- Email: [email protected]
Pump size optimization in microchip lasers
I April 1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
136 (1997) 405-409
Pump size optimization in microchip lasers F. Sanchez I, A. Chardon * GIS02.
Luhorutoire
d’Optronique,
ENSSAT 6. rue de Khunpont,
Received 17 September 1996; revised 4
B.P. 447.22305
Lonnion Crdex, France
November 1996; accepted 6 November 1996
Abstract We investigate the influence of the transverse size of the pumping light both on the threshold and the efficiency of microchip lasers. A general method is presented and applied in the case of one-dimensional geometry. It is shown that for positive cavity detuning, the laser efficiency always decreases when the pump size increases while for negative cavity detuning, it is optimized for a particular pump width. These results are interpreted in terms of index guiding or anti-guiding effects depending on the sign of the cavity detuning.
1. Introduction Diode-pumped solid-state lasers are very interesting because it is possible to build-up compact and low cost lasers in the infrared and visible spectral ranges. Optimization of their output power or spatial characteristics is needed for many applications [I -91. Optimization of the transverse size of the pump beam for maximum power extraction in solid-state lasers or amplifiers has been previously studied [4-81. The case of a four-level amplifying medium inserted in a resonant cavity (typically a planoconcave cavity) end-pumped with a laser diode has been investigated in Refs. [6,8]. The optimization consisted basically in evaluating the overlapping integral of the pump and the laser transverse distributions. In this configuration, the transverse widths of the pump and of the laser beams were independent: indeed, the transverse field distribution of the laser is mainly determined by the geometrical properties of the empty cavity and does not depend on the pump profile (if thermal effects and self-focusing or defocusing effects are neglected). The main result in these studies is that on-axis pumping provides the lowest threshold and the highest slope efficiency. In Ref. [4] the authors considered a three-level medium in a plano-concave mi-
I
E-mail: [email protected]. * E-mail: [email protected].
crochip laser where both pump beam and laser mode have two-dimensional Gaussian intensity profiles in the transverse (x, y) plane with independent beam radius. Pump beam divergence was also taken into account. They found that the minimum value for the threshold power and the maximum of the slope efficiency are obtained when the average dimensions of the pump beam and the laser mode become equal. These results can be physically interpreted in terms of integral overlap between the pump and laser mode. The main difference compared to a four-level laser material is that for a pump size much lower than the mode size, the laser signal is strongly absorbed in the non-pumped volume. This fact explains the poor performances achieved with narrow pump beams. Additional results on three-level flat-flat microchip lasers are given in Ref. [IO]. The author identifies that the mode shaping is driven by the off-axis losses (aperture-guiding). As a consequence, the mode-spot radius is fairly insensitive to both pump power and shape. The obtained results cannot be applied to the real case of end-pumped four-level flat-flat microchip lasers for the following reasons: (i) the empty plane-plane cavity does not sustain confined modes, (ii) transverse mode structures (in particular the fundamental mode width) are highly dependent on both the pump profile and the pump width. Therefore the pump size optimization in microchip lasers differs from that of a classical solid-state laser and needs a specific study. In this paper we are interested in investigating the influence of the pump size on both the threshold pump
0030-4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)00726-2
406
F. Sunchr;.
A. Chardon/
Optics
power and the power laser efficiency (i.e. the slope of the output power versus input pump power characteristic) of microchip lasers. This problem can be studied directly from the following differential nonlinear equation which describes the steady-state for the laser field in the cavity [I l-131: Vz-iS+S,--fi,+Pf(r)
8= I +6;+lZ12
136 (1997)
405-409
which plays a key role in the optimization threshold and the laser efficiency.
2. Theoretical
of both the
method
The first step consists in finding the transverse modes (in particular the fundamental one) at threshold. This is performed by solving Eq. (I) without the saturation term:
6,+i
[
Communication.s
0.
I
(‘1 Eq. (1) is valid under the mean-field approximation and disregards any thermal effect. 8 is the electric field distribution, p represents the pumping parameter (more precisely the on-axis pump intensity), T is the transmission of the output mirror, 6, and S, are respectively the laser field frequency and the longitudinal cavity mode frequency detunings, both referenced to the atomic line center. The function fir) represents the pump beam profile which is characterized by its width. Eq. (1) also assumes a homogeneously broadened four-level medium. In Refs. [I I, 121 the saturation factor of Eq. (I) contains an additional factor in front of the 6, terms. For convenience this term is taken as unity in this paper. Although thermal index effects can be important in some cases [14], we neglect them in this paper and focus our attention on gain guiding and index guiding or anti-guiding effects resulting from the pump profile. Note that thermal guiding is not predominant in the mode formation in three-level microchip lasers [IO]. For a given pump profile, Eq. (I) allows to calculate the field distribution for any pumping parameter and to further evaluate the laser output power. Nevertheless, the saturation term makes the problem nonlinear and fully numerical methods have to be used. We propose here an alternative method allowing us to achieve analytical (in one-dimensional geometry) and simple numerical results. Owing to the nonlinearity of Eq. (I), iterative methods are needed to obtain the solution which depends on the pumping parameter p (in fact (1) is not an eigenvalue problem, except at threshold, because of nonlinear saturation). We assume in this paper that the transverse distribution of the pump and the laser remain the same for pumping rates very close to (but above) the laser threshold. As a matter of fact, it has been recently reported in Ref. [7] that the mode-width in an Er:Yb:glass microchip laser undergoes a very limited reduction (I IO%) when the pump power is increased up to twice the threshold. Similar results are reported in Ref. [IO]. With this hypothesis the problem can be solved in two steps: (i) evaluate the transverse mode distributions at threshold using the adapted Eq. (I) and (ii) calculate the laser efficiency using the oscillation condition derived in the next section. The theoretical method is presented in Section 2 and the results obtained in one-dimensional geometry in Section 3. A particular attention has been given to the study of the influence of the cavity detuning
The unknowns in (2) are the field profile Z’(r), /3 and 6,. Eq. (2) has confined solutions only for particular values of p and S, which correspond respectively to the threshold (&) and the detuning of the transverse modes. For a given mode, the threshold pump power is given by: (3) Above threshold ( /3 > &,). the incident pump power (P) is calculated with (3) by replacing &, by /3. Solutions of Eq. (2) are written as: K:‘(r) = \II,oe(r).
(4)
e(r) represents the laser field distribution and I, the on-axis intensity. Our hypothesis is that e(r) does not vary very much when p is just above the threshold. In order to calculate the slope of the laser characteristic for a given transverse mode order, we modify (I) as follows: we multiply both sides of (I) by 8 - and integrate over the transverse plane. The real and imaginary part of the obtained expression are given by [12]:
(5)
(6) The conservation relation (5) means the oscillation tion: gain equals losses. Relation (6) corresponds mode-pulling phenomenon. Inserting (4) in (5) yields:
condito the
We now derive relation (7) versus the pumping parameter and then evaluate the expression at threshold (when I, = 0). This allows one to obtain the derivative of the on-axis intensity at threshold:
(8) Following our hypothesis we have neglected of e(r) versus /3 in (8).
the variations
F. Sunche:. A. Churdon/
The output p,,J
power
is defined
Optics Cnmmunicurions 136 (1997) 405-409
407
as:
P) = 7% P)lle(r)12dS.
(9)
The slope of the laser characteristic at threshold (or the laser efficiency) is defined as and calculated with:
SE-
d Pour dPout (jf(r)dS))’ dP P!h dD Blh
I
=-
I
= “1;
s%‘. th
o.oL
(‘0)
0
a
1 5
IO
w
where
I5
20
P
Fig. 1. Evolution of the threshold pump power as a function of the pump width for the TEM, and the TEM, transverse modes. T = 0.02 and 8, = 0. Note that s is proportional to I/P,, and to a geometrical factor (ZY) which is related to the overlapping between the pump and the lasing mode. This geometrical factor is similar to that derived in Ref. [4] for independent Gaussian pump and lasing mode. However a great difference exists between the model of Ref. [4] and ours. Our model is closer to the practice where the field distribution strongly depends on the pump profile. In addition, the cavity detuninp is taken into account in this paper while it is not introduced in Ref. [4]. Lastly, we consider a four-level amplifying medium whereas a three-level is considered in Ref. [4]. Therefore, one can expect that the resufts wilf be different. Our analysis gives the slope efficiency at threshold, Experimental results show that the characteristic in microchip lasers is quasi-linear [4, IO. 151. The slope at threshold is therefore (with a good approximation) a measure of the laser efficiency.
3. Results in one-dimensional
geometry
In this section we apply the general method presented in Section 2 to the particular case of a one-dimensional geometry considering a pump profile for which analytical results for the transverse modes can be obtained [ 121: f(x)
= sech2(2x/a),
For each mode N, the laser detuning. the threshold and 7) are expressed as: 6 = (2N+
l)2T/(a’z)+4N(N+
L
l)/a2+6,
I +i+T/2
Prh=(l+Q(:+V2),
(‘3)
T
n=F(l+ZN)+i$L i.
8 l+2N’
where ; is the real positive root of a fifth-order polynomial equation not given here (for a complete study see Ref.
Lf21). Let us now consider the evolution of the threshold pump power ( Pth) versus the pump width for TEM, and TEM, modes. Fig. I gives the results obtained for 6, = 0. Note that we have assumed that higher-order modes operate independently of each other. The threshold for the fundamental mode presents a minimum in some range of pump widths (in the vicinity of n’r,= 6). The discrimination factor between the TEM, and the TEM, mode increases for narrow pump widths. It is worth to recall that the threshold of the second lasing mode must be calculated when the medium is saturated by the fundamental mode and hence the discrimination factor that we deduce from Fig. 2 is under-evaluated (this is due to the fact that the gain medium is higher without saturation than with satura-
(11)
where a represents the normalized pump width. Note however that the parameter a is not the half-width at 1/e2 of the pump intensity distribution. The latter is hap= 0.8287~. The pump width and the .r-coordinate are both normalized versus (lA,,/27r)‘/2, where I is the length of the active medium and A, the laser wavelength. The transverse modes are given by [ 121: eN(x)
I = ~cosh(2x,a~12q
HN[tanWx/a>l+
(‘2)
where n is a complex number and HN a polynomial order N: H,(X)= I and H,(X)=X.
of
Fig. 2. Fundamental mode (solid line) and pump (dashed line) profiles for the lowest threshold operation. T = 0.02 and 6, = 0.
F. Sunche~.
408
0
IO
w
A. Chardon
15
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Communiccttions
136
f 1997)
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P
Fig. 3. Evolution of the threshold of the fundamental mode as a function of the pump width for several cavity detunings.
tion) [I I]. Lastly, the different thresholds tend towards a common value for very large pump sizes (in this case the problem falls in the plane-wave approximation). Fig. 2 gives the mode and pump profiles for the optimal value of rvp (nb = 6). Note that the optimization is achieved for pump sizes lower than the mode-width contrary to the results of Ref. [4]. We expect this difference is due to the fact that. in our model the pump and mode shapes (and widths) are dependent while in Ref. [4] they are independent and Gaussian. Moreover, Ref. [4] treats the case of a three-level system. Fig. 3 represents the evolution of the threshold pump power for the fundamental mode as a function of the pump width for different cavity detunings. This figure points out that negative cavity detunings strongly increase the laser threshold for narrow pump width. For positive cavity detunings, the threshold is an increasing function of M’~_In this case the threshold is optimized for a pure on-axis pumping. For 6,~ 0 the threshold presents a minimum value. Finally, for very large pump widths, a zero cavity detuning ensures a minimum threshold operation. The different evolutions of the fundamental mode threshold can be physically interpreted in terms of overlapping between the pump beam and the fundamental mode profiles. Indeed, Fig. 4 shows the evolution of the fundamental mode-width (defined as the half-width at I/e2 of the intensity) as a function of the pump width for various cavity detunings. Let us first consider the case of negative cavity detunings. For small pump sizes, the resulting mode-width is much greater, due to index anti-guiding effect, and the losses of the mode in the unpumped regions increase the laser threshold. Conversely, for high pump sizes, the mode-width is lower and the threshold increases because of the larger pumped section. In the case of positive cavity detunings, the index guiding effect leads to a more confined fundamental mode, especially for narrow pump widths, thus resulting in a lowering threshold. For large pump sizes. the same comments as for negative detunings apply for positive detunings.
W P
Fig. 4. Evolution of the fundamental mode width versus the pump width for different cavity detunings. The previous threshold evolutions are difficult to compare to previously published theoretical works for four-level laser materials 16.81 because none of them includes either inter-dependent pump and lasing mode or index guiding (or anti-guiding) effects. However, inspection of Fig. 4 shows that for positive cavity detunings, the mode-width slightly depends on the pump size. This is due to the index guiding effect. This situation is very close to that investigated in Refs. [6,8] where the mode and the pump widths were independent. Not surprisingly, the results are the same: on-axis pumping provides the lowest threshold and the highest laser efficiency as we will see below. Conversely, for negative cavity detunings, the mode-width undergoes large variations versus the pump size and our results considerably depart from previous analyses [6.8]. Let us now consider the evolution of the laser efficiency versus the pump width. The results are shown in Fig. 5 for several cavity detunings. There is again an asymmetry for positive and negative cavity detunings. For positive cavity detunings, the laser efficiency is optimized for a pure on-axis pumping, while for negative detunings, the efficiency exhibits a maximum for a particular pump width lower than that optimizing the threshold. The global evolution of the laser efficiency can be easily interpreted in view of the threshold evolution: recall that the slope is inversely proportional to the threshold pump power. The slope is also proportional to the geometrical factor .Y (it is
Fig. 5. Variations of the laser efficiency versus the pump width for several cavity detunings.
F. Sunchr:.
A. Chcrrdon
/ Optics
worth to note that the factor 5 is not the simple integral overlap between the pump and the lasing mode). The evolution of 3’ versus the pump width (not shown here) is responsible for the shift of the \t3,-value which optimizes the threshold, towards lower values. Another consequence of the index anti-guiding effect, occurring for negative cavity detunings, is that the maximum achievable efficiency is slightly lower than that achievable for positive detunings.
Communicution.s
136 (1997)
405-409
which leads to a more expanded narrow pump widths.
409
fundamental
mode for
Acknowledgements We are very grateful to Professor valuable comments on the document.
G. Stephan
for his
References 4. Conclusions In this paper, we have investigated the influence of the pump size on both the threshold and the efficiency of microchip lasers. The method was based on the evaluation of the slope of the laser characteristic at threshold. Our results have revealed the importance of the cavity detuning on the choice of the pump width which optimizes the threshold or the laser efficiency. The analysis performed in one-dimensional geometry shows essentially two features. For positive cavity detunings. the efficiency is optimized for a pure on-axis pumping which, in turn, ensures the lowest threshold pump power. This results from the index guiding effect which leads to a more confined fundamental mode: the mode-width does not significantly vary versus the pump width. On the other hand, for negative cavity detunings. the threshold presents a minimum for a particular value of the pump width. The efficiency exhibits a maximum but for a lower pump width. In practice, there is some range of pump sizes for which the efficiency is high and the threshold is low, even if any is optimized. These results are the consequence of the index anti-guiding effect
[I] T.Y. Fan and R.L. Byer. IEEE J. Quantum Electron. QE-24 (1988) 895. [2] V. Couderc, 0. Guy, A. Barthelemy. C. Froehly and F. Louradour, Optics Lett. I9 ( 1994) I 134. [i] P. Nachman, J. Munch and R. Yee, IEEE J. Quantum Electron. QE-26 (1990) 3 17. [4] P. Laporta. S. Longhi, S. Taccheo and 0. Svelte, Optics Comm. 100 (199.3) 31 I. [5] C. Pare, Optics Comm. I23 (1996) 762. [6] P. Laporta and M. Btussard, IEEE J. Quantum Electron. QE-27 (1991) 2.319. [7] S. Taccheo, P. Laporta, S. Longhi and C. Svelte, Optics Lett. 20 ( 1995) 889. [8] D.G. Hall, R.J. Smith and R.R. Rice, Appl. Optics I9 (1980) 3041. [9] D.G. Hall, Appl. Optics 20 (1981) 1579. [IO] T.Y. Fan, Optics Lett. 19 (1994) 554. [I I] G. Harkness and W.J. Firth, J. Mod. Optics 39 (1992) 2023. (121 S. Longhi, J. Opt. Sot. Am. B II (1994) 1098. [I31 F. Sanchez and A. Chardon. J. Opt. Sot. Am. B I3 (December 1996). [l4] C. Pfistner. R. Weber, H.P. Weber, S. Merazzi and R. Gruber, IEEE J. Quantum Electron. QE-30 (1994) 1605. [ 151 J.J. Zayhowski and A. Mooradian, Optics Lett. I4 (1989) 24.
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
136 (1997) 405-409
Pump size optimization in microchip lasers F. Sanchez I, A. Chardon * GIS02.
Luhorutoire
d’Optronique,
ENSSAT 6. rue de Khunpont,
Received 17 September 1996; revised 4
B.P. 447.22305
Lonnion Crdex, France
November 1996; accepted 6 November 1996
Abstract We investigate the influence of the transverse size of the pumping light both on the threshold and the efficiency of microchip lasers. A general method is presented and applied in the case of one-dimensional geometry. It is shown that for positive cavity detuning, the laser efficiency always decreases when the pump size increases while for negative cavity detuning, it is optimized for a particular pump width. These results are interpreted in terms of index guiding or anti-guiding effects depending on the sign of the cavity detuning.
1. Introduction Diode-pumped solid-state lasers are very interesting because it is possible to build-up compact and low cost lasers in the infrared and visible spectral ranges. Optimization of their output power or spatial characteristics is needed for many applications [I -91. Optimization of the transverse size of the pump beam for maximum power extraction in solid-state lasers or amplifiers has been previously studied [4-81. The case of a four-level amplifying medium inserted in a resonant cavity (typically a planoconcave cavity) end-pumped with a laser diode has been investigated in Refs. [6,8]. The optimization consisted basically in evaluating the overlapping integral of the pump and the laser transverse distributions. In this configuration, the transverse widths of the pump and of the laser beams were independent: indeed, the transverse field distribution of the laser is mainly determined by the geometrical properties of the empty cavity and does not depend on the pump profile (if thermal effects and self-focusing or defocusing effects are neglected). The main result in these studies is that on-axis pumping provides the lowest threshold and the highest slope efficiency. In Ref. [4] the authors considered a three-level medium in a plano-concave mi-
I
E-mail: [email protected]. * E-mail: [email protected].
crochip laser where both pump beam and laser mode have two-dimensional Gaussian intensity profiles in the transverse (x, y) plane with independent beam radius. Pump beam divergence was also taken into account. They found that the minimum value for the threshold power and the maximum of the slope efficiency are obtained when the average dimensions of the pump beam and the laser mode become equal. These results can be physically interpreted in terms of integral overlap between the pump and laser mode. The main difference compared to a four-level laser material is that for a pump size much lower than the mode size, the laser signal is strongly absorbed in the non-pumped volume. This fact explains the poor performances achieved with narrow pump beams. Additional results on three-level flat-flat microchip lasers are given in Ref. [IO]. The author identifies that the mode shaping is driven by the off-axis losses (aperture-guiding). As a consequence, the mode-spot radius is fairly insensitive to both pump power and shape. The obtained results cannot be applied to the real case of end-pumped four-level flat-flat microchip lasers for the following reasons: (i) the empty plane-plane cavity does not sustain confined modes, (ii) transverse mode structures (in particular the fundamental mode width) are highly dependent on both the pump profile and the pump width. Therefore the pump size optimization in microchip lasers differs from that of a classical solid-state laser and needs a specific study. In this paper we are interested in investigating the influence of the pump size on both the threshold pump
0030-4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)00726-2
406
F. Sunchr;.
A. Chardon/
Optics
power and the power laser efficiency (i.e. the slope of the output power versus input pump power characteristic) of microchip lasers. This problem can be studied directly from the following differential nonlinear equation which describes the steady-state for the laser field in the cavity [I l-131: Vz-iS+S,--fi,+Pf(r)
8= I +6;+lZ12
136 (1997)
405-409
which plays a key role in the optimization threshold and the laser efficiency.
2. Theoretical
of both the
method
The first step consists in finding the transverse modes (in particular the fundamental one) at threshold. This is performed by solving Eq. (I) without the saturation term:
6,+i
[
Communication.s
0.
I
(‘1 Eq. (1) is valid under the mean-field approximation and disregards any thermal effect. 8 is the electric field distribution, p represents the pumping parameter (more precisely the on-axis pump intensity), T is the transmission of the output mirror, 6, and S, are respectively the laser field frequency and the longitudinal cavity mode frequency detunings, both referenced to the atomic line center. The function fir) represents the pump beam profile which is characterized by its width. Eq. (1) also assumes a homogeneously broadened four-level medium. In Refs. [I I, 121 the saturation factor of Eq. (I) contains an additional factor in front of the 6, terms. For convenience this term is taken as unity in this paper. Although thermal index effects can be important in some cases [14], we neglect them in this paper and focus our attention on gain guiding and index guiding or anti-guiding effects resulting from the pump profile. Note that thermal guiding is not predominant in the mode formation in three-level microchip lasers [IO]. For a given pump profile, Eq. (I) allows to calculate the field distribution for any pumping parameter and to further evaluate the laser output power. Nevertheless, the saturation term makes the problem nonlinear and fully numerical methods have to be used. We propose here an alternative method allowing us to achieve analytical (in one-dimensional geometry) and simple numerical results. Owing to the nonlinearity of Eq. (I), iterative methods are needed to obtain the solution which depends on the pumping parameter p (in fact (1) is not an eigenvalue problem, except at threshold, because of nonlinear saturation). We assume in this paper that the transverse distribution of the pump and the laser remain the same for pumping rates very close to (but above) the laser threshold. As a matter of fact, it has been recently reported in Ref. [7] that the mode-width in an Er:Yb:glass microchip laser undergoes a very limited reduction (I IO%) when the pump power is increased up to twice the threshold. Similar results are reported in Ref. [IO]. With this hypothesis the problem can be solved in two steps: (i) evaluate the transverse mode distributions at threshold using the adapted Eq. (I) and (ii) calculate the laser efficiency using the oscillation condition derived in the next section. The theoretical method is presented in Section 2 and the results obtained in one-dimensional geometry in Section 3. A particular attention has been given to the study of the influence of the cavity detuning
The unknowns in (2) are the field profile Z’(r), /3 and 6,. Eq. (2) has confined solutions only for particular values of p and S, which correspond respectively to the threshold (&) and the detuning of the transverse modes. For a given mode, the threshold pump power is given by: (3) Above threshold ( /3 > &,). the incident pump power (P) is calculated with (3) by replacing &, by /3. Solutions of Eq. (2) are written as: K:‘(r) = \II,oe(r).
(4)
e(r) represents the laser field distribution and I, the on-axis intensity. Our hypothesis is that e(r) does not vary very much when p is just above the threshold. In order to calculate the slope of the laser characteristic for a given transverse mode order, we modify (I) as follows: we multiply both sides of (I) by 8 - and integrate over the transverse plane. The real and imaginary part of the obtained expression are given by [12]:
(5)
(6) The conservation relation (5) means the oscillation tion: gain equals losses. Relation (6) corresponds mode-pulling phenomenon. Inserting (4) in (5) yields:
condito the
We now derive relation (7) versus the pumping parameter and then evaluate the expression at threshold (when I, = 0). This allows one to obtain the derivative of the on-axis intensity at threshold:
(8) Following our hypothesis we have neglected of e(r) versus /3 in (8).
the variations
F. Sunche:. A. Churdon/
The output p,,J
power
is defined
Optics Cnmmunicurions 136 (1997) 405-409
407
as:
P) = 7% P)lle(r)12dS.
(9)
The slope of the laser characteristic at threshold (or the laser efficiency) is defined as and calculated with:
SE-
d Pour dPout (jf(r)dS))’ dP P!h dD Blh
I
=-
I
= “1;
s%‘. th
o.oL
(‘0)
0
a
1 5
IO
w
where
I5
20
P
Fig. 1. Evolution of the threshold pump power as a function of the pump width for the TEM, and the TEM, transverse modes. T = 0.02 and 8, = 0. Note that s is proportional to I/P,, and to a geometrical factor (ZY) which is related to the overlapping between the pump and the lasing mode. This geometrical factor is similar to that derived in Ref. [4] for independent Gaussian pump and lasing mode. However a great difference exists between the model of Ref. [4] and ours. Our model is closer to the practice where the field distribution strongly depends on the pump profile. In addition, the cavity detuninp is taken into account in this paper while it is not introduced in Ref. [4]. Lastly, we consider a four-level amplifying medium whereas a three-level is considered in Ref. [4]. Therefore, one can expect that the resufts wilf be different. Our analysis gives the slope efficiency at threshold, Experimental results show that the characteristic in microchip lasers is quasi-linear [4, IO. 151. The slope at threshold is therefore (with a good approximation) a measure of the laser efficiency.
3. Results in one-dimensional
geometry
In this section we apply the general method presented in Section 2 to the particular case of a one-dimensional geometry considering a pump profile for which analytical results for the transverse modes can be obtained [ 121: f(x)
= sech2(2x/a),
For each mode N, the laser detuning. the threshold and 7) are expressed as: 6 = (2N+
l)2T/(a’z)+4N(N+
L
l)/a2+6,
I +i+T/2
Prh=(l+Q(:+V2),
(‘3)
T
n=F(l+ZN)+i$L i.
8 l+2N’
where ; is the real positive root of a fifth-order polynomial equation not given here (for a complete study see Ref.
Lf21). Let us now consider the evolution of the threshold pump power ( Pth) versus the pump width for TEM, and TEM, modes. Fig. I gives the results obtained for 6, = 0. Note that we have assumed that higher-order modes operate independently of each other. The threshold for the fundamental mode presents a minimum in some range of pump widths (in the vicinity of n’r,= 6). The discrimination factor between the TEM, and the TEM, mode increases for narrow pump widths. It is worth to recall that the threshold of the second lasing mode must be calculated when the medium is saturated by the fundamental mode and hence the discrimination factor that we deduce from Fig. 2 is under-evaluated (this is due to the fact that the gain medium is higher without saturation than with satura-
(11)
where a represents the normalized pump width. Note however that the parameter a is not the half-width at 1/e2 of the pump intensity distribution. The latter is hap= 0.8287~. The pump width and the .r-coordinate are both normalized versus (lA,,/27r)‘/2, where I is the length of the active medium and A, the laser wavelength. The transverse modes are given by [ 121: eN(x)
I = ~cosh(2x,a~12q
HN[tanWx/a>l+
(‘2)
where n is a complex number and HN a polynomial order N: H,(X)= I and H,(X)=X.
of
Fig. 2. Fundamental mode (solid line) and pump (dashed line) profiles for the lowest threshold operation. T = 0.02 and 6, = 0.
F. Sunche~.
408
0
IO
w
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15
/ Optics
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136
f 1997)
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20
P
Fig. 3. Evolution of the threshold of the fundamental mode as a function of the pump width for several cavity detunings.
tion) [I I]. Lastly, the different thresholds tend towards a common value for very large pump sizes (in this case the problem falls in the plane-wave approximation). Fig. 2 gives the mode and pump profiles for the optimal value of rvp (nb = 6). Note that the optimization is achieved for pump sizes lower than the mode-width contrary to the results of Ref. [4]. We expect this difference is due to the fact that. in our model the pump and mode shapes (and widths) are dependent while in Ref. [4] they are independent and Gaussian. Moreover, Ref. [4] treats the case of a three-level system. Fig. 3 represents the evolution of the threshold pump power for the fundamental mode as a function of the pump width for different cavity detunings. This figure points out that negative cavity detunings strongly increase the laser threshold for narrow pump width. For positive cavity detunings, the threshold is an increasing function of M’~_In this case the threshold is optimized for a pure on-axis pumping. For 6,~ 0 the threshold presents a minimum value. Finally, for very large pump widths, a zero cavity detuning ensures a minimum threshold operation. The different evolutions of the fundamental mode threshold can be physically interpreted in terms of overlapping between the pump beam and the fundamental mode profiles. Indeed, Fig. 4 shows the evolution of the fundamental mode-width (defined as the half-width at I/e2 of the intensity) as a function of the pump width for various cavity detunings. Let us first consider the case of negative cavity detunings. For small pump sizes, the resulting mode-width is much greater, due to index anti-guiding effect, and the losses of the mode in the unpumped regions increase the laser threshold. Conversely, for high pump sizes, the mode-width is lower and the threshold increases because of the larger pumped section. In the case of positive cavity detunings, the index guiding effect leads to a more confined fundamental mode, especially for narrow pump widths, thus resulting in a lowering threshold. For large pump sizes. the same comments as for negative detunings apply for positive detunings.
W P
Fig. 4. Evolution of the fundamental mode width versus the pump width for different cavity detunings. The previous threshold evolutions are difficult to compare to previously published theoretical works for four-level laser materials 16.81 because none of them includes either inter-dependent pump and lasing mode or index guiding (or anti-guiding) effects. However, inspection of Fig. 4 shows that for positive cavity detunings, the mode-width slightly depends on the pump size. This is due to the index guiding effect. This situation is very close to that investigated in Refs. [6,8] where the mode and the pump widths were independent. Not surprisingly, the results are the same: on-axis pumping provides the lowest threshold and the highest laser efficiency as we will see below. Conversely, for negative cavity detunings, the mode-width undergoes large variations versus the pump size and our results considerably depart from previous analyses [6.8]. Let us now consider the evolution of the laser efficiency versus the pump width. The results are shown in Fig. 5 for several cavity detunings. There is again an asymmetry for positive and negative cavity detunings. For positive cavity detunings, the laser efficiency is optimized for a pure on-axis pumping, while for negative detunings, the efficiency exhibits a maximum for a particular pump width lower than that optimizing the threshold. The global evolution of the laser efficiency can be easily interpreted in view of the threshold evolution: recall that the slope is inversely proportional to the threshold pump power. The slope is also proportional to the geometrical factor .Y (it is
Fig. 5. Variations of the laser efficiency versus the pump width for several cavity detunings.
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worth to note that the factor 5 is not the simple integral overlap between the pump and the lasing mode). The evolution of 3’ versus the pump width (not shown here) is responsible for the shift of the \t3,-value which optimizes the threshold, towards lower values. Another consequence of the index anti-guiding effect, occurring for negative cavity detunings, is that the maximum achievable efficiency is slightly lower than that achievable for positive detunings.
Communicution.s
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which leads to a more expanded narrow pump widths.
409
fundamental
mode for
Acknowledgements We are very grateful to Professor valuable comments on the document.
G. Stephan
for his
References 4. Conclusions In this paper, we have investigated the influence of the pump size on both the threshold and the efficiency of microchip lasers. The method was based on the evaluation of the slope of the laser characteristic at threshold. Our results have revealed the importance of the cavity detuning on the choice of the pump width which optimizes the threshold or the laser efficiency. The analysis performed in one-dimensional geometry shows essentially two features. For positive cavity detunings. the efficiency is optimized for a pure on-axis pumping which, in turn, ensures the lowest threshold pump power. This results from the index guiding effect which leads to a more confined fundamental mode: the mode-width does not significantly vary versus the pump width. On the other hand, for negative cavity detunings. the threshold presents a minimum for a particular value of the pump width. The efficiency exhibits a maximum but for a lower pump width. In practice, there is some range of pump sizes for which the efficiency is high and the threshold is low, even if any is optimized. These results are the consequence of the index anti-guiding effect
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