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Chapter 11. Laser Oscillation : Power and Frequency

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(1)

Nonlinear Optics Lab . Hanyang Univ.

Chapter 11. Laser Oscillation : Power and Frequency

Power & Frequency of Single mode continuous wave (cw) laser ?

11.2 Output Intensity : Uniform-Field Approximation

Assumptions :

(1) Homogeneous broadened gain medium

(2) Mean-field (uniform-field) approximation

(3) All loss processes are independent of the cavity intensity (4) Steady-state (temporal) or CW operation

1) Gain

(10.5.8) , => rr I

L I c

L g

cl (1 )

) 2

( 1 2

0

I rr gt

gl  

 (1 )

2 ) 1

( 1 2

2 1

1rr

: The gain is clamped on the threshold gain for cw(steady-state) oscillation.

(2)

Nonlinear Optics Lab . Hanyang Univ.

2) Output intensity

(10.12.10) or (10.11.14) => g( )1

I(g) 0(I())

/Isat gt

 



 

 

() () sat 0( ) 1 gt

I g I

I

Output intensity : Ioutt1I() If =>r1r21 I()I()



 

 

 ( ) 1

2

1 sat 0

out

gt

I g t

I

) 2 (

) 1 1 2 (

1 t s

r l

gt l



 

 

 2 ( ) 1 2

1 sat 0

out

s t

l I g

t

I

=> : A given medium, laser intensity depends on how the laser cavity (t or s) is chosen.

(3)

Nonlinear Optics Lab . Hanyang Univ.

11.3 Optimal Output Coupling

1) Optimal mirror transmission coefficient

s ls g

t  

opt

2

0

(  )

    

0

2

sat ou

max

ou

I I g ( ) l s / 2

I

t topt

t

t

 

0





topt

out

t I

) 0 (

2 2

1 1 2

2 1

2 0

0

topt t sat

sat

s t

l tI g

s t

l

I g

2) Maximum output intensity

If

l g s

) 2

0(

: Lasing is possible when : scattering loss coefficient

l s s g

l t l r

gt opt opt opt

2 ) ) (

2 ( ) 1 1 2 ( ) 1

( 0

2 , )

0(

l

g s

  I

max

g

0 21

I

sat

l

out

( )

 

: theoretical upper limit of laser intensity

(4)

Nonlinear Optics Lab . Hanyang Univ.

2) Maximum output intensity – another approach

N N I

dt

h dN ( ) ( 2 1)

emission stimulated

2

(10.12.8) =>

( ) ( )

emission stimulated

2 ( )

g I I

dt h dN

( ) ( )

sin2 ,

( ) ( )

2 

I I kz I I I

I : uniform field approx.

(11.2.5) =>

 

t

t t

t

g g

g I I g

g g I g

g I

I

g 







( ) 1 ( )

) 1 ) (

( )

(( ) ( )sat 0sat 0sat 0

If g0()gt

 

0 sat )

( )

( ( )

) (

, gI I gI : maximum intensity per unit volume

Maximum intensity extracted from the medium of length l ;

  I

21out max

g

0

( 

21

) I

sat21

l

: (11.3.5a)

(5)

Nonlinear Optics Lab . Hanyang Univ.

3) Input-to-output power conversion efficiency

3-level system, (10.7.12) =>

1 31

sat 21 0 max

) (

N P h

I e g

 

(10.11.12) =>

21 21 21

21 0

) )(

) (

(

P

N

g    P T

(10.11.7) =>

) ( 2

) (

21 21 21

21  

h P

I sat 31 1

21 21

max

2

) (

N P h

h N

e P

T

 

In the case of strongly saturated case,

N

T

N

N 2

1

2

1

 

31 21 31

21 21

max

 

P

e P

: quantum efficiency

(6)

Nonlinear Optics Lab . Hanyang Univ.

11.4 Effect of Spatial Hole Burning

Standing wave inside the cavity, (10.2.9) => g( ) 1

2I( )g/0I(sat)

sin2 kz

(11.3.6) =>

kz I

I

kz I

g

kz I

dt g h dN

2 sat

2 0

2 emission

stimulated 2

sin ) / 2 ( 1

sin )

( 2

sin ) ( 2

 

 

 

 

: Power per unit volume, at the point z, extracted from the medium

by stimulated emission.

The rate at which the field gains energy should equal the rate at which it losses energy ;

l gI dz g I 0l I Ikzsat dz 2 kz 2

0 0 1 (2 / )sin

2 sin

( ts ) I

()

12

( ts ) Ig

t

l I

For kl1,

sat

sat

0 sat 2

2

/ 2 1 1 1 2 sin

) / 2 ( 1

sin

I I I

I l kz I

I

dz kz

l

sat sat 0

/ 2 1 1 1

I g

I g I

I

t

 

(7)

Nonlinear Optics Lab . Hanyang Univ.

Put, sat

I x 2I

1

x

g x g

t

2 0

4 1 2

2 1 2

1 2sat 0 0

t

t g

g g

g I

x I : Disired solution is the one with minus

sign since should be equal to 1 when g0/gt=1.

x

4 1 2

2 1 2

1  2

sat

0

 

0

t

t

g

g g

g I

I

 

 

   

 16

1 2

4

1

0

sat 0

t

t

g

g g

I g I

 

 

   

 16

1 2

) ( 4

1 )

( 2

0 sat 0

out

t

t

g

g g

I g

I t  

Output intensity :

I

out

 ( t / 2 ) I

: The effect of spatial hole burning is to reduce the output intensity

(8)

Nonlinear Optics Lab . Hanyang Univ.

11.5 Large Output Coupling

Our analysis of output power thus far has assume that the output coupling is small( ), and we have also assumed time averaged intensity I(+) and I(-) are independent of z.

We will now allow arbitrary output coupling and therefore allow the possibility

I

(+)

and I

(-)

may vary with z.

2 1

1rr

Ignoring the spatial hole burning, (11.2.4) => ( ) 0( ) sat )]

( )

( [ ) 1

( I z I z I

z g

g

  (10.4.3) => ( ) ( )( )

) (

z I z dz g

dI

) ( ) ( ( )

) (

z I z dz g

dI

( ) ( )

( ) ( ) ( ) ( )0

dz I dI

dz I dI

I dz I

d

C z

I z I

e

i, ; ()( ) ()( )constant 

/ , 1

1

sat ) ( )

( 0 )

( ) (

I I C I

g dz

dI

I

 

sat

) ( )

(

0 )

( )

( /

1 1

I I C I

g dz

dI

I

 

 

(9)

Nonlinear Optics Lab . Hanyang Univ.

)2

( ) ( sat )

( sat )

( ) (

) ( )

( ) ( sat )

0 (

1 1 1 1

I dI I

dI C I I

dI

I dI I C

I dz I

g

) (

) 0

( ( )2

) ( sat

) (

) 0 (

) ( sat

) (

) 0

( ( )

) (

0 0

) (

) (

) (

) ( )

(

) (

1

L I I

L I I L

I I L

I dI I

C

I dI I

dz dI g

 





) 0 ( 1 )

( 1

) 0 ( )

1 ( )

0 (

) ln (

) ( )

( sat

) ( )

( sat )

( ) ( 0

I L I I

C

I L I I

I L l I

g

 





) ( 1 )

0 ( 1

) ( )

0 1 (

) 0 (

) ln (

) ( )

( sat

) ( )

( sat )

( ) ( 0

L I I

I C

L I I I

I L l I

g

=>

) ) (

) (

( ( ) 2 ( )

)

( rI L

L I L C

I Cr2

I()(L)

2

) 0 ) (

0 ) (

0

( ( ) 1 ( )

)

(

rI

I I C

) ( )

0

(

2 1 ( )

)

(

r r I L

I

) ( )

0

(

1 2 ( )

)

(

r r I L

I

=>

(10)

Nonlinear Optics Lab . Hanyang Univ.

 





 

1 2 sat 2

) (

2 sat 1

) (

2 1 0

) (

) 1 ( ln 1

r r r

I L I

r I r

L I

r l r

=> g

 

0 1 2

2 1 2

1

sat 1

2 1 2

1 2

2 1 0

sat )

(

) ln 1

( ) (

1 ) ln (

r r l

r g r r

r

I r

r r r

r r

r r l

g L I

I

 

 

 

Output intensity :

0 1 2

2 1 2

1

1 1

2 2

sat

) ( 2 )

( 1 out

) ln 1

( ) (

) ( )

0 (

r r l

r g r r

r t r

r t r

I

L I

t I

t I

 

 



 

When r1=1, t1=s1=0, r=r2, t=t2, s=s2, and t+s<<1

sat 20 1

2 out 1

s t

l I g

t

I

(11.2.11) : small output coupling

(11)

Nonlinear Optics Lab . Hanyang Univ.

<Total two-way intensity>

r r I

I

L I

L I

2 1 ) 0 ( )

0 (

) ( )

(

) ( )

(

) ( )

(

 

(11.5.10) => ( )

) ) (

( ( ) 2 ( )

)

( r I L

L I L C

I

) ( )

0 ( )

0

( 1 ( ) 1 2 ( )

)

( rI rr I L

I

(11.5.12) =>

(11.5.13) => I()(0) r2 r1I()(L)

r r r

1

1 ,

2

Total intensities are comparable at the two mirrors for reflectivity as low as 50%.

(12)

Nonlinear Optics Lab . Hanyang Univ.

11.6 Measuring Small-Signal Gain and Optimal Output Coupling

Eq. (11.2.11) for output intensity or its generalization (11.5. 18) has been shown

experimentally to be quite accurate, because the spatial hole burning effect is usually negligible in gas lasers.

In general the small signal gain and the saturation intensity are difficult to calculate accurately, because the puping and decay rates of the relavant atomic levels may not be well known.

=> Experimentally measured !

(13)

Nonlinear Optics Lab . Hanyang Univ.

<Maximal loss method to measure g0>

- The cavity loss is varied by inserting a reflecting knife-edge into the cavity - The cavity loss is increased until the laser oscillation ceases.

- (11.2.4) => g0(=gt) : loss just when the laser oscillation ceases.

<Simultaneous measurement method>

- Scattering coefficient, s=Pin/P+ - Effective output coupling : t+s - t=Pout/P+ : known => s=tPin/Pout - Ptotal=Pin+Pout : total output power

- Determine s-value for which Ptotalis maximum

=> topt=sopt + t., Ptotal=(Pin+Pout)topt

- Small signal gain : s-value at which laser oscillation stops.

(14)

Nonlinear Optics Lab . Hanyang Univ.

11.7 Inhomogeneously Broadened Laser Media

In an inhomogeneously broadened gain medium the different active atoms have different central transition frequencies 21.

# Small signal gain : Doppler broadened lineshape

 

       

] /

2 ln ) (

4 exp[

) (

2 ln 4

2 exp ln 4 1 8

) 8 (

) (

2 2 21 21

0

2 2 21 2

/ 1 0

2

0 2

0

D

D D

g A N

S A N

g







The gain coefficient is obtained by integrating the contributions from the different frequency components, each of which saturates to a different degree depending on its detuning from the cavity mode frequency .

2 21 21 2

21

0 ( ) /( ) 1 ( / )

) 1 ( )

(

2 1

 

 

d g

g sat

(15)

Nonlinear Optics Lab . Hanyang Univ.

sat 0

1

) ) (

(

I I g g

 

  



 ( ) ( )2

2

* k vzdvz21a

a x

dx

2

* 2

where, Isat h sat

21



The gain saturation set in more slowly as the intensity I is increased in the case of inhomogeneous broadening medium.





  

 

  ( ) 1

2

2 sat 0

out

gt

I g

I t

Output intensity :

cf) homogeneous medium, (11.2.9)



 

 

 ( ) 1

2

sat 0 out

gt

I g

I t

(16)

Nonlinear Optics Lab . Hanyang Univ.

11.8 Spectral Hole Burning and the Lamb Dip

<Spectral hole burning>

Spectral packets : The atom group with the central transition frequency of 2121c The gain for spectral packets with frequency 21~(field frequency) is saturated more strongly than others : spectral packets with frequency detuned from  by much more than the homogeneous linewidth, i.e., |21-|>>21, are hardly saturated at all.

Spectral hole burning (Bennet hole)

(17)

Nonlinear Optics Lab . Hanyang Univ.

<Lamb dip>

Suppose the cavity mode frequency   the center frequency of the Doppler gain profile i) The traveling-wave field propagating in the +z direction will strongly saturate the

spectral packet of atoms with Doppler-shifted frequencies ’21=.

: The Doppler effect has brought these atoms into resonance with the wave. Therefore, those atoms have the z component of velocity given by

 

c

1 v

c or v

ii) Similarly, the traveling-wave field propagating in the -z direction will strongly saturate those atoms with the z component of velocity given by

c v

(18)

Nonlinear Optics Lab . Hanyang Univ.

=> The standing wave cavity field will burn two holes in the Doppler line profile.

When the mode frequency is exactly at the center of the Doppler line, the two holes merge together. => The field can now strongly saturate only those atoms having no z component of velocity. =>

The output power exactly at resonance will be lower than slightly off resonance. : Lamb Dip.

(19)

Nonlinear Optics Lab . Hanyang Univ.

11.9 Cavity Frequency and Frequency Pulling

Cavity mode frequency : m

L mc

2

In general,

) (

) (

2 l L l n

mc

  where, l : gain medium length, n: refractive index.

or,

n

 

m

L

l 1 where,

L mc

m2

: bare(passive) cavity mode frequency

(3.3.22), (3.3.25) =>

   



  g

n

21 21 21

14

for homogeneous broadening medium

 

 

21 21

21

4 ) (

4 ) (





 

cg l L

L l

g

c m

put,

L l cg

c



4 )

( : cavity bandwidth

21 21

21 





 

 

c

m c

  

c vv21 v21 m

or : frequency pulling

(20)

Nonlinear Optics Lab . Hanyang Univ.

<Frequency pulling and gain>

In most lasers, 21c

 

21 21

21 21

21 21

21

21 1

1

v v

v v v

v

c m m

c m

c c

m c

















 

 

D c m m

D c m m

v v

 

 

 

21 21

88 . 1

2 ln 4

(homogeneous broadening)

(inhomogeneous broadening)

D

21

c

m



  

*

c gt

L cl

  and 4

: The larger the threshold gain gt, the greater the frequency pulling for fixed gain linewidth (21 or D).

(21)

Nonlinear Optics Lab . Hanyang Univ.

<Mode spacing>

21 21 ) 21

(









 

 

c

m m c

 

21 21

21 ) 1

( ) 1 (

/ 1

1

2  







 

c c

m m m

m

L c

: The effect of frequency pulling is to reduce the mode spacing from c/2L.

(22)

Nonlinear Optics Lab . Hanyang Univ.

11.11 Laser Power at Threshold

Laser power near the threshold ? => spontaneous emission.

(10.5.7) g q

L q cl Lg

cl dt dq

t

 ( )

sat 2 1 2

1

) ( 2 q

q N

P N PN N

N

T

T

 

 

 (10.11.12)

2 ( )

) (

21 21 1

2 P

N N P

N T

(Mean-field approx.)

 21 P

  cg q

L q l

L N c l

dt dq

t

 ( ) 2 1

Including the spontaneous emission :

(23)

Nonlinear Optics Lab . Hanyang Univ.

Steady-state solution :

2 2

) ( ) (

N g

q N

t  

define,

t T

t N

y N N

x N

2 and

x q x

 

1 sat sat

1

1 q q

y q

q N

x NT t

 

 

0 )

1

( sat

sat

2   

q q y q q y

sat 2

sat

) 4 1 2 (

) 1 1 2 (

1

q y y

q y

q     

gt

g y0

(11.11.1) => N2NT

<1 : below the threshold

>1 : above the threshold

1

y (far above threshold)



 

 

sat 0 1 gt

q g q

: (11.2.5), (11.2.9)

1

y (near threshold)

: (11.2.5), (11.2.9)

(10.11.8), (10.11.10), P>>21

 

PV f

e

qsat 02m 1 21

2  



(24)

Nonlinear Optics Lab . Hanyang Univ.

In many lasers, PV~103s-1, f~1, 21~10GHz => qsat~1010

1

y

 

q threshold 21qsat 4 qsat qsat 105 (very low power)

cf)

 

q thermal

eh kT 1

1 (ex) 6328 A He-Ne laser,

 

q thermal 0.023

 

thermal

)

(q t q

<The rate of change of q with y>





 

2 sat

sat sat

2 1

/ 4 ) 1 (

/ 2 1 1

q y y

q q y

dy q d

10 2

sat 1 2 1 threshold

10





q

dy q d

Extremely rapid rise in the cavity photon number at the point y=1 (threshold).

(25)

Nonlinear Optics Lab . Hanyang Univ.

11.12 Obtaining Single-Mode Oscillation

1) Short cavity length

L g

c

2

g

Ex) g~1500 MHz (He-Ne laser) => c/2L>1500MHz => L<10 cm (low power)

2) Homogeneous broadening medium

(26)

Nonlinear Optics Lab . Hanyang Univ.

3) Fabry-Perot etalon / Grating / Prism

=> Selective transmission

Ex) Fabry-Perot etalon

- resonance frequency : , 1, 2, 3,...

cos

2

m

nd m c

m

 

1 2ndcos c

m

m  

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