Nonlinear Optics Lab. Hanyang Univ. Chapter 12. Multimode and Transient Oscillation Pulsed oscillation : Relaxation oscillation, Q-switching, Mode locking

Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.

Chapter 12. Multimode and Transient Oscillation

Pulsed oscillation : Relaxation oscillation, Q-switching, Mode locking ?

12.2 Rate Equations for Intensities and Populations

(10.5.8) =>

IggL

cl

Irrl

IgL

cl

dt

dI

t

)(

)1(2

1)( 21

Iggc t )(

Ll(10.5.14), 1=2=0, A => 21

12211

22212

)(

)(

KNh

Ig

dt

dN

KNh

Ig

dt

dN

))(()()(8

)( 1212

2

NNSNNA

g


Assume, N1<<N2 => g() ~ ()N2

222122

2

)(

)(

KNINhdt

dN

IcgINcdt

dIt

12.3 Relaxation OscillationSteady-state solution ;

)(

)(

2

212

t

t

gN

g

KhI


Perturbation method ;

22 NN

II

2N

I

(12.2.5) => IcgINcIdt

dt2

IcgININc

IcgINcdt

d

t

t

22

2

02 IgcINc tsince

IccIc

dt

d0

Similarly,

t

t

g

K

h

g

dt

d 2


02

02

2

dt

d

dt

d

where tgK /2 Ih

gc t20

Sol) )cos()( 2/ tAet t where4

22

0

Intensity :

)cos(2/ tAeII t

))(/(

22

0210 t

rggc

T

210

1

g

gtr

I

IrT

2/~ te

t

homework


12.4 Q Switching

Q switching : A way of obtaining short, powerful pulses

Sudden switching of the cavity Q(loss) from a low(high) value to high(low) value.

Principle of Q switching : Suppose we pump a laser medium inside a very lossy

cavity. Laser action is precluded even if the upper level population N2 is

pumped to a very high value (nearly small signal value). Suddenly we lower

the loss to a value permitting laser oscillation. We now have a small-signal

gain much larger than the threshold gain for oscillation.

<Qualitative explanation>

tcgN

Ny

Nch

Ix t

tt

,, 2Define,

(12.2.5a) => xydt

dx)1(


Pumping and spontaneous decay of N2 during the pulse interval is negligible,

since the pulse is short enough.

(12.2.5b) =>

xyd

dy


12.5 Methods of Q Switching

- Rotating mirrors

~ 10,000 rpm

- Electro-optical switching

- Saturable absorber (Passive Q-switching)

; saturate the absorption (bleaching)


12.7 Phase-Locked Oscillators

Mode Locking : Locking together of the phases of many cavity longitudinal modes.

=> Even shorter laser pulse than can be achieved by Q switching.

<Phase-locked harmonic oscillator>

Displacements of N harmonic oscillators with equally spaced frequencies ;

)sin()( 00 txtx nn

where,

2

1,....,2

2

1,1

2

1,

2

10

NNNN

n

nn

The sum of the displacements ;

n

N

Nnn txtxtX

2/)1(

2/)1(00 )sin()()(

n

tin

n

ti

n

tnti eexex )(0

)(0

0000 ImIm


)2/sin(

)2/sin(2/)1(

2/)1( y

Nye

N

N

iny

)sin()(

)2/sin(

)2/sin()sin(

)2/sin(

)2/sin(Im)(

000

000

)(0

00

txtA

t

tNtx

t

tNextX

N

ti

# Peaks : NtAN max)( at ,....2,1,0,)2

(

mmTmtm

# Temporal width :N

T

NN

2

# Maximum total oscillation amplitudes equals to N times the amplitude of a single oscillator

# This maximum amplitudes occur at intervals of time T

# This temporal width of each spike get sharper as N is increased


12.8 Mode Locking

<Shortest pulse length>

The maximum number of longitudinal modes : gg v

c

L

Lc

vM

2

2/

The shortest pulse length :g

M vcM

L

12min

Examples)

nsec1sec101700

1116min

Dv1) He-Ne laser, MHz1700~Dv

sec10~ 11min

2) Ruby laser, Hz10~ 11Dv

sec10~ 12min

3) Dye laser, Hz10~ 12Dv


<Mode-locked laser oscillation>

Electric field of m-th longitudinal mode :

)sin(sinˆ)sin()(ˆ),( εε mmmmmmmmmm tktztz E

where,

,.....3,2,1, mL

cmckmm

,....3,2,1, mL

mkm

Assume, the mode fields all have the same magnitude and polarization, and also 0ε 0m

m

mmm

m tzktzEtzE sinsin),(),( 0ε=> Total electric field in the cavity ;

LcnMLnMk mm /)(,/)( where,

n

N

N

L

ctznM

L

ctznM

L

ctnM

L

znMtzE

)()(cos

)()(cos

2

1

)(sin

)(sin),(

0

2/)1(

2/)1(0

ε

ε


Lctz

LctzN

L

ctzM

Lctz

LctzNe

eL

ctznM

LctziM

N

Nn

LctznMiN

N

2)/(sin

2)/(sin)(cos

2)/(sin

2)/(sinRe

Re)()(

cos

/)(

2/)1(

2/)1(

/)()(2/)1(

2/)1(

Lctz

LctzN

L

ctzM

L

ctznM

n 2/)(sin

2/)(sin)(cos

)()(cos

Similarly,

)2)/(sin

2)/(sin)(cos

2)/(sin

2)/(sin)(cos

2),( 00

0εLctz

LctzNctzk

Lctz

LctzNctzktzE

Lctz

LctzNtzAN 2/)(sin

2/)(sin),()(

)(cos),()(cos),(2

),( 0)(

0)(0ε

ctzktzActzktzAtzE NN

where,


),()( tzAN

has maxima occurring at ,....2,1,0),2( mLmctz

cavity round trip time


12.9 AM Mode Locking

)sin(sin),( ε mmmmm tzktzE

mεAM(amplitude modulation) : is modulated periodically )cos1(0εε tm

modulation index

zktttzE mmmm sin)sin()cos1(),( 0ε

zkttt mmmmmmm sin]})sin[(])sin[(){sin( 220ε

If = (m+1-m=c/L), each mode becomes strongly coupled to its nearest-neighbor modes, and it turns out that there is a tendency for the modes to lock together in phase.


12.10 FM Mode Locking

FM(frequency modulation) : The phase of the fieldis modulated periodically

)cossin(sin),( ε ttzktzE mmmmm

)cossin()cos(

)coscos()sin()cossin(

tt

tttt

mm

mmmm

)2cos()()1(2)()coscos( 21

0 kxJxJx kk

k

])12cos[()()1(2)cossin( 120

kxJx kk

k

)(xJ n : Bessel function of the first kind of order n

]})5cos[(])5){cos[((

]})4sin[(])4){sin[((

]})3cos[(])3){cos[((

]})2sin[(])2){sin[((

]})cos[(])){cos[((

)sin()(

)cossin(

5

4

3

2

1

0

mmmm

mmmm

mmmm

mmmm

mmmm

mm

mm

ttJ

ttJ

ttJ

ttJ

ttJ

tJ

tt

Lc/ control !


12.11 Methods of Mode Locking

1) Acoustic loss modulation (AO modulator)

acoustic wave

diffraction

# A standing wave in a medium induces the refractive index variation ; xktatxn ss sin)sin(),(

# Diffraction angle ;sn

2

sin

# Modulation frequency ; sst 21)sin( )/(2 Lcs control !

ex) L ~ 1 m => ~ 9x108 rad/s

=> s=/4 = 75 MHz (quartz oscillator)


2) Electro-optical phase modulation (Pockels cell)

aEnn 0Refractive index of electro-optic medium ;

aEwhere, : applied electric field

zc

nt

zEc

zc

nttz a

00

00

cosx̂

cosx̂),(E

ε

ε

zEc a

Vc

where,

3) Saturable absorbers

Absorption coefficient of saturable absorber ;satII

aa

/10

satIIaa /00

Suppose that there are two oscillating cavity modes ;

)sin(sin)sin(sin),( 22221111 εε tzktzktzE


Intensity :

)}sin()sin(sinsin2

)(sinsin

)(sinsin{

),(),(

22112121

222

222

2

112

122

10

20

εεεε

ttzkzk

tzk

tzkc

tzEctzI

])cos[(

])cos[()sin()sin(2

2121

21212211

t

ttt

)]cos(sinsin2

sinsin[),(

212121

222

2122

12

εεεε0

tzkzk

zkzktzI c

212121 ,,#

Time averaged intensity :

Intensity is modulated

=> Absorption coefficient can be also modulated

Documents

Nonlinear Optics Lab. Hanyang Univ. Chapter 12. Multimode and Transient Oscillation Pulsed oscillation : Relaxation oscillation, Q-switching, Mode locking