.

ORAZIO SVELTO

PRINCIPLES of LASERS

F o u R T h E d i i N

TRANSIATJON FROM ENGHSIH b y D . N . K O S L O V , S . B . S O Z I N O V A N C I K . G . A D A M O V I C H ;

EdiTEdbyT. A . S H M A O N O V

SAINT-PETERSBURG M O S C O W KRASNODAR 2008

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2008

32.86-5 3 43

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3 43 / . . . . . . 4-. .: , 2008. 720 : . ( . ). ISBN 978-5-8114-0844-3

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32.86-5

06-02-30013

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, 2008 ,

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, 2008 Translation from the Engl ish language edition:

Principles of Lasers, 4 t K e d . B y Orazio Svelto Copyright 1998 P lenum Publ ishing Corporation, being a part of Springer Science + Business Media

A l l Rights Reserved

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(dN2/dt)sp N2. , :

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1. 13

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( ), ( ). t8p = 1/ , -

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dN2\ N2

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^f-\t=-^N2, (1.1.4)

(dN2/dt)st 2 - 1 , W21 , ( ). , (1.1.2), W21 () - 1 . , , W21 ,

. , :

W21 = G21F, (1.1.5) F , a G21 , ( , , , ,

) .

(1.1.4) ,

( ) W 1 2 ,

dt = - W l 2 # i , (1.1.6)

(dN1/dt)a 1 2 , a Nt 1. W12 , (1.1.5):

14 . .

WL2 = G12F, (1.1.7)

a 1 2 ( ), .

,

21 1 2 . X X ,

W21 = W12 , , a 2 1 = a i 2 * 1 2 gx- 2- ,

S 2 ^ 2 1 = l ^ l 2 > (1.1.8) ,

^ 2 1 = ^ 1 2 - (1.1.9) , ,

(. . 1.1): () 2 1, ; ()

2 -> 1, ; () , 1 - 2. , ,

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1.2.

1 2 Nx N2 . z F (. 1.2), dF dz ( . 1.2)

, . S .

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, SdF.

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, SdF

F

F+dF

w *

dz

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dF

dz

1. 15

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. (1.1.4) (1.1.6), : SdF = (WJ1N2 -- Wi2Nx)(Sdz)9 Sdz . , -

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(. . gx = g2). (1.1.5) (1.1.7)

dF = oF[N2 - NJdz. (1.2.1) 21 = 12 = ,

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(1.2.1) , N2 > Nt (. . dF/dz > 0) , N2

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(, ). (1.2.1) (. . ) exp{a|W 2 ~~ I . (. 1.3) R1 2 , Lt. F ,

1 2, F\ 1 ,

F' = Fexv{o(N2 - Nx)l) (1 - Lt)R2 exp{a(iV 2 - )/} (1 - ^. F = F' , , RtR2(l - 1^)2{2(2 - N^1} = 1. , , N = (N2 - Nx) , , ,

,

Nc = - [ InR X R 2 + 21n(l - )]/2. (1.2.3) (1.2.3) ,

= -In = -1(1 -

), (1.2.4) 2 = -1 2 = ( 1 - 2 ) , (1.2.46)

Y - - l n ( l - L ; ) , (1.2.4) 12 ( ). (1.2.4) (1.2.3)

Nc = y/al, (1.2.5)

1. 17

Y = [2Y; + ( Y i + y 2 ) ] / 2 . .2.6)

, yi9 (1.2.4), . , Lt 1, .

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1. 19

N2 = N1 = Nt/2. , ,

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(dN2/dt)p : (dN2/dt)p = Rp, (1.3.1)

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= 1. , , , = 1,22 , = 2/

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(1.4.1), = 1, - .

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1. 25

. 1.8

-

Dc L

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Dc , . Dc = 100 X = 1 . (1.4.1) Qd = - 2 , , , L = 100 , , D = Dc + 2QdL = 2QdL = 2 . ,

, D, , Dc.

I(P)a[(E1(t) + E2(t)]2, Ex(t) ,

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) = C[1 0sin(co* + ) + 2 0sin(co* + 2 )] 2 , , 10 20

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= (0 2 = 2() (., , . 1.6). , , () ) , ) . ,

< )> = [ 2 0 (sin 2 (* + )> + | 0 (sin 2 (* + 2 )> + +21020 (sin(co* + )sin(co* + 2))] = = [(?0 / 2 ) + ( | 0 / 2 ) + 1020((2 - ) ) ] .

, , ^*) 2(*),

(*) = \\fi(t) + kL1 2(*) = (0 + kL2, \ 2 , a L x L2 -

26 . .

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v|/ 1(f) \\f2(t) . , , (0 ~ 1(0> , , (cos (2 - )), , . :

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6 ' . , cos0 - , ' dS , ', . . cos0

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( ).

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= 4/(2)0) 2. (1.4.4)

(1.4.5)

28 - 0 -

. 1.10 ( )

9. ()

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( )

(1.4.4), 1

= (/4)(>//) 2 . , 1

D. 1

, D DL. :

Ip = n(NA)2B, (1.4.6)

NA = sin[tan~ 1()L/2/)] = (DL/2f) . (1.4.6) , .

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1. 29

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1.2. 1.1 , , .

1.3. 1 2 . 1.1 (2 - ), .

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1.4. = 300 N2/Nx 1/. v .

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1.5. ^ = ^ ^ 2 = 0,5, Lx = 1%. . I = 7,5 , = 2,8 10~1 9 2 , .

1.6. (X = 694 ) 1 . , ,

. ( 384 .)

1.7. , -, ( Labs, 107/109, 100 ) (X = 546 ) 95 / 2 . 1 (X = 514,5 ), -.

1. 31

2

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E(t) H(t) , :

| 2 ^ | 2 } , (2.2.1)

, ( ) .

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p v , v. p v : pvdv v v + dv. , p v :

P = / M v - (2.2.2)

Iv ,

, . , Jv p v, :

Iv = (c/4n)Pv, (2.2.3) , & . , Iv , p v, ,

, , v

. p v .

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2. 33

. , v

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v. p v > ", , (2.2.3), II > / , . . 1 2. ,

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p v = p v . p v(v, )

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2.2.1.

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V * E - i . 0 . O , (2.2.4)

V 2 , .

:

= 0, (2.2.5)

. 2.1

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34 . .

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= (9 , z)E(t) (2.2.6) (2.2.6) (2.2.4),

V 2 u=J _ d?E

u 2 ' dt2 '

9 , z9 t,

,

, -k2. (9 9 z) E(t):

V 2 u = -fc2u, (2.2.7)

^ - = -(cnk)2E. (2.2.76) (2.2.76) :

= 0cos (* + ), (2.2.8) 0

co = cnk. (2.2.9) E(t) (2.2.8), (2.2.6) :

(9 9 z91) = 0(9 9 )/(* + ). (2.2.9) , (9 9 z)

(9 9 z)9 .

.

(2.2.7), , ,

(2.2.5). , :

= excoskxx suikyy sinkzz9 (2.2.10)

= eysmkxx cos kyy sinkzz9 (2.2.106) u2 = ezsinkxx sinkyy coskzz (2.2. 10B)

(2.2.7a) 9 9 29

:

k2+k2+k2=k2. (2.2.11) ,

(2.2.5). ,

2. 35

(2.2.10) , , = 0, = 0, z = 0. , , , z = 0, (2.2.5)

=

= 0, (2.2.10) (2.2.106) , , 2 = 0. (2.2.5) , :

19 . , , , z = L, (2.2.5)

=

= 09 (2.2.10) (2.2.106) , , 2 = 1 (2.2.12). Z, , ,

9 .

, 19 . , kx9 ky kz (2.2.12), , (2.2.9) (2.2.11), :

,

I, . , ,

9 2 .

,

, V = 0, , (2.2.10), exkx + eyky + e2k2 = 0. :

= 0, (2.2.14) , 9 z

9 2 kx9 ky kz . (2.2.14), ,

9 2

. , Z, ( ) , , (2.2.14), , .

. , \|/ ,

, , ,

, . . = ^; +

\|/. , Z, ,

.

iV(v), 0 v. , -

kx = ln/2a9 ky = /29 k2 = nn/L9

(2.2.12) (2.2.126) (2.2.12b)

(2.2.13)

36 . .

2n/L_

I > . . 1 . 1

k k* 0 2nv/cn. (2.2.12) , kx9 ky kz ,

,

. 2.2. kx9 ky kz ,

,

. ,

,

(/2, /2, /L). , k9 0 (2nv/cn)9 1/8 (2nv/cn)9 , (/2, /2, n/L). , , k , :

14 ( 2 \ 8 83 [

) grcv3-

. 2.2

,

. 2 .1.

.

iV(v) = 2

7 1

2 2 L

-V, (2.2.15)

V .

2.2.2. -

p v, pv ( 9 , ). pv = dN(v)/Vdv, (2.2.15) :

Pv 1 dN V dv

8 2 (2.2.16)

pv, p v

(), , . .:

Pv=pv{E). (2.2.17) () , . dp

2. 37

, + dE, dp = C exp [-(E/kT)]dE, ,

00 $Cexv[-(E/kT)]dE = l.

, () :

)Eexp[-(E/kT)]dE () = * = kT 4 1

00 jexv[-(E/kT)]dE

(2.2.16) (2.2.18) :

(2.2.18)

Pv = J

. (2.2.19)

-

-.

. , , (2.2.19) ,

(. (2.2.2)). , , , (2.2.3), Iv v 2 ,

, v > ( ). ,

.

,

X X . ,

hv, v , a h , . ,

, , .

,

v 0 , (2.2.18), :

= nhv, (2.2.20) . , ,

,

nhvexp[-(nhv/kT)] ^ exp(hv/kT)-l' ^ exn(hv/kT\-l' (2.2.21)

exp[-(rc/iv/A!T)] =0

38 . .

[ 1 6 / 3 ]

at

. 2.3 p v (v , )

v

[ 1 0 1 4 ]

(2.2.18). , hv < kT (2.2.21) (2.2.18). (2.2.16), (2.2.17) (2.2.21) :

_ 8 2 hv p v

3 e x p ( * v / * T ) - l ' (2.2.22)

,

, h 6,62 10" 3 4 . . 2,3 p v v .

,

= ^ 1 = * (2.2.23) w

hv exp(hv/kT)-l () . v = 4 10 1 4 , , hv 1 . 300 kT = (1/40) , (2.2.23) , () = ( - 40). , ,

.

0,

( 10 1 0 , . 7.1 . 7).

2.2.3.

,

, (2.2.20), , , . -

(2.2.18) (2.2.21), ,

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(1904) , , ,

2. 39

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(1927). , , [2] . ,

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, ,

, v

. Ex(v, t) H^r, t) ,

(2.2.1),

V . , , ^ ( r , t) Hy(r, t) [2] . , Ex(r, t) y(r, t), , qx , , , .

,

qx . , , ,

qx9 . Ex(r, t) Hy(r9 i) ,

,

(2.2.4).

, ,

.

,

. ,

, , ,

, :

k , am . . ,

,

.

- , , kpx/29 ,

qx /2. , -

(2.2.24)

E = (kpl/2)Hql/2m), (2.2.25)

, ,

40 . .

,

\{\x{H2)/2)dV. , .

, ,

,

. ,

:

E = {l/2)hv + nh\, (2.2.26) .

,

. ,

, , (2.2.25),

qx, , , . ,

, , , (2.2.1),

.

. ,

,

( v) (2.2.26), (. (2.2.20)), , . ,

. ,

(. 2.2.1) ,

. ,

0 , . ,

(2.2.26) - , ,

, . 2.4. , , (2)9 (2) . .

, (hv/2) . (2.2.24)

E = (\pdv)-(hv/2)9 (2.2.27) , .

, -

(2)(2) , . ,

.

h\

hv

i hv/2 . 2.4

2. 41

2.3.

,

(. . ), (. . ). , (. . ) . .

,

, ,

, ,

. ,

( ) .

2 3 1

, ,

2 2, , , 1

(. . 1.1). , , :

(, t) = ^ / (2.3.1)

vi/ 2(r, t) = u2(r)exv[-j(E2/h)t], (2.3.16) u12(r) 1 2, - , ( ), h = h/2n. 2 - 1

:

= a^OVi + a2(t)^2, (2.3.2)

2 , . ,

\

\2 |2|2 , t 1 2. , - , :

k l 2 + k l 2 = i , ( 2 . . )

\

\2 |2|2. , ,

JLI. :

\x = -\e\y\2rdV, (2.3.4)

42 . .

; . (2.3.4) , , e|y| 2dV ,

dV , d\x = -(e\\\f |2 dV)r. (2.3.2) (2.3.4), (2.3.1) :

\ = [ |

|2| |2 dV + [ \2\2\2 |2 dV +

J

(2.3.5) + \ ^^^ exp j((o0t) + a[a2u{u2 exp[- j((o0t)] jlV,

* , 0 = (2 - Ex)/h. (2.3.5) , jli \iosc, 0, :

Pose = [2

^21 (0*)], (2.3.6) Re . 21

21 = \u2eruxdV. (2.3.7)

21 - , .

(2.3.6) , 2 1 08, 0,

^!, (2.3.7). , .

,

.

:

*W = cos(co0* + ) = [ exp (yo 0 OL

0 ,

, Re , ' , \'0 = 0('). 1

,

:

fi= 3 , (2.3.8) 12rce0cd

1 , , ,

,

, .

= [*] 1 / 2 , * , (. . *, * * , ).

2. 43

=| |=| 0 | , ,

. (2.3.8) , [i = 2\a1a2^i2i\y . . 2a1a2\i2i. , , :

=;\1\2\2\2, (2.3.9)

/ ,

1 6 3 2 (2.3.10)

~ 03 '

|jn| = |21| j i 2 1 . , ,

, ,

, ,

:

Tjf = (2.3.11)

= \1\21 + \2\22. (2.3.12) (2.3.3), (2.3.12) :

=

+ hv0\a2\2, (2.3.13) v 0 = (2 - Ex)/h .

(2.3.9), (2.3.10) (2.3.13) (2.3.11) :

d\a2 \ I % 121 "2 \2=~(1-1 2 \2) | 2 | 2 , (2.3.14)

xsp = hv0 / ' ,

3 / i sqC 3 (2.3.15) s p 16* 3v 3rc|uf '

( ) 2 ,

.

1 (2.3.15) :

1 - tanh

(2.3.16)

t0 , | 2 (0) | 2 . , (2.3.16)

1

. . .

44 . .

| 2 (0) | ; 1-tanh 2 (2.3.17) sp J _

| 2(0)| 2 ( , 1) t0. . 2.5 | 2()| 2 | 2 (0) | 2 = 0,96. , | 2 (0) | 2 t0 (2.3.16), . . . , , | 2 (0) | 2 = 0,8, | 2()| 2 -

. 2.5

| 2 () | 2

= xspPr/hv0.

;

-

. 2.5 , t = 0 0,8. | 2()| 2 (2.3.16). | 2()| 2, , (2.3.11) (2.3.13),

Pr = - hv0d\a2\2/dt. , = TspPr/hv0, . 2.5. , | 2()| 2 :

\a2(t)\2 = |a 2 (0) | 2 exp[-(f/x s ; , )] (2.3.18) |a 2 (0) | 2

,

( ) = 1 / :

1 6 3 ^ | 1 2 (2.39) 3he0c3

, Nt , ,

: N2(t) = Nt\a2(t)\2 N2(0) = AT f|a2(0)|2. (2.3.18) Nt9 : N2(t) = 2(0) (-t/i8p) (1.1.2), , .

2.3.2. -

-

,

, ,

.

, , |a 2 (0) | 2 = 1. (2.3.17) , t0 = . ,

. , |a 2 (0) | 2 = 1, | (0)| 2 = 0, (2.3.14) , d\a2\2/dt = 0. -, ,

(0) = 0 \xosc9 (2.3.6), . ,

, , .

. , ,

, \2\ 1 t = 0. ,

, |

|2, 1. (2.3.6) , , 0. (. . ) , 1. , \2\2 , .

, .

,

, :

1. \2\2 (2.3.16) , (. . \2\2 < 1) (. (2.3.18)).

2. (. . |a 2 (0) | 2 = 1), (), .

46 . .

,

[5, 6]: 1. ,

\2\2 ( --) . , (2.3.18) | 2 (0) | 2 .

2. 8 (2.3.15).

3.

=

- hv0d\a2\2/dt, 8.

, , --

-

(. . 2.5). 1 ,

.

,

-

.

.

(2.3.2), , .

,

. , ,

, ,

- , . ,

,

, .

, ,

. ,

. , ,

, .

= . , ,

,

- . , ,

1

, .

v 0 , .

,

, ( ) .

2. 47

2.2.3, (2) (2) ^ nj/ = ( ). ,

, , , -

. , ,

.

2.3.3.

(2.3.19) , , 0, , |] 0. , , ,

. ,

|| = 0, = 0, . , -

,

( ). .

, . . || = 0. |uj = |21|, (2.3.7) , , 2

, (. . ). 1 , (2.3.7) - , 2()() u2(-r)(-eY)ui(-r)dV. , ((-) = = ()) ((-) = -()), .

- , (2.3.7) , = 0. , (

, , 2 ), (2.3.7) - . , , || 0, .

, , -

,

( ) , ( , ). 2

1 , /() ( ), /(-) = ),

( ), /(-) = -/(). 2 ,

(. . , ) . , , , , .

48 . .

2.1.

.

,

, (2.3.19) X = c/v = 500 = , ( = 0,1 ). , = 108 "1 (. . xsp = 10 ). 105 , xsp = 1 . , (2.3.19) ,

.

,

,

.

, (, X ^ 5 ), ^ (10-100 ),

.

2.4.

, , ,

. ,

W12 W21 (. (1.1.4) (1.1.6)). , , , ,

. ,

, ,

, -

.

,

,

.

,

, .

2.4.1.

, t ^ 0 ,

(2.3.2) \

(0)\2=1\2(0)\2 = 0.

2. 49

/ '.

'

E(t, t) ( ), - ( ) . ^

:

(0, t) = E 0 s in (G )0 , (2.4.1) . ,

,

.

(2.4.1) ( ). ,

0.

, = -,

. ',

, :

' = = - E0sin cot. (2.4.2)

H'(t), . s 0,

. , t > 0 |aj(^)| 2 la^O)! 2 = 1, | 2(0| 2 . a2(t), , ,

,

,

t = 0.

| 2(0| 2> ^ , : \a2(t)\2 = ^ \ ^ 2 1 1 2 E2d(v-v0)t, (2.4.3)

v = /2, v 0 = 0/2, 5 - , 0 0 , |21| 21, (2.3.7). (2.4.3) , t > 0 |a 2 ( t) | 2 . ,

,

( ) W$ Wtf =d\a2\2 /dt, (2.4.4)

50 . .

(2.4.3):

H ? 2 a = ^ b 2 i l 2 - E o 2 8 ( v - v 0 ) . (2.4.5)

, , (2.4.4), ,

; sa ( . single atom ), W12.

, , t > 0 (2.3.2). , t > 0 08, (2.3.6). , ax(t) a2(t) , \08 . , , . .

(0) = 1 2(0) = 0, ,

. , ,

[3] . (2.4.5)

.

= 20*/2, (2.4.6)

, 0

,

w s = | 2 p 8 ( v-

V o )-

( 2-

4-

7 )

, W(2a - . , W = v v 0 W12 = v = v 0 , . . .

,

. ,

, v

( ) , v 0 , , . . ,

, v = v 0 .

( ), , ,

. ,

, ,

: (2.4.7) , 5-

2. 51

, v = v 0 # , . . , /

J8(v-v 0 )dv = l, g(y - v 0 ) , v v 0 , , . . ,

] * ( v - v 0 ) d v = l, :

_ 2 1 7iAv0 l + [ 2 ( v - v 0 ) / A v 0 ] 2 ' ( 2 ' 8 )

Av 0 - . , Wf2a :

Wsa - - 12 ~ 2

2

1^21 I 2 P g ( v - V o ) . (2.4.9) 3n 2s 0ft 2 [g(v - v 0 ) A v 0 ]

(v - v 0 ) / ( A v 0 / 2 ) . 2.6. (2.4.8) , , (FWHM . Full Width at Half Maximum ), Av 0 . g(v - v 0 ) v = v 0 ,

g(0) = 2/rcAv0 = 0,637/Av 0. (2.4.96) , (2.4.8), . . ,

(2.4.8) [3] .

W*g / .

/ = /, (2.4.10) A t e ( v - v 0 ) A v 0 ]

, (2.4.9) ,

H 2 i l 2 - t e ( v - v 0 ) . (2.4.11)

. 2.6

_ _ 2 2 _ _ 1 2 3ne0ch2

.

,

( (2.3.2) (2.3.1)) ' ( (2.4.2)) .

,

| 2()| 2

52 . .

|a i()l 2 (. ) . ,

| 2 (0)| 2 = 1 , , | 1 (0) | 2 = 0. ,

1 2. , ,

( ) Wfg (2.4.5) . (2.3.7) , | 1 2 = 2 1 |12| = |2|. ,

W^=W2f, (2.4.12) ,

(. (1.1.8)). , (2.4.9)

(2.4.11), :1

Wsa = - ^ \ ^ \ 2 p * ( v - v 0 ) , (2.4.13)

W" = - ^

2 ^ ( v - v 0 ) , (2.4.136)

= |12| = | | i 2 i l (2.4.12), W s a = W s a =W.

2.4.2.

(2.4.13) (2.3.19) , Wsa ||2. , , . ,

( )

2,

1 : 3 ( 2 . 4 . 3 ) , ( 2 . 4 . 5 ) , ( 2 . 4 . 7 ) , ( 2 . 4 . 9 ) ,

(2 .4 .11) , (2 .4 .13) (2 .4 .136) , ( ) . :

W(\. -So I2) =1 M I2 Eg

.

. , , W8a = 0, .

, , 1 2 .

, ,

.

,

,

, (2.4.11). ,

(- - ) . , ,

,

, .

We Wm. , :

,

. ,

.

, (2.4.5), We (^ 0) 2 (0)2, 0 ,

\ie .

, Wm ( jn m B 0 ) 2 ( 0 ) 2 , 0 , \

(3 ((3 == 9,27 10" 2 4 2 ) . , :

(We/Wm) = (0/$0)2 = (/$)2 = 10 5. (2.4.14) (2.4.14) , 0/0 = ( ), , = 0,05 . ,

.

2.4.3.

2.4.1

,

. Nt ,

.

54 . .

, v 0 ,

( ). Wh , :

Wh(v - v 0 ) = W*4v - v 0 ) . (2.4.15) , ,

, dN2/dt, :

(dN2/dt) = WhNt. (2.4.16) Wh , . . F = I/hv, oh:

Wh/F. (2.4.17) (2.4.13) (2.4.17) , ah

22 H 2 v ( v - v 0 ) .

(2.4.19)

3ne0ch^] ' * v ' * ' - (2.4.18) , ,

. 1.2, (2.4.16) (2.4.17), , 2 , (. ( 1

'2 ) )

' dF = -ahNtFdz. (2.4.19)

. , ,

, ,

(. 2.7). S ,

dz (. . 1.2) NtSdz, , aaNtSdz. (dF/F) dz , ,

. 2.7

S

(dF/F) = -(oaNtSdz/S). (2.4.20) (2.4.20) (2.4.19) ,

=

,

,

,

.

,

Vo v 0 ( ).

2. 55

g*(v'0 - v 0 ) , , dNt = Ntg*(v'0 - v0)dv'0 , VQ H V O +dvo. (2.4.16) d(dN2/dt)9 dNt9

d(dN2 /dt) = WhdNt = Wh (v - )Ntg * (v'0 - v 0 Wo> Wh(v-v'0) V Q .

(dN2/dt) = Nt \wh(v - )g *(v[> - v 0 Wo . (2.4.21) (2.4.21) (2.4.16) , Win9

= lwh(v-v'0)g*(vo-v0Wo. (2.4.22) (2.4.17) GIN GIN = Win/F. (2.4.22) F (2.4.17), :

(2.4.18) (2.4.23) :

2 2

2 v f t ( v - v o ) . 3ns0ch)r~> o t v w / ' (2.4.25)

(2.4.25) gt(v - v 0 ) ,

gt= lg*(x)g[(v-v0)-x]dx, (2.4.26)

^ = ( V Q - V 0 ) . 2 l n 2 (2.4.27)

A [ * ( v - v 0 ) A v * ]

[ ^ * ( v ~ v o ) ^ v o ] ( V - V 0 ) / ( A V Q / 2 ) . 2.8. , FWHM, (2.4.27) A V Q , v = v 0

,

(2.4.27), .

,

= GIN.

22 Sne0ch

|| 2 V ^ ( V - V Q ) . (2.4.29)

[^/2~

. 2.8

2. 57

W = GF :

W = 3 ^ ^ | 2 p ^ ( V - V o ) ' (2-4.30)

= (nil) = (nFhv/c) .

. (2.4.12) ,

(2.4.29) (2.4.30).

, (2.4.29) (||2, gt v 0 ) v . , (v - v 0 ) ,

.

. , ,

Nx iV 2, (2.4.19) :

dF = -G(N1 - N2)Fdz. (2.4.31) 1 (1.2.1), gi = g2. () .

,

OL = G(N1-N2). (2.4.32) Nx > iV 2 , , . (2.4.29), :

2 a =

3 ^ h ( N l - N M 2 v g ' ( v - V o ) - (2.4.33)

, ,

, , , .

,

. , (2.4.31) (2.4.32) :

dF = -aFdz. (2.4.34) , ,

I, [F(l)/F(0)] = exp(-aZ). , -

58 . .

.

, Nx N2, (2.4.32). , Nx N2 (1.2.2), NT = NX + N2H .

. , ,

1 . , , , 1 , , kT.

, N2> Nl9 , (2.4.32), , , ,

, .

gy :

; .

( ), [9] . , ,

,

.

.

, ,

.

, , p v

(2.2.22). , , , .

,

1 2 2 1.

21 12 ( ) , N{ N2 1 2. , :

g = a(N2-N1), (2.4.35)

2.4.4.

W21=B21p, .2 =

(2.4.36) (2.4.37)

AN* +B2lPvo = ^^ N{. (2.4.38)

2. 59

:

/ = exp(-/*v 0/kT). (2.4.39) (2.4.38) (2.4.39) ,

=

P v o Bi2exp(hv0/kT)-B21 * (2.4.40)

(2.4.40) (2.2.22), v = v 0 , :

1 2 = 2 1 = , (2.4.41) A Snhvln3 (2 4 42) B e 3 '

(2.4.41) , .

, (2.4.41) (2.4.12), . ,

(2.4.42) , .

(2.4.30), , .

pv

2.5.

, ,

.

. ,

, , -

.

. ,

,

.

(. . ) , .

, ,

gt(v - v 0 ) : 1. ,

.

v, .

(2.4.33) , vgt(v - v 0 ) . gt(v - v 0 ) , v 0 , , v0gt(v - v 0 ) . , gt(v - v 0 ) v .

2. ,

gt(v - v 0 ) , . , ,

- , .

,

, .

2.5.1.

.

, ,

. ., .

.

\\fx \|/2 (. (2.3.1)) . ,

\iosc (. (2.3.6)) . ,

2. 61

. 2.9

E(t), ,

.

(

10 7 . )

.

, ,

,

, \08. ,

, ,

. 2.9, . ,

. ,

v' v' 4- dV dp = pvdV, , , . . (2.4.7), :

92

d W l 2 = S n 2 o h 2 1^21 Pv 'S(v ' -V 0 )dv ' . (2.5.1)

(2.5.1) , :

Wl2=sXlh2^2112 j p v ' 8 ( v , - v o ) r f v ' . (2.5.2) pv :

p v >=ptf(v ' -v) f (2.5.3) (. (2.4.6)), g(v' - v) pv. , , = jp v dv' , (2.5.3) , g(V - v)

+ 0 0

f t f ( v ' - v ) d v ' = l . J (2.5.4)

-00 (2.5.3) (2.5.2) 8-, :

62 . .

2.4.1, , W12 5(v - v 0 ) (2.4.7) g(v - v 0 ) . , (2.5.4)

+00

J t f ( v - v o ) d V = l . (2.5.6) -00

g(v' - v ) .

(. 2 . 9 ) , , , . ,

= [(-/

) ] /

. (2.5.7) pxdx , + dx. ,

(2.5.7) , (). , ,

= o f c * = V (2.5.8)

, ,

.

, . 2 .9 ,

/>

, (2 .5 .7) . , , ,

.

^ ' - ) = 2 [ 1 + 4 1 ( , _ ) 2 ] . ( 2 . 5 . 9 )

(2.5.5) (2.5.9) v' v 0 . :

g ( v - v 0 ) = 2 T c 1 , (2.5.10)

. ,

(. . 2 . 6 ) , (2.4.8) , 2

,

Av 0 ,

0 = 1 / . (2.5.11)

2. 63

2.2. He-Ne .

.

= l/uth, I , vth , uth = (SkT/M)1/2, , / ,

, . Ne = 0,5 ( He-Ne ), (2.5.12), ^ 0,1

= 0,5, (2.5.11), Av 0 = 0,64 . , , , , Av 0 . ,

, (Av0/p) 1 / , Ne. , ,

, = V T c . , , v = 5 10 1 4 , 5 108. , . 2.9 ,

, .

2.3. Nd:YAG.

.

.

,

,

. . 2.10 Nd.YAG, . ( - 1 ) , ,

.

1 , 300

Av 0 = 4 1 = 120 Nd.YAG Av 0 = 11 1 = 330 .

.

1 v (. . - 1 )

w v/, ( / ) . , v v = cw.

( ) . = c/v = 1/w, ( 1 ) .

, :

(2.5.12)

64 . .

,

, .

. , , 2.3.2, , . .

, . ,

-

- .

, ,

. --

[10] , g(v - v 0 ) , (2.5.10)

2xsp, xsp . ,

,

(FWHM) Av0=l/2nxsp. (2.5.13)

, ,

, ,

exp(-t/xsp), ,

E(t) = exp(-t/2xsp) cos co0t. ( {E2(t)))9 (-/ 8 ),

.

,

E(t), , ,

(2.5.13).

2.4. .

Avnat . |] = , = 0,1 , X = 500 ( ), 2.1 , xsp = 10 . (2.5.13) Avnat = 16 . , Avnat, = 1/89 , vfj. ( - ).

2. 65

0 100 200 300

[ ]

. 2.10

,

,

N d : Y A G

2.5.2.

,

( ).

.

,

. - ,

, .

, ,

, , . ( .) ,

g*(v'0 - v 0 ) , . . (2.4.27). A V Q (FWHM) ,

, .

2.5. . N d 3 + . -

X = 1,05 AvJ = 5,4 , . . 40 Nd.YAG (. 2.3). , .

,

,

. , v

2 , 2 . -

, ,

, v' = v [ l - (u 2 /c ) ] , . , ,

2 > 0, v' < v, . , , v'

v 0 , . . v [ l - ( 2 / ) ] = v 0 . :

v = v 0 / [ l - ( u 2 / c ) ] , (2.5.14) : ,

, V Q ,

V o = v 0 / [ l - ( u 2 / c ) ] , (2.5.15)

66 . .

v 0 . ,

, ,

v V Q , . . v = V Q , (2.5.14) (2.5.15) , ,

, 2.5.

*(vo ~ v 0 ) , , pvdvz ,

2 2 + duz, :

1 / 2

*=[) (2.5.16)

|2|

2.1

1 - 1 0

5+10 / - 3 0 0 - 1

- 1 0 - 1

50 - 1

- 5 0 0 - 1 1-500 - 1

,

Avj Av 2 , Av = Av x + Av 2 . Av x Av 2 , Av = (Av 2 + Av | ) 1 / 2 . ,

.

, , , [11], -. , (, Ne), . .

. 2.1 .

,

xsp = 10 , , Avnat = 10 . , ,

isp = 1 , , Avnat = 1 , ,

.

, (

= 0,1 ) , , A v c = 1/ = 100 " 1. ,

; AvJ, , .

300 - 1 0,5 - 1 , , , Nd: Y A G .

2.6.

,

,

.

,

68 . .

.

, ,

, , :

1. . , ,

/ . ,

,

.

2. - .

, - , ,

, ,

, .

2.6.1.

[12], *

, :

*+-++ + , (2.6.1) *.

,

,

. (2.6.1) NB* ( ) :

dN -jf- = -kB*ANB*NA, ( 2.6.2)

NA , kB*A , .

, . . kB*A , , (, 0 2 ),

.

,

(, 238- He-Ne ). (2.6.2) :

Wnr=kB.ANA. (2.6.3) (2.6.2) (2.6.3) :

(dN2 ) = N2 (2.6.4) I dt J

'

2. 69

, , N2 ( ) *, - ,

, -

,

= (1/Wnr). , (2.6.2) , ,

(2.6.1), : + - * + - , (2.6.5)

( , ). , (2.6.2) :

(dNB. /dt) = -kB.ANB.NA + kBANBNA, (2.6.6) kBA , .

kBA kB*A, .

:

.

1 , (2.6.6)

:

kB.ANB.NA=kBANBNA. (2.6.7) :

NB* = NB (-2/kT), , . (2.6.7) :

* 2 =KBA exp(AE/kT), (2.6.8) , k (2.6.1) , (2.6.5). ,

kT. , (2.6.8), kB*A ^>kBA. , (2.6.8) , , ,

,

, -

, ,

, -

1 , 2.4.4 ,

,

.

70 . .

* * * *

. 2.11 ()

, ()

. ,

k ,

.

, ,

, . .

NB* NB, ^*^* ^ ^BANE, (2.6.6) (2.6.2). ,

(2.6.4) , > ,

.

(, (010) 0 2 ) , .

*

, (

1) : * + -> + * + , (2.6.9)

=

-

(. . 2.11). ,

[13]. ,

, (2.6.9) , , kT. .

1

( . ( 2 . 6 . 5 ) ) . ,

, ( . ( 2 . 6 . 1 ) ) , , ( . ( 2 . 6 . 9 ) ) . , , , -

.

2. 71

(, Ne He-Ne N 2 0 2 0 2 ). .

,

( , . 2.116): + *-> * +-. (2.6.10)

, , , ,

, , (. . = 0) , kB*A=kBA., ^ . ^ , (2.6.9) (2.6.10). , .

,

,

.

(NA* /NA) + ; ^ ) , (2.6.12) 1

vt ,

, ,

. , WNR

^IBL^-W N (2.6.13)

1 , , ,

, .

, , .

72 . .

, , -

,

. ,

v, Wnr Wnr= (-DAE/hv), ! ) , , , * .

,

= AE/hv , . . - .

, ,

,

.

WNROT . , ,

.

,

, ,

,

, (2.6.4),

. ,

, ,

9 2.5.1, . ,

,

. ,

, , ,

. , , ,

, ,

, ,

.

2 6 2

-

,

, -

, D, , , , .

.

[14] [15]. , ,

.

, ,

R .

2. 73

\xD, . [16] , , R, ( ), ED(t), , liD/4ns0R3.

ED(t, R), , |

.

:

Hcc\ED-a\k\iid-iia\/RZ. (2.6.14) ,

, \iD . ,

( ) .

,

, R, [14]:

(2.6.14)

8 ,

, gD ,

. , , , WDA W D A \\2, (2.6.14) W D A l u x ^ l u j 2 / ^ 6 * WDA (2.6.14) R~6, (1/

) (, 1/8 l u ^ l 2 , . (2.3.15)),

(,

|.

|2, . (2.4.29)). , (2.6.14) :

DA R (2.6.146)

R0

4

-|1/6

64 5

(2.6.146) R0 , -

. , (2.6.146) , , WDA = (1/8) R = R0, WDA = 1 0 - 6 ( 1 / T s / ) ) R = 10R0. R0 .

74 . .

. 2.12

- :

() , () -, () -.

, , -

- . ,

( D) , ,

(. 2.12). , ,

D. , -

D, -, , ,

. ,

i, . 2.126 (-). , . . AE2i = . , ,

, ,

(. 2.12). , - (. cooperative up-conversion), , . . 2 = A # 2 i -

- ,

, ND NA . (2.6.146) WDA R . , ,

, ,

R . ,

,

,

, ( ). , ,

2. 75

,

:

N2(t) = 2 (0)-[(* ) + C * 1 / 2 L (2.6.15)

,

. ,

.

2.7. Yb^-.Er3^'. [17]. Yb 3 +:Er 3 +: (. 9) - Y b 3 + ,

2 F 5 / 2 , 3 + ,

4 1 1 1 / 2 (. 2.13). , ,

Y b 3 + ,

3 +. , Y b 3 +

Y b 3 + , Yb 3 + -Er 3 + .

_

4

III.

N ; N ; :

. 2.13 - :

( ) Y b 3 + 3 + Yb:Er , () Nd:YAG -, () -

3 + .

2.8.

4F3/2 Nd:YAG.

4 F 3 / 2 Nd: YAG -. i . 2.126

4 I i 5 / 2 Nd 3 + (. 2.136). ,

4 1 9 / 2 ,

4 1 1 3 / 2 41 / 2 ( . 2.13, . . 2.15). (,

4 1 1 3 / 2 - 4 1 1 1 / 2 ) 2000 "1 (, ,

), . . 4 , YAG (~450 - 1 ) . Nd 3 + YAG 1 % .

76 . .

2.9. - 3+ [17]. , ,

3 + (. 2.13), -

.

3 +,

4 I i 3 / 2 ,

41 1 5 / 2,

4 1 9 / 2 .

41 1 3 / 2. -- ,

3 +,

4 I i 3 / 2 , , . . 50% .

2.6.3.

,

(2.6.4). ,

, N2 :

dN2_ (N2 N2) ~~{1 ^ 7 / (2.6.16)

(2.6.16) : dN2/dt = -(N2/x), (2.6.17)

1 1 ^ 1 - = - + (2.6.18)

N2(t) t , (2.6.17).

iV 2 (0 = ^ 2 ( 0 ) e x p - ( f A ) , (2.6.19) N2(0) t = 0. , , (2.6.16) N2/xr , . , ,

1 v 0 , t :

P(t) = N2(t)hv0V/Tn (2.6.20)

2. 77

V . (2.6.19) (2.6.20) : P(t) = [N2(0)hv0V/xr]exp-(t/T). (2.6.21)

,

,

, , ,

.

,

t = 0 N2(0), , (2.6.21), .

,

.

,

2. (2.6.21), :

N2(0)V / ,

, ,

,

, .

, ,

,

. ,

(2.6.18),

.

2.7.

, 1 2 . ,

.

. 2.14, , 1 2 g- 2 ~ > ( . . ), , . Nx N2 1 2, a Nu N2j - .

E2,g2,N2

2.14 ,

,

,

78 . . !

2.7.1.

.

, ,

;

} = N1 [ - ( 2 - ) / ] . (2.7.1) , , 1 , ;

=/8. (2.7.2) ,

}=/2. (2.7.26) (2.7.1) (2.7.2) :

=NZ(g2/g1)exp[-(E2-E1)/kT]. (2.7.3) ,

,

. ,

Nx N2; N2 i. ,

(2.7.4)

Wji , Wtj , (1 / )

. , Wjt Wtj (2.4.30), ||2 /, |p i ;| 2 | n y J 2 . , , (2.3.7). , \\itj\ (2.3.7),

ut, /- , 2 - .

,

Wy = WV}. (2.7.5) , ,

,

, , .

,

N2j = N2/g2y (2.7.6) # 1 * = # 1 / - (2.7.66)

2. 79

(2.7.6) (2.7.4), dN9 dt g2 gi J x

, (2.7.5),

1 =

. ^ . ^ ] . ^ (2.7.7)

8\ 82 8l g2

1 1 1 1

8l 82

1 _ 1 1 (2.7.9) #2

(2.7.7) , WN2/g2

WNi/gx

.

dF dz (. . 1.2)

dF = W -^]dz. (2.7.10) 82 Si)

2 1

1 2

2i = W/(g2F), (2.7.11) ci2 = W/(glF), (2.7.116)

, , ,

ft (N1/g1), (2.7.10), (2.7.11), : dF = gFdz, g

g = a2l(N2-N1&j. (2.7.14) , a 2i 1 2 (2.7.11) (2.7.116). Nx ^> N2 ( ), (2.7.13) a = a 1 2 iVi. , N2 Nt ( ), (2.7.14) g = a 2 1 N 2 .

80 . . 0

2.7.2.

, 2 1 g2 gx , , -

, ,

( ). , , , .

, ,

. (2.7.6) :

X2j = f2jN2, (2.7.15) Nu = fiiNl9 (2.7.156)

f2j (fu) 2 ( 1), j (0 . , , :

_ g2jexp-(E2j/kT)

tmg2mexI>-(E2m/kT) (2.7.16) 1

= guexp-(Eu/kT)

U 4 (2.7.166)

1 2

, a g2m gu . ,

(, I) 1 (, ) 2. (2.7.4)

J 1 1

(2.7.17) (2.7.15), (2.7.17)

(dN2/dt) = -W^N2 +W^Nt -(N2/x), (2.7.18) Wl9 W z ^, (1/)

W^ = f2mWml, (2.7.19) Wga=fiiWlm, (2.7.196)

(1/) = | \ | \ ( / 2 / / , ) . (2.7.19) 1 1

(2.7.18), dF dz

2. 81

dF^WZM-W&NJdz. (2.7.20)

Geml ofm

(2.7.19 ) Glm = Wlm/F oml = Wml/F I . , 1 ( ), Glm = Gml. , , (2.7.20) (2.7.21), :

: ( N2 > Nx) ,

. ,

N2 = 0 Nx = Nt, Nt , (2.7.22) :

alm=afmNt. (2.7.23) , Gfm , .

2.10. = 1,064 Nd:YAG. , Nd: YAG, . 2.15. 4 F 3 / 2 > 4 1 1 1 / 2 ( = = 1,064 ), , 4

/ 2 4 I I 3 / 2 (X = 1,32 ) 4 F 3 / 2 - 4 1 9 / 2 (X = 0,94 ). X = = 1,064 , = 2, 4 F 3 / 2 , I = 3, 41/2 ( R2 -> Y3). / 2 2 = N22/N2 = N22/(N2l + iV 2 2 ) , N22 N21 4 F 3 / 2 , N2 . , (2.7.3) N22 = N21exp -- (AE/kT)> . f22 f22 = 1/[1 + exp(AE/kT)]. = 84 1 kT = 208 "1 ( = 300 ) / 2 2 = 0,4. R2 > Y 3 2 3

=

6,5 10~19 2 [21]. 2 3 R2 ^ 3 (2.7.21), |3 = /2223 = 2,8 - 1 9 2.

oe

ml=W^l/F = f2maml, tn=Wl'k/F = fu

2.11. - .

2.16.

4 2 ;

42 ( = 730 * 800 ).

4 2

2, f2T ,

4 2 , f2T = N2T/(N2E + N2T), N2E N2T

.

N2T = N2Eexp -(AE/kT), . f2T = exp-(AE/kT)/[l + exp-(AE/kT)]. = 800 "1, = 208 ' 1 ( = 300 ) GTA = 4 10~19 2 X = 704 [22], , f2T = 2,1 10~2 = 0,8 ~ 2 0 2. f2T, .. ,

. ,

, f2T . ,

, 42 ->

4 2

(1/ = 1,5 105 - 1 (

= 6,6 ), 2 - 4 2 (1/) = 666,6 1 (

= 1,5 ). (2.7.19B) (1/) = (f2E/xE) + (f2T/xT), f2E = N2E/(N2E + N2T) = l-f2r ,

2.

(1/), , = 200 = 300 . ,

( 6,6 200 )

2,

, . , ,

, .

1,32

1,06 | 4 -

1 1 / 2 0,94

\ 9/2

1,064

= 84

J ^ 2 2 TTN* 21

0,946

. 2.15 ,

X = 1,064 N d : Y A G

= 800

.= 730-800

. 2.16 ,

2. 83

2.8.

( v 0 ) I v = v 0 . , .

, / , Nx N2 . N1 > N2 ( , Nx > N2), , WNl9 , WN2, . . 1 -> 2 , 2 -> 1. , / . .

2.8.1. :

(iV\ > N2) , .

N2 ,

(. 2.17) d^

= _ W ( N 2 _ N i ) _ ^

at (2.8.1)

N9

/\\

-^ WN, WN2 N2/x

. 2.17

/^ 1. :

Nx + N2 = Nt, (2.8.2) Nt . (2.8.1) ,

AN = Nt-N2. (2.8.3) (2.8.2) (2.8.3) Nt N2 AN Nt9 (2.8.1) :

dAN dt -AN( + 2w) + Nt.

, . . (dAN/dt) = 0, :

AN = l + 2Wx'

(2.8.4)

(2.8.5)

84 . .

AN ,

(dP/dV), = (hvWAN = m ^ r x , ( 2 . 8 . 6 )

, . . Wx > 1,

(dP/dV)s = (hv)Nt/2x. (2.8.7) (2.8.7) , (dP/dV)s, ,

, , ,

(Nt/2) .

(2.8.5) (2.8.6) . , (2.4.17) W

W = oI/hv, (2.8.8) .

(2.8.5) (2.8.6), (2.8.8), :

AN 1 (2.8.9)

I/Is

dP/dV (dP/dV)s~l + (I/Isy

Is = hv/2ax

(2.8.10)

(2.8.11) ,

.

(2.8.9). , I = Is , AN = Nt/2. v = v 0 , 18 . .

,

J .

, . 2.18,

v', / ' ,

.

,

[/(v)]

W W

. 2.18

v'

I'(v')

I v (/(v) I'(v'))

2. 85

.

(2.4.33) gt(v - v 0 ) g(v' - v 0 ) , v v'. - N1 - N2 = AN (2.8.9), :

=

0

0 = -

2 2 | u f i V , v ' ( v ' - v 0 )

(2.8.12)

(2.8.13) 3ne0ch v

( ). (2.8.12) (2.8.13) , I , .

,

g(V - v 0 ) . . 2.19 v' I/Is.

,

/ = I(t), .

, ,

.

,

AN , (2.8.4) \dAN/dt\

(2.8.14) () = Nt :

AN(t) = Nt exp (2.8.15) l-(2o/hv)jI(t)dt

(2.8.15) , t ():

t

r

t I I WN, WN2 N2/x

I

3 , iV 3

2,N2

. 2.20

,

. , 3 -> 2 1 -> g , NS = NX = 0.

N2 2:

(dN2/dt) = Rp- WN2 - ( 2 / ) , (2.8.21) Rp = WpNg , 3iNg . (. . dN2/dt = 0) Ng = Nt (2.8.21) :

1 + W V

(2.8.8), (2.8.22) : ^ 2 0 No

i + ( / / i s ) '

(2.8.22)

(2.8.23)

iV 2 0 = = RpT 2 (. . / = 0),

Is = hv/ax. (2.8.24) (2.8.24) (2.8.11) , Is , ,

. 2.17. - ,

. , AN .

, ,

. 2.18, v' , . (2.4.35)

= 0, (2.8.23), g :

g = (2.8.25)

g0 =

(2.8.25) (2.8.26) , , ,

g J, .

,

I(t). ,

(2.8.21) N2 . (2.8.23) 2 (2.8.25) , / . ,

,

2 Rp N2/x WN2. , :

(dN2/dt) = -(Gl/hv)N29 (2.8.27) (2.8.8). (2.8.27) :

N2(t) = JV20exp {- [(*) / J} , (2.8.28) N20 = Rp/ 2 , T(t) t (. (2.8.16)),

Ts = hv/a (2.8.29) .

(2.8.29) (2.8.17) , 8 ,

.

: g = g0exp { - [ ( 0 / . ] } , (2.8.30)

g0 = GN20 , (2.8.26). , ,

.

2.8.3.

,

,

(. 2.16 2.17 2, ). ,

, ,

(2.4.26). ,

2. 89

v'

. 2.21 ,

,

.

~

v o ) g ( v - V o )

. 2.22

v', ,

J(v') (

)

gt(v - v 0 ) g(v - V Q ) . ,

, . 2.21. , . 2.18, / ( v ) , V Q v. , I(v) .

, . 2.22 I(v). I(v) v. ,

, . 2.21, . . . ,

, , .

, . ,

,

.

2.3 .

, ,

.

,

, .

.

2.9.

90 . .

2.9.1.

, ,

, ,

, , .

, ,

,

. ,

( . [18]). , ,

, ,

.

-,

( (2.4.29), |p|2v - v ||2 ). .

2.9.2.

, ,

, ,

(, . amplified spontaneous emission ASE).

Q , ,

, (. 2.23). G = exp[a(iV2 - iVx)Z] , , ,

, Q ( 10 4 ). ,

,

(. 2.23),

Q, , , .

(R = 1) (. 2.236), .

.

,

D

1.

R=l

. 2.23

:

( ) , () .

2. 91

:

; , ; ; , , .

,

.

. 2.23. D

1,0

0,8

< 0,6

0,4

0,2

- 2 - 1 0 1 2 ( v - v 0 ) / A v 0

. 2.24

4 = 10' 10"; 10

I I M I L L

1 2 5 20 50 100 / s l 4 ^ " ) [ G l n G l i / 2 (2.9.3)

3/2 (2.9.36)

c^GlnG] 1 ' 2

.

, a Is = hv0/

, / Is. , , . 2.23, 4, . 1 = 18 G > 1 (2.9.3) (2.936) , :

G = 4 ^ . [ l n G ] i / 2 (2.9.4) 2

= | ^ 1 / 2 (2.9.46) . ,

(. 2.236), (2.9.4), G , G 2 , Q Q'. :

G 2 = i ^ [ l n G 2 - | i / 2 (2.9.5) 2

.

G 2 = 4ii G 2 ] 1 / 2 (2.9.56) 2

2.13, . ,

, Nd: YAG, D = 6 I = 10 , = 1,82, , . 2.23. (2.9.1) (0/4) = 2,25 10"4 . Nd: YAG , = 1, (2.9.4) , 4 T O G = 2,5 104, . . OpNthl = InG = 10,12. Nd:YAG 2,8- 10~19 2 (. 2.10), , Nth = 3,6 10 1 8 3. , . 2.236, (2.9.2) (0'/4) = 5,62 10~5 , (2.9.5) , G = 6,4 102, . . .

Nth = In G/apl = 2,3 10 1 8 3. ,

2 , Q

Q', . , , Q = 21 = 9,36 10"3 Q; =2

= 2,33 10 3 .

94 . .

. 2.26,

. 2.23 GpN2l, (Q/4n) = 10"4 , -

= 1. (2.9.3),

/ > Is- ,

, / < ,

,

. 2.23, . . (I/Is) = GpN2l/2.

,

, . .

[23].

,

. 2.236,

, -

, (. 10). ,

,

. ,

,

,

, ,

11, . . ,

,

1

. -

, -

/^J/' , , 3 + Y , Erbium-Doped Fiber Amplifier).

^

-

1550 .

100

. 2.26

J, / 8 ,

GpN2l

Q = 4 10~4

1 2- 95

2.10.

, . ,

,

, = a(v - v 0 ) .

(Av 0 ) . ,

, . ,

,

, 2.7. 2.2

, Av 0 ( AVQ )

.

6G , . ,

( 10" 1 3 2 ) A V Q ( ) ( ). ,

. ,

, Nd:YAG - , GP (10~ 2 0 ^ 10~ 1 9 2 ) , ( ), - . ,

( ), - 2.2

,

. [] [ ]

[]

H e - N e X = 0,6328 3 . 10 1 3 150 10~3 1,7

+ X = 0,5145 2,5 10- 1 3 6- 10- 3 1,7

N d 3 + : Y A G X = 1,064 2,8 - 1 9 230 120

N d 3 + : X = 1,054 4 10 2 0 300 5,4

6G

= 0,570 3,2 10- 1 6 5,5 10- 3 46

3 + : 1 2 0 4

= 0,704 0,8 l ( h 2 0 300 60 = 300

T i 3 + : A 1 2 0 3

= 0,790 4 . 10- 1 9 3,9 100 | | C r 3 + : L i S A F

= 0,845 5 10 2 0 67 84 | |

96 . .

. ,

, 6G, ,

10~1 6 2 ) , ,

.

, . 2.2, , ( 3 + :1 2 0 4 ) , (Ti 3 +:A1 2 0 3 ) CnLISAF (Cr 3 + :L iSrAlF 6 ) ,

. ,

( ), , Nd: Y A G , .

2.1. V = 1 3 , = 10 X = 600 .

2.2. p v ^, , pxdX X X + dX. ^ p v.

2.3.

X. ,

,

,

= hc/ky ( ), 5[1 - ( -* / ) ] = . .

2.4.

> . 2.3,

= 2,9 10"3 ( ).

= 6000 . ?

2.5. Rx (FWHM), 330 (. . 2.10). = 2,5 10~20 2 . ( = 1,76). , 3 ?

2.6. Nd: Y A G , Y 3 A 1 5 0 1 2 (- , Y A G ) , Y 3 + N d 3 + . N d 3 + 1%, . . 1% Y 3 + N d 3 + . Y A G 4,56 / 3 .

2. 97

N d 3 + , 412. ( ) , 134, 197, 311 848 - 1 . N d 3 + , 4 1 9 / 2 .

2.7. = 1,15

Av*0 = 9 108 . - 7 .

,

.

2.8. Sx -> S0 (. 9) 6G 0,87, 5 . Slt

2.9. X = 0,633 , , Avnat 20 , a Av c = 0,64 . ?

2.10. I .

2.11. Nd.YAG 6,3 7,5 .

1,06 = 2,8 10~19 2 , 1,82.

() (, , . . ). , ,

,

.

2.12. (. 8) ( 1,--4,4'-) .

(X = 0,6943 ) 8,1 10" 1 6 2. 22 . .

2.13. ( (2.6.9) (2.6.10)), , ( = 0) kB.A = hBA*, kB*A kBA. .

98 . .

2.14. , . 2.18, , J(v) 7(v). ,

:

a / v ( (0) . V o )

l + [ 2 ( v - v 0 ) / A v o ] 2 + ( / / / s o ) ' 0 (0) (J

1. R. Reif f, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), Chap. 9.

2. W . Heitler, The Quantum Theory of Radiation, 3rd ed. (Oxford University Press, London, 1953), Sec. II. 9. . . M., 1956. . . 9.

3. . A. Lorentz, The Theory of Electrons, 2nd ed. (Dover, New York, 1952), Chap. III. . . . M.: .-. ., 1956. . .

4. J. A. Stratton, Electromagnetic Theory, 1st ed. (McGraw-Hill, New York, 1941), pp.431-438. .. . .-.: , 1948.

5. R. . PantellandH. . Puthoff, Fundamentals of Quantum Electronics (Wiley, New York, 1964), Chap. 6. P., . . .: , 1972. . 6.

6. W . Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York, 1964), Chap. 6.

7. R. H. PantellandH. E. Puthoff, Fundamentals ofQuantum Electronics (Wiley, New York, 1964), pp. 40-41, 60, 62, and Appendix 4. P., . . .: , 1972. . 2 4.

8. R. . PantellandH. . Puthoff, Fundamentals of Quantum Electronics (Wiley, New York, 1964), Appendix 5. P., . . .: , 1972. 5.

9. A. Einstein, On the Quantum Theory of Radiation, Z. Phys. 18, 121 (1917). 10. W . Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York,

1964), Chap. 5. . . .: , 1972. . 5.

11. . G. Kuhn, Atomic Spectra, 2nd ed. (Longmans, Green, London, 1969), Chap. VII. 12. Radiationless Transitions, ed. by F. J. Fong (Springer-Verlag, Berlin, 1976), Chap. 4. 13. . K. Rhodes and A. Szoke, Gaseous Lasers: Atomic, Molecular, Ionic in Laser Hand

book ed. by F. T. Arecchi and E. O. Schultz-DuBois (North Holland, Amsterdam, 1972), Vol. 1 pp. 265-324.

14. J. B. Birks, Photophysics of Aromatic Molecules (Wiley-Interscience, New York, 1970), Sec. 11.9.

15. D. L. Dexter, J. Chem. Phys. 21, 836 (1953). 16. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Sec. 9.2.

.. . .: , 1965. . 9.2. 17. W . J. Miniscalco, Optical and Electronic Properties of Rare Earth Ions in Glasses, in

Rare Earth Doped Fiber Lasers and Amplifiers, ed.by M. J .F. Digonnet (Marcel Dekker, New York, 1993), Chap. 2.

18. T. Holstein, Imprisonment of Resonant Radiation in Gases, Phys. Rev. 72, 1212 (1947).

19. R. Arrathoon, Helium-Neon Lasers and the Positive Column in Lasers ed. by A. K. Le-vine and A. J. DeMaria (Marcel Dekker, New York, 1976), Tab. 2.

20. M. H. Dunn and J. N. Ross, The Argon Laser in Progress in Quantum Electronics, Vol. 4 ed. by J. H. Sanders and S. Stenholm (Pergamon Press, Oxford, 1977), Tab. 2.

21. W . F. Krupke, M. D. Shinn, J. E. Marion, J. A. Caird, S. E. Stokowski, Spectroscopic, Optical and Thermomechanical Properties of Neodymium- and Chromium-Doped Gadolinium Scandium Gallium Garnet, J. Opt. Soc.Am. 3, 102 (1986).

22. J. C. Walling, O. G. Peterson, J. P. Jennsen, R. C. Morris and E. W . O'Dell, Tunable Alexandrite Lasers, IEEE J. Quant. Electr. QE-16, 1302 (1980).

23. L. W . Casperson, Threshold Characteristics of Mirrorless Lasers, J.Appl. Phys. 48, 256(1977).

24. O. Svelto, S. Taccheo and C. Svelto, Analysis of Amplified Spontaneous Emission: Some Corrections to the Lyndford Formula. Optic. Comm. 149, 227-282 (1998).

100 . .

3

,

3 , ,

.

,

, ,

.

3.1.

, -

,

,

,

.

[1] .

3.1.1.

: (1)

; (2) Ev, ; (3)

, (4) . ,

. ^/ , ,

3. , 101

,

(

), (AEV) () .

ft2

* (3.1.1)

h = //2, , .

,

,

; ,

, / , ,

2/2. ,

2

Up R R0 Up = k0(R - R0)2/2 (. . 3.1). AEV :

, R

. 3.1

AEV = hv0 = h V 1/2

(3.1.2)

= 12/(1 + 2 ) . , ,

-11/2

(3.1.3) = 2ko

,

,

.

AEe = k0a2/2. (3.1.4)

2 k0 (3.1.1), (3.1.3) (3.1.4),

AEV = 2(//2. (3.1.5)

Er = h2J(J + 1) / 2 , J -

102 . .

.

, J = 0 J = 1,

= 22/2 =2(/)

,

( 3 . 1 . 1 ) . ( 3 . 1 . 5 ) AEr = (m/M)1/2AEv. ( 3 . 1 . 6 )

/ = 10~ 4 , ,

. ,

,

.

, (AEe/h), (AEv/h) (AEr/h) , , ( 2 5 - 5 0 ) 1 0 3 - 1 , 5 0 0 -3 0 0 0 - 1 1 - 2 0 - 1 .

, . ,

, ,

, --

,

R . ,

R. ( ), , JR , . 3 . 1 , 1 2 . (R > ) , , . R , .

,

, ,

R. , 1 . 3 .1 0 , .

R ,

,

. ,

(, R0). , , ,

. ,

R , , -

, .

,

R. , , , 1, ,

3. , 103

2 1

i/' = 0

3 R, R0. -

2 I^IIIZIZZZZZZZZI ^ -

,= 0 i? 0,

-

- -

.

3ZZZZZ R0 1 . -

. .2 -

, , . .

.

f_ . -

. -

,

" = 0 v" = 1 hv0, (3.1.2),

k0 . , ,

3.1 0, 1, 2, 3 . . , v = 0 , , ,

(hv0/2) . 1 2 , , . 3.1 , . 3.2, . 3.1 . , ,

v" = 3 1. . 3.1 , R , ' .

,

.

R0

. ,

, ,

.

.

, . 3.1, R ,

. , , SF 6 , (. 3.3), , . ,

. 3.3 ( ^ ) , R - . ,

3.3, SF 6 , -

104 . .

. 3.3 (, S F 6 ) . ,

( , [ 2 ] )

-f v' = l

. 3.4

,

.

,

-

-

,

U

.

,

. 3.1 ,

.

,

.

,

; (. . )

Er = BJ{J+l) (3.1.7) , h2/2I, / ,

. ,

3- , 105

, .

,

v" = 0 v' = 1 , . 3.4. , ,

; , J, . . [Er(J) - Er(J - 1)] = 2BJ.

3.1.2.

N(Ee, Ev, ) - , ,

N(Ee, EV9 ) gegvgrex? {-( + Ev + Er)/kT}9 (3.1.8) Ee, EvnEr , , a ge9 gvngr (. (2.7.3)). , 3.1.1, Ev/hc 1000 - 1 ,

EJhc .

kT/hc = 209 " 1 ( = 300 ) ,

9 Ev , kT. ,

, ,

.

1

, (3.1.7) (3.1.8),

40 J . 3.5

p(J)oc(2J+ l ) e x p [ - J ( J + l)/kT]. (3.1.9) (2J + 1) , J (2J + 1)- . , , = 0,5 " 1 kT/hc = 209 - 1 ( ), . 3.5 -

1 , , ,

. (, S F 6 )

1000 - 1 ( 100 - 1 ) ;

.

106 . .

(, ). , (3.1.9) (2J 4- 1) ( J = 0) , , J

(2J + 1)

= (2kT/B)i/2. (3.1.10) ,

.

3.1.3.

,

:

-

, ( ); - ;

-

(- ); - ;

, v" = 0, ( ); -- .

-

,

. ,

-, ,

(, ). ( . ) .

,

.

, ,

-,

. ,

" = 0 , 1 , . 3.6, 1 ,

. ,

v" > 0, .

3. , 107

2

1

uv(R) uv(R)

. ,

, - -

-'. -

,

"

v'

* \uv,uv,dR\2, (3.1.11)

, R

. 3.6

. v" = ,

, , ,

(3.1.11), , uv> uv. . 3.6 v' = 2. , , ,

,

, .

, , ( , . . ,

). , v' = 2.

R, ,

' = 2. , -

(- ) , . , -.

1 -

1 -

(, 2 ) . - ,

.

2

uv* uv-dR ; -.

108 . .

= 1 , .

v" = 0 v" = 1 (. . 3.2). , , v" = 1, v" = 2 (), v" = 0 (). ,

,

-. ,

= 1 . . 3.6 , , -

, Av = 2, 3 . ., , ( ).

, -

,

,

(3.1.9) (. . 3.5). , ,

.

AJ = 1 ( A J = J" - J\ J " H J' ). -

(, v" = 0 -> v" = 1 . 3.2), v 0 ,

(. 3.7). , , - A J = + 1 . - , v 0 ,

, (. . 3.4). , , -

A J = - 1 . (3.1.7), , 2B/h. 3.7 ,

. 3.7

.

,

v 0 .

: , -

AJ = + 1 , , -

J = - 1

- -

3. , 109

(. . 3.5). , -

(, ). AJ = 0;

v 0 (Q-). , (, , v' = 1 v" = 0 . 3.4) , , . 3.7, , .

3.1. 02 = 10,6 . 0001 -> 1000 (. , 0 2 10), v 0 , , v 0 = 960,8 "1 [20]. 0 2 = = 0,387 "1 [20], , (0001) (1000) . , -

E = hv0 + BJ'(Jf + 1) - BJ"(J" + 1) = hv0 - 2BJ", (3.1.12) , , J" . j ' m

(3.1.10). , 02 , .. = 450, J M A X = 19,6. 0 2 , J' J". , ,

, J' = 19 J' = 21. J' = 21, - J" = 22 ((22)-). (3.1.12) v = v 0 - (2BJ"/h) = 943,8 "1, X = (1/943,8) = 10,6 . , , v 0 , X = 1 / v 0 = 10,4 .

(3.1.12) , - Av = 2BAJ" = 4 = 1,55 1 .

3.2. 02 . 0 2 , ( (22)) X = 10,6 , , = 450 (. 3.1). 0 2 (2.5.28), = 5 0 . : , (2.5.18), A V Q ~ v 0 , 0 2 , He-Ne (. 2.7), , v 0 17 . , ,

, 0 2 , , He-Ne .

110 . .

3.3. 02 . 0 2 , , N 2 0 2 . ,

Av = 77,58(\|/02 + 0,73 - 4v|/N2 + 0,6) ^(/) 1/ 2 (. (2.5.12) (2.5.11)), , ,

( . .). , , ( = 15 . . C0 2:N 2:He 1:1:8) = 450 , Av = 40 . 3.2 , 0 2 . 0 2 ,

(. 10), .

,

. ,

.

A J = 1. , J J - 1.

, ,

- . -

AJ = 1 ,

. ,

v"

, ,

-. - -

, , A D = 1 A J = 1 .

A J = 1.

3.1.4.

, ,

(. . 3.6). , v' = 0 -

3. , 111

(, ). 1 ,

. v' = 0 - -

(, . . 3.6). ,

-. ,

,

CD . 3.6. (, ) v" = 0 (, , ). . 3.6 , .

.

-

. -

Av = 1 / = 1. ,

(. 3.1.1), : AJ = 1. , - , ,

,

(, xsp 1/vjj). ,

-

.

,

.

, 2.6.1, .

(. collisional deactivation) ( (2.6.1)) . ,

(, ) (, (0,1,0) 0 2 ; . 10). ,

.

1 , ,

, , .

(3 .1 .8 ) . .

112 . .

. 3.8

-

(. near-resonant energy transfer) (. (2.6.9) . 2 . ) ,

kT.

0 2 ,

(0,2,0) (0,1,0) (. 10). (. in

ternal conversion) -

(. . 3.8) . (. unimolecular decay), .

,

- ,

. , ,

. ,

. 3.6, , ( v' = 0 . 3.6), ( ).

, ,

.

,

v' = 0, , ,

.

3.2.

(. bulk semiconductors), . . ,

, . -

- (. quantum-confined semiconductors) , , , , ,

,

.

3. , 113

.

.

[3].

3.2.1.

-

,

[4] : }/() =

( ) [ ; ()] , (3.2.1) uk(r) , . (3.2.1) ,

.

, ( v . valence band), , , ( . conduction band). k, , . ,

. 3.9 k :

.

)

, ; ) * * - ^

^ ,

B f t J I

.

114 . ,

. 3.9.

,

, (. . 3.9), : h2h2

= ~ , (3.2.2) 2

= h2/[d2Ec/dk2]k=0 . Ev , , (. . 3.9), :

h2k2 E v =

fr^' (3.2.26) mv = h2/(d2Ev/dk2)k=0 . ,

,

,

(. 3.96). ' ,

, , :

E^=Eg+Ec, (3.2.3) El=-Ev, (3.2.36)

Eg , . .

kx, ky kz , , , . ., , ,

, z, (3.2.2) (3.2.3), k2 = k2 +k2 +k2.

,

.

,

Lx, Ly Lx, , ,

2. :

kt = (2nl/Lt), (3.2.4) i = , , z, / . , . 3.9

, .

,

. , ,

, 11 .

. .

2

, \\fx 2 . ,

3. , 115

d

> x E.

'2

X

> . 3.10 . 3.11

() ()

V i a V i b , d

-

N d

d . d ,

,

. -, , ,

,

, ,

- 19

.

vi/^ , 180 (. 3.10). ,

. ,

, d , ,

.

N , , , iV-

N . ,

,

2 , ,

(. 3.11). , N , N . N ,

.

, , (3.2.2) (3.2.3), (3.2.4), -

116 . .

. ,

= hk (, =

2/2), ,

Eg mv. , ,

k- , :

p = ftk. (3.2.5) , (3.2.2) (3.2.3)

, ,

k. , Si Ge, , .

IIIV, GaAs, InGaAs, AlGaAs InGaAsP. , GaAs

= 0,067 0 , 0 . , IIIV , ( hh . heavy holes) (mhh = 0,46m0 GaAs), ( lh . light holes) (mlh = 0,08m 0 GaAs) -- (. 3.12). , , , ,

,

.

,

,

, s- .

,

( 1 1

. 3.11), ,

-,

,

2 --

,

, .

,

,

- V , , k = .

-

. 3.12 : ,

-

I I I - V

3. , 117

-

- . (, GaAs = 0,34 ) kT (~ 0,028 ), - ,

. ,

3.2.2, . , ,

I I I - V .

3.2.2.

,

2.2.1, p(k), & &. . 2.2 , ,

kt, p(k) , . . 4nks/S9 , , (2n)3/LxLyL29 2, , . ,

p(k) = (ksV/Sn2), (3.2.6) V = LxLyL2 . , p(k), . . , k k k + dk, , (3.2.6),

, (3.2.7) , , , ,

, .

() . pcv(E)dE = pcv(k)dk, (3.2.2) :

1 2

in2 { h2

1 (2mv

3/2

() = ^ ^ ]'\ (3.2.8)

^ > = ^ ^ ) 3 / V - ( 3 - 2 - 8 )

,

Ev (. 3.9). , I I I - V mc

: -

3.2 3.

, .

,

. , . .

,

-, .

E'(k)> ( , , ), :

E'F , . , . 3.96, . (3.2.9), E' = E'F. , f(E'F) = l/2. ErF (3.2.9), -> 0. , f(E') = 1 'E'F. , = 0 , .

, EF . , > 0

3.2.3.1.

f(E') = l + exp[(E'-EF)/kT]9 (3.2.9)

. 3.13 ) ' k. ) f(E')

f 3. , 119

f(E') ' , . 3.136. , , Eg kT, , . .

.

, . 3.9 . 3.13 ,

. ,

,

.

. , - E'F ,

.

- E'F . ,

(~10 1 8 " 3) , .

,

.

3.2.3.2.

, () -

. ( , ~1 ) , ( ~ 1 , - ). , ,

,

.

fv fc . , fc fv (3.2.9). , , . 3.9, :

) = (3.2.10) , )

l + exp[(EC-EFc)/kT]

1 l + e x p [ ( F u - E v ) / k T ] 9 (3.2.106)

EFC EFV .

, EFC vlEFV fC(EC) fV(EV) , 3.146. ,

, , , -

UEV) =

120 . .

= . , . 3.14 = , ( ) , ( ). , ,

.

(3.2.10) , . 3.96. (3.2.3) (3.2.36), :

1

l + exv[(E'c-E'Fc)/kT] 1

(3.2.11)

A + exvKK-EkVW (3.2.116) , ,

,

. , EFc nEFv (3.2.11) , .

, ,

, Ne:

Ne= \Pc(Ec)fc(Ec)dEc. (3.2.12)

Nh , fv(Ev) = l-fv(Ev) ,

3- , 121

, , ! . (3.2.106) :

^{ E v } =

1 + [(- EFv )/kT]' (3.2.13) (3.2.13) , ,

. 3.14, , ,

(. (3.2.13) (3.2.10)).

. ,

Nh : 00

Nh = jpv(Ev)fv(Ev)dEv. (3.2.14)

, ,

- ,

, N. , , N, , ,

, (3.2.12) (3.2.14), Ne = Nh = N. , (3.2.12) (3.2.8) (3.2.10) :

00

Nc = 2(2nmckT/h2)3/2, = Ec/kT, a zF = EFc /kT. , (3.2.14) (3.2.86)

. 3.15 ) , I kT

N/Nc. , )

EF/kT iV G a A s

122 . .

(3.2.13) , (3.2.15). (3.2.15) , EFc /kT N/Nc, . 3.15. , .

3.4. GaAs.

= 0,0670 mv = mhh = 0,46m0 = 300 . Nc = = 4,12 10 1 7 - 3 ^ = (mv/mc)V2Nc = 7,41 10 1 8 "3, , (3.2.15), &NV .

N N/Nc , . 3.15, EFc /kT. .

EFJkT N GaAs, , GaAs, . 3.156.

3.2.4. :

v . ,

, , ,

1 (. (2.4.2)):

' = -, (3.2.16) = E(r, t) t. :

= 0 , - *), (3.2.17) kopt , = 2nv. v ^ Eg/h,

. 2{ , W, (.23) , :

W = g - | H S P 8 ( v - v 0 ) , < 3 - 2 8 > v 0 =

\ \=\jwl(-er'E0eik^-r)^vdv\2. (3.2.19) , \\fv (3.2.19) 1 2, (3.2.1).

1 , 2,

,

,

. ,

.

3. , 123

(3.2.18) (3.2.19) . 5- (3.2.18), , v = v 0 . ,

(Ei-E[) = hv, (3.2.20) .

, \\tv exp (jkv ) \|/ exp (ikc ), (3.2.19) ,