# Download Light and Matter Ic / Licht und Materie Ic PDF

Title Light and Matter Ic / Licht und Materie Ic English 14.4 MB 337
##### Document Text Contents
Page 2

ENCYCLOPEDIA OF PHYSICS

CHIEF EDITOR

S. FLOGGE

VOLUME XXV/2c

LIGHT AND MATTER Ic

EDITOR

L. GENZEL

WITH 72 FIGURES

SPRINGER-VERLAG BERLIN HEIDELBERG GMBH

1970

Page 168

Sect. VI.11. Direct solution of the density matrix equation. 151

The following form of the functions FE (v), F"' (v) is compatible with the above
equations:

FE(v) =GE(r); F;,(v) =Ga(r); }

F,(v)=v2 G(r); F,.(v)=v1 G(r)
(Vl.11.29)

with
4r2 =v~+v~=4/1*f1. (Vl.11.30)

The system (VI.11.25) -(Vl.11.28) now reduces to the form (where Gj (r) ==
oGi(t)for)

dG%/"} =x(2GE(r) +rG~(r))+ ~ xnth(G~(r) + ~ G~(r))- }
-g N(2G(r) +r G' (r)),

dG;t> =x(2Ga(r) +rG~(r)) + ~ xnth(G~(r) ++G~(r)) + l
+ao YIIGE(r) -yuGa(r) +g(rG'(r)- (4r2 -2)G(r))-

_ (N _ 1) {2 (Ga(r} G(r}) + (Ga(r} G(r})'}
g GE(r) r GE(r) '

d~~r) =x(3G(r) +rG'(r)) + ~ xnth(G"(r) +; G'(r))-
-yJ.G(r)+g(Ga(r)- ~ ~ G~(r)- ~ ~ G~(r))-

- (N -1) g {3 ( g; ~~) + r ( ~: ~) )'}.

(VI.11.31)

(VI.11.32)

(VI.11.33)

Though these equations still have a rather complicated structure, they may be
solved exactly, except for very small corrections, in the stationary case, i.e. for
!-_G_E__ = dGa = dG = O

dt dt dt 0

Because of the extremely small value of n1,. for optical frequencies, we may
neglect in (VI.11.31) -(VI.11.33) all terms with x n1,.in a very good approximation.
(VI. H. 31) then has the exact solution

(VI.11.34)

which may be inserted in (VL11.32), (Vl.11.3)) to give

; r(G~(r)+G~(r))=(ru- ~)Ga(r)+(; (4r2 -2)-a0 yn)GE(r) (VI.11.35)

or with xfN4:.yu, a0 Yll

~ (G~(r) +G~(r)) = (2r- 2 a:Z)GE(r) + 2: Ga(r) (VI.11.36)

or in the variable z = r2
z(GEtz+Gatz) =2(Z-O"oZ) GE+2ZGa (VI.11.37)

with
Z= Nyu. G = dG

4u ' I•- dz
(VI.11.38)

and

Page 169

152 Fully quantum mechanical solutions of the laser equations. Sect. VLH.

with
(VI.11.40)

Because of
(VI.11.41)

and
(VI.11.42)

for all possible values of r2 =z~Z,. [comparing the form of the solution (VI.11.49)],
we may neglect the terms with u2Jg2N2 and get in z:

(VI.11.43)

The elimination of (GEl• +Gal•) in (VI.11-37), (VI.11.43) leads to a relation
between GE and G a:

Ga(Z -z) =(a0 Z -z(1 +0:)) GE, (VI.11.44)

(VI.11.45)

Inserting in (VI.11.45) Gal• from (VI.11.43) and Ga from (VI.11.44) we arrive at a
first order differential equation for GE (z) alone:

or, because of 0:~2

with
Z _ (1 +a0) Z _ w11 N
,.---2- -4U·

Eq. (VI.11.47) can easily be solved and gives

G (z) E "' ' {
=C e•(z -z)(•m-i).

E =0;

(VI.11.46)

(VI.11.47)

(VI.11.48)

(VL11.49)

The constant cE is determined by the trace condition Tr P = 1. The meaning of
/'-...

the parameters z, z,. is clear; because of GEl• = 0 for z = z, z gives the number {J* fJ
where the distribution function (VI.11.49) has its maximum, while zm represents
the absolute maximum of possible photon numbers, which can be present in the
lasing mode for given constants a0 , Yll• u, N. The formula (VI.11.48) for z ... may

be interpreted as follows: As level 1 is occupied by ~ (more exactly N ( 1 ~ 0'))
atoms in the stationary state, w12 ~ atoms per second make transitions into level2
by pumping. If all these atoms fall to level1 by induced emission of a photon
(and not by other processes) the rate equation for produced and absorbed photons

reads in this optimal case w12 ~ = 2u zm which agrees with (VI.11.48). It is useful
to expand the distribution function (VI.11.49)

GE(z) = cE exp [z + (z,. -z) ln (zm -z)] (VI.11.50)

about its maximum z =z (provided z > 0, which holds above threshold. We
obtain in very good approximation for all values of a0

(VI.11.51)

Page 336

Subject Index. 319

P-representation, P-Darstellung 79.
Pulse, 2:n-pulse, Puls, 2n-Puls 246--247.
-, :n-pulse, n-Puls 245-246.
Pulse propagation, simplified model, Puls-

ausbreitung, vereinfachtes Modell
243-245.

- - in laser-active material, im laser-
aktiven Material 236--246.

- -, steady state pulse, stationtirer Puls
241-243.

- -, transienfbuild-up of the pulse, An-
wachsen des Pulses 239.

Pulse shape of mode-locked pulses, Puls-
form von M ode-gekoppelten Pulsen 222.

Pulse width of mode-locked pulses, Puls-
breite von Mode-gekoppelten Pulsen 214,
222.

Pulses, ultrashort, optical, Pulse, ultrakurze,
optische 213-224.

Pulsing, condition for self-pulsing, Pulsen,
Bedingung fur Selbstpulsen 239.

Pump power at threshold (rate equations),
Pumpleistung an der Schwelle ( Bilanz-
gleichungen) 253, 254.

cesses), quantum mechanical treatment,
Pumpvorgang ( s. auch Relaxationsvor-
gtinge), quantenmechanische Behandlung
45-50.

classical treatment, (s. auch Relaxations-
vorgtinge), halbklassische B ehandlung
175, 181, 209.

Quantization of the light field, Quantisie-
rung des Lichtfeldes 24-26.

-, second, of the electron wave field,
zweite, des Elektronenwellenfeldes 27.

Quantum mechanical consistency, quanten-
mechanische K onsistenz 44-4 5.

Quasi-linearization (quantum mechanical),
Quasilinearisierung ( quantenmechanisch)
128.

Q-spoiling, Q-W ertverschlechterung 267.
Q-value, Q-Wert 10.

processes, strahlungslose Obergtinge,
s. auch Relaxationsvorgtinge.

Rate equations, Bilanzgleichungen
249-271.

- -,approximate solution for small
oscillations, Ntihrungslosung fur kleine
Oszillationen 266--267.

- -, completely homogeneous case, voll-
sttindig homogener Fall 251-255.

- -, - - -, 4-level system, 4-Niveau-
System 255.

- -, - - -, pump power at threshold,
Pumpleistung an der Schwelle 253.

- -, - - -, 3-level system, 3-Niveau-
System 252-253.

- -,derivation of rate equations, Her-
leitung der Bilanzgleichungen 247-248.

- -, field equations, Feldgleichungen 249.

Rate equations, 4-level system, Bilanz-
gleichungen, 4-Niveau-System 266.

- - in quantized form, in quantisierter
Form 287.

- -,matter equations, Materiegleichungen
250.

- -, steady state solution, stationtire
Losung 250--251.

- -, 3-level system, 3-Niveau-System
264.

- -, threshold condition, Schwellenbedin-
gung 252.

Reflexion losses, Reflexionsverluste 1 7.
Relaxation oscillations, Relaxationsschwin-

gungen 135-138, 266--267.
Relaxation processes in atoms, semiclassical

treatment, Relaxationsprozesse in
Atomen, halbklassische Behandlung 176,
180, 181, 196, 209.

- - -, quantum mechanical treatment,
quantenmechanische Behandlung
46--49, 57-58.

Reservoir 3 s.
Resonance condition for modes, Resonanz-

bedingung fur Moden 13.
Resonator, confocal, Resonator, konfokaler

19-21.
-,Fabry-Perot, Fabry-Perot- 3, 11-18

11-18.
-, more general configuration, allgemeinere

Konfiguration 22-24.
-, optical, optischer 9-24.
-, stable and unstable regions, stabile und

unstabile Gebiete 24.

inversion), Ansprechen eines Systems,
zeitunabhtingig atomares ( s. auch In-
version) 185.

inversion), Ansprechen eines Systems,
zeitabhtingig atomares ( s. auch Inver-
sion) 185.

Rotating wave approximation, umlaufende
Welle, Ntiherung 29-30, 176, 179.

Saturated inversion, time-independent, ge-
stittigte Inversion, zeitunabhtingig 140,
131, 138, 185.

- -,time-dependent, zeitabhtingig 140,
175, 180, 185.

Schrodinger equation, formal solution,
Schrodinger-Gleichung, formale Losung
298-299.

Selection rules for fields, A uswahlregeln fur
Felder 26.

Self-pulsing, condition, Selbstpulsen, Bedin-
gung 239.

Semiclassical approach, halbklassische Ntihe-
rung 173-247.

- -,systematics, Systematiken 183.
Semiclassical equations, gas laser, halb-

klassische Gleichungen, Gaslaser
194-197.

- -,solid-state laser, Festkorperlaser 51,
173-197.

Page 337

)20 Subject Index.

Sidebands, Satellitenlinien 106, 215-217.
Signal, complex analytical, Signal, komplex

analytisches 7 3.
Signals, amplification, Signale, Verstiirkung

124-125.
-,external signals applied to laser, iiuf3ere

Signale, angewandt auf Laser 124-125.
Single-mode operation, qualitative discus-

sion, Einzel-Moden-Betrieb, qualitative
Diskussion 116-120.

- -, quantitative treatment, above thresh-
old (quantum mechanical), homogeneous
line, quantitative Behandlung, oberhalb
der Schwelle ( quantenmechanisch), homo-
gene Linie 128-138.

- -, - -, - - (quantum mechanical),
inhomogeneous line, (quantenmecha-
nisch), inhomogene Linie 144-146.

- -,- -,- - (semiclassical), gas laser
(halbklassisch), Gaslaser 197-199.

- -,- -,-- (semiclassical), general
features, ( halbklassisch), allgemeine
Eigenschaften 183-184.

- -,- -,-- (semiclassical), exact
semiclassical treatment, (halbklassisch),
exakt halbklassische Behandlung
201-203.

- -,- -,-- (semiclassical), homo-
geneous line, ( halbklassisch), homo gene
Linie 185-186.

- -,- -,-- (semiclassical), inhomo-
geneous line, (halbklassisch), inhomogene
Linie 187-189, 190.

- -, - -, below threshold (quantum
mechanical), unterhalb der Schwelle
(quantenmechanisch) 120-123.

Solid-state laser, semiclassical equations,
Festkorperlaser, halbklassische Gleichun-
gen 51, 173-197.

- - (semiclassical treatment), (halb-
klassische Behandlung) 173-194, 201.

Spiking, Spitzenstrahlung 132, 134-138,
144, 266-267.

Spin-echo, Spinecho 231-232.
Spin 1f2 , equivalence with two-level atom,

Spin 1/ 2 , .ilquivalenz mit Zwei-Niveau-
Atom 31-33, 234-236.

Spin, fictions, Spin, fiktiver 234-236.
Spin motion (quantum mechanical treat-

ment), Spin-Bewegung (quantenmecha-
nische Behandlung) 229-231.

- - (classical treatment), (klassische Be-
handlung) 228-229.

Spontaneous emission, spontane Emission
88-90.

- -, coherence properties, Kohiirenzeigen-
schaften 97-99.

Spontaneous linewidth, spontane Linien-
breite 97-99.

Stability of amplitude, Stabilitiit der A mpli-
tude 134-138, 144, 238.

- of phase, der Phase 130.

Stability of polarized modes, Stabilitiit
polarisierter Moden 201.

Stationary state, method of solution, statio-
niirer Zustand, Losungsmethode
182-183.

Stimulated absorption, induzierte Absorp-
tion 88-93.

Stimulated emission, see emission, induzierte
Emission, s. Emission 4, 9(}-91,
12(}-125.

- -, coherence properties, Kohiirenz-
eigenschaften 97-99.

superstrahlende Emission, klassische Be-
schreibung 231.

stiinde 228.

Three-level laser, quantum mechanical
treatment, Drei-Niveau-Laser, quanten-
mechanische Behandlung 125-137.

Threshold condition, Schwellenwertbedingung
186, 188, 191, 252.

Transparency, self-induced, Transparenz,
selbst-induzierte 237, 246-247.

Three-mode operation, see multi-mode ope-
ration, Drei-Moden-Betrieb, s. Viel-
M oden-Betrieb.

Two-mode operation above threshold
(quantum mechanical, semiclassical),
Zwei-Moden-Betrieb oberhalb der Schwelle
( quantenmechanisch, halbklassisch)
141-144, 184-185, 189-190, 199-201.

- - below threshold (quantum mechani-
cal), unterhalb der Schwelle ( quanten-
mechanisch) 139.

- - near threshold (quantum mechani-
cal), nahe der Schwelle (quantenmecha-
nisch) 14(}-141.

Ultrashort optical pulses, ultrakurze optische
Pulse 213-224.

Uncertainty relations, Unbestimmtheitsrela-
tionen 83-88.

Unsaturated inversion, ungesiittigte Inver-
sion 121, 177.

Vacuum state of electrons, Vakuum-Zustand
der Elektronen 27.

- - of field, des Feldes 26.
van der Pol approximation, Fokker-Planck

equation reduced in the, van der Pol-
N iiherung, reduzierte F okker-Planck-
Gleichung in der 158.

van der Pol equation, van der Pol-Gleichung
144, 158.

Wave equation, Wellengleichung 174,
178-179.

Wigner distribution, Wigner- Verteilung
61-63, 80.