Download Chapter 4 Optical Resonator

Chapter 2
Optical Resonator and
Gaussian Beam optics
What is an optical resonator?
An optical resonator, the optical counterpart of an electronic resonant circuit, confines
and stores light at certain resonance frequencies. It may be viewed as an optical
transmission system incorporating feedback; light circulates or is repeatedly reflected within
the system, without escaping.
Chapter 2 Optical resonator and Gaussian beam
Contents
• 2.1 Matrix optics
• 2.2 Planar Mirror Resonators
– Resonator Modes
– The Resonator as a Spectrum Analyzer
– Two- and Three-Dimensional Resonators
• 2.3 Gaussian waves and its characteristics
– The Gaussian beam
– Transmission through optical components
• 2.4 Spherical-Mirror Resonators
– Ray confinement
– Gaussian Modes
– Resonance Frequencies
– Hermite-Gaussian Modes
– Finite Apertures and Diffraction Loss
Chapter 2 Optical resonator and Gaussian beam
2.1
Brief review of Matrix optics
Light propagation in a optical system, can use a matrix M, whose elements are A,
B, C, D, characterizes the optical system Completely ( known as the ray-transfer
matrix.) to describe the rays transmission in the optical components.
One can use two
parameters:
Chapter 2 Optical resonator and Gaussian beam
y:
the high
q:
the angle above z axis
d
q2
y2  y1  d  tgq1
q1
q 2  q1
y2
y1
For the paraxial rays
tgq  q
 y2  1 d   y1 
q   0 1  q 
 1
 2 
y2,q2
-q2
y1,q1
q
q1
y2  y1
2
q 2  y1  q1
R
y1
q
-R
Along z upward angle is positive, and downward is negative
Chapter 2 Optical resonator and Gaussian beam
Free-Space Propagation
Refraction at a Planar Boundary
1
M 
0

1 d 
M 

0 1 
Refraction at a Spherical Boundary
1

M   (n2 - n1 )
n2 R

Transmission Through a Thin Lens
 1
M  1
 f
0
n1 
n2 
Reflection from a Planar Mirror
0
n1 
n2 
0

1

Reflection from a Spherical Mirror
1 0 
M 

0 1 
Chapter 2 Optical resonator and Gaussian beam
1
M 2

R
0

1

A Set of Parallel Transparent Plates.

1
M 

0
di 
n 

1 
i
Matrices of Cascaded Optical Components
M  M N M N -1....M1
Chapter 2 Optical resonator and Gaussian beam
Periodic Optical Systems
The reflection of light between two parallel mirrors forming an optical
resonator is a periodic optical system is a cascade of identical unit system.
Difference Equation for the Ray Position
A periodic system is composed of a cascade of identical unit systems
(stages), each with a ray-transfer matrix (A, B, C, D). A ray enters the system
with initial position y0 and slope q0. To determine the position and slope
(ym,qm) of the ray at the exit of the mth stage, we apply the ABCD matrix m
times,
 ym   A
q   C
 m 
m
B   y0 
D  q 0 
ym1  Aym  Bq m
q m1  Cym  Dq m
Chapter 2 Optical resonator and Gaussian beam
ym1  Aym  Bq m
q m1  Cym  Dq m
From these equation, we have
qm 
So that
ym 1 - Aym
B
q m1 
ym 2 - Aym1
B
And then:
ym 2  2bym1 - F 2 ym
linear differential equations,
where
b
 A  D
2
and
F 2  Ad - BC  det  M 
Chapter 2 Optical resonator and Gaussian beam
If we assumed:
y m  y0 h m
So that, we have
h 2 - 2bh  F 2  0
If we defined
We have
then
h  b  i F 2 - b2
 
  cos-1 b F
b  F cos 
F 2 - b2  F sin 
h  F (cos   i sin  )  Fei
ym  y0 F m e  im
A general solution may be constructed from the two solutions with positive
and negative signs by forming their linear combination. The sum of the two
exponential functions can always be written as a harmonic (circular) function,
ym  y0 F m sin(m  0 )  ymax F m sin(m  0 )
Chapter 2 Optical resonator and Gaussian beam
If F=1, then
ym  ymax sin(m  0 )
Condition for a Harmonic Trajectory: if ym be harmonic, the cos-1b must
be real, We have condition
b 1
or
A D
1
2
The bound b  1 therefore provides a condition of stability (boundedness) of
the ray trajectory
If, instead, |b| > 1,  is then imaginary and the solution is a hyperbolic
function (cosh or sinh), which increases without bound. A harmonic solution
ensures that y, is bounded for all m, with a maximum value of ymax. The
bound |b|< 1 therefore provides a condition of stability (boundedness) of
the ray trajectory.
Chapter 2 Optical resonator and Gaussian beam
Condition for a Periodic Trajectory
Unstable b>1
Stable and periodic
Stable nonperiodic
The harmonic function is periodic in m, if it is possible to find an integer s such
that ym+s = ym, for all m. The smallest such integer is the period.
The necessary and sufficient condition for a periodic trajectory is:
s = 2pq,
where q is an integer
Chapter 2 Optical resonator and Gaussian beam
EXERCISE
A Periodic Set of Pairs of Different Lenses. Examine the trajectories of paraxial
rays through a periodic system composed of a set of lenses with alternating focal
lengths f1 and f2 as shown in Fig. Show that the ray trajectory is bounded (stable)
if
 1
M  1
 f 2
0
 1
 1 d   1
1  0 1   
 f1

d
10

f1
 1 d   


1  0 1   d
1 1
- 

 f1 f 2 f1 f 2



d
d
d 
-  (1 - )(1 - ) 
f2
f1
f2 
2d -
d2
f1
Chapter 2 Optical resonator and Gaussian beam
Home work
1. Ray-Transfer Matrix of a Lens System. Determine the
ray-transfer matrix for an optical system made of a thin
convex lens of focal length f and a thin concave lens of
focal length -f separated by a distance f. Discuss the
imaging properties of this composite lens.
Chapter 2 Optical resonator and Gaussian beam
Home works
2. 4 X 4 Ray-Transfer Matrix for Skewed Rays. Matrix methods may be generalized to
describe skewed paraxial rays in circularly symmetric systems, and to astigmatic (noncircularly symmetric) systems. A ray crossing the plane z = 0 is generally characterized by
four variables-the coordinates (x, y) of its position in the plane, and the angles (e,, ey) that
its projections in the x-z and y-z planes make with the z axis. The emerging ray is also
characterized by four variables linearly related to the initial four variables. The optical
system may then be characterized completely, within the paraxial approximation, by a 4 X
4 matrix.
(a) Determine the 4 x 4 ray-transfer matrix of a distance d in free space.
(b) Determine the 4 X 4 ray-transfer matrix of a thin cylindrical lens with focal length f
oriented in the y direction. The cylindrical lens has focal length f for rays in the y-z plane,
and no focusing power for rays in the x-z plane.
Chapter 2 Optical resonator and Gaussian beam
2.2 Planar Mirror Resonators
Charles Fabry (1867-1945),
Alfred Perot (1863-1925),
Chapter 2 Optical resonator and Gaussian beam
2.2 Planar Mirror Resonators
This simple one-dimensional
resonator is known as a
Fabry-Perot etalon.
A. Resonator Modes
Resonator Modes as Standing Waves
A monochromatic wave of frequency v has a wavefunction as
u(r, t )  Re U (r )exp(i2p vt )
Represents the transverse component of electric field.
The complex amplitude U(r) satisfies the Helmholtz equation;
Where k =2pv/c called wavenumber, c speed of light in the medium
Chapter 2 Optical resonator and Gaussian beam
the modes of a resonator must be the solution of Helmholtz equation with
the boundary conditions:
z  0
U (r )  0

z  d
So that the general solution is standing wave:
U (r )  A sin kz
With boundary condition, we have
F 
kd  qp
q is integer.
c
2d
qp
kq 
d

 q  q 1
Resonance frequencies
∵
c
 q  q , q  1, 2,...,
2d
c
 F   q - q -1 
2d
Chapter 2 Optical resonator and Gaussian beam
d
k  2p  
2p v
c
q 
The resonance wavelength is:
The length of the resonator, d = q q /2,
wavelength
Attention:
c  c0 / n
c
q
 2d
q
is an integer number of half
Where n is the refractive index in the resonator
Resonator Modes as Traveling Waves
A mode of the resonator: is a self-reproducing wave, i.e., a wave that reproduces
itself after a single round trip , The phase shift imparted by a single round trip of
propagation (a distance 2d) must therefore be a multiple of 2p.
  k 2d 
4p n
0
d
4p
d  q 2p
c
Chapter 2 Optical resonator and Gaussian beam
q= 1,2,3,…
Density of Modes (1D)
The density of modes M(v), which is the number of
modes per unit frequency per unit length of the resonator, is
M ( ) 
4
c
For 1D resonator
The number of modes in a resonator of length d within the frequency interval v is:
4
d 
c
This represents the number of degrees of freedom for the optical waves existing in
the resonator, i.e., the number of independent ways in which these waves may be
arranged.
Chapter 2 Optical resonator and Gaussian beam
Losses and Resonance Spectral Width
The magnitude ratio of two consecutive phasors is the round-trip
amplitude attenuation factor r introduced by the two mirror reflections and
by absorption in the medium. Thus:
Mirror 1
U1  hU 0   e U 0   e
- i
-i
4p nd

U3
U 0   e-i 2 kdU 0
U2
So that, the sum of the sequential reflective light with field of
U1
U0
U  U 0  U1  U 2  U3  ...  U 0 (1  h  h  h  ...) 
2
IU 
2
U0
finally, we have
p 1/ 2
F 
1- 
2
1-  e
- i 2
I

I0
(1   2 - 2 cos  )

I0
3
U0
(1 - h)
 
(1 -  ) 2  4 sin 2  
2 

I max
,
2
2
1  (2F / p ) sin ( / 2)
I max 
Finesse of the resonator
Chapter 2 Optical resonator and Gaussian beam
I0
(1 -  ) 2
Mirror 2
The resonance spectral peak has a full width of half maximum (FWHM):


  c 4p d    F F
Due to
4p d

c
We have
where
I max
I max
I

I
min
2
2
1  (2F / p ) 2
1  (2F / p ) sin (p /  F )
 F  c 2d
   q  q F , q  1, 2,...,
c
F 
2d
 
F
F
Chapter 2 Optical resonator and Gaussian beam
I max
I
1  (2 F / p ) 2 sin 2 ( / 2)
I max
1
I max 
2
1  (2 F / p ) 2 sin 2 ( / 2)
(2F / p )2 sin 2 (10 / 2)  1
sin(10 / 2)  p / 2F
   p / F
1 0
Full width half maximum is
∵

4p d
c
210  2p / F  
So that


  c 4p d    F F
Chapter 2 Optical resonator and Gaussian beam
Spectral response of Fabry-Perot Resonator
The intensity I is a periodic function of  with period 2p. The dependence of I
on , which is the spectral response of the resonator, has a similar periodic
behavior since  = 4pd/c is proportional to . This resonance profile:
I max
I
1  (2 F / p ) 2 sin 2 (p / F )
The maximum I = Imax, is achieved at the
resonance frequencies
whereas the minimum value
The FWHM of the resonance peak is
   q  q F , q  1, 2,...,
I min 
 
I max
1  (2 F / p ) 2
c
4p d
 
F
F
Chapter 2 Optical resonator and Gaussian beam
Sources of Resonator Loss
•
Absorption and scattering loss during the round trip: exp (-2asd)
•
Imperfect reflectance of the mirror: R1, R2
  R1 R2 exp(-2a s d )
2
Defineding that
we get:
ar is an effective overall distributedloss coefficient, which is used
generally in the system design and
analysis
ar  as 
 2  exp(-2a r d )
ar  as 
1
1
ln
2d R1 R2
1
1
ln
 a s  a m1  a m 2
2d R1 R2
a m1 
1
1
ln
2d R1
Chapter 2 Optical resonator and Gaussian beam
a m1 
1
1
ln
2d R1
•
•
If the reflectance of the mirrors is very
high, approach to 1, so that
The above formula can approximate
as
ar  as 
R1  1  R2  R
a m1 
1
1
ln
 a s  a m1  a m 2
2d R1 R2
1 - R1 1 - R2
1- R

 am2 
2d
2d
2d
ar  as 
1- R
d
The finesse F can be expressed as a function of the effective loss coefficient ar,
F 
p exp(-a r d / 2)
1 - exp(-a r d )
Because ard<<1, so that exp(-ard)=1-ard, we have:
The finesse is inversely proportional to the loss factor ard
Chapter 2 Optical resonator and Gaussian beam
F 
p
ar d
Photon Lifetime of Resonator
The relationship between the resonance linewidth and the resonator loss may be
viewed as a manifestation of the time-frequency uncertainty relation. Form the
linewidth of the resonator, we have
 
ca
c / 2d
 r
p / a r d 2p
Because ar is the loss per unit length, car is the loss
per unit time, so that we can Defining the characteristic
decay time as the resonator lifetime or photon lifetime
The resonance line broadening is seen to be governed
by the decay of optical energy arising from
resonator losses
Chapter 2 Optical resonator and Gaussian beam
p 
1
ca r
 
1
2p p
The Quality Factor Q
The quality factor Q is often used to characterize electrical resonance circuits
and microwave resonators, for optical resonators, the Q factor may be determined
by percentage of that stored energy to the loss energy per cycle:
Q
2p ( storedenergy)
energylosspercycle
Large Q factors are associated with low-loss resonators
For a resonator of loss at the rate car (per unit time), which is equivalent to the rate
car /0 (per cycle), so that

Q  2p  1
 (ca r /  0 ) 
  ca r 2p
Q
0

The quality factor is related to the resonator lifetime (photon lifetime)
1
 p  1 ca 
r
2p
The quality factor is related to the finesse of the resonator by
Chapter 2 Optical resonator and Gaussian beam
Q  2p 0 p
Q
0
F
F
• In summary, three parameters are convenient for
characterizing the losses in an optical resonator:
– the finesse F
– the loss coefficient ar (cm-1),
– photon lifetime p = 1/car, (seconds).
• In addition, the quality factor Q can also be used for this
purpose
Chapter 2 Optical resonator and Gaussian beam
B. The Resonator as a Spectrum Analyzer
Transmission of a plane wave across a planar-mirror resonator (Fabry-Perot etalon)
t1
r1
r2
t2
U2
T ( ) 
It
T ( ) 
Tmax
1  (2F / p ) 2 sin 2 (p /  F )
Where:
Tmax 
U1
U0
Mirror 1
Mirror 2
I
t
2
(1 -  ) 2
, t  t1t2 ,    1 2
p 1/2
F 
1- 
The change of the length of the cavity
will change the resonance frequency

 q  - qc
2d
2

d 
Chapter 2 Optical resonator and Gaussian beam
- q d
d
C. Two- and Three-Dimensional Resonators
• Two-Dimensional Resonators
ky 
q yp
d
, kz 
• Mode density
qzp
, q y  1, 2,..., qz  1, 2,...,
d
k 2  k y2  k z2  (
2p 2
)
c
the number of modes per unit frequency per unit surface of the resonator
Determine an approximate expression for the number of modes in a twodimensional resonator with frequencies lying between 0 and , assuming that
2p/c >> p/d, i.e. d >>/2, and allowing for two orthogonal polarizations per
mode number.
Chapter 2 Optical resonator and Gaussian beam
M ( ) 
4p
c2
Three-Dimensional Resonators
Wave vector space
Physical space resonator
kx 
qp
qxp
qp
, k y  y , k z  z , qx , q y , qz  1, 2,...,
d
d
d
Mode density
8p 2
M ( )  3
c
k 2  k x2  k y2  k z2  (
2p 2
)
c
The number of modes lying in the frequency interval
between 0 and v corresponds to the number of points
lying in the volume of the positive octant of a sphere of
radius k in the k diagram
Chapter 2 Optical resonator and Gaussian beam
Optical resonators and stable condition
• A. Ray Confinement of spherical resonators
The rule of the sign: concave mirror (R < 0), convex (R > 0). The
planar-mirror resonator is R1 = R2=∞
The matrix-optics methods introduced which are
valid only for paraxial rays, are used to study the
trajectories of rays as they travel inside the
resonator
z
d
Chapter 2 Optical resonator and Gaussian beam
B. Stable condition of the resonator
For paraxial rays, where all angles are small, the relation between (ym+1, qm+1) and
(ym, qm) is linear and can be written in the matrix form
R2
R1
y1
 ym1   A B   ym 
q   C D  q 
 m
 m1  
-q 1
z
y2
A B 1
C D    2

  R1
0  1 d   1
1  0 1   R22
0  1 d 
1   0 1 
q2
q0
y0
d
reflection
from a
mirror of
radius R1
reflection
from a
mirror of
radius R2
propagation a
distance d through
free space
Chapter 2 Optical resonator and Gaussian beam
A  1  2d
R2
B  2d (1  d
C 2
R1
D  2d
 2
R1
det M  Ad - BC  1  F 2
R2
R2
 ( 2d
)
 4d
R1
ym  ymax sin(m  0 )
R1 R2
 1)(2d
R1
 1)

d 
d 
b  ( A  D) / 2  2 1  1   - 1
 R1  R2 
It the way is harmonic, we need  cos-1b must be real, that is
b 1

d 
d 
b  ( A  D) / 2  2 1  1   - 1  1
 R1  R2 
for g1=1+d/R1; g2=1+d/R2

d 
d 
0  1 
 1 
 1
R1  
R2 

0  g1 g 2  1
Chapter 2 Optical resonator and Gaussian beam
resonator is in conditionally stable, there will be:

d 
d 
0  1  1    1
 R1  R2 
0  g1 g 2  1
In summary, the confinement condition for paraxial rays in a sphericalmirror resonator, constructed of mirrors of radii R1,R2 seperated by a
distance d, is 0≤g1g2≤1, where g1=1+d/R1 and g2=1+d/R2
For the concave R is negative, for the convex R is positive
Chapter 2 Optical resonator and Gaussian beam
Stable and unstable resonators
a. Planar
g2
(R1= R2=∞)
b. Symmetrical confocal
e
1
d
(R1= R2=-d)
a
c. Symmetrical concentric
stable
b
-1
0
1
g1
(R1= R2=-d/2)
d. confocal/planar
c
Non stable
(R1= -d,R2=∞)
e. concave/convex
(R1<0,R2>0)
d/(-R) = 0, 1, and 2, corresponding to planar, confocal, and concentric resonators
Chapter 2 Optical resonator and Gaussian beam
The stable properties of optical resonators
Crystal state resonators
g1 g2  0
or
g1 g2  1
a. Planar
(R1= R2=∞)
b. Symmetrical confocal
(R1= R2=-d)
Stable
c. Symmetrical concentric
unstable
(R1= R2=-d/2)
Chapter 2 Optical resonator and Gaussian beam
Unstable resonators
g1 g2  0
or
Unstable cavity corresponds to the high loss
g1 g2  1
a. Biconvex resonator
d
b. plan-convex resonator
c. Some cases in plan-concave resonator
When R2<d, unstable
R1
d. Some cases in concave-convex resonator
d
When R1<d and R1+R2=R1-|R2|>d
e. Some cases in biconcave resonator
g1 g 2  (1  d / R1 )(1  d / R2 )  0
g1 g 2  (1  d / R1 )(1  d / R2 )  1
R1  R2  d
Chapter 2 Optical resonator and Gaussian beam
Home works
1. Resonance Frequencies of a Resonator with an Etalon. (a) Determine the
spacing between adjacent resonance frequencies in a resonator constructed
of two parallel planar mirrors separated by a distance d = 15 cm in air (n = 1).
(b) A transparent plate of thickness d, = 2.5 cm and refractive index n = 1.5 is
placed inside the resonator and is tilted slightly to prevent light reflected from
the plate from reaching the mirrors. Determine the spacing between the
resonance frequencies of the resonator.
2. Semiconductor lasers are often fabricated from crystals whose surfaces are
cleaved along crystal planes. These surfaces act as reflectors and therefore
serve as the resonator mirrors. Consider a crystal with refractive index n = 3.6
placed in air (n = 1). The light reflects between two parallel surfaces separated
by the distance d = 0.2 mm. Determine the spacing between resonance
frequencies vf, the overall distributed loss coefficient ar, the finesse , and the
spectral width ᅀv. Assume that the loss coefficient as= 1 cm-1.
3. What time does it take for the optical energy stored in a resonator of finesse =
100, length d = 50 cm, and refractive index n = 1, to decay to one-half of its
initial value?
9.1-1, 9.1-2, 9.1-4, 9.1-5, 9.2-2, 9.2-3, 9.2-5 chapter 9
Chapter 2 Optical resonator and Gaussian beam
2.3 Gaussian waves and its
characteristics
The Gaussian beam is
named after the great
mathematician Karl
Friedrich Gauss
(1777- 1855)
Chapter 2 Optical resonator and Gaussian beam
A. Gaussian beam
The electromagnetic wave propagation is under the way of Helmholtz equation
 2U  k 2U  0
Normally, a plan wave (in z direction) will be
U  U 0 exp{-i(t  k  r)}  U 0 exp(-ikz ) exp(-it )
When amplitude is not constant, the wave is
U  A( x, y, z ) exp( -ikz ) exp( -it )
An axis symmetric wave in the amplitude
U  A(  , z ) exp(-ikz ) exp( -it )
frequency
  2p
Wave vector
Chapter 2 Optical resonator and Gaussian beam
z
k
2np

Paraxial Helmholtz equation
Substitute the U into the Helmholtz equation we have:
A
 A - i 2k
0
z
2
T
where
2T
2
2
 2 2
x y
One simple solution is spherical wave：
A1
2
A(r )  exp(- jk )
z
2z
 2  x2  y 2
Chapter 2 Optical resonator and Gaussian beam
A
 A - i 2k
0
z
The equation
2
T
has the other solution, which is Gaussian wave：
W0
2
2
U (r )  A0
exp[- 2 ]exp[-ikz - ik
 i ( z )]
W ( z)
W ( z)
2 R( z )
where
z
W ( z )  W0 [1  ( ) 2 ]1/ 2
z0
z0 is Rayleigh range
z0 2
R( z )  z[1  ( ) ]
z
z
 ( z )  tan -1
q parameter
1
1


-i
q( z ) R( z ) p W 2 ( z )
z0
 z0 1/2
W0  (
)  W ( z)
p
z 0
 W (0)
Chapter 2 Optical resonator and Gaussian beam
Gaussian Beam
E
Beam
radius
z
z=0
Chapter 2 Optical resonator and Gaussian beam
Electric field of Gaussian wave propagates in z direction
A0
-( x 2  y 2 )
x2  y 2
E ( x, y, z ) 
exp[
]  exp[ -ik (
 z)  i ( z)]
2
W ( z)
W ( z)
2 R( z )
Physical meaning of parameters
 Beam width at z
 Waist width
z 2 1/ 2
W ( z )  W0 [1  ( ) ]
z0
W0  W (0)
 Radii of wave front at z
 Phase factor
p W02
z0 

p W02 2
z
R( z )  z[1  (
) ]  z[1  ( 0 ) 2 ]
z
z
z
-1 z
 ( z )  arctan
 tg
2
p W0
z0
Chapter 2 Optical resonator and Gaussian beam
Gaussian beam at z=0
A0
r2
E ( x, y,0) 
exp[ - 2 ] where
W0
W0
r 2  x2  y 2
E
A0
W0
Beam width:
W ( z )  W0 [1  (
z 2 1/ 2
) ]
z0
will be minimum
A0
eW0
wave front
2 2 



 
p
W

0
lim R ( z )  lim  z 1  
   
z 0
z 0
 z   

  

-W0
W0
at z=0, the wave front of Gaussian beam is a plan surface, but the
electric field is Gaussian form
W0 is the waist half width
Chapter 2 Optical resonator and Gaussian beam
B. The characteristics of Gaussian beam
Beam
radius
z
Gaussian beam is a axis symmetrical wave, at z=0
phase is plan and the intensity is Gaussian form, at the
other z, it is Gaussian spherical wave.
Chapter 2 Optical resonator and Gaussian beam
Intensity of Gaussian beam
• Intensity of Gaussian beam
W0 2
2
I ( , z)  I0[
] exp[- 2 ]
W ( z)
W ( z)
z=0
z=z0
y
z=2z0
y
y
x
I
I0
I
I0
1
0
x
-1
0
1
W0

I
I0
1
0
x
-1
0
1
W0

1
0
-1
0
1
W0

The normalized beam intensity as a function of the radial distance at different
axial distances
Chapter 2 Optical resonator and Gaussian beam
On the beam axis ( = 0) the intensity
I
I0
1
Variation of axial intensity as
the propagation length z
W0 2
I0
I (0, z )  I 0 [
] 
z 2
W ( z)
1 ( )
z0
1
0.5
- zo
0
0
zo
z
z0 is Rayleigh range
The normalized beam intensity I/I0at points on the beam
axis (=0) as a function of z
Chapter 2 Optical resonator and Gaussian beam
p W02
z0 

Power of the Gaussian beam
The power of Gaussian beam is calculated by the integration of the optical
intensity over a transverse plane
P
1
I 0p W02
2
So that we can express the intensity of the beam by the power
2P
2 2
I (  , z) 
exp[- 2 ]
2
pW ( z )
W ( z)
The ratio of the power carried within a circle of radius . in the
transverse plane at position z to the total power is
202
1 0
I (  , z )2p d   1 - exp[- 2 ]

0
P
W ( z)
Chapter 2 Optical resonator and Gaussian beam
Beam Radius
W ( z )  W0 [1  (
z 2 1/ 2
) ]
z0
W ( z) 
W(z)
W0
z  q0 z
z0
p W02
∵ z0 

Beam
waist
2W 0
q0 
W0
q0
-z0
z0
z
The beam radius W(z) has its minimum value W0 at the waist (z=0)
reaches 2W0 at z=±z0 and increases linearly with z for large z.
Beam Divergence
1
dW ( z ) 2z p 2W02 2
2q  2

[(
)  z2 ] 2
dz
pW0

q0 

p W0
Chapter 2 Optical resonator and Gaussian beam

p W0
The characteristics of divergence angle
• z=0， 2q =0
•
2
p
W
z= 0
• z
2q =
  z0
2q =
2 / pW0
2
p W0
or
2W ( z )
2q  lim
x 
z
z0 is Rayleigh range
Define f=z0 as the confocal parameter of Gaussian beam
pW02
f  z0 

The physical means of f ：the half distance between two section of width
f
z2
2
(1  2 )
p
f
2W ( z )

2q  lim
 lim
2
z 
z 
z
z
fp
Chapter 2 Optical resonator and Gaussian beam
Depth of Focus
Since the beam has its minimum width at z = 0, it achieves its best focus at the
plane z = 0. In either direction, the beam gradually grows “out of focus.” The
axial distance within which the beam radius lies within a factor 20.5 of its
minimum value (i.e., its area lies within a factor of 2 of its minimum) is known
as the depth of focus or confocal parameter
20.5o
o
0
2zo
The depth of focus of a Gaussian beam.
Chapter 2 Optical resonator and Gaussian beam
z
2 z0 
2p W02

2f
Phase of Gaussian beam
The phase of the Gaussian beam is,
z
-1 z
 ( z )  arctan
 tg
2
p W0
z0
k2
 (  , z )  kz -  ( z ) 
2 R( z )
On the beam axis (p = 0) the phase
 (0, z )  kz -  ( z )
kz
 ( z)
Phase of plan wave
an excess delay of the wavefront in comparison with a plane
wave or a spherical wave
The excess delay is –p/2 at z=-∞, and p/2 at z= ∞
The total accumulated excess retardation as the wave travels from z = -∞
to z =∞is p. This phenomenon is known as the Guoy effect.
Chapter 2 Optical resonator and Gaussian beam
Wavefront
pW02 2
f2
R( z )  z[1  (
) ] z
z
z
Confocal field and its equal phase front
Chapter 2 Optical resonator and Gaussian beam
Parameters Required to Characterize a Gaussian Beam
How many parameters are required to describe a plane wave, a spherical
wave, and a Gaussian beam?

The plane wave is completely specified by its complex amplitude
and direction.

The spherical wave is specified by its amplitude and the location of
its origin.

The Gaussian beam is characterized by more parameters- its peak
amplitude the parameter A, its direction (the beam axis), the
location of its waist, and one additional parameter: the waist radius
W0 or the Rayleigh range zo,
Chapter 2 Optical resonator and Gaussian beam
Parameter used to describe a Gaussian beam
 q-parameter is sufficient for characterizing a Gaussian
beam of known peak amplitude and beam axis
1
1


-i
q( z ) R( z ) p W 2 ( z )

1
1

q( z ) z  iz0
q(z) = z + iz0
If the complex number q(z) = z + iz0, is known, the distance z to the beam
waist and the Rayleigh range z0. are readily identified as the real and
imaginary parts of q(z).
the real part of q(z) z is the beam waist place
the imaginary parts of q(z) z0 is the Rayleigh range
Chapter 2 Optical resonator and Gaussian beam
C. TRANSMISSION THROUGH OPTICAL COMPONENTS
a). Transmission Through a Thin Lens
Phase +phase induce by lens must equal to the back phase
kz  k
2
2R
- - k
2
2f
 kz  k
2
2R '
-
1 1 1
 R' R f
1 1 1
- 
R R' f
Notes:
R is positive since the wavefront of the incident beam is diverging and R’ is
negative since the wavefront of the transmitted beam is converging.
Chapter 2 Optical resonator and Gaussian beam
In the thin lens transform, we have
W W '
1 1 1
 R' R f
If we know W0 , z1 , f
we can get
2
p
W
W0'2  W 2 [1  (
) 2 ]-1
R '
 R ' 2 -1
- z '  R '[1  (
) ]
2
pW
The minus sign is due to the waist lies to the right of the lens.
Chapter 2 Optical resonator and Gaussian beam
W0 ' 
W
[1  (p W 2 /  R ')2 ]1/ 2
because R  z[1  ( z0 / z ) 2 ]
-z ' 
R'
1  (p R '/ W 2 ) 2
W  W0 [1  ( z / z0 ) 2 ]1/2
and
W0 '  MW0
Waist radius
Waist location
( z '- f )  M ( z - f )
2
Depth of focus
2 z0'  M 2 (2 z0 )
Divergence angle
2q 0'  M 2 (2q 0 )
，
magnification
Mr
M
(1  r 2 )1/ 2
The beam waist is
magnified by M, the
beam depth of focus is
magnified by M2, and the
angular divergence is
minified by the factor M.
where
The formulas for lens transformation
Chapter 2 Optical resonator and Gaussian beam
r
z0
z- f
Mr 
f
z- f
Limit of Ray Optics
Consider the limiting case in which (z - f) >>zo, so that the lens is well
outside the depth of focus of the incident beam, The beam may then be
approximated by a spherical wave, thus
r
z
z’
z0
0
z- f
M  Mr
and
W0 '  MW0
2W0
1 1 1
 
z' z f
2W0’
Imaging relation
M  Mr 
f
z- f
The magnification factor Mr is that based on ray optics. provides that M < Mr,
the maximum magnification attainable is the ray-optics magnification Mr.
Chapter 2 Optical resonator and Gaussian beam
b). Beam Shaping
If a lens is placed at the waist of a Gaussian beam, so z=0,
then
1
M
∵
[1  ( z0 / f ) 2 ]1/2
Beam Focusing
W0 ' 
W0
[1  ( z0 / f ) 2 ]1/ 2
z'
f
1  ( f / z0 ) 2
If the depth of focus of the incident beam 2z0, is much longer than the focal
length f of the lens, then W0’= ( f/zo)Wo. Using z0 =pW02/, we obtain
W0 ' 

p W0
f  q0 f
z' f
The transmitted beam is then focused at the lens’ focal plane as would be
expected for parallel rays incident on a lens. This occurs because the incident
Gaussian beam is well approximated by a plane wave at its waist. The spot size
expected from ray optics is zero
Chapter 2 Optical resonator and Gaussian beam
Focus of Gaussian beam
W '02 
For given f, W '02 changes as
•when
z1  f
•when
•when
•
z1  f
z1  f
z f
W02 2
z1
(1 - )  ( )
f
f
W '02 decreases as z decreases
z1  0 W0 '
•when
W02
reaches minimum, and M<1, for f>0, it is focal effect
,
W0 '
increases as z increases
the bigger z, smaller f, better focus
W '0 reaches maximum, when
focus
Chapter 2 Optical resonator and Gaussian beam
p W02
 f

, it will be
In laser scanning, laser printing, and laser fusion, it is desirable to generate the
smallest possible spot size, this may be achieved by use of the shortest
possible wavelength, the thickest incident beam, and the shortest focal length.
Since the lens should intercept the incident beam, its diameter D must be at
least 2W0. Assuming that D = 2Wo, the diameter of the focused spot is given by
2W0 ' 
4
p
 F#
F# 
f
D
where F# is the F-number of the lens. A microscope objective with small Fnumber is often used.
Chapter 2 Optical resonator and Gaussian beam
Beam collimate
locations of the waists of the incident and transmitted beams, z and z’ are
z'
z / f -1
-1 
f
( z / f - 1) 2  ( z0 / f ) 2
The beam is collimated by making the location of the new waist z’ as
distant as possible from the lens. This is achieved by using the smallest
ratio z0/f (short depth of focus and long focal length).
Chapter 2 Optical resonator and Gaussian beam
Beam expanding
A Gaussian beam is expanded and collimated using two lenses of
focal lengths fi and f2,
Assuming that f1<< z and z - f1>> z0, determine the optimal distance
d between the lenses such that the distance z’ to the waist of the final
beam is as large as possible.
overall magnification M = W0’/Wo
Chapter 2 Optical resonator and Gaussian beam
C). Reflection from a Spherical Mirror
Reflection of a Gaussian beam of curvature R1 from a mirror of curvature R:
W2  W1
1
1 2
 
R2 R1 R
f = -R/2.
R > 0 for convex mirrors and R < 0 for concave mirrors,
Chapter 2 Optical resonator and Gaussian beam
R



R1  
R1  - R
If the mirror is planar, i.e., R =∞, then R2= R1, so that the mirror reverses
the direction of the beam without altering its curvature
If R1= ∞, i.e., the beam waist lies on the mirror, then R2= R/2. If the mirror is
concave (R < 0), R2 < 0, so that the reflected beam acquires a negative
curvature and the wavefronts converge. The mirror then focuses the beam
to a smaller spot size.
If R1= -R, i.e., the incident beam has the same curvature as the mirror, then
R2= R. The wavefronts of both the incident and reflected waves coincide
with the mirror and the wave retraces its path. This is expected since the
wavefront normals are also normal to the mirror, so that the mirror reflects
the wave back onto itself. the mirror is concave (R < 0); the incident wave is
diverging (R1 > 0) and the reflected wave is converging (R2< 0).
Chapter 2 Optical resonator and Gaussian beam
d). Transmission Through an Arbitrary Optical System
An optical system is completely characterized by the matrix M of elements (A, B, C, D)
ray-transfer matrix relating the position and inclination of the transmitted ray to those
of the incident ray
The q-parameters, q1 and q2, of the incident and transmitted
Gaussian beams at the input and output planes of a paraxial
optical system described by the (A, B, C, D) matrix are related by
Chapter 2 Optical resonator and Gaussian beam
ABCD law
The q-parameters, q1 and q2, of the incident and transmitted Gaussian beams at
the input and output planes of a par-axial optical system described by the (A, B,
C, D) matrix are related by
Aq1  B
q2 
Cq1  D
Because the q parameter identifies the width W and curvature R of the
Gaussian beam, this simple law, called the ABCD law
Invariance of the ABCD Law to Cascading
If the ABCD law is applicable to each of two optical systems with matrices
Mi =(Ai, Bi, Ci, Di), i = 1,2,…, it must also apply to a system comprising their
cascade (a system with matrix M = M1M2).
Chapter 2 Optical resonator and Gaussian beam
C. HERMITE - GAUSSIAN BEAMS
The self-reproducing waves exist in the resonator, and resonating inside of
spherical mirrors, plan mirror or some other form paraboloidal wavefront mirror,
are called the modes of the resonator
Hermite - Gaussian Beam Complex Amplitude
W0
2x
2y
x2  y 2
U l ,m ( x, y, z )  Al ,m [
]Gl [
]Gm [
]  exp[- jkz - jk
 j (l  m  1) ( z )]
W ( z)
W ( z)
W ( z)
2 R( z )
where
-u 2
Gl (u )  H l (u ) exp(
),
2
l  0,1, 2,...,
is known as the Hermite-Gaussian function of order l, and Al,m is a constant
Hermite-Gaussian beam of order (I, m).
The Hermite-Gaussian beam of order (0, 0) is the Gaussian beam.
Chapter 2 Optical resonator and Gaussian beam
H0(u) = 1, the Hermite-Gaussian function of order O, the Gaussian function.
G1(u) = 2u exp( -u2/2) is an odd function,
G2(u) = (4u2 - 2) exp( -u2/2) is even,
G3(u) = (8u3 - 12u)exp( -u2/2) is odd,
Chapter 2 Optical resonator and Gaussian beam
Intensity Distribution
The optical intensity of the (I, m) Hermite-Gaussian beam is
2
I l ,m ( x, y, z )  Al ,m [
W0 2 2 2 x 2 2 y
] Gl [
]Gm [
]
W ( z)
W ( z)
W ( z)
Chapter 2 Optical resonator and Gaussian beam
Home work 2
• Exercises in English 2,3,4 about the Gaussian Beam
Chapter 2 Optical resonator and Gaussian beam
2.4 Gaussian beam in Spherical-Mirror
Resonators
A. Gaussian Modes
• Gaussian beams are modes of the spherical-mirror resonator;
Gaussian beams provide solutions of the Helmholtz equation under
the boundary conditions imposed by the spherical-mirror resonator
Beam
radius
z
a Gaussian beam is a circularly symmetric wave whose energy is confined about its
axis (the z axis) and whose wavefront normals are paraxial rays
Chapter 2 Optical resonator and Gaussian beam
2
Gaussian beam intensity:
The Rayleigh range z0
Beam width
p W02
z0 

W ( z )  W0 [1  (
The radius of curvature
Beam waist
 W 
I  I0  0  e
W ( z ) 
e
where z0 is the distance called Rayleigh range, at
which the beam wavefronts are most curved or we
usually called confocal prrameter
z 2 1/ 2
) ]
z0
z02
R( z )  z 
z
2
2
2( x 2  y 2 ) - i  k  z  x  y  -tg -1 z 




2 R 
z0 
 
W 2 (z)
minimum value W0 at the beam waist (z = 0).
  z 2 
 z z0 
z0 2
R  R( z )  z 1      z0     z 
z
  z0  
 z0 z 
 z0
W0 
p
Chapter 2 Optical resonator and Gaussian beam
B. Gaussian Mode of a Symmetrical SphericalMirror Resonator
z2  z1  d
d
R1
z02
R1  z1 
z1 2
R2
z0
- R2  z2 
z2
z1 
z1
0
z2 z
z02 
-d ( R2  d )
, z2  z1  d
R2  R1  2d
-d ( R1  d )( R2  d )( R2  R1  2d )
( R2  R1  2d ) 2
Chapter 2 Optical resonator and Gaussian beam
the beam radii at the mirrors
zi 2 1/ 2
Wi  W0 [1  ( ) ] , i  1, 2.
z0
An imaginary value of z0 signifies that the Gaussian beam is in fact a
paraboloidal wave, which is an unconfined solution, so that z0 must be real. it
is not difficult to show that the condition z02 > 0 is equivalent to
d
d
0  (1  )(1  )  1
R1
R2
Chapter 2 Optical resonator and Gaussian beam
Gaussian Mode of a Symmetrical Spherical-Mirror Resonator
Symmetrical resonators with concave mirrors that is R1 = R2= -/RI so that
z1 = -d/2, z2 = d/2. Thus the beam center lies at the center
z0 
R
d
(2 - 1)1/ 2
2 d
d R
W 
(2 - 1)1/ 2
2p
d
2
0
W12  W22 
d / p
{(d / R )[2 - (d / R )]}1/ 2
The confinement condition becomes
d
0
2
R
Chapter 2 Optical resonator and Gaussian beam
Given a resonator of fixed mirror separation d, we now examine the
effect of increasing mirror curvature (increasing d/lRI) on the beam
radius at the waist W0, and at the mirrors Wl = W2.
As d/lRI increases, W0 decreases
until it vanishes for the concentric
resonator (d/lR| = 2); at this point
W1 = W2 = ∞
The radius of the beam at the
mirrors has its minimum value, WI =
W2= (d/p)1/2, when d/lRI = 1
z0 
d
2
W1  W2  2W0
Chapter 2 Optical resonator and Gaussian beam
W0  (
 d 1/ 2
)
2p
C. Resonance Frequencies of a Gaussian
beam
2
2
k
(
x

y
)
The phase of a Gaussian beam,  ( x, y, z )  kz - tg ( z ) 
z0
2 R( z )
-1
At the locations of the mirrors z1 and z2 on the optical aixe (x2+y2=0), we have,
 z

z
 0
 (0, z2 ) -  (0, z1 )  k ( z2 - z1 ) - [ ( z2 ) -  ( z1 )]  kd -  where  ( z )  tg -1 
As the traveling wave completes a round trip between the two mirrors, therefore, its
phase changes by
2kz - 2
For the resonance, the phase must be in condition
2kz - 2  2qp ,
If we consider the plane wave resonance frequency
We have
k  2p

 q  q F   F
p
Chapter 2 Optical resonator and Gaussian beam
c
and
q  1, 2,3...
 F  c 2d
Spherical-Mirror Resonator Resonance Frequencies (Gaussian Modes)

 q  q F   F
p
1. The frequency spacing of adjacent modes is VF = c/2d, which is the same
result as that obtained for the planar-mirror resonator.
2. For spherical-mirror resonators, this frequency spacing is independent of
the curvatures of the mirrors.
3. The second term in the fomula, which does depend on the mirror curvatures,
simply represents a displacement of all resonance frequencies.
Chapter 2 Optical resonator and Gaussian beam
D. Hermite - Gaussian Modes
The resolution for Helmholtz equation An entire family of solutions, the
Hermite-Gaussian family, exists. Although a Hermite-Gaussian beam of order
(I, m) has the same wavefronts as a Gaussian beam, its amplitude distribution
differs . It follows that the entire family of Hermite-Gaussian beams represents
modes of the spherical-mirror resonator
W0
2x
2y
x2  y 2
U l ,m ( x, y, z )  Al ,m [
]Gl [
]Gm [
]  exp[- jkz - jk
 j (l  m  1) ( z )]
W ( z)
W ( z)
W ( z)
2 R( z )
 (0, z )  kz - (l  m  1) ( z )
2kd - 2(l  m  1)  2p q, q  0, 1, 2,...,
Spherical mirror resonator Resonance Frequencies
(Hermite -Gaussian Modes)
 l ,m,q  q F  (l  m  1)

p
F
Chapter 2 Optical resonator and Gaussian beam
longitudinal or axial modes: different q and same indices (l, m) the
intensity will be the same
transverse modes: The indices (I, m) label different means different
spatial intensity dependences

 l ,m,q  q F  (l  m  1)

p F
Longitudinal modes corresponding to a given transverse mode (I, m) have resonance
frequencies spaced by vF = c/2d, i.e., vI,m,q – vI’,m’,q = vF.
Transverse modes, for which the sum of the indices l+ m is the same, have the same
resonance frequencies.
Two transverse modes (I, m), (I’, m’) corresponding mode q frequencies spaced
 l ,m,q - l ',m ',q  [(l  m) - (l ' m ')]

p
F
Chapter 2 Optical resonator and Gaussian beam
*E. Finite Apertures and Diffraction Loss
Since the resonator mirrors are of finite extent, a portion of the optical power escapes
from the resonator on each pass. An estimate of the power loss may be determined by
calculating the fractional power of the beam that is not intercepted by the mirror. That is
the finite apertures effect and this effect will cause diffraction loss.
For example:
If the Gaussian beam with radius W and the mirror is circular with radius a and a= 2W, each time
there is a small fraction, exp( - 2a2/ W2) = 3.35 x10-4, of the beam power escapes on each pass.
Higher-order transverse modes suffer greater losses since they have greater spatial extent in
the transverse plane.
 In the resonator, the mirror transmission and any aperture limitation will induce loss
 The aperture induce loss is due to diffraction loss, and the loss depend mainly on the
diameters of laser beam, the aperture place and its diameter
 We can used Fresnel number N to represent the relation between the size of light
beam and the aperture, and use N to represent the loss of resonator.
Chapter 2 Optical resonator and Gaussian beam
Diffraction loss
The Fresnel number NF
a2
a2
a2
NF 


d  2 z  pW 2
Attention: the W here is the beam width in the mirror, a is the dia. of mirror
Physical meaning：the ratio of the accepting angle (a/d) (form
one mirror to the other of the resonator )to diffractive angle of
the beam (/a) .
The higher Fresnel number corresponds to a smaller loss
Chapter 2 Optical resonator and Gaussian beam
N
is the maximum number of trip that light will propagate in side resonator without escape.
1/N represent each round trip the ratio of diffraction loss to the total energy
Symmetric confocal resonator
a12
a22

 NF
2
2
p W1 p W2
For general stable concave mirror resonator, the Fresnel number for two mirrors
are:
1
a12
a12 g1
2
N F1 

[
(1
g
g
)]
1 2
p W12 d  g 2
NF 2
1
a22
a22 g 2
2


[
(1
g
g
)]
1 2
p W22 d  g1
Chapter 2 Optical resonator and Gaussian beam
Home work 3
•
•
•
•
The light from a Nd:YAG laser at wavelength 1.06 mm is a Gaussian beam of 1 W optical power and
beam divergence 2q0= 1 mrad. Determine the beam waist radius, the depth of focus, the maximum
intensity, and the intensity on the beam axis at a distance z = 100 cm from the beam waist.
Beam Focusing. An argon-ion laser produces a Gaussian beam of wavelength  = 488 nm and waist
radius w0 = 0.5 mm. Design a single-lens optical system for focusing the light to a spot of diameter
100 pm. What is the shortest focal-length lens that may be used?
Spot Size. A Gaussian beam of Rayleigh range z0 = 50 cm and wavelength =488nm is converted
into a Gaussian beam of waist radius W0’ using a lens of focal length f = 5 cm at a distance z from its
waist. Write a computer program to plot W0’ as a function of z. Verify that in the limit z - f >>z0 , the
relations (as follows) hold; and in the limit z << z0 holds.
Beam Refraction. A Gaussian beam is incident from air (n = 1) into a medium with a planar boundary
and refractive index n = 1.5. The beam axis is normal to the boundary and the beam waist lies at the
boundary. Sketch the transmitted beam. If the angular divergence of the beam in air is 1 mrad, what
is the angular divergence in the medium?
Page 41, Problems : 1,4,5,6,10,11
W0 '  MW0
M  Mr 
f
z- f
W0 ' 
W0
[1  ( z0 / f ) 2 ]1/ 2
Chapter 2 Optical resonator and Gaussian beam
 Resonance Frequencies of a Resonator with an Etalon. (a) Determine the
spacing between adjacent resonance frequencies in a resonator
constructed of two parallel planar mirrors separated by a distance d = 15
cm in air (n = 1).(b) A transparent plate of thickness d1= 2.5 cm and
refractive index n = 1.5 is placed inside the resonator and is tilted slightly to
prevent light reflected from the plate from reaching the mirrors. Determine
the spacing between the resonance frequencies of the resonator.
 Mirrorless Resonators. Semiconductor lasers are often fabricated from
crystals whose surfaces are cleaved along crystal planes. These surfaces
act as reflectors and therefore serve as the resonator mirrors. Consider a
crystal with refractive index n = 3.6 placed in air (n = 1). The light reflects
between two parallel surfaces separated by the distance d = 0.2 mm.
Determine the spacing between resonance frequencies vF, the overall
distributed loss coefficient ar, the finesse F, and the spectral width v.
Assume that the loss coefficient (as= 1 cm-1).
Chapter 2 Optical resonator and Gaussian beam