Transcript Photonic Crystals

Nanophotonics
Class 6
Microcavities
Optical Microcavities
Vahala, Nature
424, 839 (2003)
Microcavity characteristics: Quality factor Q, mode volume V
Simplest cavity: Fabry-Perot etalon
Transmission peaks: constructive interference between
multiple reflections between the two reflecting surfaces
(wavelength fits an integer number of times in cavity).
Next few slides: definition and interpretation of free
spectral range , quality factor Q, and finesse F.
Free spectral range 
Free spectral range (FSR)  is
frequency (or wavelength) spacing
between adjacent resonances.
d
R
T
n
 depends on cavity length:
1
Eq. 1: m  1  n  d
2
 m
Eq. 2: m  1  2  m  1 
1
2
2n d
1
1
1     n  d
2
m: integer; n: refractive index
2
1
 
 
12
2n d  1
12
2n d
1  2n d
The smaller d, the larger the free spectral range  !!
Interpretation of free spectral range  in the time domain:
Consider traveling wave in the cavity:
y  y0ei k0 x0t 
Optical cycle
time
Look at phase front that is at x = 0 at t = 0: k0x  0t = 0
The time t to travel a distance x is:
t
k0 x
0

2nx
0
T
nx
0
The time tRT to make 1 round trip 2d is then:
t RT  T
2nd
0
 T 0
2nd
20
0
T



0

T
Free-space
wavelength
t RT
Free spectral range  (divided by ) is a measure for
the optical cycle time compared to the round trip time
Quality factor Q
Consider the ‘ring-down’ of a microcavity:
E
1
E = Electric field at a
certain position
u = Energy density
1
 t
e 2
E t   cos 0 t 
2
1/e
0
t
2/
Optical period T = 1/f0 = 2/0
1. Definition of Q via energy storage:
  1 t 
u t    e 2   e t




Energy density decay:

dut 
 e t
dt
0
StoredEnergy
u t 
2
Q  2
 2


dut 
EnergyLostPerOptCycle


T  T
dt
2. Definition of Q via resonance
bandwidth:
0
Q

Fourier
1
E
Time domain

1
 t
e 2
E t   cos 0 t 
1 
 
2 
t
2
I   E   
2
I
  0 2   1  
2 
2
1/e
2/
Frequency domain
Lorentzian
0 0
Q

 
  
0
The two definitions for Q are equivalent !

Finesse F
Definition of F via resonance
bandwidth:
F 





1
2
This can be rewritten as:
F



 0 

T
StoredEnergy

 Q
 Q
 2
  0
0
t RT
EnergyLostPerRoundTrip
See slide on Q
See slide on FSR
F is similar to Q except that optical cycle
time T is replaced by round trip time tRT
Quality factor vs. Finesse
 Quality factor: number of optical cycles (times 2)
before stored energy decays to 1/e of original value.
 Finesse: number of round trips (times 2) before stored
energy decays to 1/e of original value.
Suppose mirror losses dominate cavity losses, then:
• Q can be increased by increasing cavity length
• F is independent of cavity length !!
This shows that Q and F are different figures of merit
for the light circulation capabilities of a microcavity
Application: Low-threshold lasing
On threshold: Pin = 16 W.
If all light is coupled into
the cavity, then in steady
state:
D = 40 m
Q = 4  107
APL 84, 1037 (2004)
F
Pcirc 
Pin
2
with F  Q
T

Q
 3105
ttr
nD
Pin = 16 W  Pcirc = 800 mW !!!
1. Ultra-high F leads to an extremely high circulating power
relative to the input power !
Application: Low-threshold lasing
2. A small mode volume Vmode leads to strong confinement
of the circulating power, and thus to a high circulating
intensity:
Vmode Vmode 500m 3
Amode 


 4 m 2
L mode
πd
125m
800mW
Pcirc
2
I circ 


20
MW
cm
Amode
4 m 2
The light circulation concept is not only useful for
lasing, but also for:
•
•
•
•
Nonlinear optics (e.g. Raman scattering)
Purcell effect
Strong coupling between light and matter
…
See also: Vahala, Nature 424, 839 (2003), and www.vahala.caltech.edu
Differences between microcavities
Practical differences are related to:
• Ease of fabrication
• Connectivity to waveguides
• Integration in larger circuits
Principle differences are related to the figure of merits:
• Free spectral range (= spectral mode separation)
• Quality factor (= temporal time)
• Mode volume (= spatial confinement)
I circ Q
One example: the cavity build-up factor

Pin
V
See next slide…
Differences between cavities
Q/V = 102
Q/V = 103
Q/V = 104
Q/V in units
(/n)3
Q/V = 106
Q/V = 106
Q/V = 105
Vahala, Nature
424, 839 (2003)
Highest Q/V: geometries useful for fundamental research
on QED (Kimble, Caltech) but not practical for devices
Critical coupling
ex
0
Decay rates (s-1):
1/ex: coupling to waveguide
1/0: internal losses
  ex   0  i   0  

T  







i



0 
 ex 0
2
If  = 0 and ex = 0, then T = 0 !!
If the intrinsic damping rate equals the coupling rate, then
100 % of the incoming light is transferred into the cavity
(perfect destructive interference at output waveguide)
For derivation, see: Kippenberg, Ph.D. Thesis, section 3.3.2
(http://www.mpq.mpg.de/~tkippenb/TJKippenbergThesis.pdf)
Sensing example: D2O detection
Subtle difference in
optical absorptions
between D2O and H2O
is magnified due to
light circulation
in cavity.
 ex   0  i    0 
Sensitivity: 1 part
per million !!!
Evanescent waves are essential for
both sensing and fiber coupling
Armani and Vahala, Opt. Lett. 31, 1896 (2006)
Summary
• Microcavities: Confinement of light to small volumes by
resonant recirculation.
• Applications: lasing, nonlinear optics, QED, sensing, etc.
• FSR, Q, Vmode, and F characterize different aspects of the
light recirculation capabilities of a microcavity.
• Different microcavity realizations (e.g. micropost,
microsphere) differ in FSR, Q, Vmode, and F.