Optics Communications 242 (2004) 233–240 www.elsevier.com/locate/optcom
Transverse mode competition in pulsed-laser pumped solid-state lasers: generation of a second pulse I. Iparraguirre
a,* ,
T. del Rı´o Gaztelurrutia a, J. Ferna´ndez
a,b,c
a
Dpto. de Fı´sica Aplicada I, E.T.S. de Ingenierı´a, Alda. De Urquijo s/n 48013 Bilbao, Spain Centro Mixto CSIC-UPV/EHU, Paseo Manuel de Lardiza´bal 3, 20018 Donostia, Spain Donostia International Physics Center, Paseo Manuel de Lardiza´bal 3, 20018 Donostia, Spain b
c
Received 23 April 2004; received in revised form 28 July 2004; accepted 10 August 2004
Abstract Effects of transverse mode competition in macroscopic nanosecond pulsed-laser-pumped confocal solid-state lasers have been observed and measured. The effect of the alignment of the resonator on the temporal structure of the laser output has been measured. For some angular positions of the mirror a second laser output pulse has been observed with a different spatial structure from the first pulse. In order to explain this phenomenon a diffraction theory which includes gain in the pumped sample and misalignment effects has been developed. Using this theory it is possible to explain the generation of the second pulse and the temporal dynamics of the laser. The fact that transverse modal competition is the cause of the effect is clearly established. 2004 Elsevier B.V. All rights reserved. PACS: 42.60; 42.25.F Keywords: Laser physics and laser sources; Diffraction
1. Introduction Transverse-mode competition is a typical effect present in many kinds of lasers. In the last few years, a number of theoretical and experimental ef-
*
Corresponding author. Tel.: +34 9 46014264; fax: +34 9 46014178. E-mail address:
[email protected] (I. Iparraguirre).
forts to study this phenomenon in solid-state lasers have been carried out. There has been research on the spatial structure of the laser beam and how this spatial structure is affected by pump inhomogeneities [1–6]. Theoretical studies on the transverse modal structure have been also developed [5–8]. Probably, vertical cavity surface emitting lasers (VCSEL) have been the object of the most intense research on transverse modal competition and its effects [9–15]. The interaction between spatial
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.08.019
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hole-burning effects and the laser profile, which may cause changes on the modal structure, has been also investigated by some authors [15]. In our work, we show the effect of transverse modal competition in solid-state pulsed-laser pumped systems in confocal resonators. We have observed that in the process of optimisation of the alignment of this kind of lasers, the temporal profile of the output laser pulse shows a complex structure for some range of orientations of the mirrors, giving sometimes rise to a second output laser pulse separated from the first. We have observed this effect in both longitudinal and transverse pumping configurations, and for many kinds of solid laser materials, including crystals and glasses, spectrally homogeneous or inhomogeneous and isotropic or anisotropic. This suggests that the substrate is not the main cause of the phenomenon. We have found references to this phenomenon in VCSEL lasers [15], where the size of the laser is comparable to the emission wavelength and the effect is difficult to control due to the particular characteristics of these lasers. The emission of a second pulse has also been obtained in TEA CO2 lasers [16], but in this case the effect is caused by the overlap between the longitudinal mode and the gain band, which is affected by thermal effects. However, we have not found any reference to this effect in macroscopic solidstate pulsed-laser pumped lasers, where, similarly to the lasers mentioned above, there is not any temporal overlap between pumping pulse and output pulse. In this paper, we present experimental data showing how the change of alignment leads to the appearance of a second laser pulse. We also present measurements to compare the spatial structure of the first and second output laser pulses. We then develop a unidimensional diffraction model for the case of transverse pumping, which includes diffractive iteration in a misaligned symmetric confocal resonator and the spatial profile of the gain in the pumped sample. Our model explains satisfactorily the temporal structure of the pulse and the generation of a second pulse, as well as the spatial structure of the pulses.
2. Experimental results Our experimental system was a symmetric confocal resonator, 125 mm long, in which different Nd3+-doped solid laser materials were pumped [17–20]. All samples were about 2 mm thick and placed in Brewster-angle configuration. The samples were pumped with a Q-switched Ti:Sapphire laser, with around 30 mJ of output energy and 8 ns pulse time, in both longitudinal and transverse pumping configurations. Depending on the concentration of Nd3+ in the sample, the pumping wavelength was set to around 800 nm, to pump the 4F5/2 level, or to around 865 nm, to pump directly the metastable 4F3/2 level. Our materials included anisotropic crystals such as K5Nd(MoO4)4 (KNM) [17] and Rb5Nd(MoO4)4 (RNM) [20], isotropic crystals such as Yttrofluorites [19], oxyfluoride glasses [18], all of them spectrally inhomogeneous, and also homogeneous materials such as Nd:YAG. In the laser experiments performed on the samples mentioned above, the shape of the pulse showed a temporal structure and even a second pulse for some alignments. This effect was observed for both transverse and longitudinal pumping. We tested for possible differences in the spectral properties of the first and second output pulses, not finding any significant difference in standard experiments, where the spectral width of the laser pulses is about 2 nm. Only in KNM and RNM autotuning experiments [20,21], where the spectral width of the pulses is about 0.1 nm, there is a spectral shift between the first and second pulse of the same order than the pulse width. On the other hand, we found that the effect depends strongly on the pumping, appearing clearly only for pump energies above a minimum energy of about 1.5 times the threshold. We present the experimental results for a Nd3+ doped stoichiometric KNM sample transversally pumped to the 4F3/2 level with approximately twice the threshold pump energy. We focus on transverse pumping, since, unlike longitudinal pumping, it is amenable to unidimensional diffractive treatment even including misalignment. Our main concern was the critical effect of alignment. When the orientation of the mirrors was optimal for a
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0.10
Intensity (Relative Units)
minimal delay between the pump and the output laser pulse, no temporal structure was observed in the output pulse. When the output mirror was tilted clockwise (see Fig. 4), the delay rose but there was still no structure in the temporal pulse shape (case a in Fig 1). When it was tilted anticlockwise a second pulse, with variable temporal delay with respect to the first one, appeared. This delay was dependent on the exact angular position of the mirror, as can be seen in Fig. 1. The spatial profiles of the two output laser pulses were compared in the near field. A fast photomultiplier with a 0.01 mm slit was placed as close as possible (about 5 cm) to the output mirror. The photomultiplier and slit system was placed in a translation stage in order to obtain the near field structure. The signal was digitalized and the areas corresponding to the first and second pulses were obtained as a function of the lateral position in the translation stage. Fig. 2 shows two pulse samples obtained in near field at two different lateral positions of the translation stage. The spatial structure of both pulses is shown in Fig. 3. These results suggest that the second output pulse is emitted because of spatial hole burning on the population inversion caused by the emission of the first output laser pulse. They also suggest that the structure of the first pulse
235
(b)
(a)
0.10
0.05
0.00 0
300
600
0
300
600
t(nS) Fig. 2. Laser output intensity for two different lateral reception positions. (a) Maximal intensity position (lateral distance 0 mm in Fig. 3); (b) 1 mm towards pump entrance side (lateral distance – 1 mm). The alignment corresponds approximately to case (c) of Fig. 1.
corresponds to the fundamental transverse mode while the structure of the second corresponds to the TEM01 mode.
3. Theoretical model We have developed a theoretical model with the aim of explaining the generation and spatial
(a)
(b)
(c)
(d)
Power (Relative Units)
0.05
0.00 0.10
0.05
0.00 0
500
1000
0
500
1000
t (ns)
Fig. 1. Laser output power for a transversally pumped Nd3+ doped stoichiometric KNM sample for four different alignments. From (a) to (d) the mirror is being tilted anticlockwise (see Fig. 4) in steps of 0.15 mrad. Time equal zero corresponds to the pump pulse time.
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(a)
(b)
because in our setup the size of the mirror is much greater than the size of the sample. We have considered a symmetric confocal resonator and have allowed the possibility of having both mirrors misaligned. The wave amplitude on a point of coordinate x2 after reflection by an unlimited mirror misaligned an angle h is given by Z 1 A2 ðx02 Þ ¼ pffiffiffiffiffiffi expð2ih0 x02 Þ A1 ðx01 Þ 2p expðix01 x02 Þ expð2ih0 x01 Þdx01 ; ð1Þ
2
where A1 ðx01 Þ and A2 ðx02 Þ are the wave amplitudes before and after reflection in the mirror, respectively. The normalized misalignment h 0 and the normalized coordinates on the sample x01 and x02 are defined as follows, sffiffiffi pffiffiffiffiffi k 0 ; h0 ¼ h kf ; x1;2 ¼ x1;2 f
Intensity (Relative units)
150
100
50
0 -2
0
2
-2
0
Lateral distance (mm) Fig. 3. Time-integrated intensity distribution of the near field laser output as a function of lateral position: (a) first pulse; (b) second pulse.
structure of a second pulse. In the experimental setup for transverse pumping, the focus of a cylindrical lens is about 2 cm away from the entrance of the pumping to the sample, in order to avoid damage. In these conditions, the pumped zone can be considered rectangular, and variables can be separated [22], with an exponential profile for the population inversion which depends only on the variable x. Typically, the diffraction integral is done from mirror to mirror, although some other more complicated possibilities are considered in the literature (see, for example [2]). In our case, to implement in a simple way the introduction of gain, we have considered diffraction from sample back to the sample after reflection by an unlimited mirror (see Fig. 4) This assumption is reasonable
with f the focal length (125 mm), k = 2p/k, and k = 1.06 lm. The normalization factor in Eq. (1) can be verified considering the case of a very large aperture. For an infinite aperture the total energy of the diffracted wave must be identical to the total energy of the original wave. It is straightforward to show that Eq. (1) verifies this property. The net gain per pass through the sample, excluding diffractive losses, can be calculated by means of the static unsaturated expression for an amplifier: GðxÞ ¼ C expðb nðxÞÞ;
ð2Þ
Fig. 4. One side of the resonator and coordinate scheme for diffraction theory. a represents the tilting axis, perpendicular to the drawing.
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where n(x) is proportional to the population inversion, C implements the non-diffractive losses of the resonator and b = rN0l, where r is the stimulated emission cross-section, l the thickness of the sample and N0 is the population inversion at the entrance of pump radiation, which depends on the pump energy. The population inversion decays exponentially in the sample: nðxÞ ¼ expðaðx x0 ÞÞ;
ð3Þ 1
where a is the absorption coefficient, 1.3 mm in stoichiometric Nd3+: KNM and x0 is the coordinate of the point of entrance of pumping to the material. We have modeled the amplification in the cavity in the following way. The diffraction integrals implement the reflection by a mirror, aligned or not. The intensity resulting from diffraction is then multiplied by the net gain (2), with the population inversion given by (3). Starting with a random amplitude, the group of operators describing the reflection by one mirror, amplification in the sample, reflection by the other mirror and further amplification is applied iteratively until an eigenfunction and the corresponding eigenvalue are found. Once the eigenfunction is known we simulate spatial hole burning emptying the population inversion proportionally to the intensity of the mode at each point, with an adequate constant [23]. A new gain profile is then deduced from the new population inversion, and this new gain profile is used in a second iteration, until a new eigenfunction and eigenvalue are found. It is straightforward to calculate the net gain of the first and second mode with the new gain profile, in which the effect of hole burning has been taken into account. If the net gain of the first mode is lower than one while the net gain of the second mode is higher than one, the possibility of generation of a second pulse in a different transverse mode becomes clear. In the above procedure, a question that remains to be treated is the determination of the limits of the diffraction integral in the iteration process. Assuming that the pumping comes from the region of negative x, the lower limit of integration is related to the lateral position of the crystal. The
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upper limit can be large, but the exponential decay of the population inversion and the subsequent decrease of amplification allow us to use a smaller upper cut-off. In our cases of interest it was sufficient to consider a total aperture of about 4 mm. It is possible to prove analytically that the eigenfunctions and eigenvalues of the resonator with one of the mirrors misaligned an angle h 0 are in direct correspondence to the solutions of an equivalent problem in which the mirrors are not misaligned but the position of the crystal is shifted to x00 h0 . More precisely we find that if Wn(x 0 ) is an eigenfunction of the aligned resonator with lower limit of integration equal to x00 h0 ; Wn ðx0 h0 Þ expðix0 h0 Þ is an eigenfunction, with the same eigenvalue, of the equivalent problem in which one of the mirrors is misaligned an angle h 0 and the lower limit is set to x00 . The outcome of our numerical computations is consistent with this result, which can be easily generalized to the case where both mirrors are misaligned. It was verified numerically that one effect of misalignment is to change the size of the region where the intensity of radiation is significantly different from zero. The approximate size of this region agrees well with an estimation from geometrical optics, which leads to a region of size 2x00 þ 2h0 . Moreover, when we restrict the integration to that limited region the analytical problem becomes tractable and the eigenfunctions become the prolate spheroidal wave functions (PSWF). For x00 ¼ a and h0 ¼ 0, the solutions are An;a2 ðx0 =aÞ, using the notation of [24]. The PSWF are real, and symmetric and antisymmetric for odd and even n, respectively. We have observed that the actual solutions of the iteration process with the complete region of integration coincide with the PSWF except for a small zone just in the limit of the region where the function is different from zero. Given the equivalence between misalignment and a shift of position of the crystal, all our further calculations have been carried out for h 0 = 0 and different lower limits for the diffraction integral. We have set b = 0.53 and C = 0.66 to implement a system working about twice above threshold. The hole burning in the population inversion has been included supposing a fall of 80% in
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Table 1 Net gains of the fundamental and second transverse modes with the initial population inversion (G1) and the emptied population inversion (G2), calculated for different pump entrance side integral limits, equivalent to different misalignments x00
G1 (TEM00)
G2 (TEM01)
G2 (TEM00)
G1 (TEM01)
1 1.25 1.5 1.75 1.825 2 2.0625 2.125 2.25 2.375 2.5
0.98 1.27 1.45 1.50 1.50 1.49 1.48 1.47 1.45 1.43 1.42
– – – – 1.00 1.09 1.12 1.13 1.14 1.13 1.09
0.50 0.70 0.85 0.91 0.92 0.92 0.92 0.92 0.92 0.92 0.91
– – – – 1.27 1.36 1.40 1.42 1.43 1.42 1.40
In the first four cases the second mode has not been obtained.
population inversion in the maximum of the fundamental mode, which corresponds also to a pump energy about twice the threshold pump energy [23]. Table 1 summarizes the outcome of our calculations. In this table G1(TEM00) and G1(TEM01) are the net gains per round-trip of the fundamental and TEM01 modes with the initial population inversion, and G2(TEM00) and G2(TEM01) are the net gains for the same modes after the depletion of the population by the fundamental mode. It is evident that G2(TEM00) must be always less than one. Our results show clearly the possibility of a second pulse for h0 and x00 such that h0 x00 > 1:825. For h0 x00 bigger than around 2.25 the initial gains G1(TEM00) and G1(TEM01) are very close, implying that efficiency of these two modes and probably other higher order ones are very similar. As a consequence, the quasisimultaneous emission of different modes becomes possible and the temporal structure of the pulse becomes more complex, on account of the great competition between the different transverse modes. Likewise, the spatial structure of the output is also expected to be a complex multimode structure. In order to obtain a dynamical simulation of the effect, we use a traveling wave model, usual in diffraction resonator problems, combined with
rate equations. The choice of the rate equations is conditioned by the nature of our problem. In particular, the use of the spatially averaged rateequations present the difficulty of the very great difference between the active length (2 mm) and the resonator length (125 mm). Moreover they do not allow the implementation of the transverse dependence of the gain. Therefore we have developed the following set of equations: dP 1;2 ðtÞ c ¼ P 1;2 ðtÞ ðg1;2 ðtÞ C 1;2 1Þ; dt 2L
ð4Þ
dnðx; tÞ c 2 ¼ B ðjW1 ðxÞj ðg1 ðtÞ 1ÞP 1 ðtÞ dt 2L 2 þ jW2 ðxÞj ðg2 ðtÞ 1ÞP2 ðtÞÞ; where P1,2 and g1,2 are the power and gain per round trip corresponding to the first and second transverse modes, respectively, the functions W1,2(x) are the normalized amplitudes of the first and second transverse mode, respectively, n(x,t) the population inversion, while x and t are the horizontal coordinate and time, c is the velocity of light, and L the length of the resonator. C1 and C2 are related to the passive losses of the first and second mode, and include diffraction losses, calculated previously, and different from each mode. The first equation of set (4) describes the temporal evolution of the power associated to two lower order modes, and the second one the evolution of the population inversion at different positions in the amplifying plate, assuming an averaged value along the z coordinate inside the plate. In both equations the gain per pass has been converted to gain per unit time using the round trip time. In the second equation we take account of the emptying of the population due to the pass of the wave through the amplifying plate. Spontaneous decay has not been included since its associated lifetime is about 100 ls17 while the total duration of the process is shorter than 1 ls. We have included in the parameter B all the normalization constants following from our choice of variables, and its value has been chosen phenomenologically with the aim of obtaining a delay between pump and output laser pulses similar to the experimental value. In particular we have set
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B = 1.5 · 1035 in order to obtain the first pulse before 200 ns, with P1 = P2 = 1 at t = 0. The initial value of the population inversion is given by Eq. (3). The gain per pass for each mode is calculated using the overlap integral Z 1 g1;2 ðtÞ ¼ Gðx; tÞ jW1;2 ðxÞj2 dx;
The solutions of the equations for different positions of the crystal are shown in Fig. 5. They show the good quality of the simulation. Comparing Fig. 1 with Fig. 5, we see that the behavior of the laser output as a function of the misalignment is simulated well, although the power of the second pulse is smaller in the simulation than in the experiment. This disagreement is probably due to the spectral hole-burning in the sample caused by the inhomogeneous spectral broadening of the emission profile of the material. Spectral hole-burning implies that the upper laser level is not emptied as effectively when the first output laser pulse is emitted, resulting in a population inversion bigger than theoretically expected for the second emission. The spectral hole-burning hypothesis is reinforced by the fact that, in the auto-tuning experiments carried out in our laboratory [20,21], we have seen that the shift of the spectrum between the two pulses is of the same order as the pulse width, about 0.1 nm. This spectral shift is explained neither by the purely modal shift between the first two transverse modes, which is about two orders of magnitude smaller nor by our diffractive theory, since it does not predict any change in the direction of the emission, which could account to a shift of wavelength in an auto-tuning experiment.
x0
where G(x,t) is the transverse gain function Gðx; tÞ ¼ expðb nðx; tÞÞ: In the calculation of the gain per pass we have used the static unsaturated gain function for two reasons. On the one hand, given the temporal and spatial parameters of our problem, it can be seen that the spatial derivative of the field envelope inside the material is about two orders of magnitude higher than the temporal derivative divided by the velocity of light. On the other hand, the time step used in our numerical calculations is by far short enough so that we can consider there is no saturation in each pass, implying that the exponential approximation is valid [25]. Time delay effects due to reflection in the mirrors have been neglected since the round-trip time in the experimental resonator, 125 mm long, is shorter than 1 ns while the total duration of the process is two or three orders of magnitude longer.
15
x'0= -1.875
x'0= -2
x'0= -2.0625
x'0= -2.125
Output power (A.U.)
10
5
0 15
10
5
0 0
100
200
300
239
0
100
200
300
t(ns)
Fig. 5. Laser output intensity resulting from Eqs. (4), for different misalignments.
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4. Conclusions
References
The effect of misalignment on the transverse mode structure of the laser output of a nanosecond pulsed-laser-transverse-pumped solid-state laser has been measured and a diffraction model has been developed in order to explain this effect. The possibility of emission of a second output pulse has been explained satisfactorily, with the first pulse emitted in the fundamental transverse mode and the second one in the TEM01 mode. As far as we know it is the first time that a second pulse with no temporal overlapping between pump and laser pulses is observed in this kind of lasers, although it has already been observed in VCSEL and TEA CO2 lasers. Diffraction theory confirms the very low diffractive losses and the low sensibility to misalignment of this kind of resonators when the Fresnel number is not too low, and shows the equivalence between the effect of misalignment of the mirrors and translation of the pumped sample. Possible changes on the spatial profile due to relaxation processes have not included in the model, since our work shows that they are not determinant for the emission of a second pulse, even though they may have influence on the dynamics of the process. The geometrically induced transverse mode competition discussed in this work has potential applications in the field of transverse mode tuning, using an adequate spatial pumping structure and the rotation of the mirrors or translation of the pumped sample. In this line, the potentiality of the longitudinal pumping scheme could be very interesting. Another interesting field would be the generation of consecutive pulses with a very easy control on the delay between them, much more versatile that a line retarder.
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Acknowledgement This work was supported by the Spanish Government MCYT (MAT2000-1135) and Basque Country University (UPV13525/2001).