Optimization of passively Q-switched lasers

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 31, NO. 11, NOVEMBER 1995

Optimization of Passively Q-Switched Lasers John J. Degnan, Member, IEEE

Abstrmt- In optimizing passively Q-switched lasers, there is a unique choice of output coupler and unsaturated absorber transmission which maximizes the laser output energy and efficiency for each three-way combination of laser gain medium, absorber medium, and pump intensity (i.e., inversion density). In the present paper, we generalize and solve the three coupled differential equations which describe the passively Q-switched laser to obtain closed form solutions for key laser parameters such as the output energy and pulsewidth. We then apply the Lagrange multiplier technique to determine the optimum mirror reflectivities and unsaturated absorber transmissions as a function of two dimensionless variables. The first variable, 2, corresponds to the ratio of the logarithmic round-trip small signal gain to the roundtrip dissipative (nonuseful) optical loss and is identical to that which was used in previous treatments to optimize the rapidly Qswitched laser. The second variable, cy, is unique to the passively &-switched laser and is equal to the saturation energy density of the amplifying medium divided by the saturation energy density of the absorber. It is largely determined by the ration of the absorber to stimulated emission cross sections, but also depends on the speed of relaxation mechanisms in the amplifying and absorbing media relative to the resonator photon decay time. Several design curves, valid for all four level amplifgng and absorbing media, are then generated. These permit the design of an optimum passively Q-switched laser and an estimate of its key performance parameters to be obtained quickly with the aid of a simple hand calculator. In the limit of large N (>lo), the design curves are virtually indistinguishable from the rapidly Q-switched m e . The curves can also be used to perform rapid tradeoff studies of Merent absorbing materials. The theory can also be applied to CW-pumped, repetitively Qswitched systems through a simple multiplicative factor for the laser gain. The theory is applied to the analysis of a passively Qswitched Nd :YAG laser previously reported in the literature and shown to give excellent agreement with the experimental results.

and presented some numerical results. Erickson and Szabo later considered the effects of the absorber lifetime [6] and the limiting case where the absorber cross section becomes very large [7]. Their theory was formulated in terms of three parameters, i.e., 1) a normalized inversion prior to Qswitching, 2) the ratio of absorber to laser cross section, and 3) an absorber lifetime normalized to the cavity photon lifetime. They demonstrated that, for typical absorber lifetimes, the results did not deviate significantly from the case where the lifetime is assumed equal to infinity [6]. A goal of more recent analyses of the rapidly Q-switched laser is the determination of the output mirror reflectivity which results in maximum laser energy or efficiency. In some applications, alternative design goals, such as shorter pulsewidth or higher peak power, may be traded off against energy and efficiency. However, all of these goals (energy, efficiency, peak power, or pulsewidth) drive the designer toward higher gain to dissipative loss ratios, and, in this operating regime, major sacrifices in energy and efficiency must be made to achieve relatively minor improvements in pulsewidth and peak power. This is illustrated in Appendix A. Because of the transcendental nature of the Q-switched laser equations, laser design has typically required the use of a computer. However, Degnan [8] recently demonstrated, by applying the Lagrange multiplier method to the optimal coupling problem for the rapidly Q-switched laser, that simple analytical solutions could be obtained for all laser parameters of interest, e.g., optimum mirror reflectivity, output energy, pulsewidth, peak power, overall efficiency, peak internal power at the mirrors, etc. These analytical solutions all depended on a single parameter, z , defined as the ratio of I. INTRODUCTION the logarithmic round-trip small signal gain to the nonuseful -SWITCHED, or so-called “giant pulse,” lasers were dissipative loss. Q-switched laser design could now be accomfirst described theoretically almost three decades ago plished quickly using a series of general design curves and a by Wagner and Lengyel [l] and extensive numerical simple hand calculator. An independent optimization theory calculations were carried out by Kay and Waldman [2] for for rapidly Q-switched lasers was published subsequently by both simple Q-switched and Pulse Transmission Mode (PTM) Zayhowski and Kelley [9]. Q-switched lasers. In the latter, often referred to as “cavityAs an alternative to active methods, lasers can also be dumped lasers, the pulse remains inside the cavity until it passively Q-switched by means of a saturable absorber. The is “dumped by an internal electrooptic Q-switch, and, with a sufficiently fast switch, this approach results in shorter passively Q-switched laser potentially offers the advantages of pulsewidths on the order of the cavity round-trip transit time. low cost, reliability, and simplicity in fabrication and operation Various authors have also considered the effects of slow since it requires no high voltages or fast electrooptic drivers. Q-switching [3] and laser media thermalization [4] on Q- In this technique, a material with high absorption at the laser switched laser parameters. Szabo and Stein [5] first derived wavelength is placed inside the laser resonator and prevents the relevant rate equations for the passively Q-switched laser laser oscillation until the population inversion reaches a value exceeding- the combined optical losses inside the cavity. These Manuscript received November 9, 1994; revised June 29, 1995. losses include the saturable absorber loss, loss provided by The author is with the Laboratory for Terrestrial Physics, NASA Goddard the output coupler (useful loss), and the dissipative loss Space Flight Center, Greenbelt, MD 20771 USA. which includes all of the residual nonuseful losses such as IEEE Log Number 9414611.

Q

0018-9197/95$04.00 0 1995 IEEE

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DEGNAN: OPTIMIZATION OF PASSIVELY Q-SWITCHED LASERS

nonideal transmissive or reflective surfaces in the resonator, scattering or absorption in the laser crystal or other internal optics, diffraction losses, and the nonsaturable loss within the saturable absorber. As the photon density builds up following achievement of a net positive inversion, the passive absorber rapidly “saturates” or “bleaches” into a high transmission state thereby Q-switching the laser. Saturable absorbers which have been used successfully include dyes [lo], bulk semiconductors [ l l ] , gases [12], and absorbing ions or color centers, either in a separate crystal [131 Or with the active laser media in a monolithic structure [14]. Passively Q-switched microlasers [15] are currently of great interest in ultrashort pulse applications, such as laser ranging [16], where there is a desire to replace more complicated and expensive modelocked oscillators with simpler and more reliable sources of high repetition rate, subnanosecond pulse, coherent radiation. In passive Q-switching, there is a unique optimal choice of output coupler and unsaturated absorber transmission which maximizes the laser output energy and efficiency for each three-way combination of laser gain medium, absorber medium, and pump intensity (i.e., inversion density). Zhang et d.[lo] recently considered the theoretical OptimiZatiOn Of a miniature NAB laser passively @switched by BDN dye. Their optimization equations depend on three parameters: 1) the logarithmic round-trip gain, 2) the dissipative loss L . and 3) a paranleter 6 equal to the ratio of tWO absorption CrOSS Sections within the BDN dye. However, the graphs presented in the latter work are not generally valid for an arbitrary choice of laser medium and absorber. Furthermore, the parameter 6 IS not Particularly well-defined under arbitrary Pumping conditions. Nevertheless, their experimental measurements Showed good agreement with energy predictions and fair agreement With pulsewidth predictions. In the Present paper, we generalize the Szabo and Stein [51 formulation slightly and Solve the three coupled differential equations which describe the Passively Q-switched laser to obtain closed form solutions for key laser parameters such as the output energy, peak Power, and pulsewidth. w e then apply the Lagrange multiplier technique to determine the optimum mirror reflectivities and unsaturated absorber transmissions as a function of two dimensionless variables. The first variable, z = 2anil/L = In ( G i ) / L :corresponds to the ratio of the logarithmic round-trip small signal gain to the roundtrip dissipative (nonuseful) optical loss and is identical to that which was used in previous treatments [8] to optimize the rapidly Q-switched laser. The second variable, a = osys/ay, is unique to the passively Q-switched laser and is largely determined by the ratio of absorber cross section, us, to the stimulated emission cross section, a, but also depends somewhat on level degeneracies and relaxation rates within the laser and absorbing media. Several design curves, valid for all four level amplifying and absorbing media, are then generated. These permit the design of an optimum efficiency, passively Q-switched laser and an estimate of its key performance parameters to be obtained quickly with the aid of a simple hand calculator. They also permit rapid tradeoff studies between different absorbing materials to be

performed. For values of a greater than 10, the design curves are virtually indistinguishable from the actively Q-switched case. In Appendix B, we use the general Q-switched equations for the nonoptimized case to analyze a previously reported Nd :YAG laser passively Q-switched by color center defects [ 131, demonstrate the use of the design curves in generating a maximum efficiency design, and compare the theoretical results with experiment. 11. LASERRATE EQUATIONS We generalize the coupled rate equations for the passively laser, first derived by szabo and Stein [51, by writing them in the form

“[

( ();

= 2 a n l - 2a,n,l, - In +L)]E (la) dt dn - = -yacq5n (1b) (it d n’ - = -ysasc&, (IC) dt where 4 is the photon density, is the instantaneous population inversion density, ns is the instantaneous population density of the SA absorbing state, a and are the laser stimulated emission and SA absorption cross sections respectively, 1 and I , are the lengths of the gain and absorbing media respectively, is the speed of light, t, = 21’1~is the roundtrip transit time of light in the resonator of optical length 1 / , R is the reflectivity of the output mirror, and L is the remaining two way dissipative (nonuseful) optical loss. The above equations differ from the original Szabo and Stein formulation [ 5 ] in two ways. First, we allow the gain medium and absorber to have different lengths. Secondly, we introduce the constants y and ys.It has been shown that the effects of level degeneracies [17], [18] and relaxation or thermalization rates in the laser medium 181 can usually be absorbed into the constant y,which is the amount by which the inversion density is reduced by the emission of a single photon. It has alternately been referred to as a “degeneracy factor” by Koechner [17], 2* by Siegman [18] or as an “inversion reduction factor” by Degnan [SI. When the degeneracies of the upper and lower laser levels =e equal and all relaxation or thermalization times are slow compared to the Q-switched pulsewidth, y has a value of 2. The quantity ys performs a similar role in defining the effect of the absorption of a single photon by the saturable absorber on the absorber population density (negative inversion). Dividing (lb) by (IC) yields a relationship between n and TI,,, i.e.,

“’”=(’)$

(2)

-

n

Ysffs

which can be solved to yield a relationship between n and ns at all points in time, i.e.,

ns = nsi

[3

a

(3)

where n i is the initial population inversion density at the start of Q-switching, n,i is the initial density of the absorbing state,

I892


and we have defined the constant ?sus D a --. Ds

(Y=

(4)

where Da = hv/ysa, are the saturation energy densities for the amplifier and absorber, respectively, and y and ys depend on the speed, relative to the resonator photon decay time, of relaxation mechanisms in the amplifying and absorbing media, respectively [4, Section 11-F, (22)]. Dividing (la) by (lb) and substituting (3) into the result yields

71‘

{

n, - n -

+

[In(3+LI 2u1

-

5yn,, 1 Ys

The maximum photon density occurs when d$/dn = 0. Thus, from ( 5 ) we obtain the following expression for the inversion density, nt, at the point of maximum power

Note that, as (Y approaches infinity in (lo), the quantity nt/n, approaches the rapidly Q-switched value [ln(l/R) L]/2a1[8]. Since a is typically much greater than unity and (nt/n,) is always less than unity, a value for (ntln,) can be easily obtained by iterative application of (IO), using the rapidly Q-switched value ( a = CO) as an initial estimate. The maximum power is now given by [8]

which can be integrated to yield

$(n)= - 1

In a passively Q-switched laser, laser action begins the moment the population inversion density crosses the initial threshold value. Thus, setting the left side of (la) equal to zero yields the initial condition

hvAl‘

[1-

(6)

pmax = -In

tr

-

The first three terms in the above equation are the usual terms for the rapidly Q-switched laser [8] while the last term is introduced by the saturable absorber. Now an expression for the total energy in the Q-switched pulse is obtained in the usual way [8], i.e.,

.=IW

huAl‘

dt P ( t ) = -In tr nf

= “2“f“f Yl n ( i ) S , .

lr

)(;

,-a,,

In

(B) (:;) - In

-

(7)

where the final inversion density, n f , is obtained by setting (6) equal to zero yielding the transcendental equation

“1. ;

n, - n f - [in($) + L ] In - - - -ns, 2uE *

Pmax

hvAl In (;){ -

nt - nt

+ In(;)

+L] 2u1

YtV

_ -1 s x n s , [ l 1

”Is

(ye]}

In(:)

(11)

[

1-

(31 -

=0

and the FWHM pulsewidth is given by the approximate expression (12) found at the bottom of the page [8].

111. OFTIMEATION OF THE PASSIVELY Q-SWITCHED LASER In Degnan’s model for the optimally coupled, rapidly Qswitched laser [8], key laser parameters can be defined in terms of a single dimensionless parameter, z , defined by zx-

(8)

where again the last term describes the effects of the saturable absorber.

E t, = -

4max

dt d ( t )

dn - hvA

(:;)

(4)

2an,l - In G i -L L

(13)

where n, is the initial population inversion density, L is the dissipative optical loss, and Gg is the round trip unsaturated small signal power gain.


1893

In order to ultimately compare the passively and actively Q-switched results, one is motivated to define the following additional variables

1

z = - In

L

(4)

and

Substituting the new variables into (7)-(9) yields the following equations for the laser output energy,

E=

( e ) x p

the constraining equation

and the initial condition z - 2,

-5

-

1 = 0.

(17)

Using the initial condition (17) to eliminate (16), we obtain the expressions

:E

in (15) and

for the laser output energy and @(p. z , ) = az(1 - e - p

-

p)

-

z s ( l - e-Op

-

Np)

= 0 (19)

for the constraint equation. We now apply the Lagrange multiplier technique [19] to determine the value of z, which maximizes the output energy for a given value of z , i.e.,

Substituting (21) and (22) into the initial condition (17) yields the optimum coupling parameter, xOpt,which maximizes the laser output, i.e.,

Plots of these parameters versus p and cy are not particularly interesting or illuminating. In order to compare passive &-switching with the rapidly Q-switched case previously analyzed by Degnan [SI, we will prefer to eliminate the intermediate variable p and plot the optimized parameters against z for different values of a . We will achieve this by calculating tabulated values of z , zs, and zopt,E,,,, and t, as a function of p and cy, matching up individual cells within the tables, and plotting the results. In Figs. 1 and 2, we plot the absorption parameter, zipt, and the coupling parameter, zopt,as a function of z and four values of a ( 2 , 3, 5 , and 10). Both parameters increase monotonically with z , and we note that, as a increases, the curves in Fig. 2 approach the rapidly Q-switched case corresponding to (1 = 00. Fig. 3(a) and (b) show how, for the cases a = 2 and 10, the initial optical losses, which exactly offset the initial gain at the outset of Q-switching, are divided in the optimized system between unsaturated absorption ( z,:pt'), output mirror coupling (xopt), and other nonuseful losses ( L ) . The unsaturated absorption component is always larger than the output coupling component in the optimized case. As the initial gain increases (larger z values), the unsaturated absorption represents a larger fraction of the loss increasing from about 30% for z = 2 to 80% for z = 100. This is more easily seen in Fig. 4 where we have plotted the ratio

and we have used the defining equations for z and z , , (13) and (14a).

Iv. where X is the Lagrange multiplier. Solving the Lagrange equations and applying the constraint equation (19) to eliminate the parameter z s , one obtains an equation defining z in terms of p and a for the optimized case, i.e., (21) found at the bottom of the page. The optimum value of the initial absorption parameter can also be expressed as a function of p and a by solving for z, in (19) which yields

LASERPERFORMANCE RAPIDLYQ-SWITCHEDCASE

COMPARISON OF WITH

For a given dissipative resonator loss L , the rapidly Qswitched laser is the most efficient. No additional energy or gain is lost to the saturation of an absorber or to unwanted optical losses arising from a slowly transitioning electrooptic, acoustooptic, or mechanical Q-switch. In a previous analysis of the optimally coupled, rapidly Q-switched laser, Degnan [SI derived the following expressions for the maximum output energy, Emax

=

AhuL 207

[ z - 1 - In z ]


1894

c

PU)

N

1

10

100

z = 2on,l

L Fig. 1. Dependence of the optimum absorption parameter, z s = 2a,n,,l,/L = - l n ( T i ) / L on z = 2un,l/L = In G g / L and cy = a,y,/ay where Go is the single pass unsaturated small signal gain, TOis the single pass unsaturated transmission of the absorber, and L is the round-trip dissipative optical loss (excluding the absorber and mirror coupler).

1 1

L

(b)

1

..

Fig. 3. Relative magnitudes of the saturable absorber loss ( z : , ~ ) , output , dissipative loss (L) in offsetting the initial mirror coupling loss ( x o p t ) and gain for the optimized case and (a) cy = 2 and (b) N = 10.

Q

X

0.8

1

10 z = _____ 2anJ L

100

Fig. 2. Dependence of the optimum coupling parameter I on z = 2an,l/L = In G i / L and cy = a,-y,/uy. The optimum mirror reflectivity is given by R,,t = exp( - z L ) where L is the round-trip dissipative optical loss.

the optimum reflectivity that maximizes the laser energy for a given value of z ,

0.6 0.5 0.4

c

u=3/

0.1 ,/ U=-

0

I

1

I

I

I

I 1 1 1 1

,

, ,

I I I I I

10

100

z = 2onil

and the pulsewidth

t, In z

1

L Fig. 4. The optimum logarithmic absorber transmission to gain ratio r = u 3 n s t l s / u n z l= -In T o / l n Go as a function of the parameters z = 2 a n , l / L = In G i / L and cy = o,y,/ay.

the rod given by From these basic parameters, one can derive expressions for additional quantities of interest to the laser engineer such as peak intracavity and extracavity power, inversion threshold density, etc. [8] In the limit of large z , the output energy for the rapidly Q-switched case approaches the total useful energy stored in

E, = lim Emax= AhvL z = - Vhvni z--00 207 Y

(28)

so we can express the laser efficiency as a function of z as well, i.e.,

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7

O0 ' I " '

IO09

-

-

~

_ _ - -

-

r -

~

08

07 06 05 04 03

i

02 01

1

1

11111

I,

, , ,,,I

,

10

1

,, , , , ,

n "

100

100

10

1 z

z = 2on,l

=

2(rnll

L

L Fig. 5. Normalized output energy of the optimally coupled laser ()-switched by a saturable absorber for n = 2, 3, 5 , and 10. As alpha increases, the curve for the rapidly ()-switched laser approaches the limiting case (0 = s) previously derived by Degnan [8]. The normalization constant is hl/.AL/%u-, for all curves.

Fig. 6. Efficiency of the optimally coupled laser (2-switched by a saturable absorber for ( I = 2. 3 , 5, and 10. As alpha increases. the curve approaches the limiting case ( ( I = x,) for the rapidly ()-switched laser previously derived by Degnan [8].

Using (13) and (14) in (15 ) and (12), the maximum laser energy, efficiency, and pulsewidth for the optimized passively Q-switched laser are now given by 10 l o

\\\\\

and (32) found at the bottom of the page where the parameter ,zt(a,p ) is obtained from an iterative solution of the equation \ \

0.1

1

10 Z =

with an initial estimate of ~ ~ ( ( 2 .= p )1 +.r(m,p). We now plot the expressions for the output energy, laser efficiency, and pulsewidth, along with the corresponding results for the rapidly Q-switched case, in Figs. 5-7. As expected, the maximum laser output energy and efficiency for the passively Q-switched laser is less than would be achieved in the rapidly Q-switched case but the difference becomes increasingly negligible for values of (1 greater than 10. For a value z = 20, laser efficiency increases from about 40% for a = 2 to about 73% for n = 10 whereas the corresponding value is about 80% for the rapidly @switched case ( a = x ) . Similarly, for low values of a , the laser pulsewidth can be substantially longer than would be predicted for the rapidly Q-switched case. Both effects occur because some portion of the initial gain is expended in saturating the absorbing medium.

100

2(Jn11 L

Fig. 7. Normalized laser pulsewidth for the optimally coupled laser @switched by a saturable absorber for (1 = 2, 3. 5 , and 10. As alpha increases, the curve approaches the limiting case ( o = x ) for the rapidly (2-switched laser previously derived by Degnan [8]. The normalization constant is t , / L for all curves where t , is the round-trip pulse transit time in the resonator and L is the round-trip dissipative optical loss.

The optimum unsaturated transmission of the saturable absorber is best obtained by reading the parameter T , , ~ )(~n ,2 ) directly off Fig. 4 and using the defining (24) to relate it to the single pass gain. Similarly, the optimum mirror coupling is obtained by reading the value of off Fig. 2 and using the equation

R,l,t(t~.z ) = ~ ~ x p - . r ~ , pz () (L~] .

(33)

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v.

CW-PUMPING AND Q-SWITCHED PULSE TRAINS

With CW pumping, the laser will passively &-switch as soon as the gain exceeds the combined saturable and unsaturable losses in the resonator. As the pump power is increased, the laser eventually reaches a threshold condition and begins to repetitively Q-switch with a time interval between pulses, r,, somewhat longer than the relaxation time of the upper laser level. Further increases in pump power do not affect the laser output energy or pulsewidth but do increase the pulse repetition rate [ 131. From a steady state analysis [20], the initial population density at the start of Q-switching is given by

where A, is the volumetric pump rate into the upper laser level and is proportional to the CW pump power, r, is the lifetime of the upper laser level, and 6 is the fraction of the population inversion left over at the start of the next pump cycle for the repetitively Q-switched laser. We can derive an expression for 6 by tracking the population inversion through the Q-switched pulse using (14c) and several relaxation mechanisms which are fast compared to the pump interval, r,, and the upper laser level decay time, 7., For example, in Rare Earth solid state lasers such as Nd : YAG, these relaxation mechanisms would include thermalization among Stark sublevels within the upper and lower laser multiplets and then relaxation of the lower multiplet. For a typical four level laser system, it can be shown [20] that

where f a is the equilibrium fractional population of the upper laser level as determined by Maxwell-Boltzmann statistics, y is the inversion reduction factor, and p is defined by (14c) and is related to the pulse energy through (7). Since 0 < p < 03, the function S can take on values between 1 (low z ) and 1- f a / ? (high 2 ) . Equation (34) can be rewritten as

passive Q-switch, and is proportional to the CW pump power. The function 1 - exp

1 - Sexp

ncw = A,T,

(37)

is the CW population inversion density, spatially averaged over the Q-switching mode volume prior to inserting the active or

[ (-2) [

Pp 1 - exp

s= ppexp

I):-(

1 - exp

(38)

and, substituting (38) and solving for r,, we obtain the following expression for the time interval between Q-switched pulses

rc = r, In

(-)

1 - 6$ 1-$

where li, is defined by (39a). The threshold pump power for Q-switching satisfies the condition ?i, = 1. The round-trip CW gain, 2 c r n ~ ~can 1 , be either measured or calculated prior to inserting the saturable absorber. From (37) and (39a), we note that the pulse repetition rates at two different pump powers, Pp and PL, are related by the expression

(

3

PpF 6,-

=

(

3

P'F y , L .

(40)

Substituting (38) into (40) and solving for 6 yields (41) found at the bottom of the page. As we shall demonstrate in Appendix B, the latter equation provides a simple experimental technique for directly determining the value of the parameter 6 and, through (35), estimating the values of the inversion reduction factor, y. and the effective area of the Q-switched laser beam, A , in CW-pumped, passively Q-switched systems.

VI. CONCLUSION We have considered the efficiency optimization of a laser passively Q-switched by a saturable absorber. Using the technique of Lagrange multipliers, a series of laser design curves have been generated that can be applied to any arbitrary combination of amplifying and absorbing media. These curves yield the optimum mirror reflectivity and unsaturated absorber transmission as a function of two dimensionless parameters.

(3 1- p; --

(-2)

always has a value less than or equal to unity and represents the reduction in initial gain resulting from a shortened pumping period, r,, and also takes into account the residual fractional population inversion density, 6, resulting from the previous Q-switch cycle. Substituting (36) into the initial condition (9), we can write

(36) where

(-2)

[ I - exp

- p; exp

(-31

(-2) [I

-

exp

(-?)I

DEGNAN: OFTIMIZATION OF PASSIVELY Q-SWITCHED LASERS

0

1

2 3 4 5 6 Coupling parameter, x

(a)

7

8

A lo), the optimized passively (l-switched laser parameters vary only slightly from the rapidly Q-switched case. The design curves define the laser output energy, pulsewidth, and laser efficiency for the optimum efficiency case and permit detailed passively Q-switched laser design to be accomplished quickly with the aid of a simple hand calculator. We demonstrate in Appendix A that maximizing efficiency is the proper design optimization philosophy for virtually all cases of practical interest, even if the primary goal is high peak power and/or short pulsewidth. Thus, one can target a particular design energy or pulsewidth and read the required values of o and/or z from Figs. 5 or 7. The appropriate choices of mirror reflectivity and saturable absorption can then be read from the

1897

corresponding curves in Figs. I and 2 or alternatively 2 and 4. For fixed values of z and fi determined by material choices and pump limitations, one can determine the maximum energy achievable and the corresponding pulsewidth, optimum mirror reflectivity, and optimum saturable absorber transmission from the same figures. The use of the design curves is illustrated for a real laser in Appendix B following a demonstration of how the general equations for the nonoptimized case may be used, in conjunction with simple experiments. to determine laser andor absorber parameters which may not be known a priori. Surface reflection and/or host crystal losses associated with the saturable absorber can usually be known reasonably well a priori and included in our initial estimate of L. The question arises, however, on how best to handle the unsaturable portion of the absorber loss which can be expressed as a fraction of the saturable loss. This can be resolved through an iterative application of Figs. 1 and 4. We first compute a value of z ignoring the unsaturable loss, go to Figs. 1 or 4 to obtain an initial estimate of the optimum absorber transmission, adjust L accordingly to incorporate the ociated unsaturable loss and recompute z (a lower value), and retum to Figs. 1 or 4 for a new optimum absorber transmission. This iterative process can be repeated until further changes in z are negligible. For the best absorbers, unsaturable losses are small ( 5 , there is no significant improvement in the pulsewidth resulting from a choice of reflectivity greater than that which provides maximum energy and efficiency. Furthermore, low values of z in general are not conducive to high output energies, short temporal pulsewidths, nor high peak powers as can easily be seen from Figs. 5, 7. and 8. This supports our earlier contention that major sacrifices in energy

1898


and efficiency must generally be made to achieve relatively minor improvements in pulsewidth and peak power and that efficiency optimization is the proper strategy for most cases of interest. One can also verify from this example that the Q = 5 curves in Figs. 2, 5, and 7 give the proper optimum coupling parameter, output energy, and pulsewidth respectively for the various values of z .

TABLE I

~~

APPENDIXB ANALYSIS OF THE MORRIS AND POLLOCK LASER Morris and Pollock [ 131 have reported on a series of fairly detailed experiments on a short, CW diode-pumped Nd : YAG laser, passively Q-switched by an absorbing Lithium Fluoride (LiF : FF) color center. The laser consisted of only three optical elements-a Nd:YAG rod, a plate mounted at Brewster’s angle and containing the absorbing LiF color center defects, and a partially transmitting ( T = 4%) end mirror having a 2 meter radius of curvature. The Nd : YAG rod was polished flat on both ends with a 30 arcsecond wedge. One end was antireflection-coated at 1064 nm, and the other end was coated for maximum reflectivity at 1064 nm and maximum transmission at the diode pump wavelength at 808.5 nm. The resonator optical length, l’, was 4.2 cm for the experiments for which detailed measurements were given. This corresponds to a round trip optical transit time t , = 0.28 ns. Prior to installing the LiF saturable absorber in the cavity, the authors operated the Nd: YAG laser in a CW mode and found a threshold pump power Pt = 175 mW. At a maximum diode pump power P,,, = 650 mW, the Nd: YAG laser produced 260 mW in a TEMoo mode with a slope efficiency of 54%. Rigrod [21] has provided the following expression, also based on a plane wave model, for the output power of a homogeneously broadened CW laser:

4F3/2 upper laser level multiplet in Nd : YAG [24], and fa = 0.41 is the Boltzmann occupancy factor for the lasing Stark sublevel [22]. Substituting the observed CW power of 260 mW at a maximum pump power of 650 mW into (B.2) and solving for the effective beam area yields Acw = 0.154 mm2. It should be mentioned that, due to the nature of (B.l), R’ can deviate from unity by 10% or more without significantly affecting the computed beam area. As an additional check on the reasonableness of this beam area, we can compute the TEMoo beam waist radius (to the l / e 2 intensity point) from resonator theory [25]. At the coated end of the rod, the waist radius has the value WO = 0.312 mm. The computed beam radius at the output coupler is only slightly larger, i.e., 0.315 mm, and hence the beam area is fairly constant throughout the resonator. The effective TEMoo beam area, given by A00 = 7rw;/2 = 0.153 mm2, is in excellent agreement with the value determined by analysis of the CW laser data. Morris and Pollock then measured the output energy, pulsewidth, and repetition rate of their passively Q-switched laser as a function of the CW diode pump power and three I,Acw(l - R ) a 2anl - In values of the internal saturable absorption transmission (TI = pcw = 2 ( a + dR)(l, (B.1) 0.948, Tz = 0.94, and T3 = 0.92 corresponding to saturable absorptions of 5.2, 6, and 8%, respectively). The output where I, is the saturation intensity, Ac\v is the effective coupler reflectivity was fixed at a value R = 0.94. The area of the CW beam in the rod, n is the spatially-averaged saturable absorber was placed at Brewster’s angle in the population density within the CW beam, R is the reflectivity cavity to minimize Fresnel reflective losses. Furthermore, the of the output mirror, and R’ is the reflectivity of the second ratio of saturable to unsaturable absorption in LiF was quoted mirror. If we assume the dissipative losses are lumped into R’ by the authors at approximately 20 to 1 suggesting that the and are small compared to unity, (B.l) can be rewritten as unsaturable portion of the absorption was limited to a few tenths of a percent. Pcw = IsAcw In - 1), (B.2) For each of the three absorbers, the laser maintained essen2 tially the same output energy and pulsewidth once the pump where P is the CW pump power. power was increased beyond its threshold value. However, The saturation intensity, taking into account thermalization the repetition rate increased approximately linearly with pump among the 4F3/2 Stark sublevels in Nd: YAG, is given by the power in agreement with the passive Q-switching theory preequation [22] sented in Section V. The measured pulse energies, repetition rates, and pulsewidths at the minimum and maximum limits hu - 3.05W I , = -of the pump power for each absorber are listed in Table I. m-,fa mm2 Using (41), the observed variation of repetition rate with where hv = 1.87 x lOP1’J is the laser photon energy, a = pump power yields an experimental value for the parameter 6.5 x 1O-l’ cm’ is the stimulated emission cross section 6 for each of the three absorbers, i.e., 6 = 0.586, 0.516, and [23], T, = 230 microseconds is the relaxation time of the 0.425 in order of increasing absorption. Furthermore, from

[

~

(dl ~

m)

(i)(g


1r

I

I

I

I

I

“Fast”Therma1ization

I899

in the central part of the pump beam where the average gain is equal to the combined saturable and unsaturable loss. This will be discussed more fully later. We have not yet used the pulsewidth data in our analysis. Equation (12) for the pulsewidth can be written in the form of (B.5) found at the bottom of the page where the subscript 1: = 1 , 2, or 3 refers to the three absorber transmissions T, and

y = 0.6

are the corresponding numerical values (0.628, 0.688, and 0.826) obtained by substituting the three experimental values for S and the corresponding observed repetition rate at the Fig. 9. Plots of the function h described by (35) in the text as a function of maximum pump power of 650 mW into (38). the laser output energy, E . The Nd : YAG value of fct = 0.41 is assumed. The central curve corresponds to the “best fit” case ( 7 = 0.636 and . I = Equation (B.5) still contains three unknowns, i.e., cy. 2m 0.063hm’) obtained for the Moms and Pollock laser and shows the three ncwl, and ( 7 1 t / n , ) .For each value of the absorber transmisdata points (indicated by diamonds) used in deriving the fit. The remaining two sion, these parameters must satisfy the constraint equation (8) curves show the limiting cases under the assumption of “slow” thermalization (7 = 2) and “fast” thermalization (7 = 0.6) in Nd : YAG. The experimental as well as (10) for the ratio of threshold population to initial results are consistent with a thermalization time in Nd : YAG on the order of population inversion density nt / n z .The latter equations can 1 or 2 nanoseconds. be rewritten in the form Output Energy (microjoules)

(14b) and (15), the three experimental pulse energies (6, 8, and 13 FJ) must satisfy the condition

( 2 m C\\ I)F,(1 - pz - e--PT )

+ 2 In (B.7a)

Thus, using the latter constraint, one can solve three equations of the form (35) for two unknowns, y and A , to obtain a best fit solution y = 0.636 and A = 0.063 mm2. The corresponding values for p are 1.031, 1.375, and 2.234 in order of increasing absorber strength and pulse energy. All best fit solutions described here were obtained using the Minerr function in the Mathcad 5.0 Plus software sold by Mathsoft. The derived value of y is consistent with other data [4] which shows thermalization occurring among Nd : YAG Stark sublevels on time scales short (on the order of a nanosecond) relative to the observed 20-32 ns pulsewidths. In Fig. 9, the residual fractional population inversion density, 6,is plotted against laser output energy in accordance with the theoretical expressions given by (35) and (B.4); the three data points, represented by diamonds, are in excellent agreement with the best fit theoretical curve. Also plotted in the figure are the limiting curves for “slow” thermalization (y = 2) and “fast” thermalization (y = 0.6). The value f a = 0.41 for Nd: YAG was used in plotting the graphs. The effective beam area during passive ()-switching is significantly smaller than for the CW case. This is due to the fact that the Gaussian pump beam is nonuniform and hence passively Q-switched lasing can only take place over an area

(B.7b) Thus, we can write a total of nine equations (three for each value of absorber transmission) and obtain a best fit solution for five unknowns, yielding (2anc\%,l)= 0.27, N = 3.095, ,nt/n, = 0.326, 0.363, and 0.376 in order of increasing absorption. The values for the dissipative loss in each case can now be calculated using (9), i.e.,

yielding L1 = 0.022, L2 = 0.021, and L3 = 0.015 suggesting that the round-trip dissipative loss in all cases was roughly constant at about 2%. Since Morris and Pollock reported long period (2 KHz) Q-switching for saturable absorptions as high as “approximately 10%” at the maximum pump level of 650 mW, the average gain over the active Q-switching area must be approximately equal to ( 2 a n ~ w l = ) 2 In( 1/0.9) ln(1/0.96) 0.02 = 0.272, in excellent agreement with our best fit result. Using the accepted Nd: YAG stimulated

+

+

1900


emission cross section [23] and our estimated values for y and a, the corresponding saturation energy densities are computed from (4) to be D, = 0.454 J/cm2 for Nd:YAG and D, = 0.147 J/cm2 for the LiF saturable absorber at these pulsewidths. The derived value of a would also imply a value ys = 0.075 based on a reported optical cross section for (F2-):LiF of 1.7 x cm2 [26]. The author is not sufficiently familiar with the relaxation dynamics of the LiF medium to comment further. The experimentalobservations, fitted model parameters, and calculated results are summarized and compared in Table I. One can easily see that, with the proper choice of parameters, our plane wave model is capable of duplicating the experimental results extremely well even though the pump beam and circulating wave are decidedly nonuniform in the radial coordinate. Let us now consider the reasonableness of our derived parameters in terms of our a priori knowledge of the Morris and Pollock laser. For a system end-pumped by a Gaussian laser beam, the pump rate as a function of the radial coordinate r and the distance y into the rod is given by [20]

numerical values from our analysis of the Q-switched laser, the total population inversion in the active Q-switching volume is N A = ncwAZ = 1.31 x 1014 corresponding to a useful stored energy E, = NAhu/y = 38.5 pJ [8]. Thus, from (B.10), we can compute the probable pump beam area for this experiment, I.e.,

A, =

-A

(B.12)

which yields a value A, = 0.083 mm2, about 30% larger than the effective area of the Q-switch beam and about 54% of the TEMoo beam area. Thus, the radius of the Gaussian pump beam was probably equal to or slightly less than 0.23 mm. This value appears to be compatible with the experimentalpumping configuration described in [13]. The analysis carried out in this appendix illustrates the manner in which some simple experiments with an arbitrary output reflector can provide the basic laser and absorber parameters required to generate an optimum design when they are not known a priori. To illustrate the use of the universal design curves, we take our maximum gain value of ( 2 0 n c ~ Z )=~ 0.27 , ~ at the maximum diode pump power of 650 MW and an internal loss where Tp is the transmission of the pump face at the pump of approximately L = 0.02 and compute z = 13.5. Reading wavelength (assumed to be near unity), 77, is the fraction of from Fig. 2, we obtain a value xopt = 3 off the a = 3 the energy in the pump band relaxing nonradiatively to the curve for this value of z . Substituting this into (33) yields upper laser multiplet (also believed to be near unity for the an optimum output reflectivity Ropt = 0.94, the same value 4F3/2 level in Nd:YAG), a p = 4.5/cm is the absorption used by Morris and Pollock in their experiments. One can now coefficient of Nd:YAG at the pump wavelength of 808.5 use (17) to calculate z,”Pt = 9.5 which, from (14a), yields nm (which implies an absorption efficiency of about 99% -1n(TOpt) = 0.095 or Topt = 0.909 corresponding to an for the 10 mm rod length), hu, is the energy of a pump optimum absorption slightly greater than 9%. Alternatively, photon, P, is the CW pump power, and A, = ~ w ; / 2 is the one could have used Figs. 1 or 4 to obtain the same value effective cross-sectional area of the Gaussian pump beam. The via a different route. The normalizing energy scale factor in total population inversion within a central area A is obtained Fig. 5 for the Morris and Pollock laser is computed to be 2.86 by substituting (B.9) into (37) and integrating over y and r pJ. From the a = 3 curve in Fig. 5, this must be multiplied yielding by an approximate factor of 7 to obtain a maximum output 1 energy of about 20 pJ in agreement with the maximum energy N A = 2 ~ 7 , dr r dy h ( r ,y) measured by Morris and Pollock with an “approximate 10%” absorber. Similarly, in Fig. 7, the normalization parameter is = Ntot[1 - e x p ( - A / A , ) ] , (B.lO) t,/L = 0.28 nd0.02 = 14 ns. Reading from the a = 3 curve at z = 13.5 yields a multiplicative factor of approximately where 1.4. The resulting pulsewidth prediction of 19.6 ns agrees Ntot = Tp71pfa7app [l - exp(-aZ)] = 2.46 x 1014 (B.ll) with the measured value of 18 ns within our ability to read hvP accurately from the curves. We also note that, at lower pump at the maximum pump power of 650 mW if we assume 77, = powers, the value of z would be decreased proportionately and T, = 1. Thus, the maximum two way average gain obtainable the optimum reflectivity would be greater than 0.94. Thus, with this maximum total population inversion over the CW at the lower pump powers, one would expect to be able to beam area, Acw = 0.154 mm2, would be (20nZ),,~ = 0.21. achieve larger pulse energies at lower repetition rates than Using the latter value in (B.l), it can be further demonstrated those obtained experimentally by Morris and Pollock. that the loss L in the CW laser was less than 1.6% prior We can also ask what this laser could produce at the maxto inserting the saturable absorber, a result compatible with imum pump power with a much better absorber by following our derived intracavity losses for the &-switched system. the same analysis but substituting the a = 00 curves. We However, such a gain would not be capable of supporting find from Figs. 5 and 7 that an ideal absorber (a = m) passive Q-switched mode operation with saturable absorp- would have produced more energy (28 pJ) in a shorter (11 tions approaching 10% as reported by Morris and Pollock. ns) pulse. The corresponding optimum mirror reflectivity and Hence, it is clear that the pump beam was focused into a absorber transmission are estimated from Figs. 2 and 4 to be spot smaller than the TEMoo mode diameter. Substituting 92.7 and 91.7%, respectively. For large a (> IO), one can use

lr

1901


(25)-(29) as an alternative to the design curves to determine [I51 J. J. Zayhowski and C. Dill, 111, “Diode-pumped passively @switched picosecond microchip lasers,” Opfics Leff., vol. 19, pp. 1427-1429, the optimized parameters. 1994. If our goal is a CW-pumped laser operating at maximum [16] J. J. Degnan, “Satellite laser ranging: current status and future prospects,” IEEE Trans. Geosci. Remote Sensing, vol. GE-23, pp. efficiency for a specific repetition rate, then, from (36), the CW 1985. logarithmic gain must be reduced by the factor F ’ ( ~ , T ~ / T ~[I71 ) 398413, W. Koechner, Solid State Laser Engineering. New York: Springerprior to computing z and proceeding with the design curve Verlag, 1976, ch. 8. [I81 A. E. Siegman, Lasers. Mill Valley, CA: Univ. Sci. Books, 1986, ch. analysis. 26.

[ 191 R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume I.

ACKNOWLEDGMENT The author is grateful to J. L. Dallas of the Photonics Branch at the NASA Goddard Space Flight Center for bringing this very interesting problem to his attention and for providing a very useful overview of the past literature on this subject. The comments of two unidentified reviewers also had a beneficial impact on the final manuscript. REFERENCES [ l ] W. G. Wagner and B. A. Lengyel, “Evolution of the giant pulse in a laser,” J. Appl. Phys., vol. 34, pp. 2040-2046, 1963. [2] R. B. Kay and G. S. Waldman, “Complete solutions to the rate equations describing Q-spoiled and FTM laser operation,” J. Appl. Phys., vol. 36, pp. 1319-1323, 1965. [3] J. E. Midwinter, ‘The theory of Q-switching applied to slow switching and pulse shaping for solid state lasers,” Brit. J. Appl. Phys., vol. 16, pp. 1125-1 133, 1965. [4] J. J. Degnan, D. B. Coyle, and R. B. Kay, “Effects of thermalization on Q-switched laser properties,” submitted to IEEE J. Quantum Electronics. [ 5 ] A. Szabo and R. A. Stein, “Theory of laser giant pulsing by a saturable absorber,” J. Appl. Phys., vol. 36, pp. 1562-1566, 1965. [6] L. E. Erickson and A. Szabo, “Effects of saturable absorber lifetime on the performance of giant pulse lasers,” J. Appl. Phys., vol. 37, pp. 4953-4961, 1966. “Behavior of saturable-absorber giant pulse lasers in the limit [7] -, of large absorber cross-section,” J. Appl. Phys., vol. 38, pp. 2540-2542, 1967. [SI J. J. Degnan, “Theory of the optimally coupled &-switched laser,” 1EEE J. Quantum Electron., vol. 25, pp. 214-220, 1989. [9] J. J. Zayhowski and P. L. Kelley, “Optimization of Q-switched lasers,’’ IEEE J. Quantum Electron., vol. 27, pp. 2220-2225, 1991. [lo] X. Zhang, S. Zhao, Q. Wang, Y. Liu, and J. Wang, “Optimization of dye Q-switched lasers,” IEEE J. Quantum Electron., vol. 30, pp. 905-908, 1994. [I I] 0. R. Wood and S. E. Schwartz, “Passive Q-switching of a CO2 laser,” Appl. Phys. Let., vol. 11, pp. 88-89, 1967. [I21 Y. Tsou, E. Gannire, W. Chen, M. Birnbaum, and R. Asthana, “Passive Q-switching of Nd:YAG lasers by use of bulk semiconductors,” Opt. Lett., vol. 18, pp. 15141516, 1993. [13] J. A. Moms and C. R. Pollock, “Passive @switching of a diodepumped Nd:YAG laser with a saturable absorber,” Opt. Left., vol. 15, pp. 4 4 W 2 , 1990. [I41 S. Li, S. Zhou, P. Wang, and Y. C. Chen, “Self Q-switched diode-endpumped Cr,Nd:YAG laser with polarized output,” Opt. Left., vol. 18, pp. 203-204, 1993.

New York: Interscience, 1953, ch. 4. [20] J. J. Degnan and J. R. Dallas, “The Q-switched microlaser: A simple and reliable alternative to modelocking,” Proc. Ninth Int. Workshop on Laser Ranging Insfrum., Canberra, Australia, Nov. 7-1 1, 1994, to be published. 1211 - - W. W. Rigrod. “Saturation effects in high - gain - lasers,” J. Appl. . . Phys., . vol. 36, 2487-2490, 1965. [22] J. J. Degnan, “Physical processes affecting the performance of high power, frequency-doubled, short pulse laser systems: Analysis, simulation, and experiment,” Ph.D. dissertation, Univ. of Maryland, College Park, May 1979, p. 74 (available from University Microfilms, Ann Arbor, MI). [23] W. F. Krupke, M. D. Shinn, J. E. Marion, J. A. Caird, and S. E. Stokowski, “Spectroscopic, optical, and thermomechanical properties of neodymium and chromium doped gadolinium scandium gallium garnet,” J. Opt. Soc. America, vol. B3, pp. 102-113, 1986. [24] H. G. Danielmeyer, M. Blatte, and P. Balmer, “Fluorescence quenching in Nd : YAG,” Appl. Phys., vol. 1, pp. 269-274, 1973. [25] H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt., vol. 5 , pp. 1550-1567, 1966. [26] Optitron Inc., Technical Data Sheet, Torrance, CA.

G.

John Degnan (M’95) received the B.S. degree in physics from Drexel University in 1968 and the M.S. and Ph.D. degrees in physics from the University of Maryland in 1970 and 1979, respectively. He is presently Head of the Space Geodesy and Altimetry Projects Office (Code 920.1) within the Laboratory for Terrestnal Physics at NASA’s Goddard Space Flight Center (GSFC) in Greenbelt, MD. Previous GSFC positions include Deputy Manager of NASA’s Crustal Dynamics Project (901) and Head of the Advanced Electro-optical Instrument Section (723) He has been employed at GSFC since 1964 when, as a cooperative student from Drexel University, he participated in the first laser ranging expenments to the Beacon Explorer B satellite. His research activities have been directed toward the development and use of lasers in ranging, communications, planetary and atmosphenc remote sensing, and medicine He is also an Adjunct Professor of Physics at the Amencan Umversity in Washington, DC where he occasionally teaches a two semester graduate course in quantum electronics Dr. Degnan is a member of the Amencan Physical Society, Optical Society of Amenca, Amencan Geophysical Union, Sigma Pi Sigma National Physics Honor Fraternity, and a Charter Member of the Laser Communications Society. He is the author of approximately 100 joumal articles and technical reports on laser design, laser applications, optical antennas, optical heterodyne detection, spectroscopy, and nonlinear optics and is included in several biographical listings of leading technologists