Download Resonators and Mode Matching

Resonators and Mode Matching
Mirrors
Laser
Resonator Mode
Black Box
that matches
laser mode (beam) to
resonator mode
Optical Resonator
References
Fundamentals of Photonics, Saleh and Teich, John Wiley and Sons, New
York, 1991
Chapter 1.4, 3, 9
Laser Electronics, 3rd Ed., Verdeyen, Prentice Hall, Englewood Cliffs, 1995
Ch. 2, 3, 5, 6
Laser Fundamentals, Silfvast, Cambridge UP, Cambridge, 1996
Ch. 10, 11
Lasers, Siegman, University Science Books, Mill Valley, 1986
errata: http://www-ee.stanford.edu/~siegman/lasers_book_errata.pdf
Ch. 11,12,14,15,16,17,19,
Gaussian beams
Paraxial wave equation in cylindrical coordinates
1 ∂  ∂ψ 
∂ψ
=0
r
 − i 2k
r ∂r  ∂r 
∂z
where ψ is the reduced field (longitudinal phase factored out)
This yields a solution for the electric field:
 

 w0


r 2 
kr 2 
−1  z 






E ( x , y , z ) = E0 
exp − 2  exp − i kz − tan    exp − i

(
)
w
z
 w ( z ) 
 2 R(z )
 
 z0  

longitudinal
phase
amplitude
where
 z 
w( z ) = w0 1 +  
 z0 
2
Determines radius as
function of z
  z0  2 
R( z ) = z 1 +   
  z  
Radius of curvature
of phase fronts as
function of z
A Gaussian beam is
a spherical wave with
an emanating from a
source at an
imaginary location
radial
phase
z0 =
πnw0
λ0
Either
z0 (Rayleigh range)
or
w0 (waist size)
is the only free parameter
Intensity
Intensity
0.2
0.1
θ = λ/πw0
0
-0.1
-0.2
-10
-5
0
w(z)
5
10
Phase Fronts
z
θ – far field
divergence angle
1.5
1
0.5
0
-0.5
-1
-10
-5
0
w(z)
5
Phase Fronts
10
z
Higher order modes/beams
I lied, the solution I gave before is only the “fundamental”, the general solution is
2
 

 2 x   2 y  w0



kr
E ( x, y , z )
r2 
−1  z 
H p 

 − 2  exp − i kz − (1 + m + p ) tan    exp − i
exp
= H m 

 



Em , p
 2 R( z ) 
 w (z ) 
 
 z0  
 w( z )   w( z )  w(z )
where Hm(u) are Hermite polynomials
m u
u d e
H m (u ) = (− 1) e
du m
m
H0 = 1
2
2
H1 (u ) = 2u ⇒ u
(
)
H 2 (u ) = 2u 2 − 1 2 ⇒ 2u 2 − 1
Lowest Mode (TEM00)
2
1
1
E
0.8
0.6
I
0
0.4
-1
0.2
-2
-1
1
2
-2
-2
-1
0
1
2
2
0.4
1
0.2
-2
-1
1
2
TEM10
0
-0.2
-1
-0.4
-2
2
2
-2
-1
0
1
2
1
1
1
0.5
-2
-1
1
-0.5
2
TEM20
0
0
-1
-1
-1
-2
-2
-2
-2
-1
-1
0
0
1
1
2
2
2
1
TEM21
0
-1
-2
-2
-1
0
1
2
ABCD (Ray) Matrices
(lightspeed review)
Represent propagation through optical elements
Æ system matrix is simply product of matrices for individual elements
1 d 
n

0 1 
 1
− 1
 f
0
1

 1 0

2
1
 R 
Propagation through a distance d in a medium with
index of refraction n.
A thin lens with focal length f
Reflection from a spherical mirror with radius R. R > 0 for
center of curvature in positive propagation direction
Propagation of Gaussian beams through optical elements
Characterize a beam by
q = z + iz0
λ0
z0
z
1
1
1
=
= 2
+
i
=
−
i
q z + iz0 z + z 20
z 2 + z 20 R( z ) πnw2 ( z )
 A B
 is
Propagation through an element characterized by matrix 
C
D


q2 =
Aq1 + B
Cq1 + D
Check for free space
q2 =
q1 + L
1
9
Beam parameter for a resonator
Beam in resonator must be self-consistent, i.e., the same after one round trip.
Determine the ABCD matrix for one round trip in the resonator
matrix depends on starting point
Solve equation
q=
Aq + B
Cq + D
which gives 2 solutions (using fact that AD-BC =1 for ABCD matrices)
1 D − A 1  A+ D 
=
±
 −1

2B
q±
B  2 
Then construct the proper matrix to propagate to other points inside or
outside the cavity
Resonance
d
Phase shift per round trip (ignoring phase
shifts on reflection)
E0
2θ = 2kd
E1
If this is m2π, then the Ei’s add
coherently, i.e. we are on resonance
(m is an integer)
E2
M1
(d is optical path length – includes index of refraction)
Spacing of resonances is called the Free Spectral Range (FSR)
FSR =
c
2d
Width of resonances:
∆ν
Q
F
τp
b
width
Quality factor
Finesse
cavity or photon lifetime
bounce number
M2
d
Resonance widths
The total field just inside
{
( +)
the first mirror is
(
ET+ = ∑ En+ = E0 1 + Γ1Γ2 e −ik 2 d + Γ1Γ2 e
n
)
− ik 2 d 2


1
= E0 
−i 2θ 
−
Γ
Γ
1
e
1 2


E0
E1
}
+K
E2
M1
Taylor series
where Γi is the complex reflection coefficient for mirror Mi
Intensity
(
)
I + z = 0+ = I 0
1
(1 −
where Ri = Γi
R1 R2
) +4
2
R1 R2 sin 2 θ
2
Transmission – need to include transmission of M1 and M2
T (θ ) =
(1 − R1 )(1 − R2 )
It
=
I i 1 − R R 2 + 4 R R sin 2 θ
1 2
1 2
(
)
Clearly transmission peak at θ = mπ
Finite width for R1R2 ≠ 1
M2
For R1,2 = R and on resonance
2
(
1− R)
T=
(1 − R )2
=1
Independent of R Æ constructive interference in forward direction,
destructive interference in backward direction
Width, set T = ½ and solve for θ
(1 − R1 )(1 − R2 )
1
=
2 1 − R R 2 + 4 R R sin 2 θ
1 2
1 2
(
)
sin θ = ±
1 − R1 R2
2(R1 R2 )
1
θ small Æ sin θ ~ θ =
∆ν 1
2
4
ω nd
c
c
= ν + −ν + =
2πnd
1 − R1 R2 

1 
 (R1 R2 ) 4 
Other measures of resonance linewidth
Quality (Q) Factor
2πmd (R1 R2 ) 4
ν
Q=
=
∆ν
λ0 1 − R1 R2
1
m – integer that specifies mode
Finesse
free spectral range π (R1 R2 ) 4
F=
=
FWHM
1 − R1 R2
1
Cavity/Photon Lifetime (how long it takes for cavity energy to decrease by 1/e)
τp =
Q
ω0
Bounce Number (number of times a photon bounces of a mirror before being lost)
1
b=
losses
losses = T1 + L1 + T2 + L2 = 2 − R1 − R2
Example
R1,2 = 0.99
d=1m
λ0 = 632.8 nm
FSR = 150 MHz
∆ν½ = 489 kHz
Q = 9.8 x 108
τp = 329 ns
b = 50