Antsiferov V.V., Smirnov G.I. Physics of solid-state lasers (ISBN 1898326177) (CISP, 2005)(179s)

Physics of Solid-State Lasers

Fig. 6.1 Diagrams of energy levels and probability of transition between them in two-level (a), three-level (b) and four-level (c) laser systems.

the upper working level m.

3. The 4-level schema (6.1c), is described by the relationships:

N = η B U η B U +γ		γ+	−1	,η γ = γ	γ +		−1	.	(6.38)
0	p p d p p	mg ng i		lmd	lm	lg i

In the initial stage of initiation of lasing when the radiation density U is not high and the level n can be regarded as not populated, the constants γ and B are expressed by the equations:

γ =η B U +γ	+γ		,	B = B ω .	(6.39)
p p	mn	mg		mnb g

In the lasing regime, radiation density increases and together with it

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Principles of lasing of solid-state lasers

the populations n, and in this case

γ = 2 η B U +γ		+γ	,	B = 2B ω .	(6.40)
d p p	mn		mg i	mnb g

The four-level schema is used in the majority of lasers in which the media are activated by the ions of rare-earth elements (in particular, aluminium–yttrium garnet and glasses activated by Nd ions).

6.4 FREE LASING

In the previous section, it was shown that the simplest model of the solidstate laser is defined by the system of equations (6.33), (6.34) for the inversion of the populations N and radiation density U. The system permits the following stationary solutions:

U = 0,	N = N0 ;	(6.41)

	σ	0			γ F		N	I
Nc =			;	Uc =		G	0	−1J.	(6.42)
	β						Nc
					B H			K

The trivial solution of (6.40) corresponds to the case of the maximum attainable excitation of the medium in the absence of lasing. The solution of (6.41) describes the usual stationary lasing in which the gain in the system is equal to the losses.

Analysis of the stability of the stationary state of relatively small perturbations ∆ U, ∆ N is carried out by means of linearisation with respect to these perturbations. In this case, the solution of the linearised system of equations is found in the form

∆ U ~ eλ t , ∆ N ~ eλ t .

(6.43)

The roots of the quadratic characteristic equation in the case of the trivial solution

λ 1 = −γ , λ 2	=µ bBN0	−σ 0 g	(6.44)
show that it	is stable at β N0 < σ 0. In the reversed situation at

β N0 > σ 0, condition (6.40) is unstable and becomes a saddle-type singular point.

The stationary state (6.41) with non-zero intensity exists if N0 > Nc. The characteristic equation for this case

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Physics of Solid-State Lasers

λ 2 +λ		γ N0		+ µσ 0 BUc		= 0		(6.45)
		Nc
and its roots are
λ 1,2	= −		γ N		γ 2 N2	− µσ	0 BUc .
				0 ±	0			(6.46)
			Nc		4 Nc 2

both roots are real and negative and

F γ N G 0 H 2 Nc

I	2	(6.47)
J	≥ µσ 0 BUc .
K

In this case, small deviations from the stationary state are accompanied by a smooth return to this state, and the inversion tracks the field almost completely. The transition process of this type is characteristic of gas lasers and, in some cases, of solid-state lasers.

At a high density of radiation in stationary regime Uc, when the equality

	4µσ	0		N		F		N I
γ <				c		G1	−	c	J ,	(6.48)
	Nc					H		N0	K
is fulfilled, the roots of equation (6.44) are complex-conjugate
λ 1, 2	= −γ~ ±νi					;				(6.49)
~	γ N0
γ =			;							(6.50)

2 Nc

L

ν = Mµσ

MN

	F	γ N0
0 BUc	− G
	H 2 Nc

O1/2

I 2 P

J . (6.51)

K PQ

Here ν is the frequency of damping pulsations of radiation density U and the inversion of populations N. The damping of the pulsations is characterised by the real part of the roots of the characteristic equation ~γ . The solution of the system of equations (6.33), (6.34) in this situation can be written in the following approximate form:

U (t ) = Uc	+ α e−γ t cosν t, α =	const;	(6.52)

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Principles of lasing of solid-state lasers

N t	= U + α	4 Nc e	− γ t cos φ ;
b g	c	µBUc	b g

F µI

φ = arctanG J.

H ~γ K

(6.53)

(6.54)

Equation (6.51) for the frequency of pulsations indicates that they are determined by the dynamic Stark effect with the resonance effect of strong radiation on the working levels of the active medium. In fact, in the limiting case of high radiation density

Uc	>>	bγ N0	/2Nc g2
		µσ		(6.55)
				0 B

the frequency of slightly damping pulsations

ν = µσ 0 BUc	,	(6.56)

is, as indicated by equation (6.31) for the Einstein coefficient Bmn = B/ 2, proportional to the Raby frequency

G =		d		Ec	,	(6.57)
				D

chatacterising the light splitting of the working levels by the resonance field.

The realisation of the condition of existence of pulsations (6.48) depends on the ratio of the constants characterising the main relaxation processes in the system (the losses of radiation in the resonator µ σ 0, the relaxation of the inversion of populations γ ), and also the pumping determining the degree of excitation of the active system N0. In particular, a ruby laser is characterised by the following values of the parameters of the

active medium [3]: γ = γ	mn ~ 300 s–1; B ~ 600 erg–1 cm3s–1; σ 0 ~ 5 ×
109 s–1; µ ~ Nc/N0 ~ 10–1;	µ BUc ~ γ . The value of radiation density in

the stationary regime Uc corresponds in this case to intensity Ic = cUc ~ 104 W/cm2, and the conditions of existence of the pulsations (6.47), determined by the dynamic Stark effect, are fulfilled. The frequency of pulsations and the damping constant of the pulsations in accordance with equations (6.50), (6.51) are evaluated as follows: ν ~ 1.2 × 10 6 s–1;

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Physics of Solid-State Lasers

γ ~ 1500 s–1. Consequently, the period of pulsations is T = 2π /γ ~ 5 × 10 –6 s, and approximately 102 oscillations take place during the characteristic time of damping of the pulsations ~γ −1 . This is in complete agreement with the experimental results.

For a laser on aluminium–yttrium garnet, activated by Nd ions, we

have γ = γ mn + γ mg ~ 4400 s–1 ~ µ BUs; σ 0 ~ 3 × 10 8 s–1; µ ~ Nc/N0 ~ 10–1. In this laser, inequality (6.48) is not fulfilled as efficiently as in

					~
the ruby laser, and the value of the damping constant γ						of the pul-
~	~ 2 × 10	4	s	–1	) if the frequency of pul-
sations rapidly increases ( γ

sations is unchanged. Therefore, in the laser with the active medium of this type, the pulsations regime changes very rapidly to the stationary generation regime. This weakening of the effect of the dynamic Stark effect on the generation process is associated with the screening of the working levels by external electron shells.

6.5 THE GIANT PULSE REGIME

Single powerful lasing pulses can be induced by means of a sharp change of the gain or losses in the laser resonator. For rapid active changes of the losses in the laser resonators, it is recommended to use mechanical or electroand magneto-optical Q-factor modulators with the switching time of the losses from microseconds to fractions of microseconds.

The lasing behaviour of a laser with active disconnection of the losses can be described by the system of equations (6.33) and (6.34). The initial inversion of the populations N (t) at the moment of disconnection of

the initial losses σ~0	at t = 0 is restricted by their value:
Nb0g ≤ σ~0 /β .	(6.58)

After a rapid decrease of the losses to the value σ 0 <σ ~0 which can be carried out using sufficiently powerful and long-term pumping at the moment when the inversion reaches the value σ~0 /β , when the gain becomes equal to the initial losses, the radiation of the inversion in lasing of the giant pulse is almost completely independent of the pumping and spontaneous transitions. Therefore, in equation (6.33) only the term BNU, describing the forced transitions, will remain for the inversion of the populations, and the development of lasing in the giant pulse regime is described by the following system of equations

dN /dt = −BUN,		g	(6.59)
b	0		(6.60)
dU /dt = µ β N −σ	U.

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Principles of lasing of solid-state lasers

This system is reduced to the following differential equation

			µF			σ 0	I
	dU = −			Gβ −			J dN,			(6.61)
			B H			N K
	whose integration at the initial conditions
		t=0 = 0,					Nb0g =	σ~0	= N0	(6.62)
	U			N	t=0 =
								β

results in a relationship linking radiation density Uc of the populations N:

Ubtg =	µβ F		σ 0			N0	I
	G N0	− N −			ln		J
				β
	B H					N K

with the inversion

(6.63)

The maximum of the giant pulse corresponds to the values U = 0, N = σ 0/β , and radiation density is

µβ F

σ

0

I

µσ

0

N0β

Umax =

G N0

−

J

−

ln

.

(6.64)

β

B

σ

B H

K

0

Taking into account the coefficient of inactive losses σ , the power of the radiation passing outside the limits of the resonator is

A = SLUbtgbσ 0	−σ g,	(6.65)

where S is the cross-sectional area of the active medium, l is its length. Substituting (6.64) into (6.65) makes it possible to determine the maximum power of radiation passing outside the limits of the resonator.

The lasing of the giant pulse is accompanied by the release of the energy

		t		Slβ
	W = SLβ	z	N t U t dt =		b	N	− N .	(6.66)
				B
	g		b g b g			0	min g
		0

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Physics of Solid-State Lasers

inside the active medium.

The value of the minimum inversion of populations Nmin remaining after the lasing of a giant single pulse, can be determined from equation (6.61) taking into account the fact that at the minimum N(t) we have dN/dt = 0 and U >> 0 in accordance with (6.56):

N0 − Nmin ≈ σβ 0 ln N0 .

Nmin

The energy of the single pulse, leaving the resonator, Eg mined in this situation by the equation:

F

σ

I

SlB

F

σ

I

Eg

= Wg G1

−

J

=

b N0

− NmingG1

−

J.

σ

B

σ

H

0 K

H

0 K

(6.67)

is deter-

(6.68)

This energy is lower than the total energy stored in the active medium

b0g	= Slβ N0 /B,	(6.69)
Wg

since the fraction W(0)g is used as a result of the inactive losses in the resonator, and the other part, expressed by the equation

b0g	= Slβ Nmin /B,	(6.70)
∆ Wg

remains in the medium because of incomplete scintillation.

The mean duration of the giant single pulse ∆ t is evaluated by the following equation, taking equations (6.62), (6.65) into account:

Eg

N

− N

L

σ

0

F

β N I O−1.

∆ t =

=

0

min

MN0

−

lnG

0

J P

(6.71)

µσ 0

Amax

N

β

H

σ 0

K Q

The estimates for ruby and Nd glass lasers with active media with the length l ~ 10 cm, cross-section S ~ 1 cm2 and the resonators with the length l ~ 102 cm show that the maximum value of the power of

the giant pulse are A		max	~ 108 W, its energy E	g	~ 1 10 J, and the duration
∆ t ~10	102 ns.

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Stochastic and transition processes in solid-state lasers

Chapter 7

Stochastic and transition processes in solid-state lasers

7.1 STATISTICAL MODELLING OF LASING

The lasing properties of solid-state lasers are determined to a large degree by stochastic processes which have been the subject of special attention in recent years [1, 2]. In this chapter, we examine the statistical model of the transition processes in lasing. The initial and non-linear stages of lasing are analysed in detail. A dependence is found between the lasing conditions and the fluctuations of the time of realisation of the transition process. In the approximation of weak saturation, the stochastic behaviour of lasing can be stimulated using the Langevin equation for the total number of lasing photons in the unit volume n, having the form [3,4]:

dn			2	b g
		= α n−γ	n + G + f		(7.1)
				t .

dt

Here G is the mean value of the lasing rate; f(t) is the part of the rate of lasing fluctuating in a random manner. In the stochastic non-linear differential equation (7.1), the function f(t) describes the δ -correlated random process, determining the properties of the latter in a Langevin source of fluctuations [3–5]:

b g	b g b g	b	g	(7.2)
f t = 0,	f t f t′ = Gnδ		t − t′ .

The origin of these fluctuations is determined by the fact that the photons can be excited by non-equilibrium radiation, and also in the absorption of the quanta of vibrations of the lattice (phonons) and of the photons of thermal radiation which are in thermal equilibrium with the active medium. The region of applicability of the balance equation (7.1) for the concentration of the photons is restricted by the limits of taking into

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Physics of Solid-State Lasers

account only the linear and quadratic [with respect to n] terms, corresponding to taking into account the process of amplification of radiation and the saturation effect. Term α n describes the lasing of photons as a result of optical pumping. α denotes the effective coefficient of amplification of radiation. The dependence of coefficient α on radiation frequency ω reflects the specific features of the heterogeneous medium, including solid-state nanostructures with quantum wells. The term –γ n2 determines the first correction non-linear with respect to the amplitude of the light field in the polarisation of the active medium taking saturation into account. The fluctuations of the number of seed photons and different random processes determine, during the multiplication period, the stochastic behaviour of the dynamics of increase of the photon concentration. This behaviour is reflected in the fact that in multiple and rapid lasings (α > 0), in comparison with the time

τ 0 =α −1

(7.3)

the increase of the photon concentration to some level below the asymptotic value

	ns	=	α	(7.4)
			γ

takes place in different periods of time.

We shall note several special features of the increase of lasing intensity resulting directly from (7.1). The point n = 0.5ns, being the inflection point of the curve n = n(t), and the first derivative of the function n(t) at this point is equal to

	dn		n=0.5n	=	α 2	.	(7.5)
	dt
			s		γ

In multiple realisation of the transition process this quantity does not change. Consequently, the slope of the curves of increase of the photons concentration n(t) in the vicinity of the value 0.5 ns should be the same; the fluctuations of the number of seed photons result in the random parallel displacement of the curves n(t) in the vicinity of the level n = 0.5 ns. For the statistical analysis of the transition processes it is convenient [6,7] to transfer from the stochastic differential equation (7.1) to the corresponding Focker–Planck equation for the photon distribu-

146

Stochastic and transition processes in solid-state lasers

tion function W(n, t):

∂ W		∂			b		g		∂ 2	b g	(7.6)
∂ t		∂ n							∂ 2
	+					α	−γ n n+ G	W =		GnW .

This transition and the criteria of applicability of the non-linear FockerPlanck equation have been examined in detail in [3–5]. Equation (7.6) is valid for not too high concentrations n because it was derived using only the first and second moments of this random quantity. It should be noted that equations of the same type were used previously when describing the gas-discharge plasma [8] and quantum fluctuations of lasing [9, 10].

The probability of finding the value of the photon concentration in the range from n to n + dn at the moment of time t is W(n, t)dt. Equation (7.6) describes the evolution of the initial distribution of the ‘seed’ number

of the photons W(n0, 0)		at a sudden variation of the lasing parameter
α	from the value α = –α	1 < 0 (below the threshold α = 0) to the value
α	= α 2 0 (above the threshold).

Function W(n, 0) can be easily found by assuming that the stationary state is established below the threshold. The solution of equation (7.6) in the form

L	−bα	1	+γ n/2gnO
Wbn, 0g = CexpM				P,	(7.7)
N			G1	Q
corresponds to the case ∂ W/∂ t = 0, where G1					is the value of param-

eter G below the lasing threshold. Since the number of photons prior to excitation is small, the role of saturation below the lasing threshold is also negligible. From (7.7) we have the exponential dependence of the normalised distribution function on n:

Wbn, 0g =	1	F	n I			G
		expG −		J;	n1 =	1	.	(7.8)
	n1
		H	n1 K			α 1

After supplying a right-angled excitation pulse, the asymptotic state of stationary lasing with the Gaussian distribution of the photons

147