The resonance frequency
The eigenmode and eigenvalue solutions are
The diffraction loss is
Elm ( x, y,0)
lm 1 lm
2
vlmn
c lm n 2L 2
dN P N P dt P t N P (t ) N P (0) exp P t I (t ) I (0) exp P
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Due to the mirror loss, internal loss, diffraction loss, the photon in the cavity can
2kL 2m
Longitudinal modes
Mode spacing/ FSR
Concentric, confocal cavities
Concentric cavity (共心腔): L=R1+R2; L=2R (Spherical waves)
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Confocal cavity (共焦腔): L=R1=R2=R
Eigenmodes in a cavity
propagating each round trip, i.e.
E ( x, y, 2 L) E ( x, y,0) exp( j 2kL) The constant
125
Inside the cavity, the electric field of the cavity mode must reproduce the its shape after
A B
131
Assume the transfer matrix of the optical cavity is C D After one-round trip propagation
r1 A B r0 C D 0 1
After n-round trip propagation
rn A B C D n
n
r0 0
To be stable, the n-round trip matrix
must not diverge as n increases. (No element increases with n).
vFWHM v Q
A high Q factor implies a low loss of the cavity, and a narrow spectral linewidth.
Examples 5.4
Chapter 5_L10
130
Stability condition
Stability condition
128
t E (t ) u (t )exp jt exp 2 p
The resonant mode has a Lorentzian
lineshape, obtained from the Fourier transform
of the electric field. The FWHM is
vFWHM =
1 2 p
Examples 5.3
Lorentzian lineshape: f ( x) a b x c
2
Q factor of the resonant cavity
For any resonant system, in particular for a resonant optical cavity, the cavity Q factor (quality factor) is defined as
Stability condition
According to Sylvester’s theorem, define an angle
132
A D cos 2
Then, the matrix
B sin n A B 1 A sin n sin (n 1) C D sin C sin n D sin n sin ( n 1)
119
Fabry-Perot cavity/resonator
Fabry-Perot cavity: the cavity is formed by two parrallel plane mirrors (plane waves).
120
The wave inside the cavity is standing wave with the amplitude at the two mirrors are zero (boundary
Unstable cavity
Ring cavity
Ring cavity allows both clockwise and anti-clockwiave propagation of waves (standing
wave). However, if one direction propagation is blocked, the wave in the cavity will be traveling wave.
condition). Thus, the cavity length L is multiples of the half wavelength:
nr L m , m=1,2,3... 2 2nr L m c vm m 2nr L vm c 2nr L

The round-trip phase shift must be zero:
1 represents the beam attenuation, due to diffraction loss
exp( j0 )
0
represents the phase change induced by the mirrors
For one round trip, the total phase change of the field is
A light beam traal resonantor, is equivalent to that passes
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through a periodic lens-guide structure in one direction. Both have the same focal length, the same length, the same aperture size.
n
To be stable, the angle must be real, that is,
A D 1 1 2
Stability condition
The matrix (one round trip) for a optical cavity is
0 1 L 1 A B 1 C D 2 / R 1 0 1 2 / R 1 2 2L 1 R2 2 4L 2 R1 R1R2 R2 0 1 L 0 1 1
1
K(x,y,x1,y1) is the propagation kernel (核) If the light source is a point, then E ( x1, y1,0)= x1 x1 ', y1 y1 '
E ( x, y,2L) exp jk (2L) K ( x, y, x1 ', y1 ')
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Stored energy Q=2 Energy lost in one period of oscillation
Stored energy=N P hv
vN P Q 2 2 v p = p dN P / dt
dN P 1 Lost energy= hv dt v
lm
Examples 5.1
Chapter 5_L9
126
Optical resonator
Eigenmodes Photon lifetime and cavity Q-factor
Photon lifetime in a cavity
not stay forever inside the cavity, it stays for a finite time, which is desvribed by the photon lifetime.
=-2kL+0
Finally, the electric field at the facet of the cavity is
The phase shift
E ( x, y,0) K ( x, y, x1, y1 )E ( x1, y1,0)dx1dy1
1
lm =-2kL+lm =-2 n
122
c vm m nr L
Ring laser gyroscope (Sagnac effect)
Chapter 5_L9
123
Optical resonator
Eigenmodes
Photon lifetime and cavity Q-factor
Eigenmodes in a cavity
The propagation of the light field in a round trip can be solved by the Huyghens-Fresnel equation:
E ( x, y,2L) exp jk (2 L) K ( x, y, x1, y1)E ( x1, y1,0)dx1dy1