principles of lasers激光原理第5章

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
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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
A light beam travels back and forth in a optical 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.
Stability condition
According to Sylvester’s theorem, define an angle
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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)
A B
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Assume the transfer matrix of the optical cavity is C D After one-round trip propagation
r1 A B r0 C D 0 1
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
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Stability condition
Stability condition
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Fabry-Perot cavity/resonator
Fabry-Perot cavity: the cavity is formed by two parrallel plane mirrors (plane waves).
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The wave inside the cavity is standing wave with the amplitude at the two mirrors are zero (boundary
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c vm m nr L
Ring laser gyroscope (Sagnac effect)
Chapter 5_L9
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Optical resonator
Eigenmodes
Photon lifetime and cavity Q-factor
Eigenmodes in a cavity
Chapter 5_L9
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Optical resonator
Eigenmodes
Photon lifetime and cavity Q-factor
Optical resonator
Optical resonator is formed by two mirrors (plane, spherical), separated by some distance. «passive » means there is no active medium inside the cavity. «resonator» means the wave osscillates inside the cavity. The resonator is laterally open, leads to the diffration loss, and thereby reduced number of lateral mode.
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
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.
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).
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
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 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
lm
Examples 5.1
Chapter 5_L9
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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.
2L 1 1 P= c i ln R1R2 c 2 i L ln R1R2 2L cT
1
Examples 5.2
Resonant modes lineshape
The electric field as a function of time inside the cavity:
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
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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:
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
Eigenmodes in a cavity
propagating each round trip, i.e.
E ( x, y, 2 L) E ( x, y,0) exp( j 2kL) The constant
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Inside the cavity, the electric field of the cavity mode must reproduce the its shape after
=-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
For one round trip propagation, the light intensity
becomes
I (2 L) I (0) R1R2 exp 2 i L t2 L 2 L / c
Round-trip propagation time
The photon lifetime of the cavity is
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