MSEG 667
Nanophotonics: Materials and Devices
5: Optical Resonant Cavities
Prof. Juejun (JJ) Hu
hujuejun@udel.edu
Optical resonance and resonant cavities

Optical resonant mode




A time-invariant, stable electromagnetic field pattern (complex
amplitude): an eigen-solution to the Maxwell equations
Discretized resonant frequencies (eigen-values), i.e. these
modes appear only at particular frequencies/wavelengths
The modal fields are usually spatially confined in a finite domain
Optical resonant cavities (resonators)

Devices that support optical resonant modes
Guided mode resonance, surface plasmon (polariton) resonance, and
spoof surface plasmon resonance all refer to coupling to propagating
modes, even though the same term “resonance” is referenced!
Resonance: a mechanical analog
The resonance frequency of a string
determines the pitch of sound it produces
An “infinite corridor”
in two mirrors
Electromagnetic waves
between two perfect
conductors (perfect mirrors)
Photon
 Interference between backand-forth reflected light
 Standing wave formation
A simple mathematical model
t 1 , r1
Field
amplitude: 1
t2, r2
a1
a2
…
…
an
1


a1  t1  exp  ikL   L   t2
2


a2  a1  r
r  r2  r1  exp  2ikL   L 
an  a1  r n 1
α = 2pK/λ, L
Transmission
coefficient

Ttot  atot
2
a
 1
1 r
2
n 
atot
a1
  ai 
1 r
i 1
when |r| < 1
Ray tracing: summation of field amplitude, taking into
account interference effect (the phase term)
A close inspection of phasor summation…
Transmission
coefficient
Phasor
Ttot  atot
2
a
 1
1 r
2
when |r| < 1
r  r1  r2  exp   L   exp  2ikL 
Eq. (1)
A vector on the complex plane
with a modulus/length ≤1
Firstly let’s look at a lossless cavity, i.e. α = 0, r1 = r2 = 1, and thus |r| = 1.
When kL ≠ Np, the vectors
have different directions…
When kL = Np, the
vectors are aligned
(resonant condition).
Finite, non-vanishing
transmitted intensity
ONLY at resonance
Transmission spectra
Ttot
Free Spectral Range
FSR = pc/L
ω
Peak FWHM = 0
A close inspection of phasor summation…
Transmission
coefficient
Phasor
Ttot  atot
2
a
 1
1 r
2
when |r| < 1
r  r1  r2  exp   L   exp  2ikL 
Eq. (1)
A vector on the complex plane
with a modulus/length ≤1
When there is loss in the cavity, |r| < 1, and Eq. (1) holds
The transmission spectra have non-vanishing
values even when the resonant condition is not met!
Quality factor Q:
Cavity finesse:
 L r

Q 0  0
 c  (1  r )
0.5
Transmission spectra
Ttot
Free Spectral Range
FSR = pc/L
FSR p  r
F


1 r
0 .5
ω
FSR: Free Spectral Range, peak separation
ω0 : resonant (angular) frequency
Δω : peak FWHM (Full Width at Half Maximum)
Peak FWHM ≠ 0!
Extinction ratio: 10·log10(Tmax/Tmin)
Standing wave modes in F-P cavities
x
t 1 , r1
y
t 2 , r2
z
1


EL  R ,1  z   t1  exp  ikz   z 
2


1


ER  L ,1  z   t1r2  exp  2ikL   L   exp  ikz   z 
2


ELR,n  z   r n1  ELR,1  z 
…
ERL,n  z   r n1  ERL,1  z 
α = 2pK/λ, L
r  r1  r2  exp   L   exp  2ikL 
Cavity field: Etot  z   EL  R ,1  z   ER  L ,1  z   EL  R ,2  z   ER  L ,2  z   ...

1
  EL  R ,1  z   ER  L ,1  z  
1 r
Standing wave modes in F-P cavities (cont’d)
N=1
N=2
N=3
N=4
N=5
…
Important concepts

Quality factor (Q-factor)
Q

FSR

2 

~
Q
2 ng L
Include the factor 2 for travelling wave cavities
Free spectral range (FSR, frequency domain)
2  p c0

FSR 
ng L

W : Energy stored in the cavity in J
Ploss : Power loss in J/s or W
FWHM should be calculated in the linear scale
Finesse
F

0
W
 0 

Ploss
Include the factor 2 for travelling wave cavities
Reference: Juejun Hu, Ph.D. thesis, Appendix I
Optical loss in cavities

Round trip loss in an F-P cavity


1  r  1  r12  r22  exp  2 L  ~ 1  r12  r22  2 L
2

Coupling loss (mirror loss): 1  r12  r2 2



Internal loss (distributed loss): 2 L



Non-unity mirror reflectance
Independent of cavity length
Absorption/scattering of light in the cavity
Loss proportional to cavity length L
Both Q and finesse scales inversely with cavity loss


If distributed loss dominates, Q is independent of cavity length
If coupling loss dominates, F is independent of cavity length
Cavity perturbation theory


Resonant frequency shift due to perturbation
Material perturbation
e

0  
0
2

 e r  E r
3
2
2
3
Sharp perturbation
e

e  e
    d r  O e
 
 e r   E r  d r
2
e
S. Johnson et al., ”Perturbation theory for
Maxwell’s equations with shifting material
boundaries,” Phys. Rev. E 65, 066611 (2002).
The frequency shift scales with field intensity
Standing wave vs. travelling wave cavities
Standing wave resonators
 PhC cavities/Fabry-Perot (FP) cavity
 Light forms a standing wave
inside the cavity
Traveling wave resonators
 Micro-ring/disk/racetrack
resonators, microspheres
 Light circulates inside the
resonant cavity
2-d PhC cavity
(top-view)
Micro-ring
F-P cavity
Microsphere
attached to a
fiber end
Micro-disk
Standing wave vs. travelling wave cavities
Standing wave resonators
 PhC cavities/Fabry-Perot (FP) cavity
 Light forms a standing wave
inside the cavity
2-d PhC cavity
(top-view)
F-P cavity
Traveling wave resonators
 Micro-ring/disk/racetrack
resonators, microspheres
 Light circulates inside the
resonant cavity
Whispering gallery mode
Acoustics
Sound wave
Optics
CW
mode
Standing wave vs. travelling wave cavities
Standing wave resonators
 Light forms a standing wave
inside the cavity
Traveling wave resonators
 Light circulates inside the
resonant cavity
Ecos  2 E0  cos  kz 
z
Ez   E0  exp  ikz 
z
Esin  2 E0  sin  kz 
z
Ez   E0  exp  ikz 
z
 Ecos 
1  1 1   Ez  



 E 
E
1

1
2
  z 
 sin 
z
+
z
=
Azimuthally symmetric travelling wave cavities support CW &
CCW travelling wave modes as well as standing wave modes;
and they are all degenerate (i.e. same resonant frequency)
z
Degeneracy lifting in travelling wave cavities
Antisymmetric mode
Breaking the cavity
azimuthal symmetry
leads to resonance
frequency splitting of
standing wave modes
Symmetric mode
Nat. Photonics 4, 46 (2010).
APL 97, 051102 (2010).
IEEE JSTQE 12, 52 (2006).
PNAS 107, 22407 (2010).
Optical coupling to cavity modes

Coupling approaches


Free space coupling: F-P cavity
Waveguide/fiber coupling: traveling wave cavities, PhC cavities


Phase matching condition: efficient coupling
External Q-factor

Energy loss due
to coupling: Qex
1
1
1
1
 

Qtot
Q Qin Qex

Extinction ratio
depends on coupling

Critical coupling
J. Hu et al., Opt. Lett. 33, 2500-2502 (2008).
Optical coupling to cavity modes
Coupling approaches


Free space coupling: F-P cavity
Waveguide/fiber coupling: traveling wave cavities, PhC cavities


Phase matching condition: efficient coupling
External Q-factor

Energy loss due
to coupling: Qex
1
1
1
1
 

Qtot
Q Qin Qex

Extinction ratio
depends on coupling

Transmission (dB)

Increase
coupling
strength
Critical coupling
Wavelength (μm)
Critical coupling
thru = 0
input
Critical coupling
 Complete power transfer:
Pthru = 0
 Occurs when Qex = Qin
 Maximum field enhancement
inside the resonator
Under coupling
 Qex > Qin
Over coupling
 Qex < Qin
Matrix representation of directional couplers
b1
b2
a2
a1
a1
Linear, lossless, unidirectional, reciprocal,
single-mode couplers
a2
a1
a2
Matrix K1
Matrix K2
Coupler
1
Coupler
2
Lossless
coupler
b2
b1
   a1 
 b1   t
 
 

 b2    * t *   a2 
2
2
where t    1
Matrix Kn
…
Coupler
n
b1
Cascadability:
b2
b  K1K 2 ...K n  a
Ch. 4, Photonics: Optical Electronics in Modern Communications, A. Yariv and P. Yeh
Coupling matrix approach for travelling wave cavities
a2
Transmission
1
b2
Lossless
coupler
5 mm
a1
0.8
0.6
0.4
0.2
0
b1
1546
1548
1550
1552
1554
Wavelength (nm)
α : waveguide loss; β : propagation constant; L : round-trip length
   a1 
 b1   t
 
 

 b2    * t *   a2 
1


a2  b2  exp  i  L   L 
2





A. Yariv, Electron. Lett. 36, 321-322 (2000).
A2  t  2 A t  cos   L 
2
b1 
2
1  A2 t  2 A t  cos   L 
2
 1

L
 2

where A  exp  
 a1
2
Coupling matrix approach for travelling wave cavities
a15
3rd
order adddrop filters
Coupler
4
a13
a16
a14
L6, 6
a11
L5, 5
a12
Coupler
3
L4, 4
Coupled resonator steady state solution:
 2 equations for each coupler: 8 total
 1 equation for each ring section: 6 total
 2 known inputs: a1, a16
 Compile the equation coefficients into
a 14-by-14 matrix
 Solve the set of linear equations

a9
a10
a7
a8
Coupler
2
a5
L3, 3
a6
L2, 2
L1, 1
a3
a1
Coupler
1
a4
a2
The algorithm can be automated to solve coupled cavities of arbitrary topology
The versatile optical resonator

Selective spectral transmission/reflection


Coherent optical feedback


Lasers
Increased optical path (interaction) length





Optical filters for WDM
Spectroscopy and sensing
Modulators and switches
Slow light: coupled resonator optical waveguide (CROW)
Cavity-enhanced photodetector
Enhanced field amplitude (photon LDOS)



Nonlinear optics
Cavity quantum-electrodynamics (QED)
Cavity optomechanics
Wavelength Division Multiplexing (WDM)




Better use of existing
fiber bandwidth
Little cross-talk
between channels
Transparent to data
format and rate
Mature technology
See what the “FiOS boy” says about WDM!
Wavelength Division Multiplexing (WDM)
Multiplexing



De-multiplexing
λ1 λ2
λ3 …

Better use of existing
fiber bandwidth
Little cross-talk
between channels
Transparent to data
format and rate
Mature technology
Ring resonator add-drop filter
λ1
λ2
•
•
•
λn
λ1
λ2
λn
…
Add-drop filter design rules:
• Low insertion loss: critical coupling, low WG loss
• Low cross-talk:
large extinction ratio, FSR >> channel spacing
• Flat response in the pass band
• B. Little et al., J. Lightwave
Technol. 15, 998 (1997).
• B. Little et al., IEEE PTL 16,
2263 (2004).
• T. Barwicz et al., JLT 24,
2207 (2006).
• F. Xia et al., Opt. Express
15, 11934 (2007).
• P. Dong et al., Opt. Express
18, 23784 (2010).
Semiconductor lasers

AlGaAs-GaAs-AlGaAs
double heterojunction
lasers
n-type AlGaAs
GaAs
Laser
output
p-type AlGaAs
+
mirror
mirror
Edge-emitting laser
Vertical Cavity Surface Emitting Lasers (VCSELs)




On-wafer testing
Single longitudinal
mode operation
Low threshold
current
Long lifetime
http://www.rp-photonics.com/vertical_cavity_surface_emitting_lasers.html
External Cavity Lasers and VECSELs
Wide wavelength tuning
range, single
longitudinal
mode
operation
Vertical External-cavity
Surface-emitting Lasers
(VECSELs)
Rev. Sci. Instrum. 72, 4477 (2001).
http://www.rp-photonics.com/external_cavity_diode_lasers.html
The strong photon-matter interaction in integrated high-Q
optical resonators make them ideal for sensing
Detection of refractive index change induced by surface binding of
biological molecular species: proteins, nucleic acids, virus particles
WGM
resonance
Specific surface binding
High Q-factor leads to superior spectral resolution and improved sensitivity
Cavity-enhanced IR spectroscopy achieves high
sensitivity and small footprint simultaneously
Single-pass spectrophotometer
Source
Receiver
Cavity-enhanced spectroscopy
Extinction ratio
change due to
presence of
absorption
Optical path length: L
Lambert-beer’s law:
I %  1  exp( L)  L
Sensitivity
Footprint
Analyst 135, 133-139 (2010).
Silicon micro-ring switch/modulator


Refractive index change in silicon via free carrier dispersion
effect: optical/electrical carrier injection
Low power consumption due to small footprint
V. Almeida et al., “All-optical control of light on a silicon chip,” Nature 431, 1081 (2004).
Q. Xu et al., “Micrometer-scale silicon electro-optic modulator,” Nature, 435, 325 (2005).
The challenges: narrow band operation &
fabrication/thermal sensitivity
2000 GHz
Q = 1,000
Si waveguide
cross-section 450
nm × 200 nm