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A numerical model of Kerr-lens mode-locking
15
October1997 OPTICS COMMUNICATIONS
ELSEVIER
Optics Communications
142 (1997) 315-321
Full length article
A numerical model of Kerr-lens mode-locking A. Ritsataki
*,
P.M.W. French, G.H.C. New
Laser Optics and Spectroscopy Group. Department of Physics, Imperial College, London SW7 2B2, UK Received
17 March 1997; accepted 3 June 1997
Abstract We have developed a numerical model of Kerr-lens mode-locking (KLM) that includes astigmatism, non-linear coupling and gain-guiding. Under the current approximations of the model, gain-guiding decreases the value of the hard-aperture mode-locking parameter 6 = (l/w)(dw/dp)l,,a. The model also allows one to study the effect of elliptical astigmatic pump beams, typically encountered in diode pumped laser systems, as the position of the pump beam waist is moved within the gain medium. 0 1997 Elsevier Science B.V.
1. Introduction Kerr-lens mode-locking (KLM) has been used extensively over the past few years to generate femtosecond pulses in solid state lasers, see e.g. Refs. [1,2]. In the simplest picture of the KLM process, self-focusing in the active medium. in combination with an intracavity aperture, causes the losses to be reduced as the radiation intensity increases: mode-locked operation is thereby favoured. Whereas mode-locking has indeed been achieved with a real aperture in the resonator [3], some systems mode-lock without an aperture present, indicating that there must be other mechanisms that favour a more tightly-focused beam. One possible candidate is the non-uniform transverse gain profile created by the pump laser which serves as a soft aperture favouring narrower beams that pass through the region of the active medium where the gain is highest. The non-uniform gain also influences the width of the beam leaving the laser gain medium, and so affects the mode profile throughout the rest of the cavity and therefore the discrimination caused by a real (hard) aperture. Hermann [4] distinguished between these distinct (though related) mechanisms, which we will refer to as “gain-aperturing” and “gain- guiding” respectively; a complete model of the KLM process should include both. It should be noted that gain-guiding dramatically extends the stability regions of otherwise unstable resonators [5,6]. A great deal of work has been performed over recent years to optimise the cavity design of mode-locked solid-state lasers, and particularly to achieve self-starting with as few optical elements in the cavity as possible [7]. A criterion for this optimisation was introduced by Cemllo et a1.[8,9] who derived a simple formula for a parameter 6 that measures the relative beam size variation at one of the cavity mirrors with intra-cavity power. A large negative value of 6 implies that a hard aperture placed at that location would offer lower loss to the narrower (higher intensity) portions of the circulating radiation; the effect that an actual aperture would have on the mode profile in the cavity [lo] is however ignored. Although useful, this approach involves a number of simplifications, which restrict its applicability; both gain-guiding and the nonlinear coupling between the mode profiles in the sagittal and tangential planes [I 1,121 were for example ignored in Refs. [8,9]. In the current work, we present a numerical model of a KLM laser that widens the applicability of the approach of Cerullo et al. [9]. As well as the self-focusing process, the model includes astigmatism (introduced by off-axis spherical
* E-mail: [email protected]. 0030.4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-4018(97)00310-6
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mirrors and Brewster-angled rods), the divergence of elliptical pump beams in the laser medium (and the non-uniform transverse gain profile this creates), nonlinear coupling between the mode profiles in the sagittal and tangential planes [11,12], and gain-guiding. However, the effect of gain aperturing (as defined above) is not considered in the present study. Although the simplicity of the analytical formula developed by Cerullo et al. [9] is lost, our model provides a more complete picture of the effects taking place in the cavity, and allows the elliptical (astigmatic) pump beams in real end-pumped diode lasers 1131 to be taken properly into account.
2. The numerical model The basic principles
and assumptions
made in our numerical model are as follows: of the intensity-dependant laser beam around the cavity in an iterative procedure that searches for a self-consistent solution to the complex beam parameter. * Only the fundamental Gaussian transverse mode is assumed to be present in the cavity and the paraxial approximation ensures that the beam preserves its Gaussian character. * The gain medium is pumped by a laser whose transverse beam profile is Gaussian; pump depletion within the gain medium and gain saturation are both neglected - Nonlinear coupling, astigmatism, and gain-guiding are all included in the model. The self-focusing process is modelled using the nonlinear ABCD matrix technique of Magni et al. [8], which is based on a quadratic approximation of the Gaussian transverse profile. We include the extension suggested by Bridges et al. [ 111 to allow for nonlinear coupling (the influence of the beam profile in one transverse plane on the refractive index seen by the profile in the other). To enable nonlinear coupling and/or gain-guiding to be included, it is necessary to adopt a split-step method in which the Kerr medium is divided into a large number of slices (typically 100). The matrix representing propagation through a thin slice of nonlinear material of refractive index n, in the presence of cross-coupling is
- ABCD matrix methods are used to track the propagation
1
-
d
1
where
“=
w,(z,)
pcrit
1 + ( mvf,/dh)2(
1 + d/R, ;)’
’
(2)
Similar equations with the indices x and _v reversed apply for the orthogonal polarisation. In these equations, d is the path length in the medium, P/P,,, the intracavity power normalised to the critical power for self-trapping, no is the linear refractive index, and A is the wavelength in the medium. The parameters w, * and R, x are the respective width and radius of curvature of the beam on entry into the slice, while w,(z,) and w&z,> are the beam sizes at the centre (z = z,) of the segment calculated in the limit of zero power (see Ref. [S]). Note that if the slice is placed at the Brewster angle, the matrix governing propagation in the tangential plane is obtained by sandwiching M in Eq. (1) between matrices representing Snell’s Law of refraction yielding
(3) -
which shows that the effect of the astigmatism on propagation in the tangential plane is to change the effective distance propagation from d/n, to d/n: where d is now the path length in the material, not the thickness of the slab. The matrix the sagittal plane is unaltered. A matrix can also be constructed to represent the effect of the non-uniform gain on the profile of the beam leaving active medium; this is the process we have termed gain-guiding, as distinct from gain-apetturing which concerns how
of for the the
A. Ritsataki et al. / Optics Communications
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same non-uniformity affects the gain efficiency. Propagation through a segment of length d and refractive index n, with a Gaussian transverse gain profile of peak value g, and width wr, can be described by a transformation matrix of the form [X61:
(4)
where A,, is the wavelength in vacuum. For this equation to be valid, the inequality d +c Z,/g, must be satisfied [5,6] where Za (= rru$/hl is the Rayleigh length for the pump beam. Within the same limits of approximation, i.e. for laser systems with short active media and small peak gain (or large pump beams), the matrix Ms can be factorised as follows
(5)
This indicates that propagation through a medium with a quadratic gain profile is roughly equivalent to a Gaussian aperture of width wn/ 6, followed by free space propagation through a distance d/n,,. The combination within the split-step framework of self-focusing with gain-guiding is accomplished simply by taking the product of the appropriate matrices, bearing in mind that the free propagation component must only be included once. For the x-dimension, the relevant combined matrix is
(6)
In this case, one can no longer take astigmatism into account by changing the effective distance of propagation tangential plane from d/n, to d/n: as in Eq. (3) above. Rather one should simply apply the matrix
[
n0 0
0
l/n0
in the
1
in the tangential plane on entry to the medium and the inverse matrix on exit. In previous treatments of gain-guiding [4-61, the pump beam was usually assumed to be circular and non-diverging, so that the resulting gain profile did not vary along the t-axis. The present model allows one to treat the pump as an elliptical, astigmatic, diverging beam and to include both pump depletion along the rod axis and the change of the small signal gain with :. For simplicity, however, the results presented below are for the case where pump depletion is neglected and go does not vary with z.
3. Numerical results Calculations were performed for the typical four-element Z-type resonator used in Refs. [9,14] and shown in Fig. 1. The Kerr-active laser medium was 2 cm long, its refractive index was 1.76, and it was positioned between the two identical spherical mirrors M, and M, both having a radius of curvature of IO cm. The folding angle 2 0 was 29.6” and the cavity was terminated by the two plane mirrors M, and M, both at 85 cm from the corresponding spherical mirror to form a symmetrical resonator. The separation z of M, and M, is the key parameter controlling cavity stability and, together with x which measures the position of the rod between the folding mirrors, has a critical influence on the KLM process. A series of calculations was performed in which the relative beam size variation at M, in the low power limit (S = (l/w)(dw/dp)],,a) was studied as a function of x and z, for increasingly sophisticated cavity models. We first simulated the KLM laser on the basis of the simplifying assumptions of Cerullo et al. [9], where nonlinear coupling and gain-guiding was neglected; in this case, 6 depends only on the characteristics of the cavity elements and their relative
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142 (1997) 315-321
M2 11
12
M3
I
M4 Fig. 1. Schematic of a typical Z-folded cavity. Spherical mirrors M, and M3 have radii of curvature and the laser medium (refractive index 1.76) is 20 mm long.
100 mm, the folding angle 2 fI is 29.6”,
positions. As a basic test, the formula for 8 quoted in [9] was compared with our own model, and the results are displayed as contour plots of S(x,z) for the tangential plane in Figs. 2(a) and 2(b) respectively, which are in encouragingly good agreement. The grey scaling runs from S = -0.5 (darkest) to 6 = + 0.5 (lightest); this implies that where the display is darkest, the spot size at M , decreases most strongly with increasing intracavity power, and so a hard aperture placed at M,
(4
5.5
5.5
_
5.0
-
E
s 4.5
x
4.0
0 g
ABOVE
5.0
E
2. x
(b)
6.0
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4.0
0.5
0.30.4-
0.4 0.5
0.2 -
0.3
0.1 -
0.2
-0.0 -
0.1
I
-0.2 -0.1 - -0.0 -0.1
I
-0.4-0.3-
3.5
3.5 11.3
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114
z@m)
11.5
11.6
11.7
11.6
z(cm)
6.0
-0.3 -0.2 5.5
I
BELOW -0.5 - -0.4 -0.5
4.0
3.5 11.3
11.4
11.5 z(cm)
11.6
11.7
11.8
11.3
11.4
11.5
11.6
11.7
11.6
z(cm)
Fig. 2. Contour plots of 6(x,z) using grey scaling between -0.5 and f0.5: (a) predictions of the analytical formula of Ref. [8] in the tangential plane; (b) the numerical model, with nonlinear x-y coupling and gain-guiding neglected, in the tangential plane; (c) and (d) the numerical model with nonlinear x-y coupling included for sagittal and tangential planes respecitively. The magnitudes of 6(x, z) significantly exceed the extremes of the grey scaling close to the confocal boundary.
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would favour mode-locked operation. For the parameter values used, the bare resonator is stable in the ranges 11.20 < z < 11.54 and 1 I.54 < ,- < 11.87 cm in the sagittal plane and 1 I.3 1 < ,- < 11.59 and I 1.59 < z < 11.93 in the tangential plane. In the tangential plane, at ; = Il.59 cm, the resonator is exactly confocal and consequently marginally stable: this is the significance of the vertical feature at this point in Fig. 2(a). It is evident that the optimal region for mode-locking is close to the confocal boundary, either on the low : side around x = 4.5 or the high z side around x = 5.2. The S-plot for the sagittal plane (not shown here) is very similar to that for the tangential plane, although the vertical feature is now centred at 2 = I1 54. Note that magnitudes of 6 close to the boundary may be much greater than 0.5. Figs, 2(c) and 2(d) show that the properties of the laser are modified very significantly when nonlinear coupling between the two planes is included. The results for the sagittal plane (frame c) and the tangential plane (frame d) are in excellent
(b)
Fig. 3. Contour plots of 6(.x.;) for increasingly strong gain-guiding: (a) and (b) large 200 +m pump beam waist for sagittal and tangential planes respectively: (c) and (d) similarly for 100 pm waist: (e) and (f) similarly for 50 pm waist. All plots have same grey scaling as in Fig. 2. Small signal gain g,, is I.0 per one-way pass (or.01 per segment).
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agreement with those in [14]. The map for the sagittal plane is evidently more complex, and tighter control on the parameters values would appear to be necessary to stay in the mode-locking region. Since the two planes are now coupled, the confocal boundary for one plane shows up as a slight vertical feature in the &plot for the other: this is most evident in Fig. 2(d) at z = 11.54 cm. With significantly different behaviour in the two planes, selecting an optimum operating region is now more difficult and will involve the choice of hard aperture which could be a horizontal or vertical slit, or an ellipse of appropriate eccentricity. Next we include gain-guiding as well as x-y coupling in the case of a circular Gaussian pump beam at 670 nm. In Fig. 3 we show how the &plots in the sagittal and tangential planes are altered when gain-guiding of increasing strength is included in the model. In Figs. 3(a) and 3(b), the effect of a small amount of gain-guiding caused by a large pump beam of 200 km radius, is shown. The main characteristics of Figs. 2(c) and 2(d) can still be seen in the sagittal and tangential plots of Figs. 3(a) and 3(b), but the confocal boundaries are starting to become less distinct and a certain narrowing in the x-range is becoming apparent. In frames (c) and (d) of Fig. 3 where the pump beam radius is 100 urn, these trends are much more pronounced, culminating in a complete break of symmetry and weakening of the mode-locking parameter in frames (e) and (f) where the pump beam radius is further reduced to 50 km. Since gain-guiding tends to increase resonator stability [5,6], it is not surprising that the vertical feature associated with cavity instability is removed when gain-guiding is included. At the same time, the horizontal symmetry apparent in frames (al and (bl is radically altered, with negative 6 values now occurring only in the upper right quadrant of the picture. The sagittal &plots shows far weaker features than those for the tangential plane, suggesting that optimum operation will be achieved by using a vertical slit. For this plane, the single mode-locking region (where 6(x. z) is strongly negative) is narrowed in x and expanded in the z-direction. In both planes, the magnitude of 6 is generally lower than in the absence of gain-guiding. From all of the above, we are led to conclude that gain-guiding works against hard-aperture mode locking and should be minimised in such systems. In real end-pumped lasers, the pump beam profile varies down the length of the laser medium as a result of diffraction and absorption. The numerical model is easily extended to incorporate these features, as well as the effect of varying the position of the beam waist within the cell. This capability will be especially important when gain-aperturing (mode-matching) effects are included in the model at a later stage. Figs. 4 and 5 were generated for a diffraction-limited pump beam of 50 p,rn in the sagittal plane and 35 p,m in the tangential plane, assuming that it propagates without absorption as a Gaussian beam (w,,(z) = wPXO(l + z%~&/w~~~)‘/~ where wpxO is the beam waist and 13~~the far-field half angle OP.r= A,/n,rr wr,,,). We note that for the case of end pumping with non diffraction-limited beams from diode lasers or one-dimensional arrays of diode lasers, propagation through the medium can be described by wr.J z> = wP,,, + @rXl11 for spatially incoherent beams
1131. Fig. 4 traces the variation in the laser beam spot size round the entire astigmatic cavity in the case where the waist of the pump beam is at the end of the rod nearer to mirror Mz. It shows in particular that, in the presence of gain-guiding, the beam size depends on the direction of propagation. Moving the pump beam waist within the rod changes the laser spot size throughout the cavity as well as the &plots based on the spot size at M,. This latter effect is demonstrated in Fig. 5 which indicates that slightly better mode-locking conditions occur when the waist is at the end of the rod nearer to M, (frame cl. There is clearly reasonable correspondence between the &plot in frame (f) of Fig. 3 (where the pump beam has a uniform 50 urn radius throughout the laser medium) and those of Fig. 5 where wP 1 and wP,, are 50 pm and 35 urn respectively. This suggests that the assumption of Fig. 3 is a fairly good approximation for diffraction-limited beams. However, it seems likely
0.9
0.8 0.7 0.6 0.5 0.4
I
I
‘s,agittal’ ‘tange”t,#
I
-_ L-Z_
I
I
I
I
I
I
,;q ..;:;_$ ,,,,,# I,. .. -_-_._~_11,:.~:.....:_.__ _.-__~_----f_=.,’ ,.. ;*I .. _ ..,I ,,,‘. / : I _ ,A’ : i _. ._ ?_,” : _/’
0.3 0.2 0.1
Tt?P? 0
20
40 60 80 100 120 140 160 180 200 round trip from mirror Ml (cm)
Fig. 4. Evolution of laser beam size when the pump beam waist is at the entry (left plane) of the rod. The two curves represent evolution in the sagittal and tangential planes. The pump beam is elliptical with beam radius 50 km in the sagittal and 35 )*m in the tangential is 1% per segment and P/P, = 0.1.
plane; R,,
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11.5
11.6
117
118
z (cm)
114
115
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142 (1997) 315-321
11.7
11.8
114
z (cm)
11.5
11.6
117
11.8
z (cm)
Fig. 5. Contour plots of 6(x_.:) for three different positions of the pump waist in the rod and with nonlinear x-v coupling and gain-guiding included: (a) pump waist at the rod end nearer M?, (b) at centre of rod, (c) at rod end nearer M1. The tangential plane is shown in all three cases. The grey scaling the same as in Figs. 2 and 3.
that the detailed pump beam characteristics will become more important when gain-aperturing incorporated into the model and pump beams that are not diffraction-limited are considered.
(mode-matching)
effects are
4. Conclusions We have developed a numerical model of KLM that takes into account self-focusing, astigmatism and gain-guiding, and demonstrated in particular how gain-guiding affects the hard-aperture mode-locking parameter 6 = (l/~~)(du~/dp)(,,=~. In the simplified case where the gain profile is approximated by a parabola, and gain saturation is ignored, we have found that gain-guiding tends to work against hard-aperture mode-locking, reducing the magnitude of 6 while making the optimum operational region narrower overall. The model enables one to study the effect of elliptical, astigmatic pump beams on the laser beam throughout the cavity, as the position of the pump beam waist is moved within the gain medium. While it has been pointed out in REf. [4]. that the parabolic approximation overestimates the gain-guiding effect, it seems probable that gain saturation also weakens the gain guiding effect. Incorporation of both gain-aperturing [4] and gain-saturation [ 15.161 into the model is presently under way. Another potentially important process that is ignored in the present work is thermal lensing, and experimental work on our in-house laser systems is in progress to assess its magnitude. These effects, together with the study of soft-apertured three-element systems. will be treated in a future pu b lication.
References [I] [Z] [3] [4] [5] [6] [7] [8] [9] [lo] [I I] [12] [ 131 [14] [15] 1161
D.E. Spence. P.N. Kean. W. Sibbett, Optics Lett. 16 (1991) 42. 100. Ch. Spielmann. P.F. Curley, T. Brabec. F. Krausz. IEEE J. Quantum Electron. 30 (1994) T. Brabec. P.F. Curley. Ch. Spielmann, E. Wintner, A.J. Schmidt, J. Opt. Sot. Am. 10 (1993) 1029 J. Herrmann. J. Opt. Sot. Am. B 1I (1994) 498. F. Salin. J. Squier, Optics Lett. 17 (1992) 1353. F. Salin. J. Squier. M. Piche, Optics Lett. 16 (1991) 1674. E. Bouma, J.G. Fujimoto, Optics Lett. 21 (1996) 134. V. Magni. G. Cerullo, S. De Silvestri, Optics Comm. 96 (1993) 348. G. Cerullo, S. De Silvestri. V. Magni, L. Pallaro, Optics Lett. 19 (1994) 807. 0. Haderka, Optics Lett. 20 (1995) 240. E. Bridges, R.W. Boyd. G.P. Agrawal, Optics Lett. 18 (1993) 2026. E. Bridges. R.W. Boyd, G.P. Agrawal, J. Opt. Sot. Am. B 13 (1996) 553. T.Y. Fan, A. Sanchez, IEEE J. Quantum Electron. 26 (1990) 3 I 1. V. Magni, G. Cerullo, S. De Silvestri, A. Monguzzi. J. Opt. Sot. Am. B 12 (1995) 476. F. Salin, J. Squier. Optics Comm. 86 (1991) 397. M. Picht. F. Salin. Optics Lett. 18 (1993) 1041.
October1997 OPTICS COMMUNICATIONS
ELSEVIER
Optics Communications
142 (1997) 315-321
Full length article
A numerical model of Kerr-lens mode-locking A. Ritsataki
*,
P.M.W. French, G.H.C. New
Laser Optics and Spectroscopy Group. Department of Physics, Imperial College, London SW7 2B2, UK Received
17 March 1997; accepted 3 June 1997
Abstract We have developed a numerical model of Kerr-lens mode-locking (KLM) that includes astigmatism, non-linear coupling and gain-guiding. Under the current approximations of the model, gain-guiding decreases the value of the hard-aperture mode-locking parameter 6 = (l/w)(dw/dp)l,,a. The model also allows one to study the effect of elliptical astigmatic pump beams, typically encountered in diode pumped laser systems, as the position of the pump beam waist is moved within the gain medium. 0 1997 Elsevier Science B.V.
1. Introduction Kerr-lens mode-locking (KLM) has been used extensively over the past few years to generate femtosecond pulses in solid state lasers, see e.g. Refs. [1,2]. In the simplest picture of the KLM process, self-focusing in the active medium. in combination with an intracavity aperture, causes the losses to be reduced as the radiation intensity increases: mode-locked operation is thereby favoured. Whereas mode-locking has indeed been achieved with a real aperture in the resonator [3], some systems mode-lock without an aperture present, indicating that there must be other mechanisms that favour a more tightly-focused beam. One possible candidate is the non-uniform transverse gain profile created by the pump laser which serves as a soft aperture favouring narrower beams that pass through the region of the active medium where the gain is highest. The non-uniform gain also influences the width of the beam leaving the laser gain medium, and so affects the mode profile throughout the rest of the cavity and therefore the discrimination caused by a real (hard) aperture. Hermann [4] distinguished between these distinct (though related) mechanisms, which we will refer to as “gain-aperturing” and “gain- guiding” respectively; a complete model of the KLM process should include both. It should be noted that gain-guiding dramatically extends the stability regions of otherwise unstable resonators [5,6]. A great deal of work has been performed over recent years to optimise the cavity design of mode-locked solid-state lasers, and particularly to achieve self-starting with as few optical elements in the cavity as possible [7]. A criterion for this optimisation was introduced by Cemllo et a1.[8,9] who derived a simple formula for a parameter 6 that measures the relative beam size variation at one of the cavity mirrors with intra-cavity power. A large negative value of 6 implies that a hard aperture placed at that location would offer lower loss to the narrower (higher intensity) portions of the circulating radiation; the effect that an actual aperture would have on the mode profile in the cavity [lo] is however ignored. Although useful, this approach involves a number of simplifications, which restrict its applicability; both gain-guiding and the nonlinear coupling between the mode profiles in the sagittal and tangential planes [I 1,121 were for example ignored in Refs. [8,9]. In the current work, we present a numerical model of a KLM laser that widens the applicability of the approach of Cerullo et al. [9]. As well as the self-focusing process, the model includes astigmatism (introduced by off-axis spherical
* E-mail: [email protected]. 0030.4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-4018(97)00310-6
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A. Ritsataki et al. / Optics Communications142 (1997) 315-321
mirrors and Brewster-angled rods), the divergence of elliptical pump beams in the laser medium (and the non-uniform transverse gain profile this creates), nonlinear coupling between the mode profiles in the sagittal and tangential planes [11,12], and gain-guiding. However, the effect of gain aperturing (as defined above) is not considered in the present study. Although the simplicity of the analytical formula developed by Cerullo et al. [9] is lost, our model provides a more complete picture of the effects taking place in the cavity, and allows the elliptical (astigmatic) pump beams in real end-pumped diode lasers 1131 to be taken properly into account.
2. The numerical model The basic principles
and assumptions
made in our numerical model are as follows: of the intensity-dependant laser beam around the cavity in an iterative procedure that searches for a self-consistent solution to the complex beam parameter. * Only the fundamental Gaussian transverse mode is assumed to be present in the cavity and the paraxial approximation ensures that the beam preserves its Gaussian character. * The gain medium is pumped by a laser whose transverse beam profile is Gaussian; pump depletion within the gain medium and gain saturation are both neglected - Nonlinear coupling, astigmatism, and gain-guiding are all included in the model. The self-focusing process is modelled using the nonlinear ABCD matrix technique of Magni et al. [8], which is based on a quadratic approximation of the Gaussian transverse profile. We include the extension suggested by Bridges et al. [ 111 to allow for nonlinear coupling (the influence of the beam profile in one transverse plane on the refractive index seen by the profile in the other). To enable nonlinear coupling and/or gain-guiding to be included, it is necessary to adopt a split-step method in which the Kerr medium is divided into a large number of slices (typically 100). The matrix representing propagation through a thin slice of nonlinear material of refractive index n, in the presence of cross-coupling is
- ABCD matrix methods are used to track the propagation
1
-
d
1
where
“=
w,(z,)
pcrit
1 + ( mvf,/dh)2(
1 + d/R, ;)’
’
(2)
Similar equations with the indices x and _v reversed apply for the orthogonal polarisation. In these equations, d is the path length in the medium, P/P,,, the intracavity power normalised to the critical power for self-trapping, no is the linear refractive index, and A is the wavelength in the medium. The parameters w, * and R, x are the respective width and radius of curvature of the beam on entry into the slice, while w,(z,) and w&z,> are the beam sizes at the centre (z = z,) of the segment calculated in the limit of zero power (see Ref. [S]). Note that if the slice is placed at the Brewster angle, the matrix governing propagation in the tangential plane is obtained by sandwiching M in Eq. (1) between matrices representing Snell’s Law of refraction yielding
(3) -
which shows that the effect of the astigmatism on propagation in the tangential plane is to change the effective distance propagation from d/n, to d/n: where d is now the path length in the material, not the thickness of the slab. The matrix the sagittal plane is unaltered. A matrix can also be constructed to represent the effect of the non-uniform gain on the profile of the beam leaving active medium; this is the process we have termed gain-guiding, as distinct from gain-apetturing which concerns how
of for the the
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142 (19971315-321
317
same non-uniformity affects the gain efficiency. Propagation through a segment of length d and refractive index n, with a Gaussian transverse gain profile of peak value g, and width wr, can be described by a transformation matrix of the form [X61:
(4)
where A,, is the wavelength in vacuum. For this equation to be valid, the inequality d +c Z,/g, must be satisfied [5,6] where Za (= rru$/hl is the Rayleigh length for the pump beam. Within the same limits of approximation, i.e. for laser systems with short active media and small peak gain (or large pump beams), the matrix Ms can be factorised as follows
(5)
This indicates that propagation through a medium with a quadratic gain profile is roughly equivalent to a Gaussian aperture of width wn/ 6, followed by free space propagation through a distance d/n,,. The combination within the split-step framework of self-focusing with gain-guiding is accomplished simply by taking the product of the appropriate matrices, bearing in mind that the free propagation component must only be included once. For the x-dimension, the relevant combined matrix is
(6)
In this case, one can no longer take astigmatism into account by changing the effective distance of propagation tangential plane from d/n, to d/n: as in Eq. (3) above. Rather one should simply apply the matrix
[
n0 0
0
l/n0
in the
1
in the tangential plane on entry to the medium and the inverse matrix on exit. In previous treatments of gain-guiding [4-61, the pump beam was usually assumed to be circular and non-diverging, so that the resulting gain profile did not vary along the t-axis. The present model allows one to treat the pump as an elliptical, astigmatic, diverging beam and to include both pump depletion along the rod axis and the change of the small signal gain with :. For simplicity, however, the results presented below are for the case where pump depletion is neglected and go does not vary with z.
3. Numerical results Calculations were performed for the typical four-element Z-type resonator used in Refs. [9,14] and shown in Fig. 1. The Kerr-active laser medium was 2 cm long, its refractive index was 1.76, and it was positioned between the two identical spherical mirrors M, and M, both having a radius of curvature of IO cm. The folding angle 2 0 was 29.6” and the cavity was terminated by the two plane mirrors M, and M, both at 85 cm from the corresponding spherical mirror to form a symmetrical resonator. The separation z of M, and M, is the key parameter controlling cavity stability and, together with x which measures the position of the rod between the folding mirrors, has a critical influence on the KLM process. A series of calculations was performed in which the relative beam size variation at M, in the low power limit (S = (l/w)(dw/dp)],,a) was studied as a function of x and z, for increasingly sophisticated cavity models. We first simulated the KLM laser on the basis of the simplifying assumptions of Cerullo et al. [9], where nonlinear coupling and gain-guiding was neglected; in this case, 6 depends only on the characteristics of the cavity elements and their relative
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100 mm, the folding angle 2 fI is 29.6”,
positions. As a basic test, the formula for 8 quoted in [9] was compared with our own model, and the results are displayed as contour plots of S(x,z) for the tangential plane in Figs. 2(a) and 2(b) respectively, which are in encouragingly good agreement. The grey scaling runs from S = -0.5 (darkest) to 6 = + 0.5 (lightest); this implies that where the display is darkest, the spot size at M , decreases most strongly with increasing intracavity power, and so a hard aperture placed at M,
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Fig. 2. Contour plots of 6(x,z) using grey scaling between -0.5 and f0.5: (a) predictions of the analytical formula of Ref. [8] in the tangential plane; (b) the numerical model, with nonlinear x-y coupling and gain-guiding neglected, in the tangential plane; (c) and (d) the numerical model with nonlinear x-y coupling included for sagittal and tangential planes respecitively. The magnitudes of 6(x, z) significantly exceed the extremes of the grey scaling close to the confocal boundary.
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would favour mode-locked operation. For the parameter values used, the bare resonator is stable in the ranges 11.20 < z < 11.54 and 1 I.54 < ,- < 11.87 cm in the sagittal plane and 1 I.3 1 < ,- < 11.59 and I 1.59 < z < 11.93 in the tangential plane. In the tangential plane, at ; = Il.59 cm, the resonator is exactly confocal and consequently marginally stable: this is the significance of the vertical feature at this point in Fig. 2(a). It is evident that the optimal region for mode-locking is close to the confocal boundary, either on the low : side around x = 4.5 or the high z side around x = 5.2. The S-plot for the sagittal plane (not shown here) is very similar to that for the tangential plane, although the vertical feature is now centred at 2 = I1 54. Note that magnitudes of 6 close to the boundary may be much greater than 0.5. Figs, 2(c) and 2(d) show that the properties of the laser are modified very significantly when nonlinear coupling between the two planes is included. The results for the sagittal plane (frame c) and the tangential plane (frame d) are in excellent
(b)
Fig. 3. Contour plots of 6(.x.;) for increasingly strong gain-guiding: (a) and (b) large 200 +m pump beam waist for sagittal and tangential planes respectively: (c) and (d) similarly for 100 pm waist: (e) and (f) similarly for 50 pm waist. All plots have same grey scaling as in Fig. 2. Small signal gain g,, is I.0 per one-way pass (or.01 per segment).
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agreement with those in [14]. The map for the sagittal plane is evidently more complex, and tighter control on the parameters values would appear to be necessary to stay in the mode-locking region. Since the two planes are now coupled, the confocal boundary for one plane shows up as a slight vertical feature in the &plot for the other: this is most evident in Fig. 2(d) at z = 11.54 cm. With significantly different behaviour in the two planes, selecting an optimum operating region is now more difficult and will involve the choice of hard aperture which could be a horizontal or vertical slit, or an ellipse of appropriate eccentricity. Next we include gain-guiding as well as x-y coupling in the case of a circular Gaussian pump beam at 670 nm. In Fig. 3 we show how the &plots in the sagittal and tangential planes are altered when gain-guiding of increasing strength is included in the model. In Figs. 3(a) and 3(b), the effect of a small amount of gain-guiding caused by a large pump beam of 200 km radius, is shown. The main characteristics of Figs. 2(c) and 2(d) can still be seen in the sagittal and tangential plots of Figs. 3(a) and 3(b), but the confocal boundaries are starting to become less distinct and a certain narrowing in the x-range is becoming apparent. In frames (c) and (d) of Fig. 3 where the pump beam radius is 100 urn, these trends are much more pronounced, culminating in a complete break of symmetry and weakening of the mode-locking parameter in frames (e) and (f) where the pump beam radius is further reduced to 50 km. Since gain-guiding tends to increase resonator stability [5,6], it is not surprising that the vertical feature associated with cavity instability is removed when gain-guiding is included. At the same time, the horizontal symmetry apparent in frames (al and (bl is radically altered, with negative 6 values now occurring only in the upper right quadrant of the picture. The sagittal &plots shows far weaker features than those for the tangential plane, suggesting that optimum operation will be achieved by using a vertical slit. For this plane, the single mode-locking region (where 6(x. z) is strongly negative) is narrowed in x and expanded in the z-direction. In both planes, the magnitude of 6 is generally lower than in the absence of gain-guiding. From all of the above, we are led to conclude that gain-guiding works against hard-aperture mode locking and should be minimised in such systems. In real end-pumped lasers, the pump beam profile varies down the length of the laser medium as a result of diffraction and absorption. The numerical model is easily extended to incorporate these features, as well as the effect of varying the position of the beam waist within the cell. This capability will be especially important when gain-aperturing (mode-matching) effects are included in the model at a later stage. Figs. 4 and 5 were generated for a diffraction-limited pump beam of 50 p,rn in the sagittal plane and 35 p,m in the tangential plane, assuming that it propagates without absorption as a Gaussian beam (w,,(z) = wPXO(l + z%~&/w~~~)‘/~ where wpxO is the beam waist and 13~~the far-field half angle OP.r= A,/n,rr wr,,,). We note that for the case of end pumping with non diffraction-limited beams from diode lasers or one-dimensional arrays of diode lasers, propagation through the medium can be described by wr.J z> = wP,,, + @rXl11 for spatially incoherent beams
1131. Fig. 4 traces the variation in the laser beam spot size round the entire astigmatic cavity in the case where the waist of the pump beam is at the end of the rod nearer to mirror Mz. It shows in particular that, in the presence of gain-guiding, the beam size depends on the direction of propagation. Moving the pump beam waist within the rod changes the laser spot size throughout the cavity as well as the &plots based on the spot size at M,. This latter effect is demonstrated in Fig. 5 which indicates that slightly better mode-locking conditions occur when the waist is at the end of the rod nearer to M, (frame cl. There is clearly reasonable correspondence between the &plot in frame (f) of Fig. 3 (where the pump beam has a uniform 50 urn radius throughout the laser medium) and those of Fig. 5 where wP 1 and wP,, are 50 pm and 35 urn respectively. This suggests that the assumption of Fig. 3 is a fairly good approximation for diffraction-limited beams. However, it seems likely
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Fig. 4. Evolution of laser beam size when the pump beam waist is at the entry (left plane) of the rod. The two curves represent evolution in the sagittal and tangential planes. The pump beam is elliptical with beam radius 50 km in the sagittal and 35 )*m in the tangential is 1% per segment and P/P, = 0.1.
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Fig. 5. Contour plots of 6(x_.:) for three different positions of the pump waist in the rod and with nonlinear x-v coupling and gain-guiding included: (a) pump waist at the rod end nearer M?, (b) at centre of rod, (c) at rod end nearer M1. The tangential plane is shown in all three cases. The grey scaling the same as in Figs. 2 and 3.
that the detailed pump beam characteristics will become more important when gain-aperturing incorporated into the model and pump beams that are not diffraction-limited are considered.
(mode-matching)
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4. Conclusions We have developed a numerical model of KLM that takes into account self-focusing, astigmatism and gain-guiding, and demonstrated in particular how gain-guiding affects the hard-aperture mode-locking parameter 6 = (l/~~)(du~/dp)(,,=~. In the simplified case where the gain profile is approximated by a parabola, and gain saturation is ignored, we have found that gain-guiding tends to work against hard-aperture mode-locking, reducing the magnitude of 6 while making the optimum operational region narrower overall. The model enables one to study the effect of elliptical, astigmatic pump beams on the laser beam throughout the cavity, as the position of the pump beam waist is moved within the gain medium. While it has been pointed out in REf. [4]. that the parabolic approximation overestimates the gain-guiding effect, it seems probable that gain saturation also weakens the gain guiding effect. Incorporation of both gain-aperturing [4] and gain-saturation [ 15.161 into the model is presently under way. Another potentially important process that is ignored in the present work is thermal lensing, and experimental work on our in-house laser systems is in progress to assess its magnitude. These effects, together with the study of soft-apertured three-element systems. will be treated in a future pu b lication.
References [I] [Z] [3] [4] [5] [6] [7] [8] [9] [lo] [I I] [12] [ 131 [14] [15] 1161
D.E. Spence. P.N. Kean. W. Sibbett, Optics Lett. 16 (1991) 42. 100. Ch. Spielmann. P.F. Curley, T. Brabec. F. Krausz. IEEE J. Quantum Electron. 30 (1994) T. Brabec. P.F. Curley. Ch. Spielmann, E. Wintner, A.J. Schmidt, J. Opt. Sot. Am. 10 (1993) 1029 J. Herrmann. J. Opt. Sot. Am. B 1I (1994) 498. F. Salin. J. Squier, Optics Lett. 17 (1992) 1353. F. Salin. J. Squier. M. Piche, Optics Lett. 16 (1991) 1674. E. Bouma, J.G. Fujimoto, Optics Lett. 21 (1996) 134. V. Magni. G. Cerullo, S. De Silvestri, Optics Comm. 96 (1993) 348. G. Cerullo, S. De Silvestri. V. Magni, L. Pallaro, Optics Lett. 19 (1994) 807. 0. Haderka, Optics Lett. 20 (1995) 240. E. Bridges, R.W. Boyd. G.P. Agrawal, Optics Lett. 18 (1993) 2026. E. Bridges. R.W. Boyd, G.P. Agrawal, J. Opt. Sot. Am. B 13 (1996) 553. T.Y. Fan, A. Sanchez, IEEE J. Quantum Electron. 26 (1990) 3 I 1. V. Magni, G. Cerullo, S. De Silvestri, A. Monguzzi. J. Opt. Sot. Am. B 12 (1995) 476. F. Salin, J. Squier. Optics Comm. 86 (1991) 397. M. Picht. F. Salin. Optics Lett. 18 (1993) 1041.