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Super-Gaussian field statistics in broadband laser emission
cm .__
1 February 1997
__ l!iB
OPTICS COMMUNICATIONS
ELSEVIER
Optics
Communications 135(1997) 157-163
Full length article
Super-Gaussian field statistics in broadband laser emission A.J. Bain *, A. Squire Department of Physics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK
Received 17 July 1996;accepted 15 October 19%
Abstract
Second harmonic and four-wave mixing studies of the above threshold oscillation of a conventional broadband Nd:YAG pumped dye laser indicate the presence of super-Gaussian field statistics. Direct transfer of fast (> 10” Hz) optical pump noise to the dye laser output is seen to occur over a wide range of pumping conditions giving a broadband light source with both fast (picosecond) and ultrafast (femtosecond) optical coherence. Keywords:
Super-Gaussian; Laser, Coherence;Statistics; Broadband, Noise 2. Optical coherence in broadband
1. Introduction
The high powers obtained with broadband laser radiation have found wide use in the study of nonlinear optics, for example in the measurement of multiphoton cross-sections [I], multiplexed coherent Raman spectroscopy of combustion processes [2-51, the measurement of ultrafast relaxation dynamics [6-91, and fluctuation spectroscopy [lO,l 11.All nonlinear processes are highly sensitive to the field statistics [ 1,121 and the conventional assumption that the latter are Gaussian for broadband fields is almost universally applied in the analysis of experimental data. Here we demonstrate for the first time that super-Gaussian field statistics hold for the output of a conventional broadband dye laser operating over a wide range of pumping conditions. For such a laser, the direct transfer of gigahertz optical pump noise, associated with the picosecond coherence properties of the frequency doubled, Q-switched Nd:YAG pump laser, to the dye laser output results in broadband light with two distinct coherence times. The influence of such dual coherence on four-wave mixing experiments can be profound. In the light of this work a reappraisal of standard models for broadband laser emission is indicated.
In contrast ‘to single frequency lasers it is well known that the output intensity of a free running broadband laser suffers intrinsic random fluctuation arising from the complex interferences between its many oscillating longitudinal modes [1,13,14]. The output statistics are Gaussian and the timescale of the fluctuations (the coherence time 7,) is inversely proportional to the lasing bandwidth. Fig. 1 shows the time series (in units of the coherence time) for the cycle averaged intensity of Gaussian distributed light generated for a Gaussian spectral profile by the procedure of Vannucci and Teich [ 151.This results in a random noise train where the intensity fluctuations about the mean ZAv are exponentially distributed according to [ 121
(1) The average width of the noise spikes correlates with the coherence time 7c of the light [l]. This behaviour is solely determined by the bandwidth Av of the source, which for a Gaussian spectral profile (typical of a broadband dye laser) is given by [13]
T==
(A+’
* Corresponding author. 0030~4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-40
18(96)00672-4
laser oscillation
J
2 W) .
-
?r
(2)
158
A.J. Bain, A. Squire/ Optics Communications 135 (1997) 157-163
Fig. 1. The computer generated time series (in units of the coherence time TJ for the cycle averaged intensity of Gaussian distributed light, assuming a light source with a Gaussian spectral profile.
The determination of 7c can be achieved in an autocorrelation measurement involving non-collinear second harmonic generation [ 16,171 where the laser pulse is split into two identical beams with one suffering a variable optical delay 7 before they are recombined in a frequency doubling crystal. The simultaneous arrival of a photon at w from both beams is required for the generation of a sum-frequency photon at 2~. The’ intensity of the generated signal measured by a slow photodetector is thus proportional to the temporal overlap of the two fields, Z(2w) aj+ml(t) -m
1(t-
7) dt.
This is directly proportional to the normal&d second-order autocorrelation function Gc2)(7) of the light [ 171,
(4) Fof a Gaussian distributed light field with a Gaussian spectral profile Gc2)(~) is given by [ 12,131 G(‘)(T) = (1 + exp[ - Tr(%-/~~)‘]},
rations using this scheme w&e investigated and with a range of laser dyes and dye mixtures, broadband Gaussian spectral outputs between 560 nm and 750 nm were achieved, with variable bandwidths giving adjustable correlation times of 66 fs to 1.3 ps. Typical output energies from the oscillator were in the region of 6 mJ. Determination of Gc2)(,r>for the dye laser output was achieved using non-collinear second harmonic generation (SHG). The basic experimental configuration for such measurements is shown in Fig. 2. A portion of the oscillator output (between 100-400 pJ) was divided into two beams of equal intensity, one beam taking a fixed optical path the other suffering a variable delay, before being recombined at an angle of ca. 4” in a 100 pm thick crystal of KD *P using two 25 cm focal length aplanatic lenses. The two beam SHG signal is spatially filtered from the fundamental and the single beam SHG signal using an iris diaphragm. Any further potential contributions from the fundamental laser inputs are removed by spectral filtering using UGll filters (Schott) and a mini-monochromator (Jobin Yvon HIOUV). The SHG signal is detected by a UV photomultiplier (Thorn EMI Q9125A) and a normalising signal (where required) is generated by directing one of the single beam SHG signals through a stack of UGll filters to a second photomultiplier (Hamamatsu R212). The variable delay between the two beams was obtained using a delay line (Photon Control PTS 16OOM)which provided 300 mm of overall travel with zero backlash. Operated by a “Nanostepper” driver unit (Photon Control) controlled by a personal computer (DEC286), this arrangement allowed the optical path difference to be varied with a resolution of 1.0 pm. The autocorrelation and reference SHG signals were averaged using a computer controlled boxcar integrator (Stanford SR250 and SR245 units) allowing the collection of a normalised SHG autocorrelation signal as a function of the delay line position and hence
(9
which for zero time delay gives the normalised second moment of the intensity fluctuation in the field as G(2)(O) = 2. In practice however values of 1.7 and above have been generally accepted as evidence of Gaussian statistical behaviour [8,14,18,19].
MO
INPUT FIELD, ---?-
3. Experimental The dye laser system used in this work employs a modified Littman oscillator [20] (Spectron Laser Systems) which was transversely pumped by a portion (ca. 20 mJ) of the 532 nm output from a 10 Hz Q-switched Nd:YAG laser (Spectron SL400). Broadband oscillation was achieved by interrupting the narrowband feedback from the diffraction grating [8]. A number of oscillator configu-
Fig. 2. The experimental configuration autocorrelation measurements.
for non-co&ear
SHG
A.J. Bain, A. Squire/Optics
Frequency (cm’)
Communications
Delay (femtoseconds)
Fig. 3. (a) The output spectmm of the broadband laser operating with rhodamine 6G. The spectrum fits well to a Gaussian function with a FWHM of 133 cm-* at a centre frequency of 17557 cm-’ (569 nm), indicating an apparent coherence time, rC, of 166 fs. (b) The autocorrelation function G(‘)(T) of the broadband rhodamioe 6G source (shown in (a)) measured by performing an SHG autocorrelation. The data have been fitted to a Gaussian function
with a FWHM of 154 fs and displays a second moment in the intensity of 1.85.
the delay I-. Typical autocorrelation ’ and spectral characteristics of the dye laser operating well above threshold are shown in Fig. 3 indicating for Rhodamine 6G apparent 154 fs Gaussian noise with Gc2)(0) = 1.85.
4. Super-Gaussian tion
field statistics in broadband
159
135 (1997) 157-163
available from this extended scan clearly shows superGaussian [19] statistical behaviour with an overall second moment Gc2)(0) of 3.6. The similarity of the broad correlation feature with that of the Nd:YAG autocorrelation, (Fig. 5a) appears to locate the picosecond fluctuations in the YAG pump laser as the source of this extra coherence. This is rigorously established by performing a sumfrequency cross correlation in a similar manner to the SHG autocorrelation measurements described above. From Fig. 5b it can be seen that this recovers a single 42 ps correlation feature similar to that seen in both autocorrelations. This confitms the picosecond fluctuations arising from the Nd:YAG pump laser as the source of the additional coherence and establishes the statistical independence of these fluctuations from those arising from intrinsic cavity mode noise in the dye laser. Furthermore, the robustness of the transferred coherence feature is established since similar behaviour has also been observed by us in a number of other gain media (see below for example). From these results the broadband field from the dye laser, E(I), can thus be described in terms of a slowly varying (nanosecond) envelope E,(r), modulated by two uncorrelated random variables u,(r) and u,(t), representing intrinsic cavity noise (typically < 200 fs) and pump modulations (- 32 ps) respectively, a(t)
=%(t)
The measured
u,(t)
UP(t).
second-order
(6) autocorrelation
function
;o
A’
for
oscilla-
From the autocorrelation and spectral measurements detailed above the output of the dye laser falls well within accepted criteria for a Gaussian light source. Evidence however of more complex coherence in the broadband output was observed by us in a number of four wave mixing experiments [21,22] indicating an additional degree of (ps) coherence in the broadband field. This is clearly resolved in an extended autocorrelation measurement of the dye laser (Fig. 4b) which shows a femtosecond correlation feature associated with intrinsic cavity mode noise superimposed on a much broader (32 ps) feature not apparent over the limited delay time of a conventional autocorrelation, as in Fig. 3b. Such behaviour cannot be inferred from the Gaussian spectral profile of the dye laser which is dominated by the intrinsic dye bandwidth. From conventional analysis (Eq. (1)) the DCM oscillation bandwidth of 250 cm-’ would indicate a single coherence time of 89 fs (Fig. 4a). Furthermore, the additional information
’ The relationship between the autocorrelation FWHM, ague, and the coherence time, TV, is given by rC = 1.064~~~~~ 1131.
.G
B
I,;,
-Irn
Al
Ii
io
Delay z (femtoseconds)
2
Delay T (picoseconds) Fig. 4. (a) A normalised sum-frequency autocorrelation measure ment of the broadband DCM dye laser centred at 637 mn with a frequency bandwidth of 250 cm-‘. (b> An extended autocorrelation measurement clearly revealing the presence of additional coherence in the broadband field. From the extended scan an overall second moment G(*)(O) = 3.6 is obtained, showing the field statistics to be clearly super-Gaussian.
160
A.J. Bain, A. Squire/Optics Communications 135 (1997) 157-163
YAG I:
0::
I::::
which is simply the product bf the autocorrelation tions for un( t) and uP( t) respectively,
func-
G”‘(r)
(10)
= Gg’(r)
This expression is identical, in form, to that predicted by Jakeman et al. [23] to describe the coherence properties of (thermal) mercury arc light randomly modulated by (kHz) dynamic scattering processes in a 50 pm thick layer of the liquid crystal MBBA. Since electronic correlation techniques were employed, however, they were unable to resolve the intrinsic ultrafast fluctuations (I; 10e9 s) in the mercury arc source and a composite autocorrelation structure, as in Fig. 4, was not observed. Assuming Gaussian statistical behaviour for both noise terms we may substitute Eq. (5) into Eq. (10) to give
(b)
3
G(')(T) =
(1+ exp[ x(1
DELAY
T (PICOSECONDS)
Fig. 5. The upper trace shows a normalised autocorrelation measurement of the Nd:YAG pump laser. The data fits to a Gaussian function giving an autocorrelation width of 40 ps with a correlation coefficient G?)(O) = 2 (exact Gaussian behaviour). The lower trace shows the cross-correlation sum-frequency signal obtained from the Nd:YAG and DCM dye laser pulses. the short delay time behaviour to Eq. (4) as
G’*‘(r)
j-+ICc(t) --m
=
s*(r)
l’*s(t)
of c(t) is given according
e(r+r) c*(t)
-cc
e(t)
E*(~+T) .s*(t)
dt
.
dt
Go’.
- rr(r/rD)*]}
+exp[
(11)
-a-(r/rr)‘]},
where ro and rP are the coherence times of the intrinsic cavity and pump noise, respectively. Calculating ro from its spectral properties (Eq. (l)), equating the pump noise with the intrinsic coherence properties of the Nd:YAG pump laser i.e. or = ryAG, and substituting into Eq. (11) we were able to closely reproduce the dye laser autocorrelation of Fig. 4b. This would imply a nearly linear transfer of the optical coherence properties from the Nd:YAG to the output of the dye laser when strongly pumped (ca. 25% pump conversion). A computer generated time series showing the intensity fluctuations for such dual coherence light about its mean ZAv is given in Fig. 6. The two Gaussian modulations u,(t) and u,(t) were chosen with coherence time rP = 257,. The greater degree of temporal fluctuations in dual coherence light about its mean is
(7) Substituting
Eq. (6) into (7) gives -I
Gc2)(r)
’
I
s
’
’
s
’
’
:
‘r
=(Uo(t)Up~~~~(t)Uo(t+T) xu,(t+
r)uG(t+
r)up*(t+
7))
x(u,(r)u,(t)u~(t)u,*(t)>-2.
(8)
Here we have assumed both ergodicity of random variables u,(t) and u,(t), in order to replace the time average with an ensemble average ( > 1121, and negligible variation in the nanosecond envelope En(t) on a timescale over which these variables fluctuate. Taking r+,(t) to be uncorrelated with u,(t) Eq. (8) becomes
Gc2'(7)
=
(u,(t)u;(t)u,(f+
7)uI;(t+
7))
2 x
(up(t)u,*(t)u,(t+T)U~(t+T))
u;(f>>’
)
($))
Fig. 6. A computer generated time series of the fluctuations of a super-Gaussian light source, with two distinct coherence times or and mu where or = 257,. The larger temporal fluctuations in Z(t) about the mean intensity IAv are clearly apparent and contrast strongly with those in Fig. 1, where the fluctuations are equivalent to u,(t) alone.
A.J. Bain, A. Squire/Optics
Communications
Fig. 7. Illustration of a composite autocorrelation measurement for broadband emission with two well separated coherence times in and or. In the experimental autocorrelations performed in this work the differences between in and TV, are more marked (Tp = (240-510) Tn).
clearly apparent and contrasts strongly the fluctuations are equivalent to u,(t)
135 (1997) 157-163
161
losses, threshold is only reached with increasingly large pump fluctuations and a more intermittent emission train results. This has a consequent reduction in self-overlap outside the pump coherence time, leading to a significantly reduced autocorrelation background. For low cavity losses, lasing follows the full fluctuations in the pump and normal Gaussian behaviour is recovered (G’,~)(T) - 1.8 1). In contrast the statistics of intrinsic cavity mode noise in the dye laser remain approximately Gaussian (G$)(O) - 1.81) under all pumping conditions. The overall statistical behaviour of the laser (GC2)(0)) is thus super-Gaussian under all operating regimes and changes in its behaviour arise solely from the pump induced modulations u,(t). The nature of the noise transfer process can be inferred from a comparison of the coherence time of u,(t), rp, to the pump laser’s coherence trme, ryAG (42 ps). A plot of rp versus the dye laser output energy is shown in Fig. 9. For linear noise transfer the pump intensity fluctuations in the dye laser, Zr( t), directly reproduce the random fluctuations in the pump laser output, IYAG(t). The coherence times rp and ryAG should therefore be equal. In Fig. 9 however, it is seen that for low pump conversion (output
with Fig. 1 where alone.
5. Pump noise transfer Following our observations of significant pump noise transfer in the above threshold operation of the dye laser, a detailed investigation of the statistical properties of the broadband output was undertaken as a function of lasing threshold. The threshold conditions were most conveniently altered by placing a range of neutral density filters in the dye laser cavity with care being taken to avoid parasitic oscillations. For a fixed Nd:YAG pump energy of 20 mJ this allowed (using Rhodamine 6G) a variation in the output energy from 200 PJ to 6 mJ (no intracavity filters present). Extended dye laser autocorrelation measurements were performed over this energy range. The statistical independence and large coherence time separation (ca. 40 ps and 142 fs) of the two noise terms allows the overall (measured) correlation function to be decomposed into picosecond and femtosecond autocorrelation functions from which the second moments of the two noise terms are easily determined (see Fig. 7). Plots of G$?(O), Gg)(O) and G(‘)(O) versus the dye laser output energy are shown in Fig. 8. Super-Gaussian statistical behaviour in the pump noise, (with a maximum second moment of G$?(O) - 12) is observed at significant cavity losses (output energies below 500 pJ). Here with rising
OUTPUT ENERGY FROM DYE LASER CAVlTY (ml)
Fig. 8. (a) Plot of the variation in C&?(O) as a function of the lasing threshold, showing distinctly super-Gaussian behaviour for the pump noise component u,(t) at low output energies tending to Gaussian behaviour (G’,z)(O)= 2) at normal oscillation. (b) A similar plot for the intrinsic femtosecond dye noise shows C@(O) to be unaffected by the pumping conditions. (c) The variation in the overall second moment of the broadband field (given by the product G$?(O)Gg)(O)) indicates super-Gaussian statistical behaviour under all pumping conditions, with the low output behaviour mirroring that of u,(t).
AJ. Bain, A. Squire/Optics Communications 135 (1997) 157-163
162
band dye laser emission are often implicitly assumed in the analysis of a number of quantitative nonlinear optical measurements 12-91. In the light of our findings an accurate characterisation of optical field coherence would appear to be an essential prerequisite of any nonlinear mixing experiment.
”
0
b
2
”
I
4
”
I
”
6
k 8
DYE LASER OUTPUT ENERGY (mJ) Fig. 9. A plot of 7p as a function of dye laser output energy. Under all pumping conditions, 7p < ~~~o, with the noise transfer approaching a linear regime (TV = 7vAG) at normal output energies.
energies below 500 p,J) the multiplicative noise transfer is highly nonlinear (TV m 20-26 ps) and tends towards but does not reach a fully linear regime even with significant pump conversion (e.g. 7p = 36 ps at 5 mJ). For low pump conversion the measurements of Gc2)(0) (Fig. SC) show an inverse output energy dependence in agreement with previous studies of below threshold noise transfer in continuous wave (c.w.) single mode dye lasers [24,25]. In such systems, close to threshold low pass filtering of pump noise is observed with linear noise transfer well above threshold [26]. Theoretical models of C.W. multimode systems predict similar near threshold behaviour [27]. Such studies involve slow &Hz-MHz), low amplitude, non-fully-mcdulated C.W. pump fluctuations. In contrast the ca. 5 ns Q-switched pump pulses used in this work are fully modulated by random Gaussian fluctuations with a frequency greater than 10” Hz. Here the dye laser is pumped rapidly and randomly above and below threshold and no filtering of the pump noise is observed *. This clearly corresponds to an operating regime far removed from previous investigations [24-291 and to our knowledge this work represents the first study of noise transfer and optical coherence in this widely used class of laser.
6. Conclusions We have shown that the output of a conventional pulsed broadband dye laser obeys super-Gaussian statistics over a wide range of pumping conditions. This is seen to arise from the direct transfer of fast (42 ps) Gaussian fluctuations in the pump laser. Gaussian statistics in broad-
* The upper lasing level population rates in organic dyes are typically on the order of lo’* s- ’ [17]. It would therefore not be unreasonable to expect the gain medium to respond to fast pump fluctuations and that this rate would represent the upper bound to its frequency response to external noise.
Acknowledgements We would like to thank EPSRC for the financial support of this work, Professor R. Loudon FRS for helpful discussions and E. White for assistance with the experimental work.
References [l] A.V. Masalov, in: Progress in Optics XXII, ed. E. Wolf (Elsevier Science, 1985) pp. 147-196. [2] D.A. Greenhalgh and S.T. Whittley, Appl. Optics 24 (1985) 907. [3] S. Kriill and D. Sandell, J. Opt. Sot. Am. B 5 (1988) 1910. [4] A.M.F. Lau and R.L. Farrow, Optics Lett. 14 (1989) 367. [5] D.R. Snelling, G.J. Smallwood, R.A. Sawchuk and T. Parameswaran, Appl. Optics 26 (1987) 110. [6] T. Kobayashi, A. Terasaki, T. Hattori and K. Kurokawa, Appl. Phys. B 47 (1988) 107. [7] T. Hattori and T. Kobayashi, J. Chem. Phys. 94 (1991) 3332. [8] M.A. Dugan and A.C. Albrecht, Phys. Rev. A 43 (1991) 3922. [9] S.A. Schaertel, A.C. Albrecht, A. Lau and A. Kummrow, Appl. Phys. B 59 (1994) 377. [lo] A.V. Masalov and L. Allen, J. Phys. B 15 (1982) 2375. [II] I.V. Jyotsna and G.S. Agarwal, Phys. Rev. A 51 (1995) 3169. [12] R. Loudon, Quantum Theory of Light (Oxford University Press, New York, 1983). [13] J.W. Goodman, Statistical Optics (Wiley, New York, 1985). [14] A.A. Griitter, H.P. Weber and R. D&ndliker, Phys. Rev. 185 (1969)629. [ 151 G. Vannucci and MC. Teich, Appl. Optics 19 (1980) 548. [16] H.A. Pike and M. Hercher, J. Appl. Phys. 41 (1970) 4562. [17] G.R. Fleming, Chemical Applications of Ultrafast Spectroscopy (Oxford University Press, New York 1986). [18] H.P. Weber and H.G. Danielmeyer, Phys. Rev. A 2 (1970) 2074. [19] M. Harris, G.N. Pearson, C.A. Hill and J.M. Vaughan, Appl. Optics 33 (1994) 7226. [20] M.G. Littman and H.J. Metcalf, Appl. Optics 17 (1978) 2224. 1211 A. Squire, Incoherent laser light as a probe of ultrafast nonlinear optics, PhD thesis, University of Essex (1996), unpublished. [22] A.J. Bain and A. Squire, Nonlinear optical correlation techniques for incoherent laser fields, to be published. [23] E. Jakeman, P.N. Pusey and J.M. Vaughan, Optics Comm. 17 (1976) 305.
AJ. Bain, A. Squire/ Optics Communications 135 (1997) 157-163 [24] R. Short, L. Mandel and R. Roy, Phys. Rev. Lett. 49 (1982) 647. [25] P. Lett, R. Short and L. Mandel, Phys. Rev. Lett. 52 (1984) 341. [26] A.W. Yu, G.P. Agrawal and R. Roy, Optics I&t. 12 (1987) 806.
163
[27] M. Beck, I. McMacltin and M.G. Raymer, Phys. Rev. A 40 (1989) 2410. [28] K.J. Phillips, M.R. Young and S. Singh, Phys. Rev. A 44 (1991) 3239. [29] D. Von der Lmde, Appl. Phys. B 39 (1986) 201.
1 February 1997
__ l!iB
OPTICS COMMUNICATIONS
ELSEVIER
Optics
Communications 135(1997) 157-163
Full length article
Super-Gaussian field statistics in broadband laser emission A.J. Bain *, A. Squire Department of Physics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK
Received 17 July 1996;accepted 15 October 19%
Abstract
Second harmonic and four-wave mixing studies of the above threshold oscillation of a conventional broadband Nd:YAG pumped dye laser indicate the presence of super-Gaussian field statistics. Direct transfer of fast (> 10” Hz) optical pump noise to the dye laser output is seen to occur over a wide range of pumping conditions giving a broadband light source with both fast (picosecond) and ultrafast (femtosecond) optical coherence. Keywords:
Super-Gaussian; Laser, Coherence;Statistics; Broadband, Noise 2. Optical coherence in broadband
1. Introduction
The high powers obtained with broadband laser radiation have found wide use in the study of nonlinear optics, for example in the measurement of multiphoton cross-sections [I], multiplexed coherent Raman spectroscopy of combustion processes [2-51, the measurement of ultrafast relaxation dynamics [6-91, and fluctuation spectroscopy [lO,l 11.All nonlinear processes are highly sensitive to the field statistics [ 1,121 and the conventional assumption that the latter are Gaussian for broadband fields is almost universally applied in the analysis of experimental data. Here we demonstrate for the first time that super-Gaussian field statistics hold for the output of a conventional broadband dye laser operating over a wide range of pumping conditions. For such a laser, the direct transfer of gigahertz optical pump noise, associated with the picosecond coherence properties of the frequency doubled, Q-switched Nd:YAG pump laser, to the dye laser output results in broadband light with two distinct coherence times. The influence of such dual coherence on four-wave mixing experiments can be profound. In the light of this work a reappraisal of standard models for broadband laser emission is indicated.
In contrast ‘to single frequency lasers it is well known that the output intensity of a free running broadband laser suffers intrinsic random fluctuation arising from the complex interferences between its many oscillating longitudinal modes [1,13,14]. The output statistics are Gaussian and the timescale of the fluctuations (the coherence time 7,) is inversely proportional to the lasing bandwidth. Fig. 1 shows the time series (in units of the coherence time) for the cycle averaged intensity of Gaussian distributed light generated for a Gaussian spectral profile by the procedure of Vannucci and Teich [ 151.This results in a random noise train where the intensity fluctuations about the mean ZAv are exponentially distributed according to [ 121
(1) The average width of the noise spikes correlates with the coherence time 7c of the light [l]. This behaviour is solely determined by the bandwidth Av of the source, which for a Gaussian spectral profile (typical of a broadband dye laser) is given by [13]
T==
(A+’
* Corresponding author. 0030~4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-40
18(96)00672-4
laser oscillation
J
2 W) .
-
?r
(2)
158
A.J. Bain, A. Squire/ Optics Communications 135 (1997) 157-163
Fig. 1. The computer generated time series (in units of the coherence time TJ for the cycle averaged intensity of Gaussian distributed light, assuming a light source with a Gaussian spectral profile.
The determination of 7c can be achieved in an autocorrelation measurement involving non-collinear second harmonic generation [ 16,171 where the laser pulse is split into two identical beams with one suffering a variable optical delay 7 before they are recombined in a frequency doubling crystal. The simultaneous arrival of a photon at w from both beams is required for the generation of a sum-frequency photon at 2~. The’ intensity of the generated signal measured by a slow photodetector is thus proportional to the temporal overlap of the two fields, Z(2w) aj+ml(t) -m
1(t-
7) dt.
This is directly proportional to the normal&d second-order autocorrelation function Gc2)(7) of the light [ 171,
(4) Fof a Gaussian distributed light field with a Gaussian spectral profile Gc2)(~) is given by [ 12,131 G(‘)(T) = (1 + exp[ - Tr(%-/~~)‘]},
rations using this scheme w&e investigated and with a range of laser dyes and dye mixtures, broadband Gaussian spectral outputs between 560 nm and 750 nm were achieved, with variable bandwidths giving adjustable correlation times of 66 fs to 1.3 ps. Typical output energies from the oscillator were in the region of 6 mJ. Determination of Gc2)(,r>for the dye laser output was achieved using non-collinear second harmonic generation (SHG). The basic experimental configuration for such measurements is shown in Fig. 2. A portion of the oscillator output (between 100-400 pJ) was divided into two beams of equal intensity, one beam taking a fixed optical path the other suffering a variable delay, before being recombined at an angle of ca. 4” in a 100 pm thick crystal of KD *P using two 25 cm focal length aplanatic lenses. The two beam SHG signal is spatially filtered from the fundamental and the single beam SHG signal using an iris diaphragm. Any further potential contributions from the fundamental laser inputs are removed by spectral filtering using UGll filters (Schott) and a mini-monochromator (Jobin Yvon HIOUV). The SHG signal is detected by a UV photomultiplier (Thorn EMI Q9125A) and a normalising signal (where required) is generated by directing one of the single beam SHG signals through a stack of UGll filters to a second photomultiplier (Hamamatsu R212). The variable delay between the two beams was obtained using a delay line (Photon Control PTS 16OOM)which provided 300 mm of overall travel with zero backlash. Operated by a “Nanostepper” driver unit (Photon Control) controlled by a personal computer (DEC286), this arrangement allowed the optical path difference to be varied with a resolution of 1.0 pm. The autocorrelation and reference SHG signals were averaged using a computer controlled boxcar integrator (Stanford SR250 and SR245 units) allowing the collection of a normalised SHG autocorrelation signal as a function of the delay line position and hence
(9
which for zero time delay gives the normalised second moment of the intensity fluctuation in the field as G(2)(O) = 2. In practice however values of 1.7 and above have been generally accepted as evidence of Gaussian statistical behaviour [8,14,18,19].
MO
INPUT FIELD, ---?-
3. Experimental The dye laser system used in this work employs a modified Littman oscillator [20] (Spectron Laser Systems) which was transversely pumped by a portion (ca. 20 mJ) of the 532 nm output from a 10 Hz Q-switched Nd:YAG laser (Spectron SL400). Broadband oscillation was achieved by interrupting the narrowband feedback from the diffraction grating [8]. A number of oscillator configu-
Fig. 2. The experimental configuration autocorrelation measurements.
for non-co&ear
SHG
A.J. Bain, A. Squire/Optics
Frequency (cm’)
Communications
Delay (femtoseconds)
Fig. 3. (a) The output spectmm of the broadband laser operating with rhodamine 6G. The spectrum fits well to a Gaussian function with a FWHM of 133 cm-* at a centre frequency of 17557 cm-’ (569 nm), indicating an apparent coherence time, rC, of 166 fs. (b) The autocorrelation function G(‘)(T) of the broadband rhodamioe 6G source (shown in (a)) measured by performing an SHG autocorrelation. The data have been fitted to a Gaussian function
with a FWHM of 154 fs and displays a second moment in the intensity of 1.85.
the delay I-. Typical autocorrelation ’ and spectral characteristics of the dye laser operating well above threshold are shown in Fig. 3 indicating for Rhodamine 6G apparent 154 fs Gaussian noise with Gc2)(0) = 1.85.
4. Super-Gaussian tion
field statistics in broadband
159
135 (1997) 157-163
available from this extended scan clearly shows superGaussian [19] statistical behaviour with an overall second moment Gc2)(0) of 3.6. The similarity of the broad correlation feature with that of the Nd:YAG autocorrelation, (Fig. 5a) appears to locate the picosecond fluctuations in the YAG pump laser as the source of this extra coherence. This is rigorously established by performing a sumfrequency cross correlation in a similar manner to the SHG autocorrelation measurements described above. From Fig. 5b it can be seen that this recovers a single 42 ps correlation feature similar to that seen in both autocorrelations. This confitms the picosecond fluctuations arising from the Nd:YAG pump laser as the source of the additional coherence and establishes the statistical independence of these fluctuations from those arising from intrinsic cavity mode noise in the dye laser. Furthermore, the robustness of the transferred coherence feature is established since similar behaviour has also been observed by us in a number of other gain media (see below for example). From these results the broadband field from the dye laser, E(I), can thus be described in terms of a slowly varying (nanosecond) envelope E,(r), modulated by two uncorrelated random variables u,(r) and u,(t), representing intrinsic cavity noise (typically < 200 fs) and pump modulations (- 32 ps) respectively, a(t)
=%(t)
The measured
u,(t)
UP(t).
second-order
(6) autocorrelation
function
;o
A’
for
oscilla-
From the autocorrelation and spectral measurements detailed above the output of the dye laser falls well within accepted criteria for a Gaussian light source. Evidence however of more complex coherence in the broadband output was observed by us in a number of four wave mixing experiments [21,22] indicating an additional degree of (ps) coherence in the broadband field. This is clearly resolved in an extended autocorrelation measurement of the dye laser (Fig. 4b) which shows a femtosecond correlation feature associated with intrinsic cavity mode noise superimposed on a much broader (32 ps) feature not apparent over the limited delay time of a conventional autocorrelation, as in Fig. 3b. Such behaviour cannot be inferred from the Gaussian spectral profile of the dye laser which is dominated by the intrinsic dye bandwidth. From conventional analysis (Eq. (1)) the DCM oscillation bandwidth of 250 cm-’ would indicate a single coherence time of 89 fs (Fig. 4a). Furthermore, the additional information
’ The relationship between the autocorrelation FWHM, ague, and the coherence time, TV, is given by rC = 1.064~~~~~ 1131.
.G
B
I,;,
-Irn
Al
Ii
io
Delay z (femtoseconds)
2
Delay T (picoseconds) Fig. 4. (a) A normalised sum-frequency autocorrelation measure ment of the broadband DCM dye laser centred at 637 mn with a frequency bandwidth of 250 cm-‘. (b> An extended autocorrelation measurement clearly revealing the presence of additional coherence in the broadband field. From the extended scan an overall second moment G(*)(O) = 3.6 is obtained, showing the field statistics to be clearly super-Gaussian.
160
A.J. Bain, A. Squire/Optics Communications 135 (1997) 157-163
YAG I:
0::
I::::
which is simply the product bf the autocorrelation tions for un( t) and uP( t) respectively,
func-
G”‘(r)
(10)
= Gg’(r)
This expression is identical, in form, to that predicted by Jakeman et al. [23] to describe the coherence properties of (thermal) mercury arc light randomly modulated by (kHz) dynamic scattering processes in a 50 pm thick layer of the liquid crystal MBBA. Since electronic correlation techniques were employed, however, they were unable to resolve the intrinsic ultrafast fluctuations (I; 10e9 s) in the mercury arc source and a composite autocorrelation structure, as in Fig. 4, was not observed. Assuming Gaussian statistical behaviour for both noise terms we may substitute Eq. (5) into Eq. (10) to give
(b)
3
G(')(T) =
(1+ exp[ x(1
DELAY
T (PICOSECONDS)
Fig. 5. The upper trace shows a normalised autocorrelation measurement of the Nd:YAG pump laser. The data fits to a Gaussian function giving an autocorrelation width of 40 ps with a correlation coefficient G?)(O) = 2 (exact Gaussian behaviour). The lower trace shows the cross-correlation sum-frequency signal obtained from the Nd:YAG and DCM dye laser pulses. the short delay time behaviour to Eq. (4) as
G’*‘(r)
j-+ICc(t) --m
=
s*(r)
l’*s(t)
of c(t) is given according
e(r+r) c*(t)
-cc
e(t)
E*(~+T) .s*(t)
dt
.
dt
Go’.
- rr(r/rD)*]}
+exp[
(11)
-a-(r/rr)‘]},
where ro and rP are the coherence times of the intrinsic cavity and pump noise, respectively. Calculating ro from its spectral properties (Eq. (l)), equating the pump noise with the intrinsic coherence properties of the Nd:YAG pump laser i.e. or = ryAG, and substituting into Eq. (11) we were able to closely reproduce the dye laser autocorrelation of Fig. 4b. This would imply a nearly linear transfer of the optical coherence properties from the Nd:YAG to the output of the dye laser when strongly pumped (ca. 25% pump conversion). A computer generated time series showing the intensity fluctuations for such dual coherence light about its mean ZAv is given in Fig. 6. The two Gaussian modulations u,(t) and u,(t) were chosen with coherence time rP = 257,. The greater degree of temporal fluctuations in dual coherence light about its mean is
(7) Substituting
Eq. (6) into (7) gives -I
Gc2)(r)
’
I
s
’
’
s
’
’
:
‘r
=(Uo(t)Up
r)uG(t+
r)up*(t+
7))
x(u,(r)u,(t)u~(t)u,*(t)>-2.
(8)
Here we have assumed both ergodicity of random variables u,(t) and u,(t), in order to replace the time average with an ensemble average ( > 1121, and negligible variation in the nanosecond envelope En(t) on a timescale over which these variables fluctuate. Taking r+,(t) to be uncorrelated with u,(t) Eq. (8) becomes
Gc2'(7)
=
(u,(t)u;(t)u,(f+
7)uI;(t+
7))
(up(t)u,*(t)u,(t+T)U~(t+T))
)
($))
Fig. 6. A computer generated time series of the fluctuations of a super-Gaussian light source, with two distinct coherence times or and mu where or = 257,. The larger temporal fluctuations in Z(t) about the mean intensity IAv are clearly apparent and contrast strongly with those in Fig. 1, where the fluctuations are equivalent to u,(t) alone.
A.J. Bain, A. Squire/Optics
Communications
Fig. 7. Illustration of a composite autocorrelation measurement for broadband emission with two well separated coherence times in and or. In the experimental autocorrelations performed in this work the differences between in and TV, are more marked (Tp = (240-510) Tn).
clearly apparent and contrasts strongly the fluctuations are equivalent to u,(t)
135 (1997) 157-163
161
losses, threshold is only reached with increasingly large pump fluctuations and a more intermittent emission train results. This has a consequent reduction in self-overlap outside the pump coherence time, leading to a significantly reduced autocorrelation background. For low cavity losses, lasing follows the full fluctuations in the pump and normal Gaussian behaviour is recovered (G’,~)(T) - 1.8 1). In contrast the statistics of intrinsic cavity mode noise in the dye laser remain approximately Gaussian (G$)(O) - 1.81) under all pumping conditions. The overall statistical behaviour of the laser (GC2)(0)) is thus super-Gaussian under all operating regimes and changes in its behaviour arise solely from the pump induced modulations u,(t). The nature of the noise transfer process can be inferred from a comparison of the coherence time of u,(t), rp, to the pump laser’s coherence trme, ryAG (42 ps). A plot of rp versus the dye laser output energy is shown in Fig. 9. For linear noise transfer the pump intensity fluctuations in the dye laser, Zr( t), directly reproduce the random fluctuations in the pump laser output, IYAG(t). The coherence times rp and ryAG should therefore be equal. In Fig. 9 however, it is seen that for low pump conversion (output
with Fig. 1 where alone.
5. Pump noise transfer Following our observations of significant pump noise transfer in the above threshold operation of the dye laser, a detailed investigation of the statistical properties of the broadband output was undertaken as a function of lasing threshold. The threshold conditions were most conveniently altered by placing a range of neutral density filters in the dye laser cavity with care being taken to avoid parasitic oscillations. For a fixed Nd:YAG pump energy of 20 mJ this allowed (using Rhodamine 6G) a variation in the output energy from 200 PJ to 6 mJ (no intracavity filters present). Extended dye laser autocorrelation measurements were performed over this energy range. The statistical independence and large coherence time separation (ca. 40 ps and 142 fs) of the two noise terms allows the overall (measured) correlation function to be decomposed into picosecond and femtosecond autocorrelation functions from which the second moments of the two noise terms are easily determined (see Fig. 7). Plots of G$?(O), Gg)(O) and G(‘)(O) versus the dye laser output energy are shown in Fig. 8. Super-Gaussian statistical behaviour in the pump noise, (with a maximum second moment of G$?(O) - 12) is observed at significant cavity losses (output energies below 500 pJ). Here with rising
OUTPUT ENERGY FROM DYE LASER CAVlTY (ml)
Fig. 8. (a) Plot of the variation in C&?(O) as a function of the lasing threshold, showing distinctly super-Gaussian behaviour for the pump noise component u,(t) at low output energies tending to Gaussian behaviour (G’,z)(O)= 2) at normal oscillation. (b) A similar plot for the intrinsic femtosecond dye noise shows C@(O) to be unaffected by the pumping conditions. (c) The variation in the overall second moment of the broadband field (given by the product G$?(O)Gg)(O)) indicates super-Gaussian statistical behaviour under all pumping conditions, with the low output behaviour mirroring that of u,(t).
AJ. Bain, A. Squire/Optics Communications 135 (1997) 157-163
162
band dye laser emission are often implicitly assumed in the analysis of a number of quantitative nonlinear optical measurements 12-91. In the light of our findings an accurate characterisation of optical field coherence would appear to be an essential prerequisite of any nonlinear mixing experiment.
”
0
b
2
”
I
4
”
I
”
6
k 8
DYE LASER OUTPUT ENERGY (mJ) Fig. 9. A plot of 7p as a function of dye laser output energy. Under all pumping conditions, 7p < ~~~o, with the noise transfer approaching a linear regime (TV = 7vAG) at normal output energies.
energies below 500 p,J) the multiplicative noise transfer is highly nonlinear (TV m 20-26 ps) and tends towards but does not reach a fully linear regime even with significant pump conversion (e.g. 7p = 36 ps at 5 mJ). For low pump conversion the measurements of Gc2)(0) (Fig. SC) show an inverse output energy dependence in agreement with previous studies of below threshold noise transfer in continuous wave (c.w.) single mode dye lasers [24,25]. In such systems, close to threshold low pass filtering of pump noise is observed with linear noise transfer well above threshold [26]. Theoretical models of C.W. multimode systems predict similar near threshold behaviour [27]. Such studies involve slow &Hz-MHz), low amplitude, non-fully-mcdulated C.W. pump fluctuations. In contrast the ca. 5 ns Q-switched pump pulses used in this work are fully modulated by random Gaussian fluctuations with a frequency greater than 10” Hz. Here the dye laser is pumped rapidly and randomly above and below threshold and no filtering of the pump noise is observed *. This clearly corresponds to an operating regime far removed from previous investigations [24-291 and to our knowledge this work represents the first study of noise transfer and optical coherence in this widely used class of laser.
6. Conclusions We have shown that the output of a conventional pulsed broadband dye laser obeys super-Gaussian statistics over a wide range of pumping conditions. This is seen to arise from the direct transfer of fast (42 ps) Gaussian fluctuations in the pump laser. Gaussian statistics in broad-
* The upper lasing level population rates in organic dyes are typically on the order of lo’* s- ’ [17]. It would therefore not be unreasonable to expect the gain medium to respond to fast pump fluctuations and that this rate would represent the upper bound to its frequency response to external noise.
Acknowledgements We would like to thank EPSRC for the financial support of this work, Professor R. Loudon FRS for helpful discussions and E. White for assistance with the experimental work.
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[27] M. Beck, I. McMacltin and M.G. Raymer, Phys. Rev. A 40 (1989) 2410. [28] K.J. Phillips, M.R. Young and S. Singh, Phys. Rev. A 44 (1991) 3239. [29] D. Von der Lmde, Appl. Phys. B 39 (1986) 201.