양자광학 - laser optics 레이저광학)optics.hanyang.ac.kr/~choh/degree/quantum...

Nonlinear Optics Lab. Hanyang Univ.

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


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

…


Einstein’s quantum theory of radiation

[light-matter interaction] * N1, 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

stimulatedemission

spontaneousemission

B12N1r B21N2rA21N2

E2

E1


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.


Einstein’s A, B coefficients

Rate equation :

0)()( 1212122122 nrnr BNBNAN

dt

dN(thermal equilibrium)

kThkTEE eeN

N //)(

1

2 12 n (Boltzman distribution of atoms)

1

18)(

/3

3

21

/

12

21

kThkTh ec

h

BeB

Ann

nnr (Planck’s blackbody radiation law)

3

3

21

212112

8,

c

h

B

ABB

n

12 NNif (population inversion)

Light amplification ! (Lasing)


Four key elements of a LASER

- Gain medium (Active medium)

- Pumping source

- Cavity (Resonator)

- Output couplerpumping laser

relaxation

relaxation

Laser light

pumping source

gain medium

cavity (resonator)

output coupler

total reflector


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


1.2 Lasers and Laser Light (Characteristics of laser light)

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


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

- Number of photons, q

bqanqdt

dq

stimulated emission

loss

- In steady state : 0 nq

tna

bn : threshold number of atoms

: Minimum(threshold) pumping condition

- Number of atoms in level 2, n

Pfnanqdt

dn

spontaneousemission

pumping

tt nfa

fbP

a

f

b

Pq 0


Spatial distribution of laser beam (Gaussian beam)

tt

HE

EH m ,

Maxwell’s curl equations

: Scalar wave equation02

22

t

EE m

Put, tiex,y,zEtzyxE )(),,,( 0 (monochromatic wave)

=> Helmholtz equation : 02

0

2

0

2

t

EE m

=>

Assume, ikzezyxE ),,(0

=> 022

2

2

2

zik

yx

Put,2/122

2

)(,]})(2

)([exp{ yxrzq

krzpi

=> q

i

dz

dp

qdz

d

q ,0)

1(

12


0)1

(1

2

qdz

d

q=> 0qzq

is must be a complex ! => q 0qAssume, is pure imaginary.

=> put, 0izzq ( : real) 0z

At z = z0,

)}0(exp{)2

exp()0(0

2

ipz

krz

Beam radius at z=0, 2/10

0 )2

(k

zw : Beam Waist

l

2

0wizq at arbitrary z,q

=>22

0

2

0

2

0

20

111

wi

Rzz

zi

zz

z

izzq

l

: Complex beam radius


q

i

dz

dp => )/(tan])/(1ln[)( 0

12/12

0 zzizzzip

=> )]/(tanexp[])/(1[

1)](exp[ 0

1

2/12

0

zzizz

zip


Wave field

)(2exp)/(tan[exp

)(exp

)(

),,( 2

0

1

2

2

00

zR

krizzkzi

zw

r

zw

w

E

zyxE

A

where,

2

0

2

0

2

2

0

2

0

2 11)(z

zw

nw

zwzw

l: Beam radius

2

0

22

0 11)(z

zz

z

nwzzR

l

: Radius of curvature of the wave front

l

2

00

nwz : Confocal parameter(2z0) or Rayleigh range


Gaussian beam

0z0wI

Gaussian profile

02w

0/2/ nwlq

spread angle :

0z

Near field

(~ plane wave)

Far field

(~ spherical wave)

z


Propagation of Gaussian beam - ABCD law

Matrix method (Ray optics)

yi

yoai

aooptical

elements

i

i

o

o y

DC

BAy

aa

DC

BA: ray-transfer matrix


1) Free space

q

r1r2

z1 z2

r2 = r1 + qd

q : constant

(paraxial ray approximation)

d

1

1

2

2

10

1

qq

rdr

q1

n1/s + n2/s’ = (n2-n1)/R

r : constant

q2 q1 n1/n2 – (1- n1/n2) (r1/R)

1

1

2

1

2

12

2

2

01

qq

r

n

n

Rn

nnr

2) Refracting surface

q2

s s’

r

n1 n2

R

…

Ray-transfer matrices


ABCD law for Gaussian beam

i

i

o

o y

DC

BAy

aa iio

iio

DCy

BAyy

aa

a

ii

ii

o

oo

DCy

BAyyR

a

a

a

)()( opticsGaussianqopticsrayRo

DCy

BAy

ii

ii

a

a

/

/

DCq

BAqq

1

12

2q1q

optical system

DC

BA

ABCD law for Gaussian beam :

0izzq

l

2

00

nwz


example) Focusing a Gaussian beam

q101w 02w

1z 2z

?

?

fz

fzzzzfz

z

f

z

DC

BA

/10

//1

10

1

1/1

01

10

1

1

21212

12

)/1(/

)/()/1(

11

2121122

fzfq

fzzzzqfzq


2

01

2

2

1

2

01

2

02

11

11

l

w

ff

z

ww

)()/()(

)(22

01

2

1

1

2

2 fwfz

fzffz

l

0201 ww - If a strong positive lens is used ; => 101

02 q

lf

w

fw

2

1

2

01 )(/ fzw l- If => fz 2

=> dfff

w

fw N

N /,2

)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.


Chapter 2. Classical Dispersion Theory

2.1 Introduction

Maxwell’s equations :t

DH,

t

B-E ,0B,0D

HμB 0 (for nonmagnetic media)

PED 0

Wave equations :

2

2

2

0

2

2

2

2

t

P

cε

1

t

E

c

1-E

(2.1.13)


2.2 The Electron Oscillator Model

)r(F),r(Er2

2

enenee

e tedt

dm

Equation of motion for the electron :

Electric-dipole approximation :

)x(F),R(Ex2

2

entedt

dm

where, xR

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

m : reduced mass

xpP NeN x),R(Ex2

2

sktedt

dm

Electron oscillator model (Lorentz model)


2.3 Refractive Index and Polarizability

x),R(Ex2

2

sktedt

dm ),R(Ex

2

02

2

tm

e

dt

d

Consider a monochromatic plane wave, )cos(Eε̂),(E 0 kzttz

)cos(/E

ε̂x22

0

0 kztme

Dipole moment : Eexp a

where, polarizability : 22

0

2 /)(

a

me

Polarization :

)cos(E/

ˆpP 0220

2

kztmNe

N


From (2.1.13),

)cos(Eˆ)(

)cos(Eˆ 00

2

2

02

22 kzt

N

ckzt

c-k

a

)()(1 22

2

0

2

22

an

c

N

ck

: dispersion relation in a medium

For a medium with the z electrons in an atom :

2/1

0

)(1)(

a

Nn : refractive index of medium

,)cos(/E

ε̂x22

0i kzt

me

i

z

i

ie1

xp

2/1

122

2

0

2/1

0

/1

)(1)(

z

i i

meNNn

a (2.3.22a)


Electric susceptibility (macroscopic parameter), :

EP 0 0/)( a N

2/1)](1[)( n

z

i im

Ne

122

0

2 1)(


2.4 The Cauchy Formula

z

i i

i

mc

Nen

122

22

2

0

2

22

41)(

ll

ll

l

From (2.3.22),

If 2

il2l

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