Post on 02-Feb-2021
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
Nonlinear Optics Lab. Hanyang Univ.
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
…
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
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.
Nonlinear Optics Lab. Hanyang Univ.
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)
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
1.2 Lasers and Laser Light (Characteristics of laser light)
Monochromaticity (단색성)- Linewidth(FWHM) : 7.5 kHz (He-Ne laser)
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
q
i
dz
dp => )/(tan])/(1ln[)( 0
12/12
0 zzizzzip
=> )]/(tanexp[])/(1[
1)](exp[ 0
1
2/12
0
zzizz
zip
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
Gaussian beam
0z0wI
Gaussian profile
02w
0/2/ nwlq
spread angle :
0z
Near field
(~ plane wave)
Far field
(~ spherical wave)
z
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
1) Free space
q
r1r2
z1 z2
r2 = r1 + qd
q : constant
(paraxial ray approximation)
d
1
1
2
2
10
1
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
r
n
n
Rn
nnr
2) Refracting surface
q2
s s’
r
n1 n2
R
…
Ray-transfer matrices
Nonlinear Optics Lab. Hanyang Univ.
Nonlinear Optics Lab. Hanyang Univ.
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
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.
Nonlinear Optics Lab. Hanyang Univ.
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)
Nonlinear Optics Lab. Hanyang Univ.
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)
Nonlinear Optics Lab. Hanyang Univ.
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
Nonlinear Optics Lab. Hanyang Univ.
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)
Nonlinear Optics Lab. Hanyang Univ.
Electric susceptibility (macroscopic parameter), :
EP 0 0/)( a N
2/1)](1[)( n
z
i im
Ne
122
0
2 1)(
Nonlinear Optics Lab. Hanyang Univ.
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