양자 광학 - Laser Optics

Nonlinear Optics Lab

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양자 광학

- Laser Optics ^{(레이저 광학) -}

담당 교수 : 오 차 환

교 재 : P.W. Miloni, J.H. Eberly, LASERS, John Wiley & Sons, 1991 부교재 : W. Demtroder, Laser Spectroscopy, Springer-Verlag, 1998

F. L. Pedrotti, S.J., L.S. Pedrotti, Introduction to Optics, Prentice-Hall, 1993 2008 봄학기

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Chapter 1. Introduction to Laser Operation

1.1 Introduction

LASER : Light Amplification by the Stimulated Emission of Radiation

1916, A. Einstein : predicted stimulated emission 1954, C. H. Townes et al. : developed a MASER

1958, A. Schawlow, C.H. Townes : adapted the principle of MASER to light 1960, T.H. Maiman : Ruby laser @ 694.3 nm

1961, A. Javan : He-Ne laser @ 1.15 mm, 632.8 nm

…

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Einstein’s quantum theory of radiation

[light-matter interaction] ^{* N}1, N₂ : No. of atoms at E₁, E₂

* r : photon density

* A₂₁=1/t₂₁ : spontaneous emission rate

* B₁₂, B₂₁ : stimulated absorption/emission coefficients

[radiative processes]

(stimulated) absorption

stimulated emission spontaneous

emission

B₁₂N₁r A₂₁N₂ B₂₁N₂r E₂

E₁

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Spontaneous & Stimulated emissions

Spontaneous emission

Stimulated emission

Phase and propagation

direction of created photon is random.

Created photon has the same phase,

frequency, polarization, and propagation direction as the input photon.

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Einstein’s A, B coefficients Rate equation :

0 ) ( )

( _{1 12}

21 2 21

2

2  N A  N B

r n

r n

dt

dN (thermal equilibrium)

kT h kT

E

E e

N e

N _{( )/ /}

1

2   2 1   n (Boltzman distribution of atoms)

1 1 ) 8

( _{3 /}

3

21 /

12

21

 

 _{h kT}  _{h kT}

e c

h B

e B

A

n n

n n 

r (Planck’s blackbody radiation law)

3 3

21 21 21

12

, 8

c h B

B A

B  n



 ^{if N}

^{ N}

(population inversion)

Light amplification ! (Lasing)

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Four key elements of a LASER

- Gain medium (Active medium) - Pumping source

- Cavity (Resonator) - Output coupler

pumping laser

relaxation

relaxation

Laser light

pumping source gain medium cavity (resonator)

output coupler total

reflector

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1) Pumping source

- Optical : Nd-YAG, Ruby, Dye, Ti:sapphire, … - Electrical : He-Ne, Ar⁺, CO₂, N₂, LD, …

- Chemical : HF, I₂, …

2) Active medium

- Gas : He-Ne, Ar+, CO₂, N₂, … - Liquid : Dye

- Solid : Nd-YAG, Ruby, Ti:sapphire, LD, …

3) Cavity or Resonator

- Resonator with total reflector : Maximizing the light amplification - Output coupler : Extracting a laser light

- Resonance condition : ml/2=L (m:integer)

Four key elements of a LASER

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1.2 Lasers and Laser Light (Characteristics of laser light)

Monochromaticity (단색성)

- Linewidth(FWHM) : 7.5 kHz (He-Ne laser)

<< 940 MHz (low pressure Cd lamp) Coherence (결맞음)

- Definite phase correlation in the radiation field at different locations(spatial) and different times(temporal)

Directionality (지향성)

- Divergence angle : f1.27l/D < q2.44l/D (diffraction limit angle)

Brightness (높은 휘도)

- Radiance : 10⁶ W/cm²-sr (4mW, He-Ne laser)

<< 250 W/cm²-sr (super-high-pressure Hg lamp) Focusability (집속도)

- Focusing diameter : d ~ f f

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1.5 Einstein theory of light-matter interaction (Laser action)

- Number of photons, q

bq

dt anq

dq  

stimulated emission

loss

- In steady state : q  n  0

n

a

n  b 

: Minimum(threshold) pumping condition - Number of atoms in level 2,

n

P fn

dt anq

dn    

spontaneous emission

pumping

t

t

f n

a P fb

a f b

q  P   0   

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Spatial distribution of laser beam (Gaussian beam)

t

t 

 





 

 



 H

E E

H  , m

Maxwell’s curl equations

: Scalar wave equation

2 0

2

2 



 

 t

E m E

Put, E(x, y,z,t) E₀(x,y,z)e^it (monochromatic wave)

=> Helmholtz equation : ₀

2 0 2 0

2 



 

 t

E m E

=>

Assume, E₀  (x, y,z)e^ikz

=> ₂ ^{2 0}

2 2

2  



 





 







 





ik z y

x

Put, ^{2 2 1/2}

2

) (

, )]}

( ) 2

( [

exp{ r x y

z q z kr

p

i   







=>

q i dz

dp q

dz d

q  1)  0,   1 (

2

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0 1)

1 (

2  

q dz

d

q ^=> q  z q⁰

is must be a complex ! =>

q Assume, q₀ is pure imaginary.

=> put, q  z iz₀ ( : real) z₀

At z = z₀,

)}

0 ( exp{

2 ) exp(

) 0 (

0 2

z ip

z    kr 



Beam radius at z=0, ₀ 2 ⁰)^1/2 ( k

w  z : Beam Waist





l

w₀² i

z q  

at arbitrary z, q

=> _{2 2}

0 2

0 2

0 2

0

1 1

1

i w z R

z i z z

z z iz

z

q 

 l

 

 

 

 : Complex beam radius

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q i dz

dp   _=> ip(z) ln[1(z / z₀)²]^1/2 itan^1(z/ z₀)

=> exp[ tan ( / )]

] ) / ( 1 [ )] 1 (

exp[ _{2 1/2} ¹ ₀

0

z z z i

z z

ip ^

 



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Wave field

 



 







 





 



 





 ^

) ( exp 2

) / ( tan [

) exp exp (

) ( )

, ,

( ²

0 1

2 2 0

0

z R i kr z

z kz

z i w

r z

w w E

z y x E

A

where,















 





 























2

0 2

0 2

2 0 2

0

2( ) 1 1

z w z

nw w z

z

w 

l : Beam radius











 



 





 























2 0 2 2

0 1

1 )

( z

z z z

z nw z

R l

 : Radius of curvature of the wave front

l

 ₀²

0

z  nw : Confocal parameter(2z₀) or Rayleigh range

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Gaussian beam

z

w0

I

Gaussian profile

2w0

/ 0

2

/ l nw

q 

spread angle :

 0 z

Near field (~ plane wave)

Far field

(~ spherical wave)

z

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Propagation of Gaussian beam - ABCD law

Matrix method (Ray optics)

y_i

y_o a_i

a_o

optical elements

 

 



 



 



 

 

 





i i o

o

y

D C

B y A

a

a ^{   C D   }

B

A

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1) Free space

q

r₁ r₂

z₁ z₂

r₂ = r₁ + qd q : constant

(paraxial ray approximation)

d



 



 



 



 



 





1 1 2

2

1 0 1

q q

r d

r

q₁

n₁/s + n₂/s’ = (n₂-n₁)/R r : constant

q₂  q₁ n₁/n₂ – (1- n₁/n₂) (r₁/R)



 



 











 

 



 



 

1 1

2 1 2

1 2

2 2

0 1

q q

r n

n R

n n r n

2) Refracting surface

q₂

s s’

r

n₁ R n₂

…

Ray-transfer matrices

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ABCD law for Gaussian beam

 

 



 



 



 

 

 





i i o

o

y

D C

B A

y

a

a

i i

o

D Cy

B Ay

y

a a

a









i i

i i

o o

o

Cy D

B Ay

R y

a a

a 

 



) (

)

(ray optics q Gaussian optics

R_o 

D Cy

B Ay

i i

i i



 

a a /

/

D Cq

B q Aq



 

1 1 2

q

q

optical system

 

 





D C

B A

ABCD law for Gaussian beam : iz0

z q  

l

 ₀²

0

z  nw

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example) Focusing a Gaussian beam

q₁

w01 w₀₂

z1 z₂

?

?



 











 



 







 





 



 



 



 





f z

f z z z z f z

z f

z D

C B A

/ 1

0

/ /

1

1 0 1 1 /

1

0 1

1 0 1

1 2 1 2 1 2

1 2

) / 1

( /

) / (

) / 1

(

1 1

2 1 2

1 1

2

2 q f z f

f z z z

z q

f q z













 



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2 01 2

2 1 2

01 2

02

1 1 1

1 



 



 



 



 

 l

w f

f z w

w

) ) (

/ (

) (

) (

2 2 01 2

1

1 2

2 f

w f

z

f z f f

z 





 

  l

02

01 w

w 

- If a strong positive lens is used ; => 1

01

02 q



l f w

w  f 

2 1

2

01 / (z f )

w l  



- If => z₂  f

=> f f f d

w

w f 2 ^N , _N /

) 2 (

2

01

02   

 l



l : f-number

; The smaller the f# fo the lens, the smaller the beam waist at the focused spot.

Note) To satisfy this condition, the beam is expanded before being focused.

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Chapter 2. Classical Dispersion Theory

2.1 Introduction

Maxwell’s equations :

t H D

t , - B E

, 0 B ,

0

D 

 



 

 















 μ H

B  ₀ (for nonmagnetic media) P

E D 



Wave equations :

2 2 2 0 2

2 2 2

t P ε c

1 t

E c

- 1

E 

 



  (2.1.13)

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2.2 The Electron Oscillator Model

) r ( F ) , r ( r E

2 2

en en e

e

e e t

dt

m d  

Equation of motion for the electron :

Electric-dipole approximation : ) x ( F ) , R ( x E

2 2

t en

dt e

m d  

where, x R

: relative coordinate of the e-n pair

: center-of-mass coordinate of the e-n pair

m: reduced mass

x p

P  N  Ne x

) , R ( x E

2 2

ks

t dt e

m d   

Electron oscillator model (Lorentz model) <refer p.30-31>

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2.3 Refractive Index and Polarizability

x )

, R ( x E

2 2

ks

t dt e

m d   _{2 0}² x E(R, )

2

m t e dt

d  



 



 





Consider a monochromatic plane wave, E(z,t)  εˆE₀ cos(



/ cos(

εˆ E

x 2 2

0

0 m t kz

e  







 





 

Dipole moment : p  ex a E

where, polarizability :

2 2 0

2 / )

(

  

a

Polarization :

) cos(

/ E p ˆ

P _{2 2 0}

0 2

kz m t

N Ne  



 





 







 

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From (2.1.13),

) cos(

ˆE ) ) (

cos(

ˆE ₀

0 2

2 2 0

2

2 N t kz

kz c c t

-k     



 



   



 a

 

 



0 2

2

2

 



 a



c N

k c  

 



 

 : dispersion relation in a medium

For a medium with the z electrons in an atom :

2 / 1

0

) 1 (

)

( 



 



 

 



 ^Na

n : refractive index of medium

, ) / cos(

εˆ E

x 2 2

0

i e m t kz

i

 









  







 ^z

i

e i 1

x p

2 / 1

1 2 2

2

0 2

/ 1

0

1 / )

1 ( )

( 









 

 



 



 





 z

i i

m e

N n N



 





 a



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Electric susceptibility (macroscopic parameter), :

E

P  

    N a (  ) / 

2 /

)]

( 1

[ )

(      n







i i

m Ne

1

2 2 0

2

1

)

(    





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2.4 The Cauchy Formula





 ^z

i i

i

mc n Ne

1 2 2

2 2 2

0 2

2 2

1 4 )

( l l

l l

 l 

From (2.3.22),

If l_i² << l²,







 



 ^z

i

i

mc i

n Ne

1

2 2 2

2 0 2

2

2 1

1 4 )

( l

l l

 l 

If we suppose further that | n²(l)1|  1



 

 



 







 







 2

1

2 2 2

2 0 2

2

1 1

8 1 1 )

( l l

l l



l  A ^B

mc n Ne

z

i

i i

(as in like a gas medium)

where, ,

8 ₁

2 2

0 2

2





 ^z

i

mc i

A Ne l











 _z

i i z

i i

B

1 2 1

4

l l

: Cauchy formula