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524 J. Opt. Soc. Am. B/Vol. 27, No. 3 /March 2010 Damon et al.

Broadband photonic arbitrary waveformgeneration using a frequency agile laser at 1.5 �m

Vianney Damon,1 Vincent Crozatier,2 Thierry Chanelière,1 Jean-Louis Le Gouët,1,* and Ivan Lorgeré1

1Laboratoire Aimé Cotton, CNRS UPR 3321, Bâtiment 505, Campus Universitaire, 91440 Orsay, France2FASTLITE, Centre Scientifique d’Orsay, Bâtiment 503, Plateau du Moulon, BP 45, 91401 Orsay, France

*Corresponding author: jean-louis.legouet@lac.u-psud.fr

Received October 7, 2009; revised January 8, 2010; accepted January 11, 2010;posted January 11, 2010 (Doc. ID 118308); published February 22, 2010

We use a pulse-compression chirp-transform algorithm to generate broadband photonic arbitrary waveforms. Aphase-locked frequency agile laser provides the needed broadband frequency scans. We theoretically show andexperimentally demonstrate that the residual laser frequency errors do not significantly alter the waveformgeneration. The experiment exhibits features such as the operation at the telecom wavelength of 1.5 �m, thecounterpropagating, crossed-polarized beam configuration, and the storage of the initial waveform in the activemedium. The last specificity gives access to large time-bandwidth product values. © 2010 Optical Society ofAmerica

OCIS codes: 320.5520, 070.1170, 300.6240.

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. INTRODUCTIONnalog processing based on filtering by rare earth dopedrystals (REIC) enjoys unrivaled capabilities in terms ofandwidth and time-to-bandwidth product. SeveralEIC-based architectures for the processing of opticallyarried broadband radar signals are today at the stage ofransfer to industrial demonstrators. Meanwhile researchor new architectures and materials is continuing. Thehirp-transform algorithm is a fully coherent optical pro-ess which performs time-to-frequency Fourier transformn real time [1]. It was first considered for spectral analy-is application [1–3] and can also be applied to widebandrbitrary waveforms generation (AWG) [4,5].The latter application relies on the strong temporal

ompression of a much lower bandwidth initial waveform.he REIC-based filter operates in the optical spectralange. Hence the radio frequency (RF) signal to be pro-essed has to be transposed on an optical carrier. The pro-essing bandwidth � is limited by the inhomogeneousinewidth of the material absorption profile at the operat-ng wavelength. This bandwidth may reach tens and evenundreds of gigahertz. In the chirp-transform algorithm,he optical carriers are frequency chirped over the band-idth.Broadband AWG with REIC technology was demon-

trated in thulium doped yttrium aluminum at the wave-ength of 793 nm in [5]. The optical carriers were pro-uced by a frequency-stabilized monochromatic laser. Therequency chirp was achieved by external electro-opticodulators (EOMs). Driving the EOMs over a large band-idth was not easy. Only 20% of the available materialandwidth could be covered, in spite of sophisticated andxpensive fast electronic means, such as ultra-fast pulseattern generators. Anyway, an optical processing proto-ol is expected to rely on photonics means and is less con-incing when depending too much on best electronics. Un-ike an external EOM, a frequency agile laser can offer

0740-3224/10/030524-7/$15.00 © 2

he straightforward, cheap, and effective full optical wayor fast broadband scans [3].

In this paper we reexamine the conditions for perform-ng broadband AWG with a chirped laser. We theoreticallynd experimentally show that the linewidth and chirp lin-arity requirements are less severe than mentioned in [4].n addition we propose a variant of the architecture, of-ering a much larger time-bandwidth product (TBP). Fi-ally, we perform the experimental demonstration in er-ium doped yttrium orthosilicate (Er:YSO), at the wave-ength of 1.5 �m, where advanced telecom equipment isvailable.

. PHOTON ECHO CHIRP TRANSFORMhe chirp-transform algorithm [6] computes the Fourier

ransform s��� of a signal s�t� according to

�s�f = rt�� = ��s�t�e−i�rt2� � ei�rt2

�, �1�

here � stands for the convolution product. The signal isrst imparted a linear frequency sweep at rate r followedy a dispersion process at rate 1/r. This yields the signalourier transform, save for a phase factor. One actuallybtains the temporal image of the signal spectrum withhe time-to-frequency rule f=rt.

The chirp-transform algorithm has been extensivelysed for RF signal processing in compressive receivers us-

ng acousto-optic (AO) and surface acoustic wave (SAW)evices [7]. The algorithm can also be implemented withhe help of stimulated photon echo (SPE) [1–3]. In theerturbation regime the SPE response can be expresseds the convolution product of the three driving fields ac-ording to

Ee�t� = E1��− t� � E2�t� � E3�t�. �2�

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Damon et al. Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. B 525

When fields E1�t� and E2�t� are chirped at rates r1 and2, respectively (see Fig. 1), their product E1

��−t� � E2�t�perates as a dispersive (or compressive) filter on therobe E3�t�, making this field undergo dispersion at rate/r3=1/r2−1/r1. If E3�t� is chirped at rate −r3 and modu-

ated by a RF signal s�t�, the echo power displays theower Fourier transform of s�t� in accordance with Eq. (1).The compression effect simply reflects the fact that the

pectrum of a long monochromatic pulse is a narrow fea-ure. Displaying this feature in the time domain may re-ult in a short pulse. Alternatively, multiplying s�t� by ahirped field can be regarded as a way to spread the spec-rum of the signal, making this spectrum cover the entire

interval swept by the chirped carrier. The response inhe time domain can be made as short as �−1, which is2/r3 times shorter than the � /r3-long probe field. Therobe pulse and the material response must be containedltogether within the atomic coherence lifetime, whichimits the compression effect.

Many REIC offer an inhomogeneous broadening in theulti-gigahertz range, as required to process fast signals.s illustrated in [5], the main difficulty is the productionf fast, broadband, phase-coherent optical chirps. In pre-ious works [1–3], we programmed the dispersive filterver � with the help of two successive pulses E1�t� and2�t� with opposite chirp rates −2r and 2r. The signal car-

ier E3�t� was chirped at rate −r. We directly scanned arequency agile laser to generate the desired chirp rates.

e have developed a specific self-heterodyning servo-loopo improve the linearity and spectral purity of the sweeps8]. Indeed, the frequency error on the optical chirp muste kept less than the inverse duration of the pulses. Theulse duration, � / �2r� or � /r, falls in the microsecondange, to be consistent with the atomic coherence lifetime,nd the locked laser satisfies the condition on the fre-uency error quite easily.One may question the use of microsecond pulses for

toring the dispersive filter E1��−t� � E2�t�. Indeed the pro-

rammed filter persists for the typical 10 ms population

ν1

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ig. 1. (Color online) Chirp transform with photon echo: instan-aneous frequency and intensity of the driving fields as a func-ion of time. The dispersive filter is engraved by the first twoverlapping pulses. Their frequency is scanned at rates r1 and r2,ver a spectral interval of width �. The probe pulse is chirped atate −r3 over the same interval. The delay of the first two excita-ions is a function of frequency that varies from T= ��1−�2� /r2 at1 to T+� /r3 at �1+�. The characteristic time evolution of theaterial response is given by the inverse spectral span �−1. The

gure illustrates the response to a rectangular probe of duration/r3. With respect to the engraving fields, the response under-oes a �2 /r2 compression.

ifetime of the REIC bottleneck state. Programmingulses in the millisecond range can engrave the filterore efficiently at lower intensity. Temporally overlapped

ulses can achieve such a long storage. Indeed, with r1lose to r2, one can simultaneously stretch the engravingtep and satisfy the short probe pulse condition, with r3r1 ,r2. However, millisecond pulses seem to impose a

ery demanding condition on laser stability. With fre-uency errors smaller than the inverse duration of theulses, laser control in the sub-kilohertz range would beequired. We shall address this issue in the next section.

The fields E2�t� and E3�t� play symmetric roles in Eq.2). Therefore the signal to be processed, s�t�, can be car-ied by either E2�t� or E3�t�. However the two proceduresre not equivalent. As already noticed, the filter E1

��−t�� E2�t� is stored in the atomic level population, with aypical lifetime of 10 ms. The probe field E3�t� is just fil-ered by this spectral structure. Hence, transposing s�t�n E3�t� means that the stored operator E1

��−t� � E2�t� onlyontains the dispersive function. One can use that filter torocess many optically carried signals, without any delayetween successive readings.On the contrary, when s�t� is carried by E2�t�, one stores

he signal itself in the REIC where it survives for theopulation lifetime. As a result, one must wait long beforerocessing another signal. Continuous processing is notossible. However this option should not be rejected atnce. Remember that the TBP is conserved throughouthe processing. The upper limit of the TBP is given by theroduct of � and the coherence lifetime T2. Let the en-raving pulses temporally overlap, with r3�r1 ,r2. If s�t�s carried by E3�t�, the processor TBP-capacity is given byhe product of the probe duration, � /r3, and the�t�-bandwidth. To take full advantage of the ��T2 TBP-apacity, one needs a s�t�-bandwidth of r3T2. The sameBP can be reached with a r2 /r3 times smaller�t�-bandwidth if s�t� is carried by E2�t�. This attractiveeature, illustrated in Fig. 2, may balance the low repeti-ion rate drawback.

The reduction in s�t�-bandwidth can also be viewed asn increase in the compression factor. Reflecting the de-ice’s ability to broaden the signal bandwidth, this factors defined as the ratio of the output � bandwidth, and thenput s�t�-bandwidth. Hence, for a given � value of theBP, the compression factor reads as �2 / �r3�� or �2 / �r2��epending on whether s�t� is carried by E3�t� or E2�t�. Theompression factor is r3 /r2 times larger when s�t� is car-ied by E2�t� instead of E3�t�.

. IMPACT OF LASER FREQUENCY ERRORSs noticed above, it has been considered that the laser

requency errors should be kept smaller than the inverseulse duration. With 1 ms-long programming pulses, aub-kilohertz frequency precision would be required.aintaining so small frequency errors is out of reach of

vailable frequency agile lasers. However, to derive thisery strict condition, one implicitly assumes that the dif-erent fields are uncorrelated. This is not necessarily truehen E1�t� and E2�t� are overlapped with r1�r2. Indeed

he two fields can be simultaneously derived from a single

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526 J. Opt. Soc. Am. B/Vol. 27, No. 3 /March 2010 Damon et al.

ource, the chirp difference r1−r2 being provided by, e.g.,n AO modulator. Then the critical parameter is no longerhe inverse field duration.

To be specific, let the frequency agile laser scan begin atime t=0. The AO shifter makes the instantaneous fre-uencies of fields E1�t� and E2�t�, respectively, evolve as1�t�=�1+r1t and �2�t�=�2+r2t. During engraving, antom at frequency � successively interacts with the twoelds at times ��−�1� /r1 and ��−�2� /r2. The interactionith the field depends on the frequency errors that accu-ulate during the delay � /r3+�1 /r1−�2 /r2. This time in-

ν1

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ν1

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r1

r2 - r3

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ig. 2. (Color online) TBP and modulation bandwidth. The sig-al to be compressed, s�t�, is transposed as a single sideband onhe optical carrier with the help of a �0-bandwidth modulator.epending on whether s�t� is carried by E3�t� (upper box) or by2�t� (lower box), the available TBP is given by ��0 /r3 or by�0 /r2. For a given TBP, the required modulator bandwidth is

2/r3 smaller in the latter case.

2

2

4

∆τ1/

2

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∆τ

I e(τ

,Γ)/

I e(0

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0 1

1

-1 0 1

1

-1

∆τ

40

2

4

Γ∆/r

3

(a)

(b) (c)

ig. 3. (Color online) Temporal compression and laser frequencyrrors. (a) SPE response normalized width �1/2 (solid line) andormalized inverse coherently scanned interval �� /r3 (dashed

ine) as a function of �� /r3. [(b) and (c)] Temporal profiles of thePE response at �� /r =0 and 4, respectively.

3

erval varies from T= ��1−�2� /r2 to T+� /r3 as the laser iscanned from �1 to �1+� and can remain much smallerhan the duration of the engraving fields.

Our chirped laser is self-stabilized with an opticalhase lock loop (OPLL) [8]. Most of the 1/ f technical noises suppressed. We have observed that the surviving fre-uency fluctuations exhibit a white noise spectrum be-ween 0 and 1 MHz [9]. The average response intensityan be calculated easily within the frame of this simplerequency white noise model. The calculation is detailedn the Appendix. We assume that none of the participat-ng fields carries any additional signal s�t�. In otherords, each one of E1�t�, E2�t�, and E3�t� reduces to a

ingle, monotonously chirped monochromatic field. Inhese conditions, the material response reduces to aingle brief emission around time t= t3+T. As a function ofhe quiescent laser linewidth � / �2��, the average tempo-al profile of this short pulse reads as

Ie�,�� = Ie�0,0�e−�T2

2 + �2���2�1 − e− cos�2���

− e−sin�2���

2��� , �3�

here = t− t3−T, and =�� /r3.In the absence of frequency errors, when �=0,

e� ,�� /Ie�0,0� reduces to �sin���� / �����2, which corre-ponds to the squared Fourier transform of a rectangularulse [see Fig. 3(b)]. The 1/2 full width at half maximumquals 0.89�−1.

As � is increased, the frequency errors affect the re-ponse in two different ways. First the temporal profilee� ,�� is stretched as � gets larger than the inverse probeulse duration. This reflects the loss of coherence betweentoms with different transition frequencies. One mightxpect that 1/2 tends to the inverse width of a coherentlycanned interval given by � /r3. On the contrary, 1/2 var-es much more slowly, as illustrated in Fig. 3. In otherords, the response coherently combines contributions

rom a spectral interval much broader than r3 /�. The fre-uency errors are more significantly reflected in the decayactor e−�T that affects the response intensity as a whole.inally, the pulse compressor should work properly pro-ided �T�1. In the following we experimentally explorehe chirped laser capabilities and limitations in thisulse-compression context.

. EXPERIMENTAL. Pulse-Compression Setuphe pulse-compression experiment takes place in 0.005%t. Er3+:YSO at 1.5 �m. At this wavelength a lot of opti-al telecom devices are available such as erbium-doped fi-er amplifiers, pig-tailed modulators, couplers, low-costbers, etc. The setup is depicted in Fig. 4. The laser beam

s split in two arms that counterpropagate to the crystal.he fields E1�t� and E3�t� follow the same path in the op-osite direction to the other engraving field E2�t�. The sig-al s�t� is carried by E2�t�. As noticed in Section 2, this is

ess demanding in terms of modulation bandwidth thanransposing the signal on E �t�. With the counterpropa-

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Damon et al. Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. B 527

ating arrangement, one gets rid of spurious echoes thatould emerge from the intermixing of the optically carriedignal components. The engraving fields are cross-olarized. The SPE response is extracted by a polarizingube and collected on the photodiode PD. The Er3+:YSOrystal sits in a helium bath cryostat. The beams are di-ected along the b axis in the 7.5 mm-thick crystal. The1�t� [respectively, E2�t�] beam waist at the crystal is5 �m (respectively, 110 �m). The E1�t� [respectively,2�t�] power is 7.1 mW (respectively, 5.2 mW). The Kram-rs degeneracy is lifted by a vertical magnetic field of2.2 T. The crystal is birefringent and the optical coher-

nce lifetime T2 of erbium ions is optimized when the op-ical axes D1 and D2 are oriented at 45° from the mag-etic field [10]. In these conditions of temperature andagnetic field size and orientation, T2�150 �s [3]. Since

he crystal neutral lines must be slanted at 45° from theertical magnetic field, half-wave plates are needed toatch their direction with the polarizing cube orienta-

ion.The chirped laser frequency time diagram is given in

ig. 5. To chirp the frequency of the engraving pulses, onecans the laser over 1.5 GHz in 1 ms, which corresponds

Er:YSO

1,7 K – 2,2 T

AO1

laser

Amp

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PD

TE

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E1(t)

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ig. 4. (Color online) Experimental setup. The chirped laseream is split in two arms that counterpropagate to ther3+:YSO. On one arm, the AO modulator AO2 transposes theignal on E2�t�. On the other arm, the shifter AO1 takes care of1�t� and E3�t� time ordering and provides the chirp rate differ-nce r1−r2. E1�t� and E3�t� are cross-polarized with E2�t�. ThePE response is detected on photodiode PD.

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z

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z

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18 MH

z

80 MH

z

ig. 5. (Color online) Laser frequency time diagram. The AOodulation operates on top of the laser direct scan to provide the

arious chirp rates that are needed for engraving and probing.oth AOs offer a bandwidth of �35 MHz centered at 80 MHz.he 17 MHz-wide lower side of AO2 bandwidth is used to trans-ose s�t� on E2�t�. The 18 MHz-wide upper side of AO1 bandwidths used to generate the small r1−r2 chirp rate difference neededo match r .

3

o r2=1.5�1012 Hz/s. This is the baseline on top of whichO1 and AO2 operate, making E1�t� and E2�t� simulta-eously available from the same source.The probe chirp rate is obtained as the laser is driven

ack to initial frequency in 15 �s. This corresponds to r31015 Hz/s. The OPLL does not react instantaneously to

he chirp rate change. To account for this latency time, weeduce the working spectral interval to �1.2 GHz. Thismposes a waiting time of 300 �s between the engravingnd the probing steps. The AO device AO1 is opened to let3�t� probe the engraved spectral interval.Both AOs offer a bandwidth of �35 MHz centered at 80Hz. On the path followed by E2�t�, one uses the 17 MHz-ide lower side of the AO2 bandwidth to transpose s�t� on2�t� (see Fig. 5). The upper side of the AO1 bandwidth issed to generate the small r1−r2�r2

2 /r3=2.251010 Hz/s chirp rate difference that is needed to match

3. This represents a frequency variation of 18 MHz overhe explored spectral interval, which indeed matches thepper side of the AO1 bandwidth.The maximum initial frequency shift of E1�t� and E2�t�

mounts to 17 MHz. This corresponds to an initial delay�11 �s, consistent with the optical coherence lifetime2=150 �s.The experiment is controlled by an arbitrary waveform

enerator [(AWG) Tektronix 5004]. This device drivesO1, feeds the signal s�t� to AO2, and controls the laserhirp. The arbitrary waveform generator delivers a volt-ge ramp to the intra-cavity EOM through a high voltagemplifier and simultaneously acts as a matched local os-illator to the OPLL [8].

. Resultss a demonstration of the processing capabilities, we gen-rate a sequence of ten shorts pulses by feeding tenvenly spectrally spaced monochromatic waveforms to2�t� through AO2. All the waveforms are applied during.8 ms, while the laser is scanned over 1.2 GHz. The re-ulting spectral grating is engraved over the erbium ionbsorption profile. With a full width at half maximum of.68 GHz, the ion spectral distribution is not uniform overhe 1.2 GHz-wide laser scanning range. By optimizing thePE intensity, we can center the absorption profile withinhe scanned interval, as illustrated in Fig. 6. Each mono-hromatic waveform gives rise to a pulse of about 1 ns du-ation. The temporal profile of each pulse should be giveny the power Fourier transform of the absorption profilever the scanned interval.

In a first experiment, the ten monochromatic wave-orms at the input of AO2 are spread over an interval of.54 MHz. This gives rise to a 1 �s-long burst of tenulses (see Fig. 7). The pulse duration is close to 1 ns asxpected, which represents a 106 compression with re-pect to the input waveform. To observe the decay as aunction of T, we increase the spectral spacing of theonochromatic waveforms. As shown in Fig. 8, the burst

s stretched to 10 �s, with the same pulse duration. Thisorresponds to an available TBP of 104, which repre-ents �1/10 of the material maximum capacity in theresent magnetic field conditions. The response decayuring the burst partly reflects the atomic coherence re-axation at 6.7�103 s−1 rate. The data fit in Fig. 8 leads to

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528 J. Opt. Soc. Am. B/Vol. 27, No. 3 /March 2010 Damon et al.

� laser frequency error rate of at most 1.1�105 s−1,hich corresponds to a quiescent laser linewidth of20 kHz. According to Eq. (3), the SPE intensity depends

n =�� /r3 expressing the fraction of � that is spannedy the probe during the laser coherence time. With theeasured � value, one gets �1.1. As can be seen in Fig.

, with such a -value, the pulse width is not affected byhe laser frequency errors. Hence, despite the linewidtheing much larger than the inverse engraving pulse du-ation, the processing capabilities are not significantly af-ected, except for faster decay, and therefore smallerignal-to-noise ratio on long time scales.

In [8] we performed a self-heterodyning measurementf the free running laser coherence time. This timeeaches 180 �s at 5 mW, which corresponds to a linewidthf 2 kHz. From the analysis of the servo error signal, welso estimated the locked-chirp laser frequency error to65 kHz, on a time scale of a few tens of microseconds,ith a chirp rate of 6�1014 Hz/s. In the present experi-ent, the 20 kHz quiescent laser linewidth, as deduced

-4 -2 0 2 4

frequency (GHz)

0.9ns

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inte

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(a.u

.)

time (ns)

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dist

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tion

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ig. 6. (Color online) Upper box: experimentally observed spec-ral distribution of the absorbing ions. The shaded area repre-ents the 1.2 GHz-wide scanning range of the laser. Lower box:ower Fourier transform of the spectral distribution over the 1.2Hz-wide working interval.

0.0 0.5 1.00.0SP

Ein

tens

ity(a

.u.)

time (µs)

0.00.13 0.14 0.15 0.16

1.3ns

SPE

inte

nsity

(a.u

.)

ig. 7. (Color online) Generation of a 1 �s-long burst of tenulses. The lower box displays the first pulse with a time scaleagnification of �35. The pulse duration is close to the 1 ns ex-

ected value.

rom the SPE response decay, appears to be intermediateetween these two linewidths. This point should be clari-ed.

. CONCLUSIONlocked-chirp frequency agile laser can be stable enough

o deliver the millisecond-long engraving pulses for broad-and waveform generation using the chirp-transform al-orithm. This important result emphasizes the possibilityo resort to simple photonic means to synthesize a com-lex wideband RF signal. We show theoretically and dem-nstrate experimentally that the laser frequency errorso not significantly alter the signal compression proce-ure. A 104 time-bandwidth product (TBP) has been ob-ained experimentally. Experiments are conducted at theelecom wavelength in Er3+:YSO. However this materialffers a limited bandwidth. To reach a bandwidth of sev-ral tens of gigahertz, alternative materials, such asr3+:LiNbO3, should be investigated.

Note: The research reported in this paper was per-ormed under the smart and enthusiastic guidance of Ivanorgeré. Ivan passed away off Brazil on June 1, 2009, in

he crash of the AF447 flight, on his way back from a sci-ntific meeting with Brazilian colleagues.

PPENDIX A: SPE RESPONSE WITH LASERREQUENCY ERRORSccording to Eq. (2), the statistical average of the SPE re-ponse intensity reads as

Ie =�1

�1+�

d��1

�1+�

d���E1����E2���E2

�����E1�����

��E3���E3������e2i���−���t, �A1�

here the probe pulse is assumed to be statistically inde-endent of the engraving pulses. The sum runs over the

time (µs)

0 5 100.0SP

Ein

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0.00.79 0.80 0.81

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inte

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.)

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ected value. The dashed (respectively, dotted dashed) line rep-esents the theoretically predicted decay accounting for the 6.7103 s−1 relaxation rate of the optical dipole alone (respectively,

ccounting for both the optical dipole relaxation and the laserrequency errors at rate �=1.1�105 s−1).

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Damon et al. Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. B 529

pectral interval �, where the excitation is uniformly dis-ributed. The frequency errors are assumed to be de-cribed by a stochastic phase ��t�. An external modulators used to shift E1�t� with respect to E2�t�. This operationoes not affect the stochastic phase. Therefore the fieldsan be described as

Ej�t� = Ej0e2i��jt+i� jrjt2+i��t�, �A2�

here 1,2=1, 3=−1, and 1/r3=1/r2−1/r1. In the frame ofhe white frequency noise model, the statistical average oftwo-time phase factor can be written as

�ei��t�−i��t��� = e�−1/2���t−t��. �A3�

ccording to this model, E1����E2��� is decorrelated from

˜2�����E1���� in Eq. (A1), as soon as ��−��� is larger than

he frequency interval that is spanned by the chirpedaser during the coherence time �−1. Therefore, underhe assumption that ��r2�−1, the four-field aver-ge �E1

����E2���E2�����E1����� reduces to �E1

����E2�����E2

�����E1����� over most of the integration domain. Weerform the calculation under this assumption and willheck its validity in the present experiment.

According to Eqs. (A2) and (A3), the two-field averageeads as

�E1����E2���� = E10

� E20 dt dt�

�e2i����−�1�t−��−�2�t��−i��r1t2−r2t2�−1/2��t−t��.

�A4�

ith the change in variables,

t,t� → u = t − t�, v =r1t − r2t�

r1 − r2,

he expression reduces to

�E1����E2���� = � r3E10

� E20e−i���2 − �1�2/�r2−r1�

� dfe−i��/r3��� − �1 − f + r3T�2L�f�, �A5�

here �=1/ r3�r2−r1�, T= ��1−�2� /r2, and

L�f� =�

�2�f�2 + ��/2�2 .

n quite the same way one obtains the spectral autocorre-ation function of the probe field:

E3�����E3����

= r3−1�E30�2 df�ei��/r3���� − �3 + f��2−��� − �3 − f��2�L�f�

�A6�

ith the help of Eqs. (A5) and (A6), one can rearrange theix-field product as

�E1����E2�����E1����E2

�������E3�����E3����

= �2�E10�2�E20�2�E30�2e−2i���−����t3+T�

� df df�L�f�L�f��

�e−i��/r3��f2−f�2�+2i����−�1�/r3+T�f−2i�����−�1�/r3+T�f�

� df�L�f��e2i���−���f�/r3, �A7�

here t3= ��3−�1� /r3. Assuming �2�r3, we can neglect theuadratic phases. Hence we are simply left with the fol-owing product of Lorentzian Fourier transforms:

�E1����E2�����E1����E2

�������E3�����E3����

= �2�E10�2�E20�2�E30�2e−2i���−����t3+T�

�e�−�/2r3����−�1+r3T�+���−�1+r3T�+��−����. �A8�

ubstitution of Eq. (A8) into Eq. (A1) leads to

Ie = �2�E10�2�E20�2�E30�2r3T

�+r3T

d�r3T

�+r3T

d��

�e2i���−���e−��+��+��−�����/�2r3�, �A9�

here = t− t3−T. Integration is easily performed. Ex-ressing the average response intensity as a function of nd �, one finally obtains

Ie�,�� = Ie�0,0�e−�T2

2 + �2���2�1 − e− cos�2���

− e−sin�2���

2��� , �A10�

here =�� /r3.

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