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ABCD matrix analysis of propagation of gaussian beams through Kerr media
FULL LENGTH ARTICLE L
Oplics Communications 96 ( 1993 ) ~48-355 North-l-tolland
OPTICS COMMUNICATIONS
F u l l length article
ABCD matrix analysis of propagation of gaussian beams through
Kerr media Vittorio Magni, Giulio Cerullo and Sandro De Silvestri Centro di Elettronica Quanti~tica e Strumentazione Elettronica del CNR, Dipartimento di Fisica del Politecnico, Piazza L. da Vinci 32, 20133 Milan, Italy
Received 20 July 1992; revised manuscript received 15 September 1992
A new formalism and a nonlinear ABCD matrix, derived from the aberrationless theory of self-focusing, are introduced to treat the propagation of gaussian beams in materials with Kerr nonlinearity. The analysis of propagation of gaussian beams shows that the effect of the Kerr medium can be interpreted, to the first order in the beam power, as due to a lens and propagation over a negative distance. The latter effect has been named self-shortening. The formalism has been applied to find the optimal focusing condition for the maximum nonlinear effects and to the calculation of gaussian modes in resonators containing a Kerr medium.
I. I n t r o d u c t i o n
Self-focusing is a well known optical p h e n o m e n o n that can be observed when a light b e a m propagates in a m e d i u m whose refractive index linearly varies with the intensity ( K e r r nonlinearity) and has been deeply analyzed experimentally and theoretically (for a review see refs. [ 1-3 ] ). Recently self-focusing has been successfully exploited as a mode-locking mechanism to generate ultrashort laser pulses in various solid state lasers [ 4 - 9 ] . In these lasers the modulation of the transverse profile o f the resonant modes induced in a Kerr m e d i u m is used to produce the power d e p e n d e n t losses responsible for m o d e locking. To analyze the optical modes in these lasers and their d e p e n d e n c e on the circulating power, numerical models [10,1 1] and a distributed lens model [ 12] have been developed. On the basis of the aberrationless theory of self-focusing, in which the beam is assumed to m a i n t a i n its gaussian profile during propagation in the Kerr m e d i u m , a new formalism has been recently p r o p o s e d to readily treat the transf o r m a t i o n o f the complex b e a m p a r a m e t e r in the nonlinear m e d i u m [ 13 ]. In this Letter we introduce a " n o n l i n e a r A B C D ray m a t r i x " to treat the gaussian b e a m p r o p a g a t i o n in a Kerr m e d i u m . The b a c k g r o u n d o f the formalism is 348
the aberrationless theory. The nonlinear matrix allows, as an alternative to the m e t h o d presented in ref. [ 13 ], the extension of the well assessed " A B C D law" o f gaussian beam transformation in linear optical systems [14,15] to more general systems containing Kerr media. We also show that the modifications o f spot size and radius o f curvature in a nonlinear Kerr m e d i u m , where the refractive index increases with intensity, are equivalent, to the first order in the b e a m power, to those produced by a positive lens plus a propagation over a negative distance. Both the lens dioptric power and the length of propagation d e p e n d on the incident b e a m parameters. The positive lens accounts for the expected effects o f self-focusing. The negative propagation corresponds to an effect that we call self-shortening, which should not be equivocated with the much smaller variation o f optical path length affecting the axial phase shift. The conditions of optimal focalization of the b e a m to achieve the m a x i m u m nonlinear effect have been analyzed and formulated in a simple equation. The application of the new formalism can greatly simplify the analysis o f resonators used for the previously m e n t i o n e d new modelocking technique, moreover the concept of selfshortening along with self-focusing can provide a substantial advantage for a clear physical insight. As
0030-4018/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
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an illustrative example, the TEMoo mode of a resonator typically used with Ti:sapphire lasers is calculated as a function of the circulating power inside the resonator.
count of the conservation of beam power P, which is given by P=~ctonoU2/4. To solve eq. (3) we approximate the nonlinear term I u l 2 by a parabola
lu12~3U2{l_ 2r2 ~ -
4 w2 \
3we]
(5)
"
2. Aberrationless theory background The electric field of a linearly polarized, sinusoidally time varying beam propagating along the z axis can be written as
E= ½{u exp[i(~ot-kz] +c.c.} ,
(1)
where w is the angular frequency, k = noO~C(c being the light velocity in vacuo and no the linear refractive index), u the complex field amplitude slowly varying with z, and c.c. stands for complex conjugate. The Kerr medium is described by an intensity dependent refractive index n: (2)
n=no+½n 2 l u l 2 .
Within the paraxial approximation of the scalar diffraction theory the propagation of the beam is described by the equation [ 1 6 ] VTU-2ik~
ctZ
+ k 2 n-L l u l 2 u = 0 ,
This approximation is similar to that used in the aberrationless theory of self-focusing [ 1 ], but it is not the second order expansion of I u l 2. In fact we chose to approximate l u12 by an interpolating parabola c~+c2r2, such that the coefficients Cl and c2 minimize the weighted square mean error i(C
l ul 2) 21 ul 2 2zcrdr.
+C2 r 2 -
0
The validity of this approximation will be discussed later. Inserting eqs. (4) and ( 5 ) in eq. ( 3 ) gives four ordinary differential equations: one is related to the power conservation and is identically satisfied, the other three can be solved for w(z), R(z) and ~o(z). The solutions can be written as
k~now~] k
(3)
(6)
no
_(wj'~ 2 R(z) kw(z)] 1
where VT is the transverse laplacian acting on the coordinates orthogonal to the z axis. A number o f numerical and approximated analytical solutions of eq. (3) have been thoroughly discussed in the literature [ 1-3, 17 ]. In particular it has been shown that selfsimilar solutions exist which retain a constant shape except for a scale factor that changes with z. For moderate nonlinearity the profile of the "fundamental solution" approaches a gaussian function [ 2 ]. For the purpose o f this work we consider the propagation of a cylindrically symmetric gaussian beam with the approximation that the gaussian profile is maintained during the propagation. For such a beam the complex field can be written as
where wl, Rl, and ~01 are the initial values for z = 0 and 2 is the vacuum wavelength. The parameter Pc is the critical power for self-focusing and is given by
u(r, z) = (U/w) exp [ - (r/w) 2- ikr2/2R + iq~] ,
Pc = c%22/(27rn2) .
(4) where r is the distance from the z axis, U is a real constant amplitude factor, while the spot size w, the radius of curvature R, and the phase shift ~0are functions of z only. The amplitude U is a constant on ac-
x
1
z +z
+R~
~o(z)-~o,=
1
P -
,
(7)
1-3P/2Pc ~arctan( 7rnow2(z) l_xfF27~ ~L \2R(z)x/1-P/PJ
(
- arctan 2R, x/1/1--P/~JJ '
(8)
(9)
For n 2 = 0 (i.e., for P c = ~ ) eqs. ( 6 ) - ( 8 ) reduce to the expressions for the propagation of a gaussian beam in a homogeneous medium of refractive index no [ 14 ]. Equations (6) and (7), which give the spot size and the radius of curvature, can also be ob349
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tained, following the formalism of ref. [13], from the expressions for free space propagation after multiplying the term ~/(gn0w~ ), i.e. the imaginary part of the 1/q parameter o f the input gaussian beam, by the scaling factor V/1 - P / P c . Note, however, that the axial phase shift does not obey to such a transformation and cannot be derived with that formalism. When P > Pc the spot size can vanish after a certain propagation distance, which depends on the initial conditions Wl and R~. If P = P c and the initial wavefront curvature is fiat (R ~= oo ) the beam propagates in a guided mode without variations of spot size and radius of curvature (self-trapping). Various expressions for the critical power, which are obtained with different approaches and may differ from eq. (9), can be found in the literature [1-3,12,16,17]. In particular, by comparison with the numerical and analytical results reported in ref. [2], it can be seen that eq. (9) agrees with the expression of the minimum power o f a gaussian beam for which all the beam power can collapse into a point after a certain propagation, and that Pc is equal to 1.074 times the m i n i m u m power o f self-trapping. An expression for critical power similar to eq. (9) has been obtained also in ref. [17] by means of variational calculations. Moreover the formalism described in the following sections does not depend on the expression of the critical power, so that different expressions could also be used. As an example, the formula for the critical power given in ref. [ 13 ] contains an adjustable correction factor, which can be used for a precise fitting of experimental data [eq. (9) agrees with the expression given in ref. [ 13 ] if the correction factor is 4, which is within the specified range ]. These considerations suggest that the results presented here can be applied even for high beam power (P~,P~). Note that if we had approximated l ul 2 by the second order series expansion, we would have obtained a value of critical power four times smaller.
3. ABCD formulation
It is convenient to recast eqs. (6), (7) in a form consistent with the standard formulas for gaussian beam transformations. For a gaussian beam of spot size w and radius of curvature R in a material of re350
15 February1993
fractive index n~ one defines a complex radius of curvature, q, as [ 14,15 ]
1/q= l / R - i 2 / ~ n l w 2 .
(10)
If q~ is the complex radius at the input of a linear optical system characterized by a ray matrix
the complex radius, q2, at the output is given by ("ABCD law") [14,15]:
q2 = (Aq, + B ) / ( C q l + D ) .
(11)
We consider now an optical system made of a nonlinear material of length d with flat faces perpendicular to the z axis surrounded by a medium with refractive index 1. As in the case of lens-like media [14], we take into account the refraction at interfaces by multiplying the real radius of curvature by no at the input and by dividing it by no at the output. The transformation of the complex radius, calculated by using eqs. (6) and ( 7 ), can by put in a form consistent with the ABCD law by defining the ray transfer matrix of the Kerr media as M=x//1 - ?
-7/[(1-7)de]
'
where de = d/no is the effective length of the medium for P = 0 and y=
[ 1+ -1( 4 \ 2d~
< f l -'vPT"
2gWo2J_]
(13)
In eq. (13) Wc is the spot size at the center of the medium and Wo is the spot size at the beam waist calculated as i f P / P ~ = 0 . This parameter can also be expressed in terms of the complex radius of curvature at any arbitrary plane. Since the elements of the matrix in eq. (12) depend on the beam power and complex radius, the ray matrix is, as it must be, nonlinear. For P/Pc = 0 the matrix reduces to that pertaining to a linear dielectric block [ 18 ]. By inspection of eq. (13), it is apparent that 7~P/Pc. It can also easily be shown that for 7>~ 1 the beam is (ideally) focused into a point (vanishing spot size) inside the medium. In this case our approximation is no longer valid, therefore we only consider cases where y< 1. In order to evidence the nonlinear effects and sep-
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arate them from the linear ones corresponding to the linear propagation through a length de and represented by the matrix
('0 we factor out the matrix M as follows: M=(;
de/2) M k ( ~
de/2),
(14)
15 February 1993
the matrix of a combination of these two elements, can be approximated by
if the term z / f c a n be neglected as a second order term for flarge and z small. Therefore, the first order matrix Mk can be viewed as a superposition of the effect of a lens of focal length fk and a propagation through a negative distance Zk given by
where Mk = 1/ lx~_7(1._--7:~
-,d¢/4")1 - ?/2,1"
(15)
The matrices on the left and right hand side of the product in eq. (14) correspond to a linear propagation through a medium of length d / 2 and refractive index no. The matrix Mk is nonlinear and reduces to the identity matrix for P/Pc= ?= 0. The Kerr nonlinearity is therefore lumped into the matrix Mk and separated from linear propagation. Obviously this model only allows the calculation of the beam parameters at the output plane, not inside the medium. Finally, for 7 small, Mk can be expanded at the first order: Mk~
(
1 -7~de
- 71dd4)
"
(16)
Note that the determinant of this matrix is unity only if the term in 72 is neglected. This circumstance arises from the fact that the elements of the matrix in eq. (16) are only the first order expansion of the elements of the matrix in eq. (15), whose determinant is indeed unity, as for usual A B C D matrices. In the expressions derived by using eq. (16) only the zero and first order terms in ~ should be retained as significant, since eq. (16) is valid only for low power (low 7). Equation (16), however, allows to derive the following useful physical interpretation of the nonlinear Kerr matrix. Since the matrix of a thin lens of focal length f i s
(_1 01) 1/f
'
and that of a propagation through a distance z is
:)
1/fk=~ld~ ,
(17)
Zk = - ~ d e / 4 .
(18)
The focal length fk represents the self-focusing, the negative distance represents the self-shortening. The effective length de for linear propagation is thus shortened by Zk due to the nonlinear effect. It should be noted that this variation of length is not related to the axial phase shift which is given by eq. (8). Note that, if the nonlinear coefficient n2 is negative (i.e., Pc<0 and y < 0 ) the focal length fk is also negative (self-defocusing), whereas the variation in propagation distance become positive ("self-lengthening"). Although the equivalent nonlinear focal length and shortening have been obtained by a suitable, but not unique, factorization of the matrix in eq. (12), it can be shown that the same result can be obtained calculating starting from eq. (12), the focal length and the shift of the principal planes [ 14 ] of the system represented by eq. (12). To understand physically the origin of self-shortening consider the simple case of an input beam with beam waist at the input face of the Kerr medium [i.e., 1/R~ = 0 in eqs. ( 6 ) (8) ] and a power P=Pc. As shown by eqs. (6) and (7) in this case the spot size and the radius of curvature do not change along propagation since the beam propagates in a self guided mode. The transformation operated by the Kerr medium is therefore an identity transformation, which corresponds to a propagation through a vanishing distance. For a given beam power both 1/fk and Zk reach a maximum when y reaches its maximum value equal to P/Pc. The optimum focusing condition, obtained form eq. (13), is Wc Wo =2de/ ( 2~z ) .
(19)
351
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In particular, when the beam waist Wo is in the center of the material we obtain we= % = x/2dJ27r. In this case the length of the material is equal to the confocal parameter of the gaussian beam. It might be noted that the operator given by eqs. (12), (15), and (16) is not mathematically a true matrix operator since the elements depend on the input beam parameter, but can nevertheless be used in the same way as the matrices to calculate the beam transformations, proceeding as follows. G i v e n an input beam one first calculates the spot size w~ in the center of the Kerr block and the beam waist w0 with the standard ABCD linear propagation technique assuming zero beam power P/P~. With these data one calculates the factor 7 by using eq. (13) for the real beam power and builds the n o n l i n e a r matrix using eq. (15) or (16). The matrix of Kerr m e d i u m can then be combined, simply by matrix multiplication, with the ABCD ray matrix of other linear elements following the Kerr m e d i u m to treat the propagation of gaussian beams in complex optical systems and calculate the complex beam radius at any plane after the Kerr element. Finally it should be remembered that, if one wants to use eq. (16) instead of the complete matrix given by eq. (15), which is possible to obtain approximations to the first order in the beam power, only the terms up to the first order in 7 (i.e., P/P~) must be retained as significant. To check our calculations, the results have been compared with the numerical solutions of eq. (3). Figure 1 shows such a comparison in case of optimal focalization for different beam powers. For P/P~ = 0.3 the matrix formalism gives results in excellent agreement with the exact numerical solution both for intensity and phase. For P/P~ = 0.6 some discrepancies begin to arise, however the field substantially maintains a gaussian profile. For a close comparison we have interpolated the exact field given by the numerical solutions, u,, with a gaussian field, Ug, as given by eq. (4), that m i n i m i z e s the m e a n square error i ]U,--Ug[ 2 27crdr. 0
The interpolated spot size, radius of curvature, and axial phase shift are plotted in fig. 2 as a function of P/P~ and compared to the corresponding results of 352
3 •
15 February 1993 i
i
2I
ia )
'
i
,
i
,
(b)
1
~c ~ 2 ',. 0.6 __=
,
0.3 0.3 =u - 1
o
1
Q_
a:
0
-2 --,_3
0
I
'
i
i
Ii
410
0 40 80 2O Radial Distance (iJm)
80
1 20
Radial Distance (pro)
Fig. 1. Comparison of numerical solutions of paraxial wave equation (dashed lines) and results of matrix formalism (solid lines). The effective length of Kerr medium is de= 15 ram. The incident gaussianbeam has a wavelengthof 1.064 p,m and, for P~ Pc=0, a beam waist in the center of the medium with a spot size of 50.q ~tm (optimum focalization). The parameter near the curves is the normalized power P/Pc. 1.2
-0.0 radius .
~3 o 0.8
p sp°t
" ~ ' ~
-0.6
-u
s
:~ o ~
--1.2 u)
0.6
:~
o_ u3
-1.8
• 0.4
Eo
z 0.2 0.0 0.0
-2.4 ,
,
0.2 0.4 0.6 0.8 Normalized Power P/Pc
.0
-3.0
Fig. 2. Parameters of a gaussian beam after the propagation in a Kerr medium as a function of the power P/Pc. The conditions are those of fig. 1. The beam spot size and radius of curvature (left scale) are divided by the valuesfor P/Pc= 0. The axial phase shift (fight scale) is plotted as differencebetween the actual value and that for P/Pc=O.The solid lines corresponds to the matrix calculation while the dots are the values obtained by a gaussian interpolation of the numerical solutions• matrix calculation. The substantial agreement between the results confirms the validity of our approximated theory. The self-shortening, the self-focusing, and the effects of beam focalization can be assessed from fig. 3, where the normalized induced shortening and induced dioptric power (i.e., - 2zJde and d,,/2fk, both
OPTICS COMMUNICATIONS
Volume 96, number 4,5,6 0.12
,
i
,
0.5
• o
i
0.4
,
i
,
E
0
•
i
P/P==0.2
t3u 0 g 0.08
~5
0. o J~ 03
0.04
o z 0"000.0
0.5
' 1.~0 ' 1.J5 ' 2.'0
Normalized
Beam
Fig. 3. Normalized induced shortening,
' 2.'5
' 3.0
Waist - - Z k / 2 de
and dioptric
power 2dJfk as a function of the normalized spot size at the beam waist of the incident gaussian beam, W o / ~ . The parameter near the curves is the ratio of the effective distance of the beam waist (for P=0) from the center of the Kerr medium to the effective length d~ of the material. The effective distance is given by the real distance divided by no when the waist is inside the material, or by d~/2 plus the distance from the face of the material when the waist is outside. equal to ),/2 ) for P/Pc = 0.2 are shown as a function o f the spot size at the b e a m waist (for P/Pc=O) o f the incident gaussian beam. The curves in fig. 3 correspond to different distances o f the gaussian beam waist from the center o f the K e r r m e d i u m . W h e n the waist is inside the material both shortening and dioptric power reach always the same m a x i m u m in correspondence to the o p t i m a l focalization o f the gaussian beam. If the b e a m waist is outside the material it is impossible to satisfy eq. ( 19 ) for o p t i m u m focalization. Note that the m a x i m u m shortening, given by - ( d e / 4 ) P / P c , can a m o u n t to a significant fraction o f the length dc o f the material.
4. Applications As an example o f application o f this formalism we first m e n t i o n the Z-scan experiments [ 19 ] where a nonlinear m e d i u m is scanned across the focus o f a gaussian b e a m and the variation o f the b e a m spot size at a reference plane are m e a s u r e d to d e t e r m i n e the nonlinear coefficient o f the material. W i t h o u t going here into details we point out that experiments p e r f o r m e d with arbitrary long samples o f nonlinear
15 February 1993
material can be easily interpreted on the basis of the new formalism. In particular, the results could be easily understood physically and more correctly analyzed by considering not only a lens effect but also a shortening (or lengthening, if r/z<0) of the propagation distance. Moreover, as it will be shown elsewhere, the m a x i m u m nonlinear effect on the b e a m spot size is observed only when the o p t i m u m focusing condition given by eq. (18) is satisfied. A second relevant application is the mode calculation in resonators containing a Kerr m e d i u m . In this case one has to apply the formalism as follows. F o r any value o f 7 one calculates, using the matrix Mk as a linear one, the self consistent spot size w and radius o f curvature R in a reference plane inside the resonator, obtaining w = w (7) and R = R ( 7)- Starting from that reference plane one calculates, assuming linear propagation, the spot size in the center o f the Kerr m e d i u m , Wc and the corresponding beam waist Wo. By using eq. (13) the beam power can be expressed as a function o f 7: P / P c = P ( 7 ) / P c • The dependence of the spot size w on the power P is given, in a parametric form, by a set o f two equations [i.e., w = w(7) and P = P ( 7 ) ] with 7 as a parameter. These equations can be readily used to plot the spot size in any reference plane as a function o f the power. As an illustrative example consider the resonator shown in fig. 4a, which is a typical configuration for self modelocked T i : s a p p h i r e femtosecond lasers. The resonator is m a d e by two plane end mirrors and contains a Ti: sapphire rod between two focusing mirrors represented as lenses. Figure 4b shows the spot size on the left m i r r o r as a function of the circulating power in the resonator for different values, x, o f the distance between one face of the rod and the corresponding lens. This resonator is optically stable for P/Pc=O if 36.64 m m < x < 3 9 . 0 2 mm. If the resonator is initially stable only one value o f self consistent spot size is found for any value of P/Pc and the spot size decreases with increasing power, producing a fast saturable absorber effect when an aperture is placed near the left mirror. On the other hand, if the resonator is initially unstable, it can be driven in a stable configuration at high enough power. Note that in this case two values o f self consistent spot size are found for a given P/Pc: the resonator becomes therefore bistable. This circumstance m a y be also relevant for self m o d e locking o f solid state lasers. 353
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given by eq. (19) with the beam waist in the center o f the material.
B 2o5 f=50
2.0
loo 3o0
5. Conclusions
f=50
Xmm l
i
i
i
(b)
1.2 m 0.8 "6 0.4
%0
1
i
o.;
i
o.'6
i
i
,.o
Normolized Power P / P : Fig. 4. (a) Typical resonator used for femtosecond mode-locked
Ti:sapphire lasers: K is the Kerr medium (Ti:sapphire, refractive index 1.76), the dimensions are in mm. (b) Behavior of the spot size on the left mirror as a function of the normalized circulating power in the resonator for different distances x. A crucial p a r a m e t e r to characterize the nonlinear losses p r o d u c e d by the variation o f spot size in a resonator initially stable is the slope of the curve in fig. 4b near P/Pc=O (the higher the slope, the higher the nonlinear effect on the laser d y n a m i c s ) . By using our matrix formalism, we have found, as we will show elsewhere, that the m a x i m u m (achievable with an o p t i m i z e d resonator) relative variation o f the spot size with respect with the circulating power is directly related to the stability p a r a m e t e r gig2 [ 15 ] o f the resonator for P = 0, namely:
I;
dw ] =+ 1 d(PTPc) p : o - 4x/g, g 2 ( 1 - g ~ g 2 )
(20)
This equation evidences that, to obtain large spot size variation with power, the resonator configuration must be very close to a stability limit, g~g2 = 0 or 1. To reach the m a x i m u m slope for a given gig2 it is necessary to satisfy the o p t i m u m focusing condition 354
15 February 1993
In this work we have introduced a new formalism, based on the aberrationless theory o f self focusing, to treat the propagation of a gaussian b e a m in a material presenting Kerr nonlinearity. We have shown that the Kerr m e d i u m can be represented by an ABCD ray transfer matrix whose elements depend on the parameters of the input beam. The main advantage of this new formalism is that the matrix can be c o m b i n e d by matrix multiplication to other matrices associated to linear elements to treat the propagation in complex optical systems. The calculation o f the output b e a m proceeds along two steps by using the standard ABCD law: given an input beam, one first determines the elements o f the nonlinear matrix, then calculates the output b e a m by applying the matrix. The results o f matrix calculation satisfactorily agree with the exact numerical solution of the nonlinear wave equation even for b e a m power close to the critical power. By using the new formalism we have pointed out that the effects o f the Kerr m e d i u m can be interpreted, to the first order in the b e a m power, as those due to a lens representing the self-focusing, plus those due to a propagation through a negative distance. This second effect has been n a m e d self-shortening. As an illustrative example of application we have analyzed the dependence of the TEMoo m o d e on the circulating power in a typical resonator used in self m o d e locked T i : s a p p h i r e lasers. In particular, we have pointed out that the m a x i m u m first-order variation of the mode size, which is responsible for mode locking, is directly related to the gig2 stability parameter of the resonator and that a bistable behavior could be observed in resonators initially unstable at low power. We believe that the new formalism can find significant applications mainly in the analysis, design and o p t i m i z a t i o n o f femtosecond self-mode locked solid-state lasers.
Note added in proof After submission of this paper the authors noticed
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OPTICS COMMUNICATIONS
t h a t t h e p o s s i b i l i t y o f d e r i v i n g a n o n l i n e a r m a t r i x was p r e v i o u s l y m e n t i o n e d b y P.A. B e l a n g e r a n d C. P a r e , in A p p l . O p t i c s 22 ( 1 9 8 3 ) 1293.
Acknowledgements The authors thank Dr. G.P. Giuliani for several helpful discussions and suggestions.
References [ 1 ] S.A. Akhmanov, R.V. Khokhlov and A.P. Sukhorukov, in: Laser Handbook, eds. F.T. Arecchi and E.O. Schulz-Dubois (North-Holland, Amsterdam, 1972) Chap. E3. [2] J.H. Marburger, in: Progress in quantum electronics, Vol. 4, eds. J.H. Sanders and S. Stenholm (Pergamon, Oxford, 1977) p. 35-110. [3 ] Y.R. Shen, The principles of nonlinear optics (Wiley, New .York, 1984) Chap. 17; Y.R. Shen, in: Progress in quantum electronics, Vol. 4, eds. J.H. Sanders and S. Stenholm (Pergamon, Oxford, 1977) p. 1-34. [4] L. Spinelli, B. Couillaud, N. Goldblatt and D.K. Negus, in: Conf. on Lasers and electro-optics 1991 (Opt. Soc. Am., Washington, 1991 ) p. 583.
15 February 1993
[ 5 ] U. Keller, G.W. 'tHoofl, W.H. Knox and J.E. Cunningham, Optics Lett. 16 (1991) 1022. [6] F. Krausz, Ch. Spielmann, T. Brabec, E, Wintner and A.J. Schmidt, Optics Lett. 17 (1992) 204. [ 7 ] G. Gabetta, D. Huang, J. Jacobson, M. Ramaswamy, E.P. Ippen and J.G. Fujimoto, Optics Lett. 16 ( 1991 ) 1756. [8] G.P.A. Malcom and A.I. Ferguson, Optics Lett. 16 ( 1991 ) 1967. [9] D.E. Spence, J.M. Evans, W.E. Sleat and W. Sibbett, Optics Lett. 16 (1991) 1762. [10] F. Salin, J. Squier and M. Pich6, Optics Lett. 16 (1991) 1674. [ 11 ] M. Pich6, Optics Comm. 86 ( 1991 ) 156. [ 12 ] M. Sheik-Bahae, A.A. Said, D.J. Hagan, M.J. Soileau and E.W. Van Stryland, Opt. Eng. 30 ( 1991 ) 1228. [13] D. Huang, M. Ulman, L.H. Acioli, H.A. Haus and J.G. Fujimoto, Optics Lett. 17 (1992) 511. [ 14] H. Kogelnik, Bell System Tech. J. 44 ( 1965 ) 455. [ 15 ] A.E. Siegman, Lasers (Oxford Univ. Press, Oxford, 1986) Chap. 20. [ 16 ] A. Yariv, Quantum electronics, third Ed. (Wiley, New York, 1989) p. 482-489. [ 17 ] M. Desaix, D. Anderson and M. Lisak, J. Opt. Soc. Am. B 8 (1991) 2082. [18] A.E. Siegman, Lasers (Oxford Univ. Press, 1986) p. 585. [ 19] M. Sbeik-Bahae, A.A. Said, T.H. Wei, D.J. Hagan and E.W. Van Stryland, IEEE J. Quantum Electron. QE-26 (1990) 760.
355
Oplics Communications 96 ( 1993 ) ~48-355 North-l-tolland
OPTICS COMMUNICATIONS
F u l l length article
ABCD matrix analysis of propagation of gaussian beams through
Kerr media Vittorio Magni, Giulio Cerullo and Sandro De Silvestri Centro di Elettronica Quanti~tica e Strumentazione Elettronica del CNR, Dipartimento di Fisica del Politecnico, Piazza L. da Vinci 32, 20133 Milan, Italy
Received 20 July 1992; revised manuscript received 15 September 1992
A new formalism and a nonlinear ABCD matrix, derived from the aberrationless theory of self-focusing, are introduced to treat the propagation of gaussian beams in materials with Kerr nonlinearity. The analysis of propagation of gaussian beams shows that the effect of the Kerr medium can be interpreted, to the first order in the beam power, as due to a lens and propagation over a negative distance. The latter effect has been named self-shortening. The formalism has been applied to find the optimal focusing condition for the maximum nonlinear effects and to the calculation of gaussian modes in resonators containing a Kerr medium.
I. I n t r o d u c t i o n
Self-focusing is a well known optical p h e n o m e n o n that can be observed when a light b e a m propagates in a m e d i u m whose refractive index linearly varies with the intensity ( K e r r nonlinearity) and has been deeply analyzed experimentally and theoretically (for a review see refs. [ 1-3 ] ). Recently self-focusing has been successfully exploited as a mode-locking mechanism to generate ultrashort laser pulses in various solid state lasers [ 4 - 9 ] . In these lasers the modulation of the transverse profile o f the resonant modes induced in a Kerr m e d i u m is used to produce the power d e p e n d e n t losses responsible for m o d e locking. To analyze the optical modes in these lasers and their d e p e n d e n c e on the circulating power, numerical models [10,1 1] and a distributed lens model [ 12] have been developed. On the basis of the aberrationless theory of self-focusing, in which the beam is assumed to m a i n t a i n its gaussian profile during propagation in the Kerr m e d i u m , a new formalism has been recently p r o p o s e d to readily treat the transf o r m a t i o n o f the complex b e a m p a r a m e t e r in the nonlinear m e d i u m [ 13 ]. In this Letter we introduce a " n o n l i n e a r A B C D ray m a t r i x " to treat the gaussian b e a m p r o p a g a t i o n in a Kerr m e d i u m . The b a c k g r o u n d o f the formalism is 348
the aberrationless theory. The nonlinear matrix allows, as an alternative to the m e t h o d presented in ref. [ 13 ], the extension of the well assessed " A B C D law" o f gaussian beam transformation in linear optical systems [14,15] to more general systems containing Kerr media. We also show that the modifications o f spot size and radius o f curvature in a nonlinear Kerr m e d i u m , where the refractive index increases with intensity, are equivalent, to the first order in the b e a m power, to those produced by a positive lens plus a propagation over a negative distance. Both the lens dioptric power and the length of propagation d e p e n d on the incident b e a m parameters. The positive lens accounts for the expected effects o f self-focusing. The negative propagation corresponds to an effect that we call self-shortening, which should not be equivocated with the much smaller variation o f optical path length affecting the axial phase shift. The conditions of optimal focalization of the b e a m to achieve the m a x i m u m nonlinear effect have been analyzed and formulated in a simple equation. The application of the new formalism can greatly simplify the analysis o f resonators used for the previously m e n t i o n e d new modelocking technique, moreover the concept of selfshortening along with self-focusing can provide a substantial advantage for a clear physical insight. As
0030-4018/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
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OPTICS C O M M U N I C A T I O N S
an illustrative example, the TEMoo mode of a resonator typically used with Ti:sapphire lasers is calculated as a function of the circulating power inside the resonator.
count of the conservation of beam power P, which is given by P=~ctonoU2/4. To solve eq. (3) we approximate the nonlinear term I u l 2 by a parabola
lu12~3U2{l_ 2r2 ~ -
4 w2 \
3we]
(5)
"
2. Aberrationless theory background The electric field of a linearly polarized, sinusoidally time varying beam propagating along the z axis can be written as
E= ½{u exp[i(~ot-kz] +c.c.} ,
(1)
where w is the angular frequency, k = noO~C(c being the light velocity in vacuo and no the linear refractive index), u the complex field amplitude slowly varying with z, and c.c. stands for complex conjugate. The Kerr medium is described by an intensity dependent refractive index n: (2)
n=no+½n 2 l u l 2 .
Within the paraxial approximation of the scalar diffraction theory the propagation of the beam is described by the equation [ 1 6 ] VTU-2ik~
ctZ
+ k 2 n-L l u l 2 u = 0 ,
This approximation is similar to that used in the aberrationless theory of self-focusing [ 1 ], but it is not the second order expansion of I u l 2. In fact we chose to approximate l u12 by an interpolating parabola c~+c2r2, such that the coefficients Cl and c2 minimize the weighted square mean error i(C
l ul 2) 21 ul 2 2zcrdr.
+C2 r 2 -
0
The validity of this approximation will be discussed later. Inserting eqs. (4) and ( 5 ) in eq. ( 3 ) gives four ordinary differential equations: one is related to the power conservation and is identically satisfied, the other three can be solved for w(z), R(z) and ~o(z). The solutions can be written as
k~now~] k
(3)
(6)
no
_(wj'~ 2 R(z) kw(z)] 1
where VT is the transverse laplacian acting on the coordinates orthogonal to the z axis. A number o f numerical and approximated analytical solutions of eq. (3) have been thoroughly discussed in the literature [ 1-3, 17 ]. In particular it has been shown that selfsimilar solutions exist which retain a constant shape except for a scale factor that changes with z. For moderate nonlinearity the profile of the "fundamental solution" approaches a gaussian function [ 2 ]. For the purpose o f this work we consider the propagation of a cylindrically symmetric gaussian beam with the approximation that the gaussian profile is maintained during the propagation. For such a beam the complex field can be written as
where wl, Rl, and ~01 are the initial values for z = 0 and 2 is the vacuum wavelength. The parameter Pc is the critical power for self-focusing and is given by
u(r, z) = (U/w) exp [ - (r/w) 2- ikr2/2R + iq~] ,
Pc = c%22/(27rn2) .
(4) where r is the distance from the z axis, U is a real constant amplitude factor, while the spot size w, the radius of curvature R, and the phase shift ~0are functions of z only. The amplitude U is a constant on ac-
x
1
z +z
+R~
~o(z)-~o,=
1
P -
,
(7)
1-3P/2Pc ~arctan( 7rnow2(z) l_xfF27~ ~L \2R(z)x/1-P/PJ
(
- arctan 2R, x/1/1--P/~JJ '
(8)
(9)
For n 2 = 0 (i.e., for P c = ~ ) eqs. ( 6 ) - ( 8 ) reduce to the expressions for the propagation of a gaussian beam in a homogeneous medium of refractive index no [ 14 ]. Equations (6) and (7), which give the spot size and the radius of curvature, can also be ob349
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tained, following the formalism of ref. [13], from the expressions for free space propagation after multiplying the term ~/(gn0w~ ), i.e. the imaginary part of the 1/q parameter o f the input gaussian beam, by the scaling factor V/1 - P / P c . Note, however, that the axial phase shift does not obey to such a transformation and cannot be derived with that formalism. When P > Pc the spot size can vanish after a certain propagation distance, which depends on the initial conditions Wl and R~. If P = P c and the initial wavefront curvature is fiat (R ~= oo ) the beam propagates in a guided mode without variations of spot size and radius of curvature (self-trapping). Various expressions for the critical power, which are obtained with different approaches and may differ from eq. (9), can be found in the literature [1-3,12,16,17]. In particular, by comparison with the numerical and analytical results reported in ref. [2], it can be seen that eq. (9) agrees with the expression of the minimum power o f a gaussian beam for which all the beam power can collapse into a point after a certain propagation, and that Pc is equal to 1.074 times the m i n i m u m power o f self-trapping. An expression for critical power similar to eq. (9) has been obtained also in ref. [17] by means of variational calculations. Moreover the formalism described in the following sections does not depend on the expression of the critical power, so that different expressions could also be used. As an example, the formula for the critical power given in ref. [ 13 ] contains an adjustable correction factor, which can be used for a precise fitting of experimental data [eq. (9) agrees with the expression given in ref. [ 13 ] if the correction factor is 4, which is within the specified range ]. These considerations suggest that the results presented here can be applied even for high beam power (P~,P~). Note that if we had approximated l ul 2 by the second order series expansion, we would have obtained a value of critical power four times smaller.
3. ABCD formulation
It is convenient to recast eqs. (6), (7) in a form consistent with the standard formulas for gaussian beam transformations. For a gaussian beam of spot size w and radius of curvature R in a material of re350
15 February1993
fractive index n~ one defines a complex radius of curvature, q, as [ 14,15 ]
1/q= l / R - i 2 / ~ n l w 2 .
(10)
If q~ is the complex radius at the input of a linear optical system characterized by a ray matrix
the complex radius, q2, at the output is given by ("ABCD law") [14,15]:
q2 = (Aq, + B ) / ( C q l + D ) .
(11)
We consider now an optical system made of a nonlinear material of length d with flat faces perpendicular to the z axis surrounded by a medium with refractive index 1. As in the case of lens-like media [14], we take into account the refraction at interfaces by multiplying the real radius of curvature by no at the input and by dividing it by no at the output. The transformation of the complex radius, calculated by using eqs. (6) and ( 7 ), can by put in a form consistent with the ABCD law by defining the ray transfer matrix of the Kerr media as M=x//1 - ?
-7/[(1-7)de]
'
where de = d/no is the effective length of the medium for P = 0 and y=
[ 1+ -1( 4 \ 2d~
< f l -'vPT"
2gWo2J_]
(13)
In eq. (13) Wc is the spot size at the center of the medium and Wo is the spot size at the beam waist calculated as i f P / P ~ = 0 . This parameter can also be expressed in terms of the complex radius of curvature at any arbitrary plane. Since the elements of the matrix in eq. (12) depend on the beam power and complex radius, the ray matrix is, as it must be, nonlinear. For P/Pc = 0 the matrix reduces to that pertaining to a linear dielectric block [ 18 ]. By inspection of eq. (13), it is apparent that 7~P/Pc. It can also easily be shown that for 7>~ 1 the beam is (ideally) focused into a point (vanishing spot size) inside the medium. In this case our approximation is no longer valid, therefore we only consider cases where y< 1. In order to evidence the nonlinear effects and sep-
OPTICS COMMUNICATIONS
Volume 96, number 4,5,6
arate them from the linear ones corresponding to the linear propagation through a length de and represented by the matrix
('0 we factor out the matrix M as follows: M=(;
de/2) M k ( ~
de/2),
(14)
15 February 1993
the matrix of a combination of these two elements, can be approximated by
if the term z / f c a n be neglected as a second order term for flarge and z small. Therefore, the first order matrix Mk can be viewed as a superposition of the effect of a lens of focal length fk and a propagation through a negative distance Zk given by
where Mk = 1/ lx~_7(1._--7:~
-,d¢/4")1 - ?/2,1"
(15)
The matrices on the left and right hand side of the product in eq. (14) correspond to a linear propagation through a medium of length d / 2 and refractive index no. The matrix Mk is nonlinear and reduces to the identity matrix for P/Pc= ?= 0. The Kerr nonlinearity is therefore lumped into the matrix Mk and separated from linear propagation. Obviously this model only allows the calculation of the beam parameters at the output plane, not inside the medium. Finally, for 7 small, Mk can be expanded at the first order: Mk~
(
1 -7~de
- 71dd4)
"
(16)
Note that the determinant of this matrix is unity only if the term in 72 is neglected. This circumstance arises from the fact that the elements of the matrix in eq. (16) are only the first order expansion of the elements of the matrix in eq. (15), whose determinant is indeed unity, as for usual A B C D matrices. In the expressions derived by using eq. (16) only the zero and first order terms in ~ should be retained as significant, since eq. (16) is valid only for low power (low 7). Equation (16), however, allows to derive the following useful physical interpretation of the nonlinear Kerr matrix. Since the matrix of a thin lens of focal length f i s
(_1 01) 1/f
'
and that of a propagation through a distance z is
:)
1/fk=~ld~ ,
(17)
Zk = - ~ d e / 4 .
(18)
The focal length fk represents the self-focusing, the negative distance represents the self-shortening. The effective length de for linear propagation is thus shortened by Zk due to the nonlinear effect. It should be noted that this variation of length is not related to the axial phase shift which is given by eq. (8). Note that, if the nonlinear coefficient n2 is negative (i.e., Pc<0 and y < 0 ) the focal length fk is also negative (self-defocusing), whereas the variation in propagation distance become positive ("self-lengthening"). Although the equivalent nonlinear focal length and shortening have been obtained by a suitable, but not unique, factorization of the matrix in eq. (12), it can be shown that the same result can be obtained calculating starting from eq. (12), the focal length and the shift of the principal planes [ 14 ] of the system represented by eq. (12). To understand physically the origin of self-shortening consider the simple case of an input beam with beam waist at the input face of the Kerr medium [i.e., 1/R~ = 0 in eqs. ( 6 ) (8) ] and a power P=Pc. As shown by eqs. (6) and (7) in this case the spot size and the radius of curvature do not change along propagation since the beam propagates in a self guided mode. The transformation operated by the Kerr medium is therefore an identity transformation, which corresponds to a propagation through a vanishing distance. For a given beam power both 1/fk and Zk reach a maximum when y reaches its maximum value equal to P/Pc. The optimum focusing condition, obtained form eq. (13), is Wc Wo =2de/ ( 2~z ) .
(19)
351
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In particular, when the beam waist Wo is in the center of the material we obtain we= % = x/2dJ27r. In this case the length of the material is equal to the confocal parameter of the gaussian beam. It might be noted that the operator given by eqs. (12), (15), and (16) is not mathematically a true matrix operator since the elements depend on the input beam parameter, but can nevertheless be used in the same way as the matrices to calculate the beam transformations, proceeding as follows. G i v e n an input beam one first calculates the spot size w~ in the center of the Kerr block and the beam waist w0 with the standard ABCD linear propagation technique assuming zero beam power P/P~. With these data one calculates the factor 7 by using eq. (13) for the real beam power and builds the n o n l i n e a r matrix using eq. (15) or (16). The matrix of Kerr m e d i u m can then be combined, simply by matrix multiplication, with the ABCD ray matrix of other linear elements following the Kerr m e d i u m to treat the propagation of gaussian beams in complex optical systems and calculate the complex beam radius at any plane after the Kerr element. Finally it should be remembered that, if one wants to use eq. (16) instead of the complete matrix given by eq. (15), which is possible to obtain approximations to the first order in the beam power, only the terms up to the first order in 7 (i.e., P/P~) must be retained as significant. To check our calculations, the results have been compared with the numerical solutions of eq. (3). Figure 1 shows such a comparison in case of optimal focalization for different beam powers. For P/P~ = 0.3 the matrix formalism gives results in excellent agreement with the exact numerical solution both for intensity and phase. For P/P~ = 0.6 some discrepancies begin to arise, however the field substantially maintains a gaussian profile. For a close comparison we have interpolated the exact field given by the numerical solutions, u,, with a gaussian field, Ug, as given by eq. (4), that m i n i m i z e s the m e a n square error i ]U,--Ug[ 2 27crdr. 0
The interpolated spot size, radius of curvature, and axial phase shift are plotted in fig. 2 as a function of P/P~ and compared to the corresponding results of 352
3 •
15 February 1993 i
i
2I
ia )
'
i
,
i
,
(b)
1
~c ~ 2 ',. 0.6 __=
,
0.3 0.3 =u - 1
o
1
Q_
a:
0
-2 --,_3
0
I
'
i
i
Ii
410
0 40 80 2O Radial Distance (iJm)
80
1 20
Radial Distance (pro)
Fig. 1. Comparison of numerical solutions of paraxial wave equation (dashed lines) and results of matrix formalism (solid lines). The effective length of Kerr medium is de= 15 ram. The incident gaussianbeam has a wavelengthof 1.064 p,m and, for P~ Pc=0, a beam waist in the center of the medium with a spot size of 50.q ~tm (optimum focalization). The parameter near the curves is the normalized power P/Pc. 1.2
-0.0 radius .
~3 o 0.8
p sp°t
" ~ ' ~
-0.6
-u
s
:~ o ~
--1.2 u)
0.6
:~
o_ u3
-1.8
• 0.4
Eo
z 0.2 0.0 0.0
-2.4 ,
,
0.2 0.4 0.6 0.8 Normalized Power P/Pc
.0
-3.0
Fig. 2. Parameters of a gaussian beam after the propagation in a Kerr medium as a function of the power P/Pc. The conditions are those of fig. 1. The beam spot size and radius of curvature (left scale) are divided by the valuesfor P/Pc= 0. The axial phase shift (fight scale) is plotted as differencebetween the actual value and that for P/Pc=O.The solid lines corresponds to the matrix calculation while the dots are the values obtained by a gaussian interpolation of the numerical solutions• matrix calculation. The substantial agreement between the results confirms the validity of our approximated theory. The self-shortening, the self-focusing, and the effects of beam focalization can be assessed from fig. 3, where the normalized induced shortening and induced dioptric power (i.e., - 2zJde and d,,/2fk, both
OPTICS COMMUNICATIONS
Volume 96, number 4,5,6 0.12
,
i
,
0.5
• o
i
0.4
,
i
,
E
0
•
i
P/P==0.2
t3u 0 g 0.08
~5
0. o J~ 03
0.04
o z 0"000.0
0.5
' 1.~0 ' 1.J5 ' 2.'0
Normalized
Beam
Fig. 3. Normalized induced shortening,
' 2.'5
' 3.0
Waist - - Z k / 2 de
and dioptric
power 2dJfk as a function of the normalized spot size at the beam waist of the incident gaussian beam, W o / ~ . The parameter near the curves is the ratio of the effective distance of the beam waist (for P=0) from the center of the Kerr medium to the effective length d~ of the material. The effective distance is given by the real distance divided by no when the waist is inside the material, or by d~/2 plus the distance from the face of the material when the waist is outside. equal to ),/2 ) for P/Pc = 0.2 are shown as a function o f the spot size at the b e a m waist (for P/Pc=O) o f the incident gaussian beam. The curves in fig. 3 correspond to different distances o f the gaussian beam waist from the center o f the K e r r m e d i u m . W h e n the waist is inside the material both shortening and dioptric power reach always the same m a x i m u m in correspondence to the o p t i m a l focalization o f the gaussian beam. If the b e a m waist is outside the material it is impossible to satisfy eq. ( 19 ) for o p t i m u m focalization. Note that the m a x i m u m shortening, given by - ( d e / 4 ) P / P c , can a m o u n t to a significant fraction o f the length dc o f the material.
4. Applications As an example o f application o f this formalism we first m e n t i o n the Z-scan experiments [ 19 ] where a nonlinear m e d i u m is scanned across the focus o f a gaussian b e a m and the variation o f the b e a m spot size at a reference plane are m e a s u r e d to d e t e r m i n e the nonlinear coefficient o f the material. W i t h o u t going here into details we point out that experiments p e r f o r m e d with arbitrary long samples o f nonlinear
15 February 1993
material can be easily interpreted on the basis of the new formalism. In particular, the results could be easily understood physically and more correctly analyzed by considering not only a lens effect but also a shortening (or lengthening, if r/z<0) of the propagation distance. Moreover, as it will be shown elsewhere, the m a x i m u m nonlinear effect on the b e a m spot size is observed only when the o p t i m u m focusing condition given by eq. (18) is satisfied. A second relevant application is the mode calculation in resonators containing a Kerr m e d i u m . In this case one has to apply the formalism as follows. F o r any value o f 7 one calculates, using the matrix Mk as a linear one, the self consistent spot size w and radius o f curvature R in a reference plane inside the resonator, obtaining w = w (7) and R = R ( 7)- Starting from that reference plane one calculates, assuming linear propagation, the spot size in the center o f the Kerr m e d i u m , Wc and the corresponding beam waist Wo. By using eq. (13) the beam power can be expressed as a function o f 7: P / P c = P ( 7 ) / P c • The dependence of the spot size w on the power P is given, in a parametric form, by a set o f two equations [i.e., w = w(7) and P = P ( 7 ) ] with 7 as a parameter. These equations can be readily used to plot the spot size in any reference plane as a function o f the power. As an illustrative example consider the resonator shown in fig. 4a, which is a typical configuration for self modelocked T i : s a p p h i r e femtosecond lasers. The resonator is m a d e by two plane end mirrors and contains a Ti: sapphire rod between two focusing mirrors represented as lenses. Figure 4b shows the spot size on the left m i r r o r as a function of the circulating power in the resonator for different values, x, o f the distance between one face of the rod and the corresponding lens. This resonator is optically stable for P/Pc=O if 36.64 m m < x < 3 9 . 0 2 mm. If the resonator is initially stable only one value o f self consistent spot size is found for any value of P/Pc and the spot size decreases with increasing power, producing a fast saturable absorber effect when an aperture is placed near the left mirror. On the other hand, if the resonator is initially unstable, it can be driven in a stable configuration at high enough power. Note that in this case two values o f self consistent spot size are found for a given P/Pc: the resonator becomes therefore bistable. This circumstance m a y be also relevant for self m o d e locking o f solid state lasers. 353
Volume 96, number 4,5,6
OPTICS COMMUNICATIONS (o)
given by eq. (19) with the beam waist in the center o f the material.
B 2o5 f=50
2.0
loo 3o0
5. Conclusions
f=50
Xmm l
i
i
i
(b)
1.2 m 0.8 "6 0.4
%0
1
i
o.;
i
o.'6
i
i
,.o
Normolized Power P / P : Fig. 4. (a) Typical resonator used for femtosecond mode-locked
Ti:sapphire lasers: K is the Kerr medium (Ti:sapphire, refractive index 1.76), the dimensions are in mm. (b) Behavior of the spot size on the left mirror as a function of the normalized circulating power in the resonator for different distances x. A crucial p a r a m e t e r to characterize the nonlinear losses p r o d u c e d by the variation o f spot size in a resonator initially stable is the slope of the curve in fig. 4b near P/Pc=O (the higher the slope, the higher the nonlinear effect on the laser d y n a m i c s ) . By using our matrix formalism, we have found, as we will show elsewhere, that the m a x i m u m (achievable with an o p t i m i z e d resonator) relative variation o f the spot size with respect with the circulating power is directly related to the stability p a r a m e t e r gig2 [ 15 ] o f the resonator for P = 0, namely:
I;
dw ] =+ 1 d(PTPc) p : o - 4x/g, g 2 ( 1 - g ~ g 2 )
(20)
This equation evidences that, to obtain large spot size variation with power, the resonator configuration must be very close to a stability limit, g~g2 = 0 or 1. To reach the m a x i m u m slope for a given gig2 it is necessary to satisfy the o p t i m u m focusing condition 354
15 February 1993
In this work we have introduced a new formalism, based on the aberrationless theory o f self focusing, to treat the propagation of a gaussian b e a m in a material presenting Kerr nonlinearity. We have shown that the Kerr m e d i u m can be represented by an ABCD ray transfer matrix whose elements depend on the parameters of the input beam. The main advantage of this new formalism is that the matrix can be c o m b i n e d by matrix multiplication to other matrices associated to linear elements to treat the propagation in complex optical systems. The calculation o f the output b e a m proceeds along two steps by using the standard ABCD law: given an input beam, one first determines the elements o f the nonlinear matrix, then calculates the output b e a m by applying the matrix. The results o f matrix calculation satisfactorily agree with the exact numerical solution of the nonlinear wave equation even for b e a m power close to the critical power. By using the new formalism we have pointed out that the effects o f the Kerr m e d i u m can be interpreted, to the first order in the b e a m power, as those due to a lens representing the self-focusing, plus those due to a propagation through a negative distance. This second effect has been n a m e d self-shortening. As an illustrative example of application we have analyzed the dependence of the TEMoo m o d e on the circulating power in a typical resonator used in self m o d e locked T i : s a p p h i r e lasers. In particular, we have pointed out that the m a x i m u m first-order variation of the mode size, which is responsible for mode locking, is directly related to the gig2 stability parameter of the resonator and that a bistable behavior could be observed in resonators initially unstable at low power. We believe that the new formalism can find significant applications mainly in the analysis, design and o p t i m i z a t i o n o f femtosecond self-mode locked solid-state lasers.
Note added in proof After submission of this paper the authors noticed
Volume 96, number 4,5,6
OPTICS COMMUNICATIONS
t h a t t h e p o s s i b i l i t y o f d e r i v i n g a n o n l i n e a r m a t r i x was p r e v i o u s l y m e n t i o n e d b y P.A. B e l a n g e r a n d C. P a r e , in A p p l . O p t i c s 22 ( 1 9 8 3 ) 1293.
Acknowledgements The authors thank Dr. G.P. Giuliani for several helpful discussions and suggestions.
References [ 1 ] S.A. Akhmanov, R.V. Khokhlov and A.P. Sukhorukov, in: Laser Handbook, eds. F.T. Arecchi and E.O. Schulz-Dubois (North-Holland, Amsterdam, 1972) Chap. E3. [2] J.H. Marburger, in: Progress in quantum electronics, Vol. 4, eds. J.H. Sanders and S. Stenholm (Pergamon, Oxford, 1977) p. 35-110. [3 ] Y.R. Shen, The principles of nonlinear optics (Wiley, New .York, 1984) Chap. 17; Y.R. Shen, in: Progress in quantum electronics, Vol. 4, eds. J.H. Sanders and S. Stenholm (Pergamon, Oxford, 1977) p. 1-34. [4] L. Spinelli, B. Couillaud, N. Goldblatt and D.K. Negus, in: Conf. on Lasers and electro-optics 1991 (Opt. Soc. Am., Washington, 1991 ) p. 583.
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[ 5 ] U. Keller, G.W. 'tHoofl, W.H. Knox and J.E. Cunningham, Optics Lett. 16 (1991) 1022. [6] F. Krausz, Ch. Spielmann, T. Brabec, E, Wintner and A.J. Schmidt, Optics Lett. 17 (1992) 204. [ 7 ] G. Gabetta, D. Huang, J. Jacobson, M. Ramaswamy, E.P. Ippen and J.G. Fujimoto, Optics Lett. 16 ( 1991 ) 1756. [8] G.P.A. Malcom and A.I. Ferguson, Optics Lett. 16 ( 1991 ) 1967. [9] D.E. Spence, J.M. Evans, W.E. Sleat and W. Sibbett, Optics Lett. 16 (1991) 1762. [10] F. Salin, J. Squier and M. Pich6, Optics Lett. 16 (1991) 1674. [ 11 ] M. Pich6, Optics Comm. 86 ( 1991 ) 156. [ 12 ] M. Sheik-Bahae, A.A. Said, D.J. Hagan, M.J. Soileau and E.W. Van Stryland, Opt. Eng. 30 ( 1991 ) 1228. [13] D. Huang, M. Ulman, L.H. Acioli, H.A. Haus and J.G. Fujimoto, Optics Lett. 17 (1992) 511. [ 14] H. Kogelnik, Bell System Tech. J. 44 ( 1965 ) 455. [ 15 ] A.E. Siegman, Lasers (Oxford Univ. Press, Oxford, 1986) Chap. 20. [ 16 ] A. Yariv, Quantum electronics, third Ed. (Wiley, New York, 1989) p. 482-489. [ 17 ] M. Desaix, D. Anderson and M. Lisak, J. Opt. Soc. Am. B 8 (1991) 2082. [18] A.E. Siegman, Lasers (Oxford Univ. Press, 1986) p. 585. [ 19] M. Sbeik-Bahae, A.A. Said, T.H. Wei, D.J. Hagan and E.W. Van Stryland, IEEE J. Quantum Electron. QE-26 (1990) 760.
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