this situation, while not justifiable in many experiments, was, in our case, absolutely essential. In our experiments, photon counting was performed on the PC beam. For that purpose the PC output is directed onto a special purpose photomultiplier tube (PMT). From the photon counting distributions thus obtained, the value of the normalized second factorial moment of the photon number operator, g2, is evaluated. For coherent-state light generated by an ideal laser,g 2 = 1 and the photon counts are Poisson distributed. For a single-mode chaotic state, g2 = 2 and the photon counts are Bose-Einstein distributed. For nonclassical states such as two-photon coherent states [11 ] or squeezed states [12] exhibiting sub-poissonian photon statistics, g2 could be less than 1. Weak classically-random fluctuations on a single-mode light beam tend to increase its g2 value according to
--g2(1 +x2),
I0 °
1 June 1984
I
I
I
}
I 0
I 1 I I 2 3 NUMBER OF PHOTONS n
I
,o-'
10 -2
I 4
Fig. 4. P h o t o n counting distribution o f the PC beam. n is the n u m b e r o f primary photo-electrons counted per pulse with probability P(n). n !P(n) is plotted versus n on a semilog scale. g2 = 1.00 -+ 0.03 for the measured distribution.
where gF is the normalized second factorial moment with fluctuations and x = oE/E gives the relative pulse energy fluctuation in terms of the mean energy ff and the standard deviation oE of the fluctuation. Thus the value of x 2 directly determines the accuracy to which g2 values can be estimated. On the basis of this analysis, therefore, our system is capable of determining g2 to better than 0.2% accuracy. That is, departures o f g 2 value from 1 by as little as 0.002 are measurable. The ultimate sensitivity is actually limited by the photon counting system as described elsewhere [6]. Fig. 4 shows the photon counting distribution obtained for the PC pulses employing the above setup; the measured g2 valus is 1.00 consistent with theoretical predictions. Detailed analysis of the photon counting statistics of the PC signal have already been published [13,14]. To the best of the author's knowledge, this is the first measurement of the quantum behavior of light that has been generated through the third-order nonlinear interaction in an isotropic medium. Our understanding of the classically-random fluctuations has played a key role in these measurement. This research was supported by Office of Naval Research Contract N00014-81 -K-0662.
(2) 187
Volume 50, number 3
OPTICS COMMUNICATIONS
References [1] H.P. Yuen and J.H. Shapiro, Optics Lett. 4 (1979) 334; M.D. Reid and D.F. Wails, Optics Comm., to be pubfished. [2] D.M. Bloom, P.F. Liao and N~P. Economu, Optics Lett. 2 (1978) 58. [3] D. Grischkowsky, N.S. Shixen and R.J. Bennett, Appl. Phys. Lett. 33 (1978) 805. [4] D. Grischkowsky, in: Physics of quantum electronics, Vol. II, eds. S.F. Jacobs, M. Sargent III, J.F. Scott and M.O. Scully (Addison-Wesley, Reading, 1975) p. 437452. [5] S.N. Jabr, L.K. Lam and R.W. Hellwarth, Phys. Rev. A24 (1981) 3264.
188
1 June 1984
[6] R.S. Bondurant, P. Kumar, J.H. Shapiro and M.M. Salour, Optics Lett. 7 (1982) 529. [7] M.G. Littman, Optics Lett. 3 (1978) 138. [8] D.S. King and R.R. Cavanagh, Optics Lett. 8 (1983) 18. [9] F.J. Duarte, IEEE J. Quant. Electron. QE-19 (1983) 1345. [ 10] P. Kumar and R.S. Bondurant, Appl. Optics 22 (1983) 1284. [11] H.P. Yuen, Phys. Rev. AI3 (1976) 2226. [12] D.F. Walls, Nature 306 (1983) 141. [13] P. Kumar, R.S. Bondurant, J.H. Shapiro and M.M. Salour, in: Coherence and quantum optics, Vol. V, eds. L. Mandel and E. Wolf (Plenum Press, New York, 1983) p. 43-49. [14] R.S. Bondurant, P. Kumar, J.H. Shapiro and M. Maeda, submitted to Phys. Rev. A.