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

Physics of Solid-State Lasers

pulses with the intensity lower than the threshold of single-pulse failure. In the presence of absorbing inclusions in the medium, this may be caused by the build-up of irreversible changes in the medium. In the absence of inclusions, the build-up effect is also observed but the reasons for this have not as yet been determined [49].

All the previously mentioned mechanisms also operate on the surface of the active medium. However, because of a considerably higher concentration of defects (in comparison with the concentration in the volume of the medium) in the subsurface layer there is a large scatter of the values of the breakdown threshold from specimen to specimen; in most cases, the thresholds of laser failure of the surfaces are lower (by a factor of 2 or more) than the threshold of volume failure [49].

It has been confirmed reliably that when self-focusing of radiation in the medium is avoided and the energy of the light quanta is smaller than the half width of the forbidden band, and there are no spatialtime fluctuations of laser radiation (single-mode and single-frequency radiation), the threshold of natural breakdown is a constant of the medium that does not depend on the radiation parameters and for K8 glass its value is 1013 W/cm2 [49].

5.8 NEW OPTICAL CIRCUITS OF SOLID-STATE LASERS

Recently, new circuits and design of solid-state lasers have been developed ensuring high power and energy of radiation to be obtained with a high degree of spatial and time coherence. In solid-state lasers, a large part of the pumping energy is not converted to lasing radiation and is transferred to thermal energy, including heating of the active element. Thermal energy is removed from the active element from its surface by liquid flows. Consequently, thermal gradients and thermo-optical strains form in the active medium and lead to the distortion of the radiation wavefront, passing through the active medium.

The application of a circuit with the mechanical removal of the heat from the excitation channel makes it possible to eliminate the thermo-optical strains of the active medium [50] and obtain a high mean lasing power (> 1 kW) at the diffraction divergence of radiation (< 1 mrad). The application of thin flat sheets instead of round bars [51] under the condition of the homogeneity of pumping has greatly increased the lasing energy parameters with a significant improvement of the spatial characteristics of radiation.

In Ref. 52, the authors proposed a laser circuit with a plate-shaped active element with the zigzag passage of radiation through the plates. Consequently, it was possible to compensate the thermal optical distortions, induced in the active rod, by pumping radiation. The pumping radiation is induced in a plate-shaped active elements by bifocal lens with

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Increasing the lasing efficiency of solid-state lasers

birefringence [49]. In circuits with the zigzag passage, these distortions are compensated and added up together during the passage of the beam from one plane to another with the opposite sign. Consequently, compensation results in high spatial and angular characteristics of radiation. The output radiation power in optimised, compact circuits is higher than 0.5 kW, with a high beam coherence [53].

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Physics of Solid-State Lasers

Chapter 6

Principles of lasing of solid-state lasers

6.1 QUANTUM KINETIC EQUATION FOR THE DENSITY MATRIX

In a semiclassic approximation in which the quantum fluctuations of the radiation field are ignored, the resonance interaction of the atom (or molecule) with the electromagnetic wave E(r, t) can be described by the Schrödinger equation

D	∂Ψ		= HΨ	(6.1)
	∂
		t

for the wave function Ψ (r, ξ , t), where r is radius-vector of the centre of inertia of the particles; ξ is the population of its internal co-ordinates. The particle energy is determined by the eigenvalues of the Hamiltonian

0	b	r, ξ , t	g	(6.2)
H = H	+ DV

which is represented by the sum of the operator of the energy of the non-perturbed electron shell and the operator of interaction with external

fields D V (r, ξ , t).

In the statistical examination of the effect of the environment on the examined system, it is efficient to transfer from the wave function to the density matrix

ρ = Ψ		b	r′,ξ	g	b	rξ,	g
	*							(6.3)

which makes it possible to describe by a simple procedure the completely and partially defined quantum mechanics state.

The Neuman equation for the density matrix

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								Principles of lasing of solid-state lasers
iD	∂ρ		=		H, ρ		,	(6.4)
	∂
		t

obtained from (6.1) is, as is well known, the most general form of the quantum mechanics description of the evolution of different systems [1, 2]. In accordance with the principles of quantum theory, the calculations of the mean quantum mechanics values of the physical quantities are carried out using the equation

	hdr,	(6.5)
A= Spξ z Aρ crξ ξr

where is the operator corresponding to quantity A.

A

When describing the interaction by internal variables and after averaging with respect to the degrees of freedom of the environment, excluding the field with a resonance interaction with the emission particle E(r, t), equation (6.4) is greatly simplified and assumes the following form

∂ρ jl	= −i		V, ρ			+ R ,	(6.6)
∂ t					jl	jl

where j, l are the indices of the energy levels. The interaction of the electromagnetic wave E (r, t) with the examined particle is described in this case by the operator Vjl(r, t), whereas the operator Rjl takes into account the averaged-out effect of the environment on the particle which is usually assumed to be a stationary random process.

In the model of the relaxation constants, the quantum kinetics equations (6.6) have the following form

F

∂

+ Γ

Iρ

= qδ

− i

Vρ ,

.

∂

H

t

K

G

jl J jl

j jl

jl

(6.7)

Here Γ jl are the elements of the relaxation matrix in the case in which their correlation time is short in comparison with the characteristic time of variation of ρ jl(t). The diagonal elements of the density matrix determine the population of the energy levels of j; non-diagonal elements ρ jl characterise the correlation of the states j and l; the quantities qj take into account the excitation of levels j.

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Physics of Solid-State Lasers

For the simplest two-level model of the active particles, the system of equations for the elements of the density matrix, corresponding to the m–n transition, has the following form

∂

∂ t

∂

∂ t

∂

∂ t

+ Γ

q

+

2 Re(iV

*

),

ρ =

ρ

mn

m mm

m

mn

+ Γ

q−

*

+)

Aρ

,

ρ =

2 Re (iV ρ

mm

n

nn

n

mn

mn

mn

(6.8)

=

iV

ρ(

− ρ

),

+ Γ ρ

nn

mn

mn

mm

here Γ m, Γ n are the total widths of the working levels m, n; Γ is the constant of polarisation relaxation, induced by the radiation resonant in respect of m–n; Amn is the Einstein probability of spontaneous decay. Equations of this type were obtained for the first time by Bloch for describing the precession of the nucleus spin in the magnetic field. From the physics of nuclear magnetic resonance, investigators transferred to the quantum electronics of the concept of transverse and longitudinal relaxation T2 = Γ –1, T1 = Γ –m1 (and Γ m = Γ n), respectively.

In the electric dipole approximation, the operator of interaction with

the electromagnetic field can be described by the equation
V = −	dmnEeiω mnt		(6.9)
mn	D

where dmn is the matrix element of the operator of the electric dipole moment d; ω mn is the Bohr frequency of the m–n transition. The polarisation of the medium P is the total dipole moment of the unit volume, and can be calculated from the following equation:

P = Spbdρ g

(6.10)

where the spur is calculated from the variables of the entire ensemble of the particles in the unit volume. In particular, the following relationship for polarisation corresponds to the two-level approximation:

c mn mn	h	(6.11)
P = 2 Re d ρ	eiω mnt .

The formal solution of the last equation of the system (4.8) has the following

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Principles of lasing of solid-state lasers

form

t
ρ mn = − idDmn −∞z e−Γ bt−t′g−iω mnt′Ebt′gNbt′gdt′.	(6.12)

N = ρ mm −ρ nn

is the difference of the populations of the working levels. Consequently

P = i		dmn		2	zt		e=Γ bt−t′g+iω mn bt−t′g − e−Γ bt−t′ g−ωi mn bt−t′ g		Ebt′gNbt′gdt′	(6.13)
		D
					−∞

After double differentiation of these equations with respect to time, we obtain the following equation for the polarisation vector:

d2 P

dP

2

2

2ω mn *

b

gh

+ 2Γ

c

+ω

mn h

P = −

mnc

mn

(6.14)

dt2

dt

D

+ Γ

d

d

E t

.

This equation, together with the equation for the difference of the populations N(t) and the Maxwell equations for the electromagnetic field E, can be used for describing the laser generation process in a self-consistent manner.

6.2 EQUATIONS FOR THE ELECTROMAGNETIC FIELD

The electromagnetic field in a laser resonator can usually be represented in the form of two running waves:

E (z, t ) =	e−iω t	∑ Es (t )eiskz + k.s.,	(6.15)
2		s±1

where Es(t) are the slowly changing functions of time; the index s indicates the direction of propagation of the running wave with the frequency ω and the wave vector k = ω /c. In a laser with a ring resonator, the lasing conditions form when Es ≠ E–s. For a laser with a Fabry– Perot resonator we have: Es = E–s.

The field E(z, t) is governed by the Maxwell equations which can be written in the following convenient form:

ε	∂ 2 Eβ	− c2	∂ 2 E	α	+ σ		∂ Eβ	+ 4π	∂ 2 P	= 0,
									α
						αβ ∂ t
αβ	∂ t2		∂ z2						∂ t2	(6.16)
α ,β	= x, y, z.

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Physics of Solid-State Lasers

Here εαβ is the tensor of dielectric permittivity; for the medium with no optical activity εαβ = δ αβ . All losses in the resonator are taken into account by introducing the tensor of effective conductivity σ αβ . Consequently, it is the necessary to solve the boundary problem because the description of the losses by ohmic conductivity gives the same results.

The vector of polarisation of the active medium P is expressed by the density matrix using equation (6.7), which makes it possible to represent it in the form identical to (6.15):

Pbz, tg =	e−iω t	∑ Pseiskz.	(6.17)
	2 s=±1

Substituting (6.15) and (6.70) into (6.16), and carrying out averaging with respect to high-frequency oscillations, we obtain an equation for slow amplitudes of the field:

L

∂

i

dσαβ c

2

2

2

i +

σ αβ

O

Mεαβ

+

k

−εβ a

ω

PEsβ

= 2π ωi Psβ .

(6.18)

N

∂ t

2ω

2

Q

Subsequently, we examine the case of linearly polarised radiation and σ αβ = σδ αβ . Consequently, the equations can be greatly simplified.

6.3 MODELLING OF LASER SYSTEMS

The effect of an illuminating filter or some other non-linear medium under the effect of coherent radiation is modelled quite easily in the two-level approximation. The main equations, describing the behaviour of the twolevel system in the field of the running wave, taking into account equation (6.18), have the following form

∂ P

− ω −ω

iP + Γ

P =

d

2

s

iNE

,

∂ t

b

mng s

s

3D

s

∂ N

= γ b N0

− Ng +

1

*

*

cEs

Ps

− Es Ps

h,

∂ t

2D

s	∂ Es	+	1		∂ Es	+
	∂ z		c
					∂ t
γ	= Γ	m =Γ			n N0

σ	ω	ω−	r			π ω2		i
	E i			E		=			P	,
2c		c
	s				s			c s

= qm − qn .

γ

(6.19)

(6.20)

(6.21)

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Principles of lasing of solid-state lasers

Here ω r denotes the eigen frequency of the resonator which can differ both from the frequency of resonance transition ω mn and from the frequency of the electromagnetic field ω ; d is the reduced matrix element of the dipole moment.

The spatial heterogeneity of the inversion of populations might become evident in the field of coherent counter waves. At not too high intensities it is accompanied by the oscillator spatial modulation of the inversion of the populations of the type

N (z, t )N	(z, t ) +	N	2	(z, t )e2ikz + k.s.	,	(6.22)
1

where N1(z, t) is the component of the inversion of the populations, slowly changing along the z axis; N2 (z, t) is the amplitude of the rapidly changing part of the inversion of the populations, reflecting the presence of its spatial modulation. The system of self-consistent equations in this case has the form

∂

t

+ Γ − iω −ω

P= i

d

2

E N

+

E

−s

N

, s= ± 1;

b

mn g

s

3D b

s 1

2 s g

(6.23)

N

=

N s

= +1;

N

=

N* , s = −1;

2 s

2

2 s

2

(∂ + γ ) N = Nγ

+

i

E

*

P− E P

*

;

t

1

0

∑ c s

s

s s h

(6.24)

2D s=±1

b∂ t+ γ gN2 s =

i

cEs* P− s−

E− s Ps* h;

(6.25)

2D

L

σ

− ibω

−ω

O

+ cs∂ zEs = π2

iwPs.

M∂ t+

r gPEs

(6.26)

N

2

Q

The systems of equations (6.90)–(6.21) and (6.23)–(6.26) are analysed taking into account the boundary conditions, determined by laser geometry, and the initial conditions.

We examine a ring-shaped running wave laser on the condition that the rate of polarisation relaxation is high:

∂ t	Ps	<< Γ Ps.	(6.27)

In this case, it may be assumed that the polarisation of the medium tracks the changes of the field:

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Physics of Solid-State Lasers

Ps

=

d

2

iNEs

, Ω = ω −ω

mn.

(6.28)

3D Γ

−Ωi

′

In this case, the ring-shaped running wave laser can be described by the balance equations:

b∂ t+ γ gN= N0γ − BUs

N;

(6.29)

∂ + σ U + sc∂ U= β

NU ;

(6.30)

b t

g s

z s

s

Us

=

Es

2

, B = 2Bmnbω g, β =

BmnDω

,

4π

c

where Us is the density of radiation equal to the amount of energy of the electromagnetic field in the unit volume of matter;

Bmnbω g =	4π d 2 Γ	(6.31)
	3D2 Γ 2 + bω −ω mn g2

is the spectral Einstein coefficient characterising the probability of forced transition; BUs is the quantity proportional to the probability of forced transition; BN describes the resonance amplification (N > 0) or absorption (N < 0) in the medium.

The density of radiation Us in J/cm3 is linked with the intensity of radiation Is (W/cm2) and with the number of photons in the unit volume ns in cm–3 by the following equation:

Us	=	Is	= nsDω .	(6.32)
		c

In a laser with counter waves, if the periodic population grating does not form, it may be assumed that N2 = 0, N = N1, and the lasing behaviour at ω = ω r = ω mn in the balance approximation can be described by the equation:

b∂ t+ γ gN= N0γ − BNU, U= U+1+ U−1 (6.33) and by two equations of type (6.30) at s = ±1. For the processes that

136

Principles of lasing of solid-state lasers

are slow in comparison with the duration of double passage of radiation along the resonator T, these equations transform into a single equation for the total density of radiation U:

t	b	0 g	(6.34)
∂ U= µ β N− σ		U.

Here σ 0 is the coefficient of laser radiation losses; µ is the degree of filling of the resonator by the active medium. The system of equations (6.33), (6.34) corresponds to the simplest model of the solid-state laser and makes it possible to examine important relationships governing generation.

The systems of the energy levels of the active media enable the inversion of populations to be developed between two levels in the generation channel using external energy sources. For controlling lasing, in addition to the active medium, it is necessary to introduce laser media with the non-linear dependence of the difference of the populations of two levels of the intensity of laser radiation. Figure 6.1 shows the schema of the energy levels in cases of laser media that are of greatest interest for practice.

For the displayed two-, threeand four-level systems with the working m–n transition on the condition that polarisation tracks the resonance field of radiation, the behaviour of laser lasing is described by the balance equations (6.33), (6.34). However, in these schemes, the relaxation constant of the inversion of populations γ , the initial value of the inversion N0 and coefficient B show different dependences of the probability of spontaneous decay Ajl, the probabilities of non-optical transition ν jl and the probability of forced transitions.

1. For the two-level system (Fig. 6.1a), we have

	γ = A +ν , N = −q , B = 2B ω ,			(6.35)
	mn mn 0	n	mnb g

where the coefficient is described by equation (6.31).

2. A ruby laser operates on the basis of the three-level schema (6.1b). Taking into account the probability of forced absorption BpUp in the pumping channel n → l, the parameters of equation (6.33) have the following form:

γ =η

B U

p

+γ

mn

,η γ=

γ

γ +

lmg

−γ1

,

α = ν

ji

+

ji

;

(6.36)

p

lmb

ln

ji

0

− q

nd

η

BU

p

−γ

mn id

p

p

γ+

mn i

−1

(6.37)

.

N

η

B U

Quantity η denotes the fraction of particles falling from the level l on

137