Chapter 12. Multimode and Transient Oscillation

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang 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) =>

 

_



 





I g L g

cl

I r l r

I L g

cl dt

dI

 t





 



  



) (

) 1

2 ( ) 1

( _{1 2}



^{g  g}



^I_

c  _t

 ( ) L

l (10.5.14), ₁=₂=0, A => ₂₁

1 2

21 1

2 2

21 2

) (

) (

K h N

I g dt

dN

K h N

I g dt

dN





















 







) )(

( ) ( ) 8 (

)

( _{2 1 2 1}

2

N N S

N A N

g      



 

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

Assume, N₁<<N₂ => g() ~ ()N₂

2 2

21 2 2

2

) (

) (

K N

I h N

dt dN

I cg I

N dt c

dI

t



















 











12.3 Relaxation Oscillation

Steady-state solution ;

) (

) (

2

21 2







 



t t

N g

g h K I





 



  



Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

Perturbation method ;

















2

2 N

N I I

N2

I







 _

(12.2.5) =>



^I_ ^

^ 

^c

^ 

^{N }

^ 

^I_ ^

^ 

^cg



^I_ ^

^ 

dt d

t 2

    



^{  }

 

^











 

































I cg I

N I

N c

I cg I

N dt c

d

t t

2 2

2

2 _ _ 0

N I cg I

c _t

since



  

 

I c c

I dt c

d   ⁰

Similarly,

 

 



t t

g K h

g dt

d ₂







Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

2 0

2 0

2   

     

dt d dt

d

where  K /₂ g_{t }



  I

h g c _t

2 

0

Sol)

 ( t )  Ae

^t/2

cos(  t   )

where

4

2 2 0

 

  

Intensity :

)

2

cos(

/

 







 I  Ae

^

t 

I

^t

) )(

/ (

2 2

0

0 21 t

r c g g

T   









21 0

1 

 

g g_t

r 

I



I

 ^Tr

2

~ e

^t/

t

homework

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

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>

t N cg

y N N

ch

x I _t

t t





 

^{ , 2 ,} Define,

(12.2.5a) => y x dt

dx  ( 1)

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

Pumping and spontaneous decay of N₂

during the pulse interval is negligible, since the pulse is short enough.

(12.2.5b) =>

d xy

dy  



Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

12.5 Methods of Q Switching

- Rotating mirrors ~ 10,000 rpm

- Electro-optical switching

- Saturable absorber (Passive Q-switching)

; saturate the absorption (bleaching)

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

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(

)

(t  x₀



t 



₀

x_{n n}

where,

2 ,...., 1 2 2

, 1 2 1 , 1 2

1

0

 

 

 

 











N N

N n N

n  n



The sum of the displacements ;

 

^











n

N N

n

n t x t

x t

X

2 / ) 1 (

2 / ) 1 (

0

0 sin( )

) ( )

(  



 



 



 



 



^{  }



^



^

n t in n

t i n

t n t

i x e e

e

x₀Im ^{(0 0 )} ₀Im ^{(0 0)}

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

) 2 / sin(

) 2 /

2 sin(

/ ) 1 (

2 / ) 1

( y

e Ny

N N

iny 



^





) sin(

) (

) 2 / sin(

) 2 / ) sin(

sin(

) 2 / sin(

) 2 / Im sin(

) (

0 0 0

0 0 0

) (

0 ^{0 0}



















 







 





 







 ^ 

t x

t A

t t t N

x

t t e N

x t X

N

t



i

# Peaks : A_N(t)_max  N at ^{(2 )  ,  0,1,2,....}

 m  mT m

t_m 

# Temporal width :

N T

N N 

 

 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

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

12.8 Mode Locking

<Shortest pulse length>

The maximum number of longitudinal modes : ^g v_g c

L L

c

M v  

 2

2 / The shortest pulse length :

g

M cM v

L

 



 2 1

min 



Examples)

nsec sec 1

10 1700

1 1

1

min 6 

 

 _

vD

 

1) He-Ne laser, v_D~1700MHz 

sec 10

~ ¹¹

min

 

2) Ruby laser, v_D~10¹¹Hz 

sec 10

~ ¹²

min

 

3) Dye laser, v_D~10¹²Hz 

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

<Mode-locked laser oscillation>

Electric field of m-th longitudinal mode :

) sin(

ˆ sin )

sin(

) ˆ (

) ,

( _m

ε

_{m m m m}

ε

_{m m m m}

m zt  z  t  k  t

E

where,

,...

3 , 2 , 1

, 



 m

L m c c k_m

m

 

,....

3 , 2 , 1

, 

 m

m L k_m 

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

ε

₀ ^m0





^



m

m m

m

m z t k z t

E t

z

E( , ) ( , )

ε

₀ sin sin

=> Total electric field in the cavity ;

L c n M L

n M

k_m(  )/ , _m(  ) / where,







 



  

 

 



 ^ 





n N

N

L

ct z n M L

ct z n M

L ct n M L

z n t M

z E

) (

) cos(

) (

) cos(

2 1

) sin(

) sin (

) , (

0 2 / ) 1 (

2 / ) 1 ( 0

ε ε









Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

L ct z

L ct z N L

ct z M L

ct z

L ct z e N

L e

ct z n M

L ct z iM

N N n

L ct z n M i N

N

2 )/

( sin

2 )/

( sin ) cos (

2 )/

( sin

2 )/

( Re sin

) Re (

) cos(

/ ) (

2 / ) 1 (

2 / ) 1 (

/ ) ( ) ( 2

/ ) 1 (

2 / ) 1 (



 



 



 





 







 

 















 





 

















L ct

z

L ct

z N L

ct z M L

ct z n M

n sin ( )/2

2 / ) (

sin ) cos (

) (

) cos(



 



 



 

 



 ^{  }_

Similarly,



 







 

 

 



 )

2 )/

( sin

2 )/

( )sin

( 2 cos

)/

( sin

2 )/

( )sin

( 2 cos )

,

(

ε

⁰ _{0 0}

L ct z

L ct z ct N

z L k

ct z

L ct z ct N

z k t

z

E 







L ct

z

L ct

z t N

z A_N

2 / ) (

sin

2 / ) (

) sin ,

)(

(



 









^{( , )cos ( ) ( , )cos ( )}



) 2 ,

(_{z t}

ε

⁰ _A^{( )} _{z t k0 z ct A}^{( )} _{z t k0 z ct}

E  _N^   _N^ 

where,

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

) ,

)(

( z t

A_N^ has maxima occurring at ^{z ct  m(2L), m 0,1,2,....}

cavity round trip time

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

12.9 AM Mode Locking

) sin(

sin )

,

(

ε

_{m m m m}

m z t k z t

E   

ε

m

AM(amplitude modulation) : is modulated periodically

 ε

m

ε

₀(1^cos^t) modulation index

z k t

t t

z

E_m

( , ) ^ ε

₀

( 1 ^{ } cos ^ ) sin( ^

_m

^{ }

_m

) sin

_m



z k t

t

t _{m m m m m m}

m

) sin[( ) ] sin[( ) ]} sin

{sin(

_{2 2}

ε

0

^{   }

^

^{     }

^

^{    }



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.

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

12.10 FM Mode Locking

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

cos sin(

sin )

,

(^{z t}

ε

^{k z t t}

E_m  _{m m m}  _m 



  

) cos sin(

) cos(

) cos cos(

) sin(

) cos sin(

t t

t t

t t

m m

m m m

m





































) 2 cos(

) ( ) 1 ( 2 ) ( ) cos

cos( ₂

1

0 

 J x J x k

x _k

k



^ k









] ) 1 2 cos[(

) ( )

1 ( 2 ) cos

sin( _{2 1}

0



  ^  _ 



 ^{J x k}

x _k

k

k

) (x

J_n : Bessel function of the first kind of order n

































































































]}

) 5 cos[(

] )

5 ){cos[(

(

]}

) 4 sin[(

] )

4 ){sin[(

(

]}

) 3 cos[(

] )

3 ){cos[(

(

]}

) 2 sin[(

] )

2 ){sin[(

(

]}

) cos[(

] )

){cos[(

(

) sin(

) (

) cos sin(

5 4 3 2 1 0

m m

m m

m m

m m

m m

m m

m m

m m

m m

m m

m m m m

t t

J

t t

J

t t

J

t t

J

t t

J

t J

t t































































L

c/







 control !

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

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 ; n(x,t) asin(_st )sink_sx

# Diffraction angle ;

ns

  sin  2

# Modulation frequency ; sin(_st)1  2_s

 2 

_s

  (   c / L )

control ! ex) L ~ 1 m => ~ 9x10⁸ rad/s

=> _s=/4 = 75 MHz (quartz oscillator)

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

2) Electro-optical phase modulation (Pockels cell)

Ea

n n  ₀   Refractive index of electro-optic medium ;

Ea

where, : applied electric field



 



 



 

 





 



  









 

c z t n

z c E

c z t n t

z _a

0 0 0 0

cos xˆ

cos xˆ

) , ( E

ε ε

z c E^a

 V

c

 

where,

3) Saturable absorbers

Absorption coefficient of saturable absorber ; _sat I I a a

/ 1

0

   a₀a₀I/I^sat Suppose that there are two oscillating cavity modes ;

) sin(

sin )

sin(

sin )

,

(^{z t }

ε

₁ ^k₁^z

^

₁^t

^

₁ ^

ε

₂ ^k₂^z

^

₂^t

^

₂

E

Nonlinear Optics Lab

Nonlinear Optics Lab . . Hanyang Univ. Hanyang Univ.

Intensity :

)}

sin(

) sin(

sin sin

2

) (

sin sin

) (

sin sin

{

) , ( )

, (

2 2

1 1

2 1

2 1

2 2

2 2

2 2 2

1 1

2 1

2 2

1 0

2 0

ε ε ε ε





































t t

z k z

k

t z

k

t z

k c

t z E c t

z I

] )

cos[(

] )

cos[(

) sin(

) sin(

2

2 1 2

1

2 1 2

1 2

2 1

1













































t t t

t

)]

cos(

sin sin

2

sin sin

[ )

, (

2 1 2

1 2

1

2 2 2

2 1

2 2

2 1

ε

ε ε

ε

0



















t z

k z

k

z k z

k t

z

I ^c









 _{2 1 2}

1 2 1, ,

#      

Time averaged intensity :

Intensity is modulated

=> Absorption coefficient can be also modulated