Sennaroglu et al. Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. B 2577

Laser beam propagation in a saturable absorber

Alphan Sennaroglu

Optoelectronics Laboratory, Department of Physics, Koc University, Istinye, Istanbul 80860, Turkey

Fatihcan M. Atay and Attila Askar

Department of Mathematics, Koc University, Istinye, Istanbul 80860, Turkey

Received December 4, 1996; revised manuscript received May 12, 1997

The effects of the saturation of absorption on the propagation of Gaussian laser beams in an absorber are in-vestigated analytically. Using a dimensionless parameter quantifying the level of saturation, we calculate themodifications in the beam parameters as perturbations from known solutions. The perturbation expansionsare carried out in two directions, covering both low- and high-saturation cases. A formula is given that esti-mates the power transmission for all saturation levels. Numerical results are provided showing the effects ofsaturation on the spatial beam-spot size variation and the power transmission. © 1997 Optical Society ofAmerica [S0740-3224(97)04310-5]

1. INTRODUCTIONThe study of transverse effects associated with laserbeam propagation in nonlinear media has attracted muchinterest in various disciplines of optics. (See Ref. 1 andthe references therein for a thorough review of previouslyconducted research in this field). In particular, detailednumerical studies have been carried out to model thepropagation of high-intensity laser beams in media sub-ject to both Kerr nonlinearities and saturation2–7 with theobjective of better understanding the phenomena ob-served in wave–material interactions and designing nu-merous novel optical devices for all-optical switching,memory, and so on.

In continuous-wave (cw) end-pumped laser gain media,transverse effects owing to nonlinearities also play an im-portant role. Even at low cw intensities where no signifi-cant beam shaping occurs owing to Kerr nonlinearities,the propagation of the pump laser beam is still affected byother nonlinear processes such as thermal loading owingto the finite heat conductivity of the crystal and the ab-sorption saturation. Because the resulting crystal trans-mission as well as the spatial beam parameters change asa function of the incident pump power, the gain and thestability of the laser resonator are inevitably affected, re-quiring a thorough understanding of these effects to de-sign efficient, well-optimized laser systems. In a previ-ous study,8 thermal loading effects inside a laser crystalwere analyzed in the absence of absorption saturation,and modifications caused in the parameters of the propa-gating Gaussian beam were calculated analytically by theuse of an iterative-perturbative method.

In this paper we analyze the propagation of a Gaussianpump beam inside a laser crystal subject to saturable ab-sorption only. Thermal loading effects as well as Kerrnonlinearities are neglected. Different from the numeri-cal approach taken in Refs. 2–7, we employ a perturbativemethod as in our earlier study8 to derive analytical ex-pressions for the modifications produced in the spatial

0740-3224/97/102577-07$10.00 ©

beam parameters and the crystal transmission as a func-tion of the incident pump power. Although numericalbeam propagation simulations that model such effects cangive accurate results for a very wide range of parameters,closed-form formulas, which are obtained by the analyti-cal treatment presented here, can serve as useful rules ofthumb for experimentalists and can be readily incorpo-rated into the design schemes of solid-state lasers. In ad-dition, analytical results reveal the effects of the perti-nent parameters, providing better physical insight intothe phenomena.

The technique employed in our study can be summa-rized as follows: The known results involving Gaussianbeams in the absence of saturation are taken as the start-ing point, a dimensionless quantity measuring the level ofsaturation is used as a perturbation parameter, and then,by calculating a correction term, we determine how theGaussian solution is modified with saturation. The re-sulting solution is then valid in cases of low saturation.To cover a wider range of saturation levels, a similarmethod is carried out also in the reverse direction,namely, by perturbing the solution corresponding to fullsaturation. The results are shown to be in excellentagreement with numerical solutions. Although the ther-mal effects are neglected in this analysis, the effects oftemperature and saturation are additive in the first-ordercorrection terms. Hence simply adding the correctionterms determined in Ref. 8 and in this paper will give thefirst-order correction terms in the more practical situationwhere both effects are present. In this way, the resultspresented here complement those in Ref. 8.

When its saturation is neglected, absorption enters thebeam propagation equations via the constant absorptioncoefficient, denoted by a0 and having dimensions of(length)21. This results in a linear wave equation, whichis known to admit a cylindrically symmetrical Gaussianbeam solution.9 A more realistic model, on the otherhand, should account for the fact that in a saturable me-dium, absorption depends on the intensity I of the electric

1997 Optical Society of America

2578 J. Opt. Soc. Am. B/Vol. 14, No. 10 /October 1997 Sennaroglu et al.

field. In the wave equation, therefore, the constant a0should be replaced with a variable a. To model its depen-dence on the field intensity, one can take the followingrelation10

a 5a0

1 1 I/Is. (1)

Here, Is is the saturation intensity level, a constant thatis a characteristic of the medium. The variable a thushas the maximum value equal to a0 , which correspondsto the case Is 5 `, i.e., the absence of saturation. For fi-nite values of Is , the net effect appears as if the absorp-tion coefficient a0 was replaced with a smaller coefficienta. The extreme case when a reaches 0 (corresponding toIs 5 0) may be thought of as the case of full saturation.Of course the value of this modified a changes from pointto point in space depending on the field intensity at thatpoint. This introduces a nonlinearity into the waveequation, making it difficult to solve in closed form. Ouraim is to determine quantitatively how the parameters ofthe Gaussian beam solution change when a includes satu-ration effects.

The basic equations of the analysis are introduced inSection 2. These equations are then solved by perturba-tion methods. Two different perturbation expansionsthat are valid for values of Is close to 0 and ` are used.To determine the ranges over which each solution is ap-plicable, numerical calculations are used. In addition tothe closed-form calculations of the first-order correctionterms, simpler approximate formulas are also given thatare applicable to weakly focused beams and are shown toagree well with the numerical calculations. We concludeby displaying graphical and analytical results showingthe effects of increasing saturation levels on the beam-spot size and power transmission.

2. BEAM PROPAGATION EQUATIONSIn steady state the propagation of the laser beam in a ho-mogenous medium is governed by the scalar wave equa-tion

¹2E 1 k2E 5 0. (2)

Here, E stands for any component of a linearly polarizedelectric field E, say, for Ex . The medium wave number kappearing in this equation is related to the vacuum wavenumber k0 by the relation

k 5 k0S n0 2 ia

2k0D , (3)

where n0 is the refractive index (assumed constant), anda is given by Eq. (1). Substituting Eqs. (1) and (3) intoEq. (2) and defining the complex constant kc by

kc 5 k0n0 2 ia0

2, (4)

one obtains

¹2E 1 S kc 1 ia0

2I/Is

1 1 I/IsD 2

E 5 0. (5)

Since the intensity I is related to the electrical field Ethrough

I 5n0

2h0uEu2, (6)

where h0 is the vacuum impedance, Eq. (5) is nonlinear inE. The following treatment considers cylindrically sym-metrical solutions, i.e., E 5 E(r, z), with r and z denot-ing the radial and axial distances, respectively. Theplane z 5 0 is taken to be the edge of the absorber wherethe beam is incident.

The case Is 5 ` (i.e., a 5 a0) corresponds to the ab-sence of any saturation of absorption, in which case Eq.(5) is linear and has the Gaussian solution

E0~r, z ! 5 @e0 exp~2ikc z !#

3 XexpH 2iFp0~z ! 1 r2kc

2q0~z !G J C, (7)

where the complex functions p0(z) and q0(z) are givenby9

q0~z ! 5 z 2 b 1 ia, p0~z ! 5 2i lnS 1 1z 2 b

ia D ,

(8)

with a and b as positive constants and e0 as a positiveamplitude term. This solution will be referred to as theunperturbed beam solution. Motivated by the form ofEq. (7), a solution E to Eq. (5) is sought such that

E~r, z ! 5 @e0 exp~2ikcz !#XexpH 2iFp~z ! 1 r2kc

2q~z !G J C,(9)

with p(z) and q(z) as complex functions to be deter-mined. Of course, in the no-saturation limit of Is → `,the functions p and q should reduce to p0 and q0 as notedin Eq. (8) above. To quantify this transition, the dimen-sionless quantity

ds 5n0e0

2

2h0Is(10)

is employed. Physically, ds is the ratio of the initial in-tensity I(0, 0) to the saturation intensity Is , or, equiva-lently, the ratio of the incident power Pi to the saturationpower (p/2)Isv0

2. In what follows, ds will be viewed as aperturbation parameter, with the value ds 5 0 corre-sponding to the absence of saturation. Our purpose inthis is to determine p(z) and q(z) as perturbation solu-tions corresponding to this parameter.

3. PERTURBATION SOLUTIONA. Low SaturationIn studying wave equation (5) it will be convenient to in-troduce the dimensionless quantities

z 5 a0z, qd~z! 5 a0q~z/a0!, pd~z! 5 p~z/a0!

(11)and furthermore define a new variable u by

1qd~z!

5u8~z!

u~z!. (12)

Sennaroglu et al. Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. B 2579

After substituting Eqs. (6), (9), and (10) into Eq. (5) andemploying the usual slowly varying envelopeapproximation9

]2E

]z2 ! kc2E,

the wave equation can be expanded in powers of ds , and,since ds is small in the low-saturation case, terms of sec-ond and higher order can be neglected. Similarly, theunknown functions pd and u are expanded in powers of dsas

pd~z! 5 p0~z! 1 ds p1~z! 1 ...,

u~z! 5 u0~z! 1 dsu1~z! 1 ... . (13)

A further expansion of the resulting expression in powersof r and equating the coefficients of r0 and r2 give thecoupled set of equations (up to first order in ds)

u09~z! 5 0, (14)

u19~z! 5i

kcexp~2z!exp$2 Im@ p0~z!#%

3 ImFkcu08~z!

u0~z! Gu0~z!, (15)

p08~z! 5 2iu08~z!

u0~z!, (16)

p18~z! 5 2iu18~z!u0~z! 2 u1~z!u08~z!

u02~z!

1i

2exp~2z!exp$2 Im@ p0~z!#%. (17)

The solution to Eq. (14) is a linear function of z,

u0~z! 5 z 1 c. (18)

(Note in Eq. (12) that multiplying u0 by a nonzero num-ber does not change the value of q; so, the coefficient of zis taken to be 1.) If c is separated into its real and imagi-nary parts as

c 5 2b 1 ia, (19)

then the parameters b and a can be interpreted as theunperturbed beam focus and the normalized Rayleighrange, respectively. Equation (16) can then be inte-grated with an arbitrary initial condition for p0 ; withoutloss of generality, p0 is taken as

p0~z! 5 2i lnS 1 1z 2 b

ia D . (20)

Equation (15) is solved next. Since k0 is much largerthan a0 , one can use the approximation

kc 5 k0n0 2 ia0

2' k0n0 . (21)

Of course care must be taken in replacing a complexquantity with just its real or imaginary part, but it turnsout in this case that this is justified. It is then possible tointegrate Eq. (15) by parts twice with zero initial condi-tions to find

u1~z! 5ak0n0

2kcIm$~z 1 c !@E1~ c ! 2 E1~z 1 c !#exp~ c !%

2a2k0n0

2kcH z

c2 ~z 1 c 1 1 !

3 @E1~ c ! 2 E1~z 1 c !#exp~ c ! 1 1 2 exp~2z!J ,

(22)

where the overbars denote complex conjugates and E1 isthe exponential integral defined by11

E1~z! 5 Ez

`

t21 exp~2t !dt.

The solution q(z) is now determined to the first order,since by Eqs. (11) and (13),

q~z ! 51a0

Fu0~z! 1 dsu1~z!

1 1 dsu18~z! G , (23)

where u0 and u1 are given by Eqs. (18) and (22), respec-tively. Finally, Eq. (17) is integrated with zero initialcondition to obtain

p1~z! 5 2iu1~z!

u0~z!1

ia2

(Im$@E1~ c !

2 E1~z 1 c !#exp~ c !%). (24)

Similar to q(z) in Eq. 23 above, the function p(z) is foundwith p(z) 5 p0(z) 1 ds p1(z), where p0 and p1 are givenby Eqs. (20) and (24).

The correction term u1 looks rather complicated com-pared with u0 . Fortunately, it is possible to derive amuch simpler expression that, under reasonable condi-tions, closely approximates u1 . The approximation willbe valid for values of z and b for which uz 2 bu is smallcompared with the parameter a. In practice, this re-quirement is usually met over the whole absorber length,provided the beam is not too sharply focused. When thisis the case, differential equation (15) can be reduced tou19(z) 5 exp(2z ), which has the solution

u1~z! 5 exp~2z! 1 z 2 1. (25)

Using Eq. (25), one obtains the approximations

qd~z! 5z 2 b 1 ia 1 ds@exp~2z! 1 z 2 1#

1 1 ds@1 2 exp~2z!#, (26)

pd~z! 5 2i lnS 1 1z 2 b

ia D 2 idsH exp~2z! 1 z 2 1z 2 b 1 ia

212

@1 2 exp~2z!#J . (27)

B. High SaturationIn Subsection 3.A the quantity ds has been treated as asmall parameter, corresponding to the case of low satura-tion. In the other extreme, one may be interested in thecase when ds is large. In the limit as Is approaches 0,which is equivalent to ds → `, wave equation (5) becomes

¹2E 1 ~k0n0!2E 5 0,

2580 J. Opt. Soc. Am. B/Vol. 14, No. 10 /October 1997 Sennaroglu et al.

which admits the Gaussian solution9

E~r, z ! 5 @e0 exp~2ik0n0z !#

3 XexpH 2iFp0~z! 1 r2k0n0

2q0~z !G J C,with p0 and q0 again given by Eq. (8). Using definition(4) of kc and employing approximation (21), we can writeE(r, z) in the same form as Eq. (7):

E~r, z ! 5 @e0 exp~2ikcz !#

3 XexpH 2iF i2

a0z 1 p0~z ! 1 r2kc

2q0~z !G J C.For large values of ds , the quantity ds

21 can be intro-duced as a small perturbation parameter, and the case ofhigh saturation can be viewed as a perturbation from theknown solution, which corresponds to ds

21 5 0. Theanalysis then proceeds in a manner similar to that dis-cussed in Subsection 3.A. Thus if we assume a solutionthat has the form of Eq. (9), wave equation (5) is ex-panded as before, this time in powers of ds

21, and the di-mensionless quantities qd , pd , and u are defined analo-gously to Eq. (11), which is then expanded as a powerseries in ds

21 so that

pd~z! 5i2

z 1 p0~z! 1 ds21p1~z! 1 ••• ,

u~z! 5 u0~z! 1 ds21u1~z! 1 ••• .

The zero-order terms are given by Eqs. (18) and (20).The equations for the first-order terms turn out to be con-siderably simpler in this case, leading to the solutions

u1~z! 5 2i

6a@z3 1 3~2b 1 ia !z2#, (28)

p1~z! 5 21

6a

z3 1 3~2b 1 ia !z2

z 2 b 1 ia

2i

6a2 @~z 2 b !3 1 3a2z 1 b3#.

As done at the end of the Subsection 3.A, simpler approxi-mate expressions can be written assuming that a is largecompared with uz 2 bu. Thus for a weakly focused beamone can take

qd~z! 5~z 2 b 1 ia ! 1

12 z2ds

21

1 1 zds21 , (29)

pd~z! 5i2

z 2 i lnS 1 1z 2 b

ia D2

i2

ds21S z2

z 2 b 1 ia1 z D . (30)

C. Comparison of the Two SolutionsWe now have two different perturbation solutions for theparameters p and q of the Gaussian beam, one for low-saturation and the other for high-saturation levels. Thenatural question is which solution to use over what rangeof ds . To that end, the two solutions are compared at a

fixed location along the absorber for positive values of ds .Figures 1 and 2 give the results for the dimensionlessquantities pd and 1/qd , respectively. The comparison isdone at a point 2 cm from the incident edge, which corre-sponds to the length L of a typical absorber. Table 1 liststhe other numerical values used. It is seen from thesefigures that the analytical solutions obtained agree wellwith the numerical solution of wave equation (9) overtheir respective ranges. Again, the figures suggest thatthe region where the two expansions are close can betaken to be the boundary between the low- and the high-saturation regimes. In this example, the approximatevalue of ds 5 1.5 separates the two regimes.

To look at a practical application, the dependence of thebeam power transmission t on the saturation level is cal-culated with both low and high solutions. [See Eq. (32)in Section 4 for the definition of t.) Both curves, as wellas the numerical solution, are plotted in Fig. 3. It is ob-served that the two solutions are close near ds 5 2. Itappears that, as a rough guide, the low-saturation solu-tions should be used for ds , 1.5 and the high-saturationsolution for ds . 2.5. Note, however, that such rangesdepend also on the other parameters; so, it is best to re-peat the comparison when the parameter values arechanged considerably. The qualitative picture, however,is expected to remain the same. It is easy to validate this

Fig. 1. Graphs of upd(L)u versus ds as calculated with the low-and high-saturation solutions, plotted together with the numeri-cal solution.

Fig. 2. Same as Fig. 1 but for u1/qd(L)u versus ds .

Sennaroglu et al. Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. B 2581

Table 1. Physical Parameters Used in Calculations

Name Symbol Value Used

Differential absorption coefficient a0 1.5 cm21

Refractive index n0 1.8Crystal length L 2 cmPump wavelength l 1.06 m

Vacuum wave number k0 5 2pl21 5.93 3 104 cm21

Normalized Rayleigh range of unperturbed beam a 5 a0pv02n0l21 8.002

analytically in the case of weakly focused beams. Usinglow-saturation approximations (26) and (27), one obtains(to second order in ds)

t low~L ! 5 exp~2a0L !$1 1 ds2@1 2 exp~2a0L !#2/2%,

whereas high-saturation approximations (29) and (30)give

thigh~L ! 5 exp~22a0L/ds!.

These are seen to be in qualitative agreement with thecurves in Fig. 3. The true solution curve can be predictedto be a monotonically increasing curve starting from thepoint @0, exp(2a0 L)# and asymptotically approaching 1 asds gets large. For example, one can take the following fit,

Fig. 3. Beam power transmission t (L) across a 2-cm length ab-sorber as a function of the saturation level ds , calculated withthe two solutions for low and high saturation, plotted togetherwith the numerical solution.

Fig. 4. Comparison of the approximate analytical power trans-mission formula [Eq. (31)] with numerical calculations.

t~L ! 5 exp~2a0L ! 1 @1 2 exp~2a0L !#exp~22a0L/ds!,

(31)as a single formula covering all saturation levels forweakly focused beams. Figure 4 shows that this simpleformula agrees impressively well with numerical compu-tations, the maximum error is less than 5%. Similarly,one can derive formulas for the other beam characteris-tics, which are valid for all saturation levels, by suitablycombining the low- and the high-saturation solutions.

4. NUMERICAL RESULTSIn this section the effects of saturation on two importantcharacteristics of beam propagation, namely, the beam-spot size and the power transmission, are investigatednumerically. Since we have determined the solutions forp and q, it is possible to determine the total power P(z0)across a cross section z 5 z0 of the Gaussian beam.Comparing this with the incident power gives the powertransmission ratio t (z) 5 P(z)/P(0) at a distance z alongthe length of the absorber. Using methods similar tothose in Ref. 8, we calculate t (z) as

t~z! 5v2~z!

v2~0 !exp~2z!exp$2 Im@ pd~z! 2 pd~0 !#%,

(32)

where v (z ) is the beam-spot size at a distance z along thelength of the absorber given by the expression

1

v~z!2 5 ImF 2kca0

2qd~z!G .

It is not hard to show that for the unperturbed beamt(z) 5 exp(2z ).

A numerical example is provided for the set of param-eter values given in Table 1. Recall that the perturba-tion parameter ds is equal to the ratio of the incidentpump power Pi to the saturation power. For the numeri-cal calculations in this section, it is assumed that thesaturation power of the absorber is 1 W, and Pi is used inplace of ds as the perturbation parameter to gain morephysical insight into the problem. With the observationsmade in Subsection 3.B, low-saturation expansions areused when the incident pump power is , 1 W, otherwisehigh-saturation expansions are employed.

Figure 5 gives the variation of the beam-spot size alongthe absorber length for different values of the incidentpower. It can be seen that both the location and the fo-cused beam size (the minima of the curves) change withthe incident pump power.

2582 J. Opt. Soc. Am. B/Vol. 14, No. 10 /October 1997 Sennaroglu et al.

Fig. 5. Beam-spot size v (z ) plotted against the normalized dis-tance z for various values of the incident pump power (v05 100m, b 5 0.75).

Fig. 6. Beam power transmission t (z ) plotted against the nor-malized distance z for various values of the incident pump power.Curves from bottom to top correspond to powers of 0.01, 1, 2.5, 5,and 10 W (v0 5 100m, b 5 0).

Fig. 7. Beam power transmission t (L) across absorber lengthplotted against the incident pump power Pi for various values ofthe unperturbed beam waist v0 (b 5 0). Note that since high-saturation expansions are used in this calculation, for Pi, 2 W, the numerical values read off from the graph are erro-neous.

Fig. 8. Beam power transmission t (L) plotted against b, the(normalized) focus location of the unperturbed beam, for variousvalues of the incident pump power (v0 5 100m).

Figure 6 gives the graph of t (z ) versus z for several val-ues of the pump power Pi . It can be seen that the powertransmission ratio increases with increasing saturation,as would be expected since the absorption coefficient is ef-fectively reduced. For small values of the pump power,neglecting saturation does not introduce too much error;however, the increase in transmission ratio is particularlysignificant for Pi larger than 2 W. The limiting profilefor the curves is of course the horizontal line t(z) 5 1,corresponding to full saturation where no power is ab-sorbed. Note that the curves obtained with the high-saturation solutions cover a very large part of the regionbetween the unsaturated and the fully saturated cases.Figure 7 shows the variation of the power transmissiont (L) across the whole absorber length with increasingpump power, again displaying the direct relationship be-tween the two. The power transmission calculations areinsensitive to the value of the parameter b, the focus lo-cation of the unperturbed beam, as shown in Fig. 8.

5. CONCLUSIONIn this study, we derived an approximate solution that isapplicable to Gaussian laser beams propagating in a satu-rable medium and is shown to agree with the numericalcalculations. Because the expansions are carried outstarting from the two extreme cases of saturation, the re-sults have practical use for a wide range of saturation lev-els in addition to their theoretical value. As noted in theintroduction, the correction terms coming from the inclu-sion of thermal effects can be added to the first-orderterms derived here, leading to a two-parameter expansionof the solution in the general case.

A natural concern that arises in perturbation solutionsis the range of values of the parameters and/or indepen-dent variables for which the solution is valid. We per-formed the calculations in this work with typical valuesthat arise in applications, as given in Table 1. Obviously,as the perturbation parameter ds or ds

21 gets smaller, theaccuracy of the perturbation solution increases, as well asthe range over which it is useful. In this context, thehigh-saturation solutions are perhaps more useful sincesmall values of ds

21 cover a much wider range of satura-tion levels compared with small values of ds . This isseen clearly in Figs. 1 and 2. The success of the presentapproximation also depends on the finite length of the ab-sorber (L 5 2 cm in the example given); limiting the in-dependent variable to a bounded domain effectivelyavoids the problem of secular terms. If one needs moreaccuracy, one could calculate second- and higher-orderterms of the perturbation expansion; calculation of thesefollows the same lines set forth in this paper. However,the first-order corrections are shown to be fairly accuratequantitatively, as well as to give qualitative informationabout the effect of saturation on the behavior of the beam.For example, the less accurate but much simpler formulasderived for weakly focused beams are shown to correctlycapture the qualitative effects of saturation on powertransmission. Experimental work in this area could ben-efit from and lead to further enhancements of the resultspresented here.

Sennaroglu et al. Vol. 14, No. 10 /October 1997 /J. Opt. Soc. Am. B 2583

Correspondence should be addressed to F. M. Atay;e-mail, fatay@ku.edu.tr.

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