Post on 30-Jun-2020
TF Krauss Romme 2014 No.1/45
Thomas F Krauss Department of Physics, University of York
thomas.krauss@york.ac.uk
Photonic Crystal Cavities
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TF Krauss Romme 2014 No.4/45 Photonic Crystal cavities
What is so special about photonic crystal cavities ?
Nonlinearity, Bistability Notomi et al., Opt. Express 13, 2678 (2005)
Trapping,Optomechanics Houdre et al., PRL 110, 123601 (2013)
Strong coupling, QIP Vuckovic et al., Opt Express 17, 18652 (2009)
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Source: Wikipedia "Optical coatings"
The reflectivity of a metal mirror
R ≈ 95-98%
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π/a!
ω!
k!
Photonic crystal bandstructure!
0! 0.05! 0.1! 0.15!0.2! 0.25! 0.3! 0.35! 0.4! 0.45! 0.5!0!0.05!0.1!0.15!0.2!0.25!0.3!0.35!
k (multiples of 2π/a)!
frequency !(multiples of c/a) [a/λ]!
W1 waveguide!
€
vg =dωdk
Operating point
n=1
vφ =c0nφ=ωk
Cross-section
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How can I make a cavity that confines light in all three directions if I only have a bandgap available in two ?
Answer: Fourier space engineering. Light line control.
+ ?
Photonic Crystal cavities
TF Krauss Romme 2014 No.8/45 Light line in 2D
In 1 dimension, the light line corresponds to the line of total internal reflection. Modes with neff>1 lie to the right, modes with neff<1 lie to the left and can radiate out.
k
ω$
kx ky
ω$ In two dimensions, one can think of this line as a cone (“light cone”). For a given frequency ω0 and in an isotropic medium, this cone becomes a circle.
kx
ky
ω0$ €
ω =cneff
k
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High Q (low loss) comes from lack of radiation within light cone. The cavity mode is designed such that it carries very little light in the light cone.
High Q cavity
Real space
Fourier space
FT explanation
TF Krauss Romme 2014 No.10/45 Fourier transform explanation
FT => x k
a) Harmonic oscillation -> delta function
FT => k
€
2π nmodeλ
€
2π ncladdingλ
k
€
2π nmodeλ
€
2π ncladdingλ
FT =>
If the mode is confined by a Gaussian envelope, its Fourier transform has minimum amplitude inside the light cone -> so very little light is lost.
b) Top hat confinement -> convolution with sinc
c) Gaussian confinement -> reduced extent in k-space
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S. Noda et al., “High-Q photonic nanocavity in a two-dimensional photonic crystal” Nature 425, p. 944 (2003).
The recipe for high Q cavities: Gaussian mode profile. Approximated here by adjusting mirror boundaries
High Q cavity
How does the Q-factor relate to reflectivity ?
TF Krauss Romme 2014 No.12/45 Q-factor vs Reflectivity
Q = 2π Energy storedEnergy lost per cycle
Q = 2π Ucav
2(1− R)Ucav
Q =π1− R
Q =mπ1− R
Assume R->1,m=1 (single mode) Assume only mirror loss Loss per single pass=1-R
for m>1 (multimode)
F = Qm=
π1− R
“Finesse”
What is the reflectivity for our cavity of Q=45,000 ?
m=3
TF Krauss Romme 2014 No.13/45 The heterostructure cavity
S. Noda et al., “Ultra-high-Q photonic double-heterostructure nanocavity”, Nature Materials 4, 207-210 (2005) Time dependence
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Q = 2π Energy storedEnergy lost /cycle
Q = 2π Energy stored
Energy lost ×TCycleΔt
The cavity Q (“Quality factor”) describes how well the cavity can store energy. High Q cavities can store a lot of light in a small space, hence increase nonlinearities; this also means that the light is stored for a long time.
Q = 2π Ucav
−dUcav
dt×TCycle
⇔−dUcav
dt=
2πQ TCycle
UcavExpress the same as a differential equation with U as the energy, and -dU/dt as the energy lost,
Ucav (t) =Ucav,0 exp−tτ
τ =Q TCycle2π
This yields the following time-dependence,
Example:, λ=1.5 µm, Q=1.2M, τ= ?
Storing light in a cavity
TF Krauss Romme 2014 No.15/45 Ultrahigh high Q cavities
€
Q =λΔλ
=1555nm
1.3×10−3nm=1.2M
€
τ =Q TCycle2π
=1.2 ×106 × 5 ×10−15
2π≈1ns
M. Notomi et al., “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity” Nature Photonics 1, pp. 49-52 (2007)
TF Krauss Romme 2014 No.16/45 Extreme Q factor cavities
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Ti→ f =2πh
f H i2ρ f
FP =34π 2
λn!
"#
$
%&3QV
τ rad,Purcell =τ radFP
The Purcell effect is based on Fermi’s Golden Rule
The strength of the transition from an initial state to a final state is the product of the matrix element < f | H | i > and the density of states in the final state ρf.
Translated into the cavity situation, a high Purcell effect can be achieved in a cavity with a high Q/V factor.
The radiative lifetime in a cavity is accordingly reduced by the Purcell factor.
Purcell effect
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λ$
Δλ$
€
Q =λΔλ
emitter
Assumptions: Cavity linewidth dominates. Emitter smaller than cavity mode Cavity and emitter are spectrally aligned.
In very simple terms, the Q/V argument is one of spectral and spatial overlap.
quantum dot
FP =34π 2
λn!
"#
$
%&3QV
Purcell effect
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“….with an estimated Purcell enhancement of 2.4 at room temperature, and 11 to 17 at cryogenic temperatures.”
Purcell effect in Er-doped overlayer
Luca Dal Negro et al., “Linewidth narrowing and Purcell enhancement in photonic crystal cavities on an Er-doped silicon nitride platform” OpEx 18, 2601 (2010)
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2) A quantum dot is placed inside a photonic crystal cavity. Why do you cool it down ?
3) An organic light emitter is placed inside a high Q cavity. Do you observe Purcell enhancement ?
Review questions
FP =34π 2
λn!
"#
$
%&3QV
4) The radiative lifetime of an emitter placed in a cavity is reduced by the Purcell effect. Does that mean the radiative efficiency improves by the same factor ?
1) Why are photonic crystal cavities better for Purcell enhancement than microring resonators ?
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M Galli et al. Opt. Express 18, 26613 (2010).
Nonlinear effects (here: Second and third harmonic generation) observed due to high intensity buildup (Icav~ Q) and far-field engineering.
Harmonic Generation
Intensity enhancement
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Icav (1-R) Icav
R Icav
R I0
I0
R(1-R) Icav
Intensity enhancement: The reflection at the first mirror RI0 and the transmission of the cavity light at the same mirror cancel out on resonance: No light is reflected back. The magnitude of the two signals has to be equal for complete destructive interference.
RI0 = (1− R)Icav ⇔ Icav =1
1− RI0
I0 Input intensity. Icav Intensity circulating in the cavity. R1=R2=R, R->1
Intensity enhancement
TF Krauss Romme 2014 No.23/45 THG and SHG in Si cavities
M Galli et al. Opt. Express 18, 26613 (2010).
TF Krauss Romme 2014 No.24/45 Silicon light source
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SOITEC website
Hydrogen in SOI
Hydrogen implantation
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Silicon Indirect band-gap gives low radiative recombination
Energy
K
Si
Si
Si Si
Si
Si
Si Si e-
h+
Defect emission mechanism
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Hydrogen implantation creates defects that overcome Δk.
Energy
K
Si
Si
Si Si
Si
Si
Si X
e-
h+
Homewood et al., �An efficient room-temperature silicon-based light-emitting diode��Nature 410, 192-194 (2001)
Defect emission mechanism
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TEM: Stefania Boninelli, Catania
Hydrogen incorporation creates line defects (“platelets”) that give rise to a compressive strain field. Compressive strain field localises carriers.
Weman & Monemar, PRB 42, 3109 (1990)
Defect emission mechanism
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So now we have a lightsource…. What can photonic crystals do to help ?
Noda et al. Nature 2003
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R. Lo Savio et al., Appl. Phys. Lett. 2011
x300
300-fold enhancement observed x 12 (Purcell) x 25 (Extraction)
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H2 Plasma (RIE)
Hydrogen Plasma
Bulk defects (SOITEC process)
Surface defects (Plasma process)
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TEM: S. Boninelli, P. Cardile, Catania
TF Krauss Romme 2014 No.33/45 PL for cavity + H2 Plasma
Hydrogen plasma treatment considerably increases defect PL emission. Cavity enhancement again adds a factor 300.
A Shakoor et al, Laser&Photonics Reviews, Jan 2013
TF Krauss Romme 2014 No.34/45 Electroluminescent device
TF Krauss Romme 2014 No.35/45 Electroluminescent operation
A Shakoor et al, Laser&Photonics Reviews, Jan 2013
TF Krauss Romme 2014 No.36/45 Electroluminescent operation
A Shakoor et al, Laser&Photonics Reviews, Jan 2013
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Stimulated emission from PbS-quantum dots in glass matrixF. Yue, J. W. Tomm, D. Kruschke, and P. Glas
LASER & PHOTONICSREVIEWS
www.lpr-journal.org Vol. 7 No. 1 January 2013
ISSN 1863-8880 Laser Photonics Rev., Vol. 7, No. 1 (January), 1–140 (2013)Now open fo
r Lette
rs and
Original A
rticles
Laser & Photonics Review
sV
olume 7
2013 N
umber 1
All-silicon photonic crystal nanocavity LED
A. Shakoor et al.
LASER & PHOTONICSREVIEWS
www.lpr-journal.org Vol. 7 No. 1 January 2013
ISSN 1863-8880 Laser Photonics Rev., Vol. 7, No. 1 (January), 1–140 (2013)Now open fo
r Lette
rs and
Original A
rticles
F. Priolo, T. Gregorkiewicz, M. Galli and T.F. Krauss , “Silicon Nanostructures for Photonics and Photovoltaics” Nature Nanotechnology January 2014
Cavity enhanced light emission
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IBM website
Optical Interconnects
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K. Debnath et al., Opt Exp 20, 27420 (2012)
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2"cascaded"PhC"p"i"n"junc.on"modulators"Modulate"each"channel"individually"Q~10,000"∆n~4e?4"
0V 2v
Cavity 1 Cavity 2
Cavity 1 Cavity 2
Photonic crystal modulators
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Very small Very low power consumption (fJ/bit)
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Cav1 Cav2 Cav3 Cav4 Cav5
Multichannel operation
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!500 Mbit/s, 0.6 fJ/bit
Comb laser source
Multichannel modulation
K. Debnath et al., Opt Exp 20, 27420 (2012)
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Novel interconnect architecture – low power modulation
Defect-based light emission
Conclusion Photonic crystals offer enhanced light-matter interaction for a number of applications; their unique advantage is the high Finesse and resulting Purcell-factor.
Enhanced harmonic generation – mW pump !!
FP =34π 2
λn!
"#
$
%&3QV
Finesse = π1− R
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2) A quantum dot is placed inside a high Q photonic crystal cavity. Why do you cool it down ? At room temperature, the qdot emission is thermally broadened (kT≈25meV), which gives an equivalent Q-value for the emitter below 100; Since the Purcell factor refers to the larger of the two linewidths (emitter or cavity), using a high Q cavity on such a relatively broad emitter is pointless.
3) An organic light emitter is placed inside a high Q cavity. Do you observe Purcell enhancement ? No. Organic light emitters typically have broadband transitions. The argument is similar as in 2). Some people have referred to wavelength-selective Purcell enhancement in this case, which is true, but since the cavity suppresses the emission off-resonance, the overall enhancement is very low.
Review questions - answers
4) The radiative lifetime of an emitter placed in a cavity is reduced by the Purcell effect. Does that mean the radiative efficiency improves by the same factor ? Not necessarily. The radiative efficiency is given by
1) Why are photonic crystal cavities better for Purcell enhancement than microring resonators ? Microrings may achieve the same Q-factor, but photonic crystal cavities achieve a much smaller volume, which leads to the higher density of states.
ηrad =τ non−rad
τ rad +τ non−radThe radiative efficiency therefore depends on the balance between radiative and non-radiative lifetimes τrad >> τnon-rad: The emitter is inefficient, but Purcell enhancement has a large impact. τrad << τnon-rad: The emitter is very efficient already, and Purcell enhancement makes little difference.