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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
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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
…
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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
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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.
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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)
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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
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Nonlinear Optics Lab. Hanyang Univ.
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Nonlinear Optics Lab. Hanyang Univ.
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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
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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
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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.
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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)
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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)
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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
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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)
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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)(
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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