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Experiments on random lasers

van Soest, G.

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Citation for published version (APA):van Soest, G. (2001). Experiments on random lasers Wageningen: Ponsen & Looijen bv

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Interferenc ee in rando m lasers

ThisThis chapter describes experiments in which the amplification of a probe beam isis used for studying light propagation in a random laser. We have performed measurementsmeasurements on speckle in the scattered and amplified light, which is discussed first.first. The second part of this chapter is about enhanced backscattering. We present thethe first high quality enhanced backscattering measurements in a random laser that cancan be driven above the laser threshold, providing experimental insight in the effect ofof the threshold on light transport. The results of these measurements are shown to bebe in quantitative agreement with the theory developed in chapter 3.

Inn the experiments described in chapter 2 we observed the fluorescence emitted by aa random laser to infer the influence of the characteristic length scales in diffuse transportt on the laser threshold. The fluorescence is generated by the system itself andd consequently details like directionality, coherence or spectral content can not bee used as experimental parameters. When studying the light transport itself, it is desirablee to have the ability to control the properties of the light source. This is most convenientlyy done by using an external beam incident on the sample. We then have aa source of diffusing light with known properties, and by measuring the scattered lightt we can investigate the transport of light in the medium. Much is known about transportt in passive disordered systems from light scattering experiments, and we cann assess the effect of amplification in multiple scattering by comparing data from randomm lasers with results for ordinary random systems.

4.11 Experimenta l consideration s

Probee beams are used in the experiments presented in this chapter, with a wavelength thatt can be amplified by the dye in the medium. The intensity of the incident probe

T" "

71 1

InterferenceInterference in random lasers

Tablee 4.1 Pulse char-acteristicss of frequency-doubledd Nd:YAG and OPO.. Specific require-mentss of die experiment determinee which is to bee the pump and which diee probe. The main advantagee of me OPO is off course its tunability, allowingg it to be used withh a variety of dyes. A benefitt of the Nd:YAG iss its coherence, making specklee measurements possible. .

Characteristic c

wavelength h max.. energy durationn (FWHM) temporall profile spatiall profile divergence e spectrall width coherencee length*

(nm) ) (mJ) ) (ns) )

(mrad) )

(cm) )

Nd:YAGG OPO

5322 410-700 1000 35 2.00 1.6-5a

gaussian n "topp hat" elliptic*

«0 .55 < l - 1 0c

^GHz^^ 0.1-5 nm 400 0.01-0.2

"Graduall increase, tuning from blue to red. *Longg axis is vertical; relatively flat top but not sharply bounded. cThee spectral width and divergence increase sharply towards the

redd end of the tuning range. d1.5xx transform limited eeDerivedDerived from the spectral width.

pulsee is chosen small with respect to the pump intensity, and preferably also with respectt to die generated fluorescence. With this provision the probe actually probes thee system instead of affecting the gain saturation.

Wee work with a setup in which the pump and probe pulses are incident simul-taneouslyy on me sample. The light in the medium then consists of amplified probe light,, in which we are interested, and fluorescence. Because we want to do our ex-perimentss at large amplification, and me maxima of the gain curve oe(X) and the fluorescencefluorescence spectrum L(k) of a dye are usually close in wavelength, the probe and fluorescencefluorescence can not be fully separated spectrally. The fast decay of the dye also precludess a distinction in time to be made. We have to tell the probe and the fluores-cencee apart in otiier ways; in this chapter that wil l be done using interference in the probee light.

Thee pump and probe pulses are provided by one laser system, an optical para-metricc oscillator (OPO) pumped by a Q-switched NdrYAG laser (Coherent Infinity 40-100/XPO).. The NdrYAG laser has a maximum pulse energy of 600 mJ at the fundamentall wavelength, 1064 nm, and a pulse length of 3 ns. The repetition rate is variablee from single shot to 100 Hz. The Nd:YAG pulse is subsequently frequency-doubledd to 532 nm (green) and tripled to 355 nm (UV). After the third harmonic generationn the energy in the green pulse is still more than sufficient, approximately 1000 mJ, to serve as a pump pulse in our experiments. The UV pulse is used to pump thee OPO, which is a light source tunable tfirough the visible part of the spectrum, withh pulse energies up to 35 mJ. If the OPO is not needed, the third harmonic can be employedd as a pump for UV absorbing materials.

72 2

4.24.2 Speckle in random lasers

Dependingg on the requirements on the pump and probe beams, we can choose too use either the second harmonic of the NdiYAG laser as a pump and the OPO as aa probe or vice versa. There are a number of characteristics in which the pulses differ ;; a summary is given in table 4.1. It shows that the properties of the OPO pulse dependd strongly on the set wavelength, and in general degrade towards the red side off the tuning range. The energy in the OPO pulse is large, but the moderate beam qualityy necessitates spatial filterin g (except perhaps for X < 480 nm), which reduces thee usable energy by a factor 50.

Forr OPO power regulation we use a set of polarizing Glan laser prisms. The firstfirst is rotated to vary the amount of power, while the second selects the transmitted polarization,, compatible with the mirror s used. A drawback of this setup is the steep intensityy change at small transmission (/ « sin4 a, where a is the rotation angle). The advantagee is its wavelength independence. For the NdiYAG we use a rotating X/2 retardationn plate mounted between two crossed Glan prism polarizers. In this case thee sensitivity for small transmissions is better (ƒ oc sin2 a).

Gettingg pump and probe pulses from the same laser system has the advantage thatt the timing of the experiment becomes particularl y simple. The jitter between thee pulses is minimal and the difference in arrival times can be easily compensated byy optical delay. We put a delay line in the pump beam path, because the probe beamm alignment is much more precarious than the pump, and might be disturbed by movingg a slightly imperfectly aligned delay line. The Nd:YAG pulse is the leading one.. Several ns of variable delay is convenient for optimization; the delay line has a lengthh of maximally 2 x 1.5 m.

4.22 Speckl e in rando m laser s

Specklee is the strongly fluctuating, grainy intensity pattern resulting from the inter-ferencee of a randomly scattered coherent wave. It can be observed in space, time and frequency.. Some statistical characteristics of the speckle pattern contain information aboutt the transport process [89]. We discuss only speckle in space, i.e. a fluctuating intensityy with angle.

Iff a coherent plane wave falls on a rough surface, a specklee pattern can be seen onn a screen positioned at some distance from the scattering object. The scattered fieldsfields at a certain position on the screen comes from all points of the rough surface, andd its random phases distributed uniforml y between 0 and 27c. The speckle is the additionn of the electric field vectors of all the N contributin g partial waves. The summationn constitutes a random walk in the complex plane, with a resulting field ^=l/v /^L J t aJ t exp(r^)[90] . .

Thee sum results in a gaussian distributio n for E, with most probable value E = 0

73 3


andd variance (\E\2) = limJV_+00(2A^)"1 E*(|Ö*I 2)- The observed intensity I = \E\2 then

followss a Rayleigh distribution:

p(/)) = <ïïe "* " (4,1)

Thee typical speckle spot size depends on the characteristic distance along the screenn on which the fields that contribute to the speckle dephase. The largest path lengthh difference at a spot on the screen is caused by the partial waves arriving from oppositee ends of the illuminated region of the scattering surface. Consequently, the typicall angular size of a speckle spot is X/d [90], if d is the diameter of the illu-minatedd region. This demonstrates that a measurement of the speckle spot size, for instancee by the autocorrelation of the speckle pattern, does not usually give infor-mationn about light inside a scattering medium. The spot size in reflection depends mainlyy on the incident beam diameter. In transmission the most influential parame-terr is the sample thickness, which determines the degree to which an incident point sourcee spreads in transport to the rear interface.

Thee field of speckle experiments in random lasers is largely uncharted territory. Thee only data available of the effect of gain on a speckle pattern produced by a probe beambeam are those of ref. 91. Refs. 92 and 93 study the related subject of coherence propertiess of the generated fight. In this section we present experimental results concerningg the intensity statistics and speckle spot size. In contrast with passive systems,, these measurements do depend on parameters of light transport. There is noo theory to compare the measurements with. We wil l give qualitative explanations off the results.

Theoreticall studies of speckle in random lasers [66,94-96] invariably investi-gatee speckle correlations. While interesting, these correlations are exprimentally nott easily accessible, for reasons that will be discussed in section 4.2.4. A further limitationn of the theoretical efforts in this field is that they all rely on a stationary formalism,, with fixed gain, so they encounter the explosion when approaching the laserr threshold.

4.2.11 Sample and setup

Spatiall speckle can only be observed if the coherence length of the light is much largerr than the maximum path length difference between partial waves contributing too speckle. In the region where the rhodamine dyes fluoresce the OPO beam has a linewidthh that is too large to produce speckle with a good contrast: its coherence is insufficiënt.. The frequency-doubled Nd: YAG can be amplified by a dye with a large gainn at 532 nm, such as Coumarin 6 (see appendix A). In dye lasers it is usually dis-solvedd in ethylene glycol, but unfortunately our Ti02 [44] colloid is not suspended

74 4


100 0

5600 0.0 wavelengthh (nm) pump fluence (mJ/mm )

Figuree 4.1 (a) Line: normalized emission spectrum of a Coumarin 6 solution in hexylene glycol,, with CW 488 nm excitation. Points: normalized emission spectra of the solution withh Ti02 scatterers, I = 10 /im; below (O) and above threshold . The pump source is aa 482 nm pulse of duration 2.6 ns. (b) Emitted intensity from the scattering solution at 527 nm,, near the maximum, as a function of pump intensity. The threshold is found to be at approximatelyy 0.22 5 mJ/mm2.

welll in this liquid. Hexylene glycol is a good alternative. The fluorescence spectrum off Coumarin 6 in hexylene glycol, cf. figure 4.1(a), is nearly equal to that in ethy-lenee glycol. Hexylene glycol also slows sedimentation of the scatterers by its high viscosity. .

Thee suspension is contained in a round plastic container with dimensions 6 mm depthh x 10 mm diameter, covered with a 4 mm thick glass window. The sample is rotatingg slowly to prevent sedimentation and dye degradation. Coumarin 6 can be pumpedd with blue light, we use the OPO tuned to 482 nm. The dye concentration iss 2 mM, providing a gain and absorption comparable to the 1 mM Sulforhodamine BB solutions of chapter 2 and section 4.3. The transport mean free path is 10 ^m, fromm enhanced backscattering (this technique will be explained in figures 4.7 and 4.8).. This sample has a rather high threshold pump intensity of/p = 0.22 mJ/mm2, determinedd from the measurements shown in figure 4.1. We could not find a reliable, durablee combination of dye and solvent that amplifies well at 532 nm, and would suspendd the colloid, with a lower threshold.

Thee speckle is recorded on a Kappa CF 8/1 FMC 8-bit CCD camera. The light fromm the sample passes through an aperture, blocking stray light, a 532 nm interfer-encee filter with a transmission FWHM of 1.0 nm to remove most of the fluorescence,

75 5


Figuree 4.2 Schematic of the setup for speckle experiments. The pump (diameter 2 mm) andd probe (diameter 0.8 mm) reach the sample simultaneously. The sample is mounted on aa motor, spinning it slowly to prevent sedimentation. The scattered and amplified probe lightt is collected on an 8-bit 752 x 582 pixel CCD camera (Kappa CF 8/1 FMC), through an aperturee (A), a 532 nm interference filter (IF) and one or more neutral density filters. The distancee between sample and camera is 10 cm.

andd a neutral density filter. The image is a single shot exposure, because the sample iss liquid so the speckle changes continuously. The probe beam diameter is 0.8 mm. Thiss produces a speckle that can be resolved well on the camera, with a large num-berr of speckle spots. The pump spot is chosen to be larger, 2 mm, to provide a large amplifyingg region for the probe light to propagate in. Figure 4.2 is a schematic of thee experimental arrangement of sample and detection.

4.2.22 Intensit y statistic s

Thee Rayleigh distribution of speckle is a very robust phenomenon. The only re-quirementss are uncorrelated phases of the scattered light and the independence of thee amplitude ak and phase tyk. It does not make a principal difference whether the scatteredd light has actually traveled inside the scattering medium or is just reflected offf the surface. Considered this way, a measurement of speckle intensity statistics doess not promise to be an effective method to obtain new information about random lasers.. It does, however, provide access to a measurement that is otherwise difficult too perform: the degree to which the incident probe is amplified by the system.

Wee make use of the coherence of the Nd:YAG frequency doubled output as a probee pulse, producing a speckle with good visibility. Figure 4.3(a) shows an image off the speckle pattern in scattered and amplified probe light from a Coumarin6/Ti02

randomm laser. The drawback of the large pump spot is a larger fluorescence compo-nentt in the image.

Thee intensity histogram of the speckle in figure 4.3(a) is shown in figure 4.3(b). Thee histogram has an unusual feature: it only starts to show Rayleigh statistics above intensityy 50. The lower intensities are incoherent fluorescence, giving each pixel an

76 6


500 100 150 200 intensityy (arb. u.)

250 0

Figuree 4.3 (a) Speckle pat-ternn on an 8-bit CCD camera off frequency-doubled Nd:YAG scatteredd and amplified by a 22 raM Coumarin 6 solution in hexylenee glycol, with Ti02 scat-tered,, £ = 10 jum. The pump pulsee from the OPO of wave-lengthh 482 nm has an energy of 0.322 raJ/mm2. (b) Intensity his-togramm of the image in (a). The Rayleighh distribution is offset by aa background of incoherent fluo-rescence.. The slope of the linear decreasee is l/(/).

offset.. The average intensity can be determined from the slope of the exponential decreasee for higher intensities. A plot of the fluorescent intensity (the offset) as a functionn of pump fluence reproduces figure 4.1 (b), providing evidence that the gain dynamicss of the system are not significantly influenced by the probe pulse.

Wee extract the average intensity of the amplified probe from the slope of the in-tensityy statistics plotted in the manner of figure 4.3. The dependence of the average intensityy of the speckle, or the intensity of the amplified probe, on pump fluence is plottedd in figure 4.4. The exposures taken at the highest pump intensities are overex-posed,, but if the average intensity can still reliably be obtained from the low intensity partt of the negative exponential it is included in the figure. A measurement in which thee intensity on the camera is attenuated using a neutral density filter shows that the linearr behavior of figure 4.4 persists up to the highest pump intensity 2.9 mJ/mm2, amountingg to an amplification factor «10.

77 7


Figuree 4.4 Average intensity de-rivedrived from Rayleigh statistics as a functionn of pump fluence, for a fixedfixed probe intensity of 59 /J/mm2. Thee amplified intensity grows lin-earlyy with pump fluence. For com-parison,, we plot the mean intensity ass a function of the probe fluence (inn units of 0.01 mJ/mm2) in the inset.. The pump intensity is high: 2.99 mJ/mm2. The lines are linear fits too the data.

- -200 0

- - 1 00 0

pumpp fluence (mJ/mm )

4.2.33 Speckle spot size

Ass discussed on page 74, a measurement of the speckle spot size yields the trans-versee dimension of the coherent source on the scattering surface. In a random laser thiss may very well depend on the pump energy, since a larger gain allows the light to spreadd further, enhancing long paths. The results of chapter 2 clearly show the rela-tionn between the transverse dimension of the amplifying volume and the threshold.

Thee spot size is measured by calculating the two-dimensional intensity autocor-relatee Gr(A6) = (/(8)/(6+A9)) of a speckle pattern as in figure 4.3(a). For a circular illuminationn spot of diameter d by purely coherent light the autocorrelation is

GI(Ae)) = ( / )2[ i+A( fAe) ] , (4.2) )

wheree AG is the angular distance between two points on the screen, and A{u) = (U(Uxx{u)/u){u)/u)22 with 7, the first order Bessel function. A(u) has the same functional dependencee as the Airy diffraction pattern of a circular aperture, and so the first zero iss expected at AG = 00 = l.22X/d, providing a measure for the speckle spot size. Thee contrast between the maximum at zero and the value at large AG is a factor 2: Gt(AGG » X/d) = (I)2, and G,(0) = 2(/)2.

Ourr diffuse light source is, however, partly incoherent due to the contribution of fluorescence.. The consequences for the intensity autocorrelate are shown in figure 4.5,, showing a cross section through the autocorrelate of the data in figure 4.3(a). Thee speckle contrast is reduced [97] to 1 + ((/)/(/T))2, where (ƒ) is the average intensityy of the amplified probe and (/T) = (/) + /F is the average total intensity, includingg the fluorescence intensity /F. For the data in figure 4.3 (/) « /F and so the contrastt in the autocorrelate is reduced to a factor 1.25. The position of the first zero inn the autocorrelate becomes hard to determine due to this lower contrast.

78 8


Figuree 4.5 A cross section through thee autocorrelate GT(A6) of the data inn figure 4.3, normalized to the aver-agee total intensity (/T). The contrast iss reduced from 2 to 1.25 due to the incoherentt background, complicating thee determination of the first zero 0O. Thee two dimensional autocorrelate it-selff is shown as the inset.

Anotherr change with respect to the fully coherent situation is the disappearance off the flat top for Gj(A0 = 0). There is no explanation for this sharpening, except per-hapss the remark in ref. 97 that the details of the autocorrelation function of speckle in partiallyy coherent light depend to a large degree on the particulars of the contribut-ingg fields. The resulting GT(A6) (= Gl (AG)) is normalized to (/T) and analyzed quantitativelyy by modelling it with a function

(/)) and (/T) are determined directly from the data, so 8C is the only free parameter in aa fit with Gm, providing a way to determine the speckle spot size. The characteristic anglee 0C is smaller than 0O by a constant factor « 1.

Thee 9C are plotted as a function of pump fluence in figure 4.6(a). The speckle spotss are found to shrink as the pump fluence is increased from 0 to 1 mJ/mm2, after whichh their size is approximately constant. The optical resolution of the imaging systemm is « 0.1 mrad. Apparently the source of diffuse light producing the speckle becomess larger if the pump fluence is larger. This is consistent with the notion thatt mainly the long paths are amplified in a random laser. If only the intensity is increased,, without actually changing the amount of amplification the speckle size is constant,, as shown in figure 4.6(b).

Withoutt the pump, the speckle size is set by the probe beam diameter of 0.8 mmm (80£), and for the highest pump energies the equivalent source size increases too approximately 1.5 times this value. For high pump fluence the speckle does not gett smaller. This is consistent with the result obtained in chapter 3 that far above thresholdd the local equilibrium gain is clamped at the local loss level, so Kg(r) does nott depend on the pump fluence.

CD D < <

O O

1.25--

1.20--

1.15--

1.10--

1.05--

1.00--

0.95--

--

--

--

--

\ \

22 3 A99 (mrad)

Gm(AG)) = 1 + III III

79 9


1.1 1

1.0' '

~\~\ 1 ' 1

.a ) ) > -- 1 n - »

oo nq CN|| U a

0) )

'SS 0.8H

"oo 0.7H d) ) Q . . CO O

0.6 6

:i i % %

* * i i hi$é'-hi$é'- *

(b) )

;H** ^ ^

00 1 2

pumpp fluence

(mJ/mm2) )

33 0.0 0.1 0.2

probee fluence

(mJ/mm2) ) Figuree 4.6 Characteristic decay angle of the autocorrelate, as measured by a fitting GT

withh Gm, given in (4.3). (a) For large pump fluence the speckle spots get significantly smaller comparedd to the case without gain, signifying that the amplification assists the spatial spread-ingg of the probe light. The probe fluence is 59 J/mm2. (b) With varying probe energy at aa pump fluence of 2.9 mJ/mm2, the speckle size is constant, demonstrating the role of the amplificationn in the effect in (a).

AA quantitative assessment of the modification of the path length distribution P(A)P(A) is not possible with the available theory. The formalism of chapter 3 takes intoo account only one spatial variable, whereas the transverse dimensions are clearly neededd for describing the lateral spreading of the probe. Even in one dimension P(A)) can not be determined, in absence of a stationary form for the diffusing density. Recallingg that the rms traveled distance in a random walk of length A is y^£A/3, we concludee that the average path length (A) becomes (1.5)2 = 2.25 times larger in this sample. .

4.2.44 Possible experiments?

Wee conclude the discussion of speckle in random lasers with a suggestion for a pos-siblee experiment, investigating intensity correlations in amplifying random media. It iss derived from the classic C{ short range speckle correlation measurement, see ref. 988 for an introduction.

Thee effect of a changing gain on the intensity correlation can be measured by Cj(A/p)) = (/(/p)/(/p + A/p)). This will reflect the change in spot size presented

80 0

4.34.3 Enhanced backscattering in random lasers

above,, but then with a method that is theoretically better controlled. We expect a changee in speckle pattern to occur while Kg(z) changes, far above threshold it should bee constant and only increase in average intensity.

( (AJfcjJJ has a weak dependence on the absorption, appreciable if 1/1% is not negligiblee compared to &^, so for small angles. Upon replacing La by -L g , which iss allowed for a weak probe that does not saturate the gain, we gauge that C1 should bee appreciably larger than the passive case for rotation angles smaller than (kLg)"

1. Thiss estimate applies to the transmission of a relatively thin sample (L « 2La for pumpp light) with two-sided pumping as proposed by Wiersma [71], so Kg is approxi-matelyy constant. The analysis of Burkov and Zyuzin [66] suggests the same relative changee due to gain for short and long range correlations.

Ann experimental obstacle is the need for a solid sample, in order to be able too correlate different speckle patterns in a well-defined measurement. We have not managedd to make a high gain solid random laser, in spite of several attempts based onn both silica glass and PMMA plastic matrices. We find that in the plastic the dye degradess too fast, while in the glass the scatterers coagulate during sol-gel synthesis. Workingg with powdered dye-doped glass is a possibility if the mean free paths re-quiredd are not too small. Another complication is the need for large dynamic range detection,, which is problematic in pulsed experiments.

4.33 Enhance d backscatterin g in rando m laser s

Inn this section we describe our enhanced backscattering measurements, probing the gainn dynamics in a random laser. Experimentally they are similar to the speckle measurements,, which is why we present both in one chapter. We focus specifically onn the laser threshold, experimentally investigating the role of the threshold for light propagation,, with a technique that allows a detailed and quantitative analysis of the results.. Enhanced backscattering (EBS) has evolved from being a subject of study inn itself [99] into a tool that can be put to use for studying transport of waves in randomm media in a very precise and quantitative manner [27]. The principle of EBS iss explained in figures 4.7 and 4.8.

Thee shape of the backscatter cone is determined by the characteristic transport distancee of the light in the medium. The exponential amplification of the intensity withh path length A in a gain medium results in a larger contribution of long light paths comparedd to light in passive material. The long light paths constitute the top of the EBSS cone: a relatively larger contribution of long paths yields a sharper and narrower EBSS line shape [70,83], This sensitivity to long paths makes EBS particularly well-suitedd for testing the alleged divergence behavior.

Wee then compare the measurements with EBS cones calculated from the dy-

81 1


Figuree 4.7 Sketch of the principle of enhanced backscattering:: a plane wave is incident on a mul-tiplyy scattering medium. Every random "path" in thee medium (gray, assumed semi-infinte) can be tra-versedd in two ways, forward and backward. These twoo waves exiting the medium are always in phase, sincee their path lengths are equal. Only outside a pathh length difference A develops, depending on the transversee distance d between both ends of the path: AA = (/sinG, where 9 is the angle with respect to thee incident direction. Each path serves as two in-phasee point sources (regardless of the longitudinal coherencee of the wave, since it interferes with itself), separatedd by a distance d, producing an interfer-encee pattern in the far field I(d; 9) <* 1 +cos(2;tA/A.). Thee contribution of long paths (with large d) varies quicklyy with 9. (Continued in figure 4.8.)

namicc random laser theory of chapter 3. This allows us to validate the theory, and providess a way to show experimentally that the explosion does not exist. We infer Kg(z)) from the results.

4.3.11 Experimenta l detail s

Ourr measurements of EBS in high gain amplifying random media are performed withh samples consisting of 220 nm diameter Ti02 [44] colloidal particles suspended inn 1.0 mM Sulforhodamine B laser dye in methanol. The samples are contained in a celll as in the speckle experiments. The cell is slowly spinning to prevent sedimenta-tion,, dye degradation and also to assist speckle averaging. When measuring EBS it iss important to average out the speckle, which has a much larger intensity variation thann the cone, and so will obscure it. A source with short coherence length produces aa speckle with less contrast, so it is easier to average. In this case the short coherence lengthh of the OPO is actually an advantage, and the use of the OPO allows us to take Sulforhodaminee B as a gain medium, which is easier to work with than Coumarin 6.

Thee setup is similar to the one used in the speckle experiment; we highlight only thee differences. The dye/Ti02 suspension is optically pumped with the frequency doubledd Nd:YAG pulse. The pump fluence range at the sample position is 0-140 /J/mm2.. The pulse repetition rate is 20 Hz. The probe pulse has a low energy of 33 fiJ/mm2, a duration of 4.4 ns, and is tuned to the maximum of the fluorescence band off the dye (590 nm). The pump and probe beams with a diameter of 3 mm coincide onn the sample front interface. Rising edges of both pulses arrive simultaneously. A

00 angl e G

82 2


2 2 paths s

i i longg i

WÊÊÊÊ WÊÊÊÊ i i

AAAAAAA A AAA A shortt i

i i i i i i » i

A A

.. V 'FWHMN. >--f*** = 0.7/Wi ^ ^

00 -15-10 -5 0 5 10 15 anglee 0 anglee 9 (mrad)

intensil l C

OO

C

D

11 1

- 1 . 4 --

-1 .2? ? Q. .

1 . 0 l

Figuree 4.8 (Continued from figure 4.7:) All these interference patterns are summed, weightedd according to the probability for a path to span a certain transverse distance d. The fringess average to a flat ("diffuse") background for all angles, except for a range of width \jki\jki around 6 = 0 where interference is always contructive. The result is an intensity distri-butionn as measured in the right hand plot, called the backscatter cone, rising to a height of maximallyy two times the diffuse background. In principle d can be infinite, which produces aa cusp at the top of the cone. In the measurement the top is rounded due to the instrument resolutionn of 1 mrad. The peak intensity relative to the background, or enhancement factor E,E, is reduced due to stray light. The information about light transport is in the distribution of traveledd distances: for plain diffusion the cone's FWHM = 0.7/U. An absorbing medium re-movess long paths and rounds the cone, while in an amplifying medium the weight of longer pathss is enhanced compared to shorter ones, making the cone narrower. It is important to realizee that E does not depend on gain or absorption, since the background and the cone are producedd by the same source, with only one path length distribution.

schematicc of the setup is shown in figure 4.9. Thee probe beam enters the sample via a beamsplitter to allow intensity measure-

mentss in the exact backscattering direction. The sample is tilted forward by « 2° too keep the specular reflection out of view of the detection. The scattered light is collectedd through an interference filter and a focusing lens on the CCD camera to recordd the EBS cone. We accumulate 51 to 204 different speckles (realizations) in eachh exposure, depending on the collected intensity. The angular resolution is 1 mrad,, limited by the probe beam divergence.

4.3.22 Results from experiment

Inn figure 4.10 an example of a measurement is shown. The image clearly shows the largerr intensity near the backscattering direction. For analysis we manually find the

83 3


probe e 5900 nm

pump p 5322 nm

too PC

V B D D AA IFLPND

Figuree 4.9 Schematic of the setup used for random laser EBS. The pump and probe (both diameterss 3 mm) arrive simultaneously at the sample. The probe is incident via a beam splitterr (BS), with a wedged shape to eliminate spurious reflections. The sample is spinning slowly.. The scattered and amplified probe light is recorded on the CCDD camera, after passing throughh the beam splitter, an aperture (A), a 589.6 5 nm interference filter (IF), a focusing lenss (L), a polarizer (P) selecting in the scattered light the incident polarization, and one or moree neutral density filters (ND). Exposure times of the camera vary from 2.56 to 10.24 s, dependingg on the amount of incident light.

centerr of the peak, since this proves to be the most reliable method, and integrate onn concentric circles around it. For visual inspection a symmetric picture is more appealing,, so we mirror the data in the 6 = 0 axis.

Fromm the backscattering cone obtained without pump, taking into account the reabsorption,, we infer that the transport mean free path I = 3 ^m. Earlier EBS ex-perimentss [83,100] have been performed with materials in which the laser threshold couldd not be reached. In our sample the laser threshold is found to be at a pump fluencee of 10 ^J/mm2 from the width of the fluorescence spectrum as a function of thee pump pulse energy.

Thee salient features of the influence of gain can be seen in figure 4.10. Both thee width and enhancement factor E become smaller with increasing pump energy. Thee enhancement factor has a (gain-independent) value of maximally 2 that is di-minishedd by angle-independent contributions to the intensity. The width is related to thee transport length: for a cone without gain it is <* t~l, and provides a measure to comparee the cones at varying pump energy. The larger fraction of long light paths at highh gain reduces the width of the EBS cone, but there is no sign of a divergence or aa sudden change in behavior at the threshold crossing. After the initial cone narrow-ing,, the width saturates far above threshold at a value that is a third of the width of thee cone without gain. The decrease of E from 1.65 to 1.25 is due to the incoherent fluorescencefluorescence component in the collected light, which becomes stronger for higher pumpp energies.

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-15 5 -55 5 15 anglee (mrad)

155 o 15 anglee (mrad)

Figuree 4.10 (a) CCD image in scattered probe light, containing the EBS cone. The sample iss described in the text, £ = 3 /urn, pump fluence 126 yuJ/mm2. The exposure is the sum of 722 shots, averaging out most of the speckle, (b) The cone derived from this image (bottom curve)) by averaging over the azimuthal angle around the top. The resulting curve is mirrored aroundd 6 = 0, and the background is normalized. For low intensities residual speckle may bee a problem, especially around the top of the cone where the amount of pixels contributing too the average is small. As an example the cone without pump is shown (top curve). E < 2 duee to single scattering, stray light, and fluorescence (cf. figure 4.3(b)).

4.3.33 Comparison with theory of chapter 3

Forr a comparison with the theory of chapter 3 we need to extract EBS cones from the calculatedd time- and position-dependent inversion data, providing the spatiotemporal gain-profilee reflected in the cone.

Thee z-dependence of Kg(z,f) in EBS can be treated with the method due to Deng etet al. [101]. It is an extension of the formalism presented in section 3.1, the ex-pansionn of a solution of the diffusion equation in eigenmodes. If the constant Kg is replacedd by one depending on z, the method can still be applied, only for general Kg(z)) the eigenfunctions <|>„(z) and eigenvalues e„ must be found numerically. Fur-thermoree the x- and y-directions are reintroduced to allow the calculation of EBS. Thee geometry is still translation invariant in the transverse dimensions, so these are Fourierr transformed to k = ksinQ. The EBS contribution to the intensity yE(0) is thenn given in terms of <|>„(z) and e„, and angle-dependent factors:

*(•)) = TL E 1 1

"" n=0 J_ + £n // dz<(»n(z)e" Jo Jo

-(v-iu)z/i -(v-iu)z/i (4.4) )

Heree v(6) = i [ l +(cosG) stantsfor92<< l/k£.

]] = 1 + O(02) and «(e) = k£(l - cos6) = O(02) are con-- 1 1--

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Thee recipe of ref. 101 is, however, a stationary description. The gain profile is staticc and also the diffusion equation it uses is time-independent. We need to work aroundd the first problem, but as a small digression we will first say some things about thee second.

Pathh length distributio n The path length distribution P(A) in a disordered mediumm can be regarded as a time-of-flight spectrum for multiply scattered light. Whatt P(A) for a random laser looks like has been a long-standing question [102]. Thee path length distribution is important for diffusive wave spectroscopy (DWS) [103],, and a useful concept in general when describing multiple scattering as a sum off light paths. It can be measured directly in time-resolved experiments. yE and P(A) forr backscattering are related by [104]:

Y E ( e) = // yE(QU)dt= P(t)e-D^dt, (4.5) JtJt{{ Jt{

wheree tf = £/c is the mean free time and P(t) — P{A/c). Whenn calculating EBS from a time-dependent diffusion equation, one can find a

formm for P(t) in terms of the eigenfunction expansion, also for an amplifying system, ass long as Kg is time-independent. From a dynamic diffusion equation we can find yE(0;r),, and identify P(A) from (4.5):

P(A)P(A) = f^e-^A/3\[LdzUz) n=00 WO

2 2 -(v-iu)z/£ -(v-iu)z/£

2 2 (v-iu)z/£ (v-iu)z/£

(4.6) )

(4.7) )

Forr long light paths (G2 <C l/k£ in the EBS cone) P(A) is angle-independent, as a pathh length distribution should be. The requirement of long light paths physically meanss that the transport must be described well by the diffusion approximation, knownn to fail near the boundary and for short paths.

Usingg this method for an amplifying medium with a model Kg(z) = Kg > 0 in a layerr 0 < z < Lz (with Lz < lla = |Lg, the critical thickness for an amplifying layer, backedd by a semi-infinite passive medium with the same transport properties), and Kg(z)) — 0 for z > Lz, we find the behavior shown in figure 4.11. From the well-

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knownn P(A) «: A -5 for a passive medium, the large A tail rises gradually towards P(A)P(A) oc A -2 when approaching Lz = L'cr, but by then the limits of the small-gain assumptionn are stretched already beyond breaking. •

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Figuree 4.11 Path length distributions in backscatteringg P(A) calculated for a thick mediumm with an amplifying layer of thicknesss Lz near the source interface. Plotted are 44 curves for different Kg, LZ/L'CI = 0, 0.47, 0.78,, and 0.97 (solid lines, bottom to top) and thee limiting curves P(A) <* A~ 2 for zero gain (L'(L'crcr —> oo) and « A" 2 for Lz —> IL. For intermediatee Kg, P(A) returns to A~z for long paths.. These reach the passive part of the system,, and are thus amplified as much as the paths off medium length, as sketched in the inset.

Inn the analysis of Kg(z), the time variation is a subtle issue. The similarity of time scaless of gain dynamics and light transport make it very difficult to solve the time-dependentt EBS cone in a varying gain-profile. Extending the method outlined above too Kg(z,f) would mean that the <))„ and e„ become time-dependent. We chose insteadd to use the averaging property by the time-integrated detection method in the experimentt to simplify the analysis.

Sincee the EBS process itself samples the medium on the time scale needed to buildd up a cone, it only senses slow variations in nx(z). The longest paths that contributee in an experiment with an angular resolution of 1 mrad have a separation betweenn entrance and exit points of d = 103A.« 600 jum. The diffusive transport time overr this distance is d2/D « 1 . 7 ns. We mimic this property by low-pass filteringg the data, and subsequently averaging nx(z,t) in time windows of length d2/D, andd use this mean inversion profile to calculate a "partial" EBS cone. The partial coness are summed, each weighted with the mean probe intensity in its window. This proceduree largely overcomes the dominance of the nearly critical Kg(z) occurring inn the relaxation oscillation: long paths, needed for the divergence to happen, do nott have the time to build up in the w 50 ps that the "supercritical" inversion lasts. Thiss demonstrates once more that the dynamic picture, although allowing for high inversionn densities, prevents the explosion. We stress that a theory that does not incorporatee the full dynamics, fails to reproduce the cones completely. In particular aa static approach predicts an extremely small width and diverging height using the samee parameters.

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400--

3 3

03 3

300--

>>200--

c c CD D

100--

-20 0 -10 0 anglee (mrad)

135uJ/mmm -

544 nJ/mm.

noo pump

10 0 20 0

Figuree 4.12 Black points: enhanced backscattering cones for pump fluences ranging from 00 to 135 ^J/mm2. Gray lines: cones calculated from the dynamic theory of chapter 3. The experimentall results are accurately reproduced by the theory, except for intermediate pump energiess (ca. 20-70 /J/mm2; one example shown) where the relaxation oscillations dominate thee temporal inversion profile.

4.3.44 Discussio n

Thee lines in figure 4.12 are obtained from the nx (z,t) found from the model of chapterr 3 with the methods of section 4.3.3. The agreement between experimental data andd theoretical description is excellent for low and high pump energies. Initially, the widthh of the EBS cone drops quickly with increasing pump pulse energy. Far above threshold,, the FWHM saturates (at w 10 mrad, depending on system parameters) duee to the pump-independence of the above-threshold nl(z).

Forr pump fluences between 20 and 70 /J/mm2 the theory deviates from the experimentall results, see figure 4.13. This discrepancy is due to the entanglement of thee time scales of transport and variation of nx (z). Since n, (z) changes faster than thee time needed for the formation of a backscatter cone, the reversibility of transport inn the medium is affected. A wave traversing the medium along a certain path experiencess a spatiotemporal gain profile that is in principle different than the profile

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4.44.4 Conclusions

40 0

T3 3 CO O

CD D c c o o Ü Ü

30 0

II 20

T T I I

-H H »l l

< < 1* *

10 0

experiment AA time-dep. theory

- II A A È A £ * "

50 0 100 0 150 0

pumpp fluence (|aJ/mm )

Figuree 4.13 The full widths at half maximumm of the backscatter cones as a functionn of pump fluence. The dashed linee indicates the threshold, obtained fromm an independent measurement. Circless are obtained from experiment, triangless from the theory in chapter 3.

seenn by the wave in the reversed path. This reduces the interference contrast in the scatteredd light, as the two waves no longer have equal amplitudes when exiting the medium.. This unbalance is especially prominent just above threshold, where the long-livedd oscillations make up an important part of the temporal gain profile. Long lightt paths are most strongly influenced by the changing nx(z). Their interference contributionn is smaller than inferred from the averaged gain profile, and the actual, measuredd EBS cone is broader than given by our theory. A simulation of dynamic EBSS backs up this explanation, showing a cone broadening of the correct magnitude duee to the inequality of interfering paths.

4.44 Conclusion s

Inn this chapter, we have reported on speckle and enhanced backscattering experimentss on high gain random lasers, consisting of a laser dye with Ti02 colloidal scattered.. The cone width becomes smaller with increasing gain, and saturates above thresholdd at a value that is three times smaller than the width of the cone withoutt gain. We find that a cone shape that fits the EBS data well is only given by a time-dependentt calculation of the population inversion nx(z) in the medium, even thoughh the experiment integrates out the temporal variations. Contrary to what is expectedd from an extrapolation of the known low-gain stationary description to the highh amplification coefficients of organic dyes, the experimental data show no sign off a divergence of the intensity.

Thee speckle experiment also reflects the saturation behavior above the laser threshold.. The speckle spots shrink with increasing amplification, due to the in-

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creasedd contribution of long light paths. This trend, too, saturates far above threshold,, where the speckles have a constant angular size of 1.5 times smaller than the speckless from a passive sample. The speckle retains its Rayleigh intensity statistics. Thee analysis of the speckle autocorrelation in terms of a coherent amplified probe onn an incoherent fluorescence background is consistent.

Thesee results show that the spatial gain profile can be investigated well with probingg techniques, allowing a quantitative study of light transport in random lasers. Thee theory developed in chapter 3 is validated by these measurements, and the interpretationn of experimental data with the help of comparisons with that theory provides insightt in the actual dynamics of a random laser, even by the stationary experiments reportedd here.

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