Open Access
Add to BibSonomy
Abstract
In two recent works, the authors have investigated the spectro-temporal properties of the random laser emission from solid state dye-doped powders in picosecond pumping regime. Each emission pulse consists, both above and below threshold, in a collection of narrow peaks of a spectro-temporal width at the theoretical limit (ΔωΔt≅1). The distribution of path lengths traveled inside the diffusive active medium by photons that can be amplified by stimulated emission explains this behavior, as demonstrated by a simple theoretical model developed by the authors. The aim of the present work is, first, to develop an implemented model that does not depend on fitting parameters, and which is compatible with the energetic and spectro-temporal properties of the material; and second, to get knowledge about the spatial properties of the emission. The transverse coherence size of each emitted photon packet has been measured; and in addition, we have shown the existence of spatial fluctuations of the emission of these materials, as our model predicts.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In two recent works [1,2], the authors have published results about the spectro-temporal properties of the random laser emission from solid state dye-doped powders (SSDRL) in picosecond pumping regime. The emission results show, regardless of the level of pumping, narrow peaks of high intensity within each pulse, and a spectro-temporal width at the theoretical limit (ΔωΔt≅1). Above the threshold, not exactly defined [1], the density of these peaks is very high and they overlap each other, whether below the threshold the peaks appear isolated and are thus separable. This behavior can be explained by a simple model developed by the authors [2], based on a distribution of paths lengths made by photons inside the diffusive active medium, and the amplification by stimulated emission.
The purpose of this third and last work on this subject is to complete the study by giving a global perspective of the performance of these systems. In this sense, the authors will implement the very simple model developed in the previous work [2], by introducing the emission and absorption spectra profiles. The integration in wavelength of the resulting equations will give us a solution that explains the energetic behavior and also the spectral properties of the emission of the SSDRL. It is important to highlight that the results are obtained with no need to resort adjustment constants. A resume of the results and consequences derived from this completed model will be presented.
Finally, the last and main goal consists in the measurement of the spatial properties of the light emitted in each pulse from the surface of the SSDRL, thus obtaining direct information about its transverse coherence size. In particular, spatial fluctuations of that size were observed under conditions of very high gain in the emission front of the SSDRL, which can be explained by the proposed model.
2. Theoretical model of the spectral profiles
The inclusion of spectral profiles and auto-absorption in the previously developed diffusive model [2], leads to the following equations
Once defined, we have verified that the model is energetically compatible with the experimental results, as Fig. 1 shows for two integration times (400 ps in red and 1000 ps in black). The model predicts the observed experimental behavior as well as the effect of the integration time used. Note that in this model, there is no need for any fitting parameter, such as the integration time or the beta parameter, which are commonly used in laser rate-equations [2,7,8].
Fig. 1. Total energy emitted by the SSDRL as function of absorbed energy (both in phot/m2). The pump beam diameter is 0.8 mm and the diffuse absorbance of the sample 60%. Red and black circles are experimental values obtained using recovery times of 400 and 1000 ps respectively. Red and black lines are obtained from the model using integration times of 400 and 1000 ps respectively.
Regarding the spectral profile of the emission P(λ), it should be keep in mind that it is narrower the greater the population inversion density N is, always within its possible values [2]. As an example, in Fig. 2(a), we can observe two emission spectral intensity profiles obtained from the model for two different population inversion densities. The narrower one corresponds to a N value close to the theoretical upper limit (1.37 × 1024 m-3), when the active material is actually lasing, which mainly occurs during the pumping time (about 30 ps), and the other spectral profile corresponds to a lower value.
Fig. 2. (a) Spectral emission intensity profiles P(λ) obtained from de model for population inversion densities N = 13 × 1023 m-3 (red) and N = 4 × 1023 m-3 (black). (b) Spectral emission intensity profiles obtained from the integration of 100 laser pulses recorded with the streak-camera, by selecting the pixel line at a time when the material is actually lasing (red), and 1 ns later (black).
The experimental emission profiles shown in Fig. 2(b), were obtained within the actual laser emission (red), and 1 ns later (black), corresponding to population inversions above and below threshold respectively. The small discrepancies between the theoretical and experimental profiles are due to the asymmetric profile of the real dye cross-section, the difficulty of fully reproducing the spectral profiles of emission and absorption, and intramolecular processes of the dye [6].
3. Spatial properties of the emission
In order to deal with the spatial properties of the emission of the SSDRL, it should be noticed that, each photon packet exiting the SSDRL [1,2] will have spatial coherence, which should lead to an average transverse coherence size.
Measurements of spatial coherence in random lasers, both organic and inorganic, has been done by some authors [9,10], and their results show in any case a low transverse coherence length well below 100 microns.
Our objective is to estimate the transverse coherence size of the SSDRL emission, i.e. the maximal distance between two points on the surface of the SSDRL which emit in phase, and also to observe whether the particular properties of its emission can affect the behavior of its transverse coherence.
The first experiment consisted of projecting the image of the SSDRL surface through a high numerical aperture (NA) lens, placing a double slit in the image plane, and observing its interference pattern. The lens of 32 mm focal length and a diameter of 50 mm had minimal spherical aberration. The lateral magnification was 13 and we employed a spectral filter (580-590 nm), as well as a long-pass filter to remove the reflected pumping radiation. The image of the interference pattern was projected on a CCD camera (Spiricon SP-503-U, Ophir) using a 30 mm focal length lens. By using double slits of 40 micron width and 250 micron distance between them, we have not observed interference. In contrast, doing the same experiment with the bulk material, interferences were observed with a fairly high degree of contrast, as observed by other authors in dye solutions under similar circumstances [10]. This must be related to a much longer diffusion length in the case of bulk.
Therefore, we decided to use single slit and observe the diffraction pattern obtained. Figure 3(a) shows the diffraction pattern of a 20 micron wide single slit, using as light source, the augmented image (×13) of the SSDRL emission surface, just like we did in the double slit measurements. The profile obtained (black line in Fig. 3(b)) is almost identical to the corresponding theoretical one. The intensity of the first secondary maximum with respect to the main one (4.5%) coincides and the zeros give maximum contrast, showing that the transverse coherence size must be quite larger than that slit width. In Fig. 3(b), we can also see the profiles of the diffraction patterns obtained using joined slits of 40, 80, and 160 micron width, observing the progressive loss of transverse coherence with respect to the 20 micron case (and almost the 40 micron case). From this figure, we could roughly estimate an average transverse coherence length in the range of 50-100 micron, 4-8 micron on the sample surface due to the employed lateral magnification. The measurement of the vertical width of the diffraction pattern of Fig. 3(a) would lead us to a similar conclusion.
Fig. 3. (a) Diffraction pattern of a 20 micron wide single slit using as source the image ×13 augmented of the SSDRL emission surface. (b) Profiles of the diffraction pattern of 20, 40, 80, and 160 micron wide single slits using the same source than (a). To get the pattern on the camera, a projection lens of 30 mm focal length was used.
On the other hand, we want to investigate if the particular properties of the emission of these lasers can give rise to the existence of emission spatial fluctuations. Note that their emission is constituted by photon packets, each caused by a spontaneous emission photon, which will output a variable number of photons, depending on the path they traveled and the population inversion density.
In order to detect these possible fluctuations, the CCD camera was placed in the image plane of the high numerical aperture lens used for the interference experiments. This allowed us to observe the image of the emitting plane with the desired lateral magnification and the same spectral filtering. To gain reliability, we have always employed minimum gain and exposure time in the detection system, ensuring a truly zero background signal and no detectable noise. Under these conditions, the neutral filters attached to the CCD camera had to be removed to detect the signal.
As a result of the numerical aperture of the optical device we employed, a diffractive resolution theoretical limit better than the pixel size (10 microns) was achieved, and the order of 108 photon packets could reach each pixel throughout each pulse [1]. Note that this number depends on the pumping and lateral magnification used and it is not appropriate to reduce it much more, because the signal would be too small to be properly detected, due to the limited sensitivity of the CCD under the imposed working conditions.
In this sense, let us consider what the model predicts under different gain conditions if we count all the photons coming to a point (pixel) from many packets, on the assumption that each packet will make a certain path following the probability distribution (3) [1,2]
being x0 the mean path length. Account taken that we can generate many photon packets travelling under a given mean population inversion, we can measure the total number of photons coming from those packets to one detector pixel throughout one pulse. If we repeat the process many times, we can find the standard deviation of the sums (total number of photons detected by one pixel) given by the sets of packages. To the extent that this deviation is different from zero, spatial fluctuations in emission would be observable, because the difference between the number of photons detected in a pixel or another of the CCD would be appreciable.To that end, we can generate packets of photons with a probability distribution of lengths given by (3) as follows
where r is a random real number between 0 and 1. The number of photons that each packet will carry is equal to which means it depends on both the amplification or gain parameter σNx0, whose value is between 0 and 1 [2], and its path length relative to the mean, whose random value will be given by expression (4).We have counted the total number of photons obtained after generating up to 108 photon packets with a given gain parameter. We have repeated the process 1000 times and we have obtained the root mean square deviation in number of photons among the 1000 sets of packet launches.
A simple analysis of the results leads to the conclusion that for amplification parameters less than 0.5, the relative root mean square deviation is less than 1%. However, for amplification parameters higher than 0.8 their root mean square deviation can be more than 25%. Therefore, according to the model, the spatial fluctuations could be observable, but only under high gain conditions.
In order to verify experimentally this consequence of the model, we have obtained images of the emitting surface with different lateral magnifications and different gain conditions. In Fig. 4, we show two images of single pulses obtained with lateral magnification 3 and very different pumping conditions: (a) 20 µJ (about threshold) and b) 1 mJ, both with a pump beam 0.8 mm in diameter.
Fig. 4. 3D images ×3 augmented of the SSDRL emission surface for pumping energies: (a) 20 µJ (about threshold) and (b) 1 mJ, both with a pump beam 0.8 mm in diameter.
The appearance of fluctuations well over threshold can be clearly observed. It is important to stress that these spatial fluctuations do not repeat their spatial structure pulse by pulse. Unlike reflected pumping, which does repeat it due to the temporal stability of the transverse phase relationships of the reflected front. In the case of random laser emission, that structure will be completely different from one pulse to another, but it will also change within each pulse, since the temporal coherence (about tenths of ps [1]) is much shorter than the pump pulse duration (about 30 ps). Consequently, it is expected that spatial fluctuations tend to disappear when adding more photons. This fact explains the strong attenuation of speckle in RL images [11,12].
But as consequence of the particular emission properties of the SSDRL, spatial fluctuations do not disappear under very high gain conditions (see Fig. 4). However, the available spatial resolution in Fig. 4 is low (about 3 micron on the SSDRL surface) and the experiment was repeated at high pumping with higher lateral magnification (×13, with a spatial resolution better than 1 micron). The pumping energy was not further increased to avoid damage to the material and at low pumping it could not be done because of lack of sensitivity of the detection system.
Figure 5 shows a partial 3D image of a single pulse obtained with lateral magnification 13 after excitation with the same pumping energy than used in Fig. 4(b) (threshold × 50). The spatial fluctuations are clearly observed, their amplitude starts to be comparable to the average signal and there is no pulse-to-pulse correlation. From the data matrices, the autocorrelation function has been extracted, and a correlation length of about 8-10 pixels was estimated, that taking into account the lateral magnification, gives us a value for the transverse coherence length of 6-8 microns, similar to the one roughly estimated in the diffraction experiments.
Fig. 5. Partial 3D image of a single pulse ×13 augmented of the SSDRL emission surface obtained with 1 mJ pump pulse of diameter 0.8 mm.
It should be noted that the observation under high gain conditions of spatial fluctuations of the same size that of transverse coherence length adds an important test to the reliability of the model.
4. Conclusions
With the set of these three works, the authors have wanted to present a study as complete as possible of the random emission of dye-based solid-state lasers in picosecond pumping regime, and at high spectro-temporal resolution.
In the first work, the randomness of the emission was demonstrated, observing always within each pulse a random structure of peaks with a spectral-time width at its theoretical limit. This detection is possible thanks to the spatial and spectro-temporal resolution of the spectrograph streak-camera system employed, which allows us to select a sufficiently low number of photon packets.
The theoretical explanation of this experimental result, pointed out in that first work, is developed and resolved in the second one, in which the proposed model is presented and tested. Likewise, the decisive effect of diffusion in the operating process of these lasers had been shown.
In this third and last work, on the one hand, the spectral issue has been completed. We have showed that the model works well both spectrally and energetically, but in addition, the spatial problem has been addressed despite the low global resolution available. It must be taken into account that in measurements with the streak-camera, we can select time and wavelength in a single point, whereas with the CCD camera used to study the spatial properties, we can select and separate all the points, but we integrate in all pulse time and partially in wavelength (580-590 nm). Only well above threshold, a high dispersion in the number of photons of the emitted packets is observed, and the existence of random spatial fluctuations in the emission of the laser is demonstrated. Their spatial autocorrelation length has been measured, and together with the results of the diffraction measurements, the transverse coherence size of the emission has been obtained, being estimated in the range of 6-8 microns.
These results provide quantitative support for the proposed model, which predicts that only under high gain conditions, deviations in the number of photons could be observable, whereas in the case of the speckle, the spatial fluctuations would not have that dependence on the gain.
In short, all the experimental evidences point to the fact that these systems work like random lasers in a very pure way, and do not seem to need any modal features to explain their operation, because the combination of amplification by stimulated emission and the path length distribution, consequence of the diffusion, is sufficient. We want to highlight the simplicity of the model, which proposes that each spontaneously emitted photon follows a path length distribution law and it is exponentially amplified depending on the specific path length it makes within the pumped volume, but above all, we must emphasize the excellent degree of approximation of the model to the experimental reality.
Funding
Euskal Herriko Unibertsitatea (GIU/21/006); Ministerio de Economía y Competitividad (PID2020-115419GB/C22).
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. I. Iparraguirre, J. Azkargorta, S. García-Revilla, J. Fernández, and R. Balda, “Spectro-temporal behavior of dye-based random lasers under picosecond pumping regime,” Opt. Express 30(6), 9674–9684 (2022). [CrossRef]
2. I. Iparraguirre, J. Azkargorta, S. García-Revilla, J. Fernández, and R. Balda, “Spectro-temporal behavior of dye-based random lasers under picosecond pumping regime: Part II,” Opt. Express 30(15), 26655–26663 (2022). [CrossRef]
3. I. Iparraguirre, J. Azkargorta, J. Fernández, S. García-Revilla, M. Barredo-Zuriarrain, and R. Balda, “Random laser model for Nd3+- doped powders and its application to stimulated emission cross-section calculations,” Opt. Express 26(23), 31018–31030 (2018). [CrossRef]
4. I. Iparraguirre, J. Azkargorta, S. García-Revilla, J. Fernández, and R. Balda, “Input/output energy in solid state dye random lasers,” Opt. Express 27(14), 19418–19425 (2019). [CrossRef]
5. J. Azkargorta, I. Iparraguirre, J. Fernández, R. Balda, S. García-Revilla, and M. Barredo-Zuriarrain, “Random laser properties of Nd3+ crystal powders,” Opt. Express 26(9), 11787–11803 (2018). [CrossRef]
6. J. Fernández, S. García-Revilla, L. D. Carlos, E. Pecoraro, M.A. Arrandiaga, and R. Balda, “Time-resolved random laser spectroscopy of inhomogeneously broadened system,” Laser & Photonics Reviews 8(3), L32–L36 (2014). [CrossRef]
7. G. van Soest and Ad. Lagendijk, “β factor in a random laser,” Phys. Rev. E 65(4), 047601 (2002). [CrossRef]
8. K.L. van der Molen, A. P. Mosk, and Ad. Lagendijk, “Relaxation oscillations in long-pulsed random lasers,” Phys. Rev. A 80(5), 055803 (2009). [CrossRef]
9. M.A. Noginov, S.U. Egarievwe, N. Noginova, H.J. Caulfield, and J.C. Wang, “Interferometric studies of coherence in a powder laser,” Opt. Mater. (Amsterdam, Neth.) 12(1), 127–134 (1999). [CrossRef]
10. B. Redding, M. A. Choma, and H. Cao, “Spatial coherence of random laser emission,” Opt. Lett. 36(17), 3404–3406 (2011). [CrossRef]
11. B. Redding, M. Choma, and H. Cao, “Specklee –free laser imaging using random laser ilunination,” Nat. Photonics 6(6), 355–359 (2012). [CrossRef]
12. M. Barredo-Zuriarrain, I. Iparraguirre, J. Fernández, J. Azkargorta, and R. Balda, “Speckle-free near-infrared imaging using a Nd3+ random laser,” Laser Phys. Lett. 14(10), 106201 (2017). [CrossRef]
