Dynamics of the Giant Pulse Laser

Lee M. Frantz

The time dependent behavior of the radiation buildup in a giant pulse laser is described analytically,neglecting cavity losses. Expressions are derived for the photon density as a function of time, and forthe delay time and the rise time of the radiation pulse. The curve of radiant energy density vs time isplotted for several representative values of the parameters involved.

Considerable success has been achieved recently inproducing very short, extremely intense pulses of lightby using a type of laser system commonly referred to aseither a giant pulse laser or a Q-spoiled laser.1 2 Thecentral idea of this system is the allowance of a very highinitial population inversion to be attained by makingthe cavity losses excessive during the pumping phase,thereby preventing the laser from going into oscillationprematurely. Once the population is inverted the con-ditions are then suddenly made appropriate for oscilla-tion, i.e., the cavity losses are made small compared tothe amplification by stimulated emission. One methodof accomplishing this is to use an internal shutter, i.e., ashutter placed between the end reflectors. It is keptclosed during pumping, then opened at the desired in-stant to bring the reflectors into play. The result ofstarting in this fashion with a highly pumped medium isthat the available stored energy is discharged ex-plosively in a typical time interval of only severalnanoseconds.

The purpose of this paper is to point out that a quiteaccurate description of the behavior of a giant pulselaser can be obtained in analytical form, if one neglectsthe effects of spectral distribution and directionality ofthe radiation as well as the cavity losses. In particu-lar, the analysis leads to simple expressions for the delaytime and the rise time of the pulse. The expressionsdisplay the dependences on the initial population in-version and on the characteristics of the medium. Theneglect of cavity losses constitutes a valid approxima-tion if the cavity lifetime is considerably greater thanthe time taken to emit the pulse of stored energy, thelatter time being of the order of 10-3 see for a ruby in-ternally shuttered laser. Otherwise this description

The author is with Space Technology Laboratories, Inc.,Redondo Beach, California.

Received 22 March 1963.

represents an upper limit of effective giant pulsegeneration. The model does not apply, of course,beyond the time taken to deposit the radiant energy inthe cavity, since over long enough intervals the cavitylosses become significant. I

Let us denote the densities of active atoms in theground and excited states by N. and N 2 , respectively,and the photon density summed over all oscillatingmodes by n. The time rate of change of the photondensity is given by

dn/dt = (N2 - Ni)nco + NVY, (1)

where c is the velocity of light in the medium, a is theresonance fluorescence cross section, and y is the spon-taneous decay rate from the excited state. The firstterm on the right-hand side of (1) is the difference of thestimulated emission and absorption rates per unitvolume, and the second term is the spontaneous emis-sion rate per unit volume. The nature of the approxi-mations involved in the "single mode" picture of Eq.(1) is discussed in detail in the Appendix. Because ourmodel permits no loss of photons from the cavity, theground- and excited-state populations are related in asimple way to the photon density:

Ni(t) = N,(O) + n(t),

N2(t) = N 2(0) - n(t).

(2a)

(2b)

The initial condition n(O) = 0 is implicit in (2). If wenow denote by A the initial population inversion A=N2(0) - N,(0) and use (2) to eliminate N, and N2 infavor of n, we reduce (1) to

-(1/2co)(dn/dt) = n2 + (-y/2ca - A/2)n - N2(0)'y/2cor. (3)

Although (3) is a nonlinear equation, the variables areseparable, and it can be easily integrated to yield

1 - exp[-(n+ - n_)2ct]n = + 1 - (n+/n-) exp [- (n+ - n_)2cat

(4)

March 1964 / Vol. 3, No. 3 / APPLIED OPTICS 417

where n+ and n. are the roots of the quadratic in (3),

n8 = /2 (A/2 - y/2ca) A 1/2 [(A/2 - y/2co.)2

+ 2N2(0)-y/2c]l/2. (5)

We can simplify (5) by taking account of the fact that,in general, the inequality y/ 2 co < A obtains. Forexample, in ruby A will be of the order of the numberdensity of chromium ions, A - 1018 cm-', while for theother quantities at room temperature we have y = 250sec', o = 2.5 X 10-20 cm 2 , and c = 1.71 X 1010cm/sec, so that y/c- - 6 X 1011 cm-3 . The in-equality holds, then, by many orders of magnitude, sowe can use the approximations,

n+= /2, (6a)

n- = -N 2(0)y/cA, (6b)

n+ - n- = /2, (6c)

and rewrite (4) as

a 1 -e-tr2 1 + Ae-t/T (7a)

r= 1/caA, (7b)

A = A2co-/2N2 (0)y. (7c)

Figure is a plot of Eq. (7) showing the radiationdensity in the cavity as a function of time for four dif-ferent initial inversions. All quantities used are appro-priate to pink ruby, including the density of activeatoms,

No- = N(0) + N2(0) + N3(0) = 1.6 X 1ol cm-3

(i.e., 0.05% concentration of Cr+++), where the quan-tity N3(0) is the initial population density in the R2fluorescent level. The number density of atoms in thisstate is always less than that in the emitting R1 state inthe proportion given by the Boltzmann distribution;the ratio N3(0)/N 2(0) = 8/9 used for Fig. 1 correspondsto a crystal temperature of -50'C. Consistent withexperimental observations, the equations discussedabove have ignored all effects of radiation from the R2level.2 The energy density plotted in Fig. 1 is relatedto the photon density n of Eq. (7) by a factor of theenergy per photon of the RI line, 2.87 X 10-19 J/photon.For pink ruby the characteristic time T given by (7b)is found to be of the order r < 1 nsec.

The curves in Fig. 1 display some of the salient fea-tures of observed outputs from internally shutteredlasers. First, the appearance of the pulse is delayedwith respect to the opening of the shutter. Second,when the buildup of radiant energy is once initiated itis extremely rapid, the available energy being depositedin times of the order of nanoseconds. Third, both thedelay time and the rise time of the pulse decrease whenthe initial population inversion is increased.

Let us define the delay time T to be the intervalfrom the moment the shutter is opened at t = 0 to the

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418 APPLIED OPTICS / Vol. 3, No. 3 / March 1964

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values of the initial population inversion.

time t = TD at which the inflection point of the curveoccurs. From Eq. (7) we find, then,

TD = T In A. (8)

The logarithmic factor is not very sensitive to theinitial populations, and for reasonable inversions inruby it is of the order of 13 to 15, i.e., TD _ 14r. Moreprecise values appear in Table I. It is generally thecase that the quantity A is much greater than unity;in ruby, for example, it is of the order A - 106 (seeTable I). As long as this inequality A >> 1 holds, itfollows from (7) that t = TD is also the time at whichhalf the asymptotic photon density A/2 is reached.

The rise time TR is defined to be the interval from theinstant at which n is equal to 1/10 of its peak valueA/2 to the instant at which it is equal to 9/10 of A/2.Provided the inequality A << 1 holds, this quantity isfound from Eq. (7) to be

1X = 4.39T.

Equation (13) provides a physical interpretation of the3-5 quantity A in Eq. (7), and also leads to the conclusion

S that the peak decay rate of excited atoms in a giante pulse ruby laser is about one million times the spontane-

3.0 ous decay rate. These rates are shown in Table I ins terms of the energy density 8 for several initial popula-

tion inversions.

The author is very much indebted to R. F. Wuerkerfor suggestions and aid in this work.

Appendix

Equation (1) is roughly the superposition of a numberof similar equations for each oscillating mode. Strictlyspeaking, y should be the spontaneous emission rateper mode summed only over oscillating modes, ratherthan the total spontaneous emission rate. How-ever, modes at all frequencies oscillate, even those inthe wings of the spectral profile; only the amplitude ofoscillation varies over the frequency spectrum. Theonly correction to y, then, should account for the factthat photons having certain propagation directions godirectly out the sides of the cavity and, therefore, do notamplify. Consistent with the spirit of this paper, inwhich directionality and all other specifically geo-metrical factors are ignored, this solid angle correc-tion has been neglected in all numerical estimates.

Now let us look into the origin of Eq. (1) in a moremathematical fashion. Let n'(v,o) be the number ofphotons in any mode corresponding to frequency v, andpropagation angle 0 with respect to the cylinder axis,and let g(v) be the Lorentzian shape factor for thenatural line,

(Al)Av/2srg() = - AP/2+ /(V - o)' + (AP/2)2

where Av is the natural line-width, and g(v) is normal-ized so that

(9)

Rise times in pink ruby, then; should ideally be of theorder of several nanoseconds. From (8) and (9) therefollows the general rule of thumb that the delay timeis an almost constant multiple of the rise time,

TD/TR = In A/4.39, (10)

the ratio being about 3 for ruby.The maximum creation rate of photons occurs at the

time t = TD. If we have A >> 1, then,

(dn/dt)m. = (A/2)(1/4r) = Aco/8. (11)

The initial rate is simply the spontaneous emnission rate(dn/dt)o = N 2(0) y. The ratio is then

(dn/dt)msx _ A 2e A

(dn/dt)o 8N2(0)y 4 (12)

fg(P)dv = 1. (A2)

Then the transition probability per unit time forspontaneous emission of a photon into a mode havingfrequency v is c7yg(')/V8rv2 , as can be verified bymultiplying by the number of modes V87rv2dv/c ina frequency interval dv and integrating to get backy, the total spontaneous emission rate. The totalemission rate into the mode in question is then

[n'(v,0) + l1](c-y/V8rv')g(v)N2V;

the absorption rate is

n'( v)(Cy/V8 rv2)g(v)NIV;

and the loss rate out the sides of the cavity and throughthe partially transmitting reflector is, by definition ofthe cavity lifetime T(0),

March 1964 / Vol. 3, No. 3 / APPLIED OPTICS 419

62.5 2

2ZLI

�d2.0 _-

I11I

I

n'(v,0)/T(O).

The equation for n'(v,6) is then

dn'(,O) n'(v,O) ± n'(vo) C3 g(v)N 2 -

dt T(O) 8srz2

+ 0c g(z')N2 . (A3)Ssrv2

Now let n(v,)dQ be the number of photons per unitvolume per unit frequency having propagation direc-tions in the solid angle dQ. It is related to n'(v,6) by

n(v,0)d12 = n'(v,0)(2v 2 dQ/c3 ). (A4)

In terms of n(v,O), Eq. (A3) can be written

dn(v,O) n(v,o) C3_-

dt T() + n(vO) 87rg(v)( -N)

+ -Y2 g(v). (A5)

Now, n of Eq. (1) is related to n (v,6) by

n = Jan(v,O)dvdQ, (A6)

where the integration is over all frequencies, and overthe solid angle Qo spanning all propagation directionsfor which there is amplification, i.e., for which thereare many cavity traversals. Integrating both sides ofEq. (A5) yields

dn , = - dQ fdvn(v, )dt - (0

+ ( - N1) 8- J dQ f dv n(v, O) + yAT2. (A7)Sir V

2 47r

Let us look more closely at the second integral on theright-hand side under two sets of conditions:

(a) When oscillation is well under way, the spectralprofile of the beam is much narrower than the naturalline-width. That is, n(v,0) is much more sharplyspiked as a function of v than is g(v)/v 2 , so we canevaluate g(v)/v2 at v = Po and take it out of the integralto give

de f dv -v n(v,O) f df2dvn(v,O) = vo n. (A8)v2 V02 ~~~~~~~~~~~Vo2

Using the well-knownlines

relation for Lorentzian shaped

* = 4 2Avpo 2/c 2 (A9)

we can now write the second term on right-hand sideof (A7) as

ca(N2 -NI)n.

(b) At t _ 0 there exist only photons resulting fromspontaneous emission, so that the spatial distributionis roughly isotropic, and the spectral distribution isLorentzian with the natural line-width. That is,

n(v,O) = (1/47r)nog(v), (A10)

where no is independent of 0 and v. The angular integra-tion is now trivial, and the frequency integral is (afterwe evaluate 1/v2 at v = vo and take it out of the in-tegral)

fg2(v)dv = l/rAv.

The second term is then

(12)co(N2 - N)no.

However, according to (A6) and (A10) we have

n = (1/4.-)no(v)dQdv = n,

(All)

(A12)

so the second term is

( 12)c(N2 -Nl)n.

Thus, except for a factor of 2 during the earlystages co'(N2 - NI)n is an excellent approximation forthe second term. If we now take T(0) to be independentof direction within the solid angle Qo, i.e., T(O) = T weobtain from (A7)

dn = [ + c(N 2 - N)] n + N2 0 T. (A13)

By ignoring the loss term 1/T and the factor Qo/4 7r, aswe have already discussed, we finally recover Eq. (1).

References1. F. J. McClung and R. W. Hellwarth, Proc. IEEE 51, 46

(1963).2. F. R. Marshall, D. H. Roberts, and R. F. Wuerker, Bull. Am.

Phys. Soc. 7, 445 (1962.3. W. G. Wagner and B. A. Lengyel, J. Appl. Phys. 34, 2040

(1963). In this work cavity losses are retained, and insteadspontaneous emission is ignored as an initiating process topermit an analytical solution. This approximation and thatof the present paper are, in a sense, complementary.

Chemical Lasers Supplement $6.00to be published January 1965

420 APPLIED OPTICS / Vol. 3, No. 3 / March 1964