1. High-Q small-V Photonic-Crystal Microcavities
control of light
Kirill Atlasov
Ecole Polytechnique Fédérale de Lausanne (EPFL)
Laboratory of Physics of Nanostructures
Lausanne, Switzerland

2. Control of light: in Nature2
I would like to begin a bit distantly, starting from Nature. So, what has Nature to do with the control of light? Actually to produce all these nice colors to our eyes and to the eyes of other peakocks, it is the photonic crystals implemented here. Manipulation of light using mainly diffraction and interference.

3. Purposes: calculus, communication, data storage
Control of Light. Us. 3
Manipulation: create, select spectrally, guide, interact, (store), detect
In a predefined way.
Purposes: calculus, communication,
data storage
What about us? People are ambitious. We would like to manipulate light in an ultimate way. Create, select it (or filter it) as precisely as possible, guide, make it interact with matter, et cetera. In a predefined way. For the purposes of calculus, telecommunication and data storage, if to put globally.
Romuald has already shown us and described this nice picture. What we have is waveguides, 3D and 2D photonic crystals, ring resonators, et cetera. Here it looks like 2D photonic-crystal microcavity. And this is a particular component that presents a means to control ultimately the light.
courtesy MIT (

4. What to control ? Microcavities: Control of Light4
What to control ?
They can be used as coupling blocks and filters in photonic circuits
Control of Spontaneous Emission of a light source
"speed up" atom-like source (QD)
Hennessy, et al, Nature 445, 896, (2007)
modify broad-band source (QWR)
The natural question is: what actually to control with microcavities? They can be used as coupling blocks and filters in photonic circuits.
More interesting, they can be used as means to control the spontaneous emission of a light source.
For example, if there is a relatively broad-band source based on recombination of carriers (here it is a semiconductor QWR), we can direct the created photons into a certain mode, which has defined spectral position and the spectral width. It is thus ultimate selection of "color" in comparison to peackocks.
Or (this is actually linked to the same effect), here is another example: atom-like source (quantum dot). We can modify the lifetime of the excited state such that it emits photons faster. Here are statistical histograms: photons emitted at certain time interval after the excitation. So, in cavity the lifetime is modified dramatically.

5. Purcell effect spont. em. directed into mode 5
Vahala, Nature, 424, 839, (2003)
emitter decays by interaction with continuum
rate ~ density of states per volume V
DoScavity > DoSfree (resonantly)
rate ~ DoS per volume V
The control of the spontaneous emission is based on the Purcell effect. In the weak-coupling regime.
If an emitter is in free space then it decays from its ecited state by interaction with vacuum continuum via coupling to all possible continuum modes. The decay rate is proportional to the density of these states per volume evaluated at the transition frequency. If an emitter is placed into a cavity, then the density of states is modified resonantly. It becomes significantly larger. Therefore the emitter decays much faster. At the same time the emission becomes also directed into the cavity mode.
What we have seen on the previous slide.
fast decay rate of emitter
spont. em. directed into mode

6. What to control ? Microcavities: Control of Light4
What to control ?
They can be used as coupling blocks and filters in photonic circuits
Control of Spontaneous Emission of a light source
modify broad-band source (QWR)
"speed up" atom-like source (QD)
Hennessy, et al, Nature 445, 896, (2007)

7. So, the figure of merit is about Q and V
Cavity Figure of merit: Q V 6
So, the figure of merit is about Q and V
What is it good for?

8. Cavity Figure of merit: Q/V. What is it good for?7
Indistinguishable photons
(identical wave packets for linear-optical
quantum computation)
Santori et al, Nature, 419, 594, (2002)
LED exhibiting laser-like
performance (β - spontaneous emission
Into cavity mode)
Yablonovitch in Confined electrons and photons:
New physics and applications, (Burstein, Weisbuch, eds)
vol. 340 NATO ASI ser.B. Plenum Press (1995)
LASER (β - spontaneous emission
Into lasing mode – low threshold)
Few QD laser: Strauf et al, PRL 96, , (2006)
For example it is good for making indistinguishable photons. If we have a single-photon emitter then it may interact with whatever vacuum mode during its lifetime. If we put it into a microcavity it will most probably interact with predefined modes, while its lifetime will be greatly shortened. This means that it won’t have time to interact much with the environement. And the wavepackets will be the same.
Owing to Purcell effect, spontaneous emission may be led into the cavity mode. Almost fully. Beta factor is of big interest.
Lasers. Actually, the spontaneous emission is destructive process for lasers, that is, the threshold is reached at higher pump; but if led into the lasing mode, the threshold goes down.
Strong coupling regime of light-matter interaction is of great interest. Energy transfer between two oscillators. Here high Q and small V are needed.
Strong coupling regime
(transfer of energy between oscillators)
Khitrova et al, Nat. Phys. 2, 81, (2006)

9. Most practical – membrane PhC
Photonic-crystal Microcavities 8
In principle, any defect of the periodicity.
John, PRL 58, 2486, (1987)
Most practical – membrane PhC

10. How does it compare to other possibilities?9
Srinivasan et al,
APL 424, 839, (2003)
Vahala, Nature 424, 839, (2003)
In PhC V < (λ/n)3
Noda et al, Nature 425, 944 (2003)
Park et al, Science 305, 1444, (2004)

11. Electrical pumping QW PhC laser
Interesting examples: LASERs 10
Few-QD laser
Strauf et al, PRL 96, , (2006)
β = 0.85
Electrical pumping QW PhC laser
Park et al, Science 305, 1444, (2004)
Park: cold-cavity Q-factor is 3500 (FDTD). ~2500 (experim). By fitting L-I curve to L-I curves from rate equations they have estimated beta = 25%
Strauf (Santa Barbara+Italy, Pavia): Used a modified L3 cavity (Q~ ). With low-density self-assembled QDs they got lasing at low temperature. Here is a non-linear L-L curve exhibiting a threshold. Power goes up while the background stays in a linear regime. The threshold is soft. It is characteristic to high beta factors. Btw the threshold by itself is very low (consistent with small active volume). The beta is estimated from the coupled rate equations that describe a laser (number of carriers and number of photons). 1/t_sp, 1/t_nr – recombination rates for spont em and non-radiative.
β = 0.25

12. Two-photon interference
Interesting examples: Indistinguishable photons 11
Two-photon interference
(QD in micropillar)
Santori et al, Nature 419, 594, (2002)
Probability that two colliding photons at beamsplitter exit via two diferent ports is low (given directly by correlation)
Two-photon interference (QD in PhC)
Laurent et al, APL 87, , (2005)
If 2 indistinguishable photos collide
at the beamsplitter they exit via the same port!!
=> 2-photon inteference
Hong et al, PRL 59, 2044, (1987)
Cannot arive then at different detectors
at the same time
The great thing is if they perfectly interfere then they come out via the same outport!!! Here: 2 photons are delayed by 2 ns. Then they come to the interferometer with 2 ns compensation branch. 3 things may happen. (1) 1st goes short arm and 2nd goes long arm -> they then do not interfere -> can be detected at +- 4ns at two different detectors: peaks 1 &5. (2) both go same way -> they do not interfere -> they are detected at +-2ns (initial delay) at two different detectors. (3) 1st goes long arm (delay is compensated) and 2nd goes short arm -> they collide and interfere ! If identical ! -> they never go to different detectors - > peak 3 vanishes!

13. (transfer of energy between oscillators)
Interesting feature: Strong Light-matter Coupling 12
(photons here)
(transfer of energy between oscillators) 1 2
Strong coupling
decay κ
decay γ
Khitrova et al, Nat. Phys. 2, 81, (2006)
If coupling strength between two same-energy oscillators exceeds the mean of their decay rates, the coupled system has two energies – their states split
Strong coupling phenomena. It’s about the transfer of energy between same-frequency oscillators. Photons, for example, in resonant cavities here. So, if the coupling strength is... states split. 1 oscilating, the other rests, then energy is transferred to the other one, and it oscillates, and back. A reversible process, whereas weak coupling is irreversible decay of an oscillator. Now the system may be a two-level system (atom) + single photonic mode (in a microcavity). The photon is emitted, circulates in cavity, and gets reabsorbed, and again. In order that the reversible process has sufficient time to take place, it is needed that the decay rates of the oscillators are less then the coupling strength.

14. The decay rate (of photon) ~ 1/Q
Interesting feature: Strong Light-matter Coupling 13
The decay rate (of photon) ~ 1/Q
And the coupling strength (vacuum field fluctuations
per volume)
Wanted:
high Q
small mode volume V
transition from Purcell enhancement
to Rabi splitting
Now, two-level system will be governed by radiative dephasing time, which is short. As for the cavity, the decay rate is inversely proportional to the photon lifetime, that is, to Q.
And the coupling strength in vacuum Rabi splitting comes about from the vacuum field fluctuations that for photon in a cavity are normalized per its mode volume. So, the coupling strength depends on V.
So, we need both high Q and small V, which we have in photonic-crystal microcavities. Here is a calculation showing how the spectrum evolves from Purcell-enhanced all the way to Rabi splitting. Once the coupling strength overcomes the loss we have the splitting, which is a direct consequence of the energy transfer.
Khitrova et al, Nat. Phys. 2, 81, (2006)

15. Interesting feature: Strong Light-matter Coupling14
1 QD in PhC
Yoshie et al, Nature 432, 200, (2004)
Rabi splitting at around zero detuning
Q = – 20000
(a=300 nm.) The strong-coupling experiments have been realized with photonic crystals and single QD. Here is one of the first. Membrane 2D PhC with the Q of upto So photoluminescence experiment. When we have to separated lines – cavity and QD – one can tune them by increasing the temperature. The cavity shifts little while the semiconductor bandgap shrinks a lot. And instead of crossing they anticross. Typical for strong coupling. Here is the splitting observed in spectra around zero detuning.
When tuned, anticrossing behavior

16. Very high-Q: Cavity geometry optimization. Q = 600 000.15
Akahane et al, Nat. Mater 4, 208, (2005)
Two different waveguides...
Bandgap -> transmission...
Photons with specific energies
can exist only in the WG 2
WG 2 stretched...
FT if Two mirrors
FT of Gaussian envelope
In real space
An interesting idea. Two different waveguides. One with pitch 410 nm, second – first a bit stretched along X. According to the band structure, what is here blue, all the frequencies can propagate. In red – no. Within the region of the WG 2 (it is short) the transmission extends to some lower frequencies. However, there is confinement along the X !! -> heterostructure principle.
At the same time the problem can be viewed in the light cone. If we have a microcavity terminated by two mirrors we’ll have a standing wave with abrupt envelope function. And the Fourier transform gives this. Leaky. If we have an ideal gaussiene as an envelope, the Fourier transform gives this and the leakage is absent. The situation gets improved already by shifting holes. But still there is mismatch to perfect gaussiene envelope here. If we now open to make it waveguide. The heterostructure principle stays as the effective index method in ridge waveguides to give confinement along X.

17. Ey Ex Q from 1500 goes upto >100 000
High Q: cavity termination adjustments. Polarization 16
odd
even Ey Ex
Intensity (arb units)
kx (2π/a)
normalized frequency f (c/a)
|Ex| k-space
ky (2π/a) -1 1
Q from 1500 goes
upto >
Actually, for L-type cavities the spectrum contains several interesting modes. If the diameter of holes is big enough. They originate from dispersion bands of W1 waveguide. From the flat parts. So far this mode was considered. So, if the holes are big this band enters the bandgap. And this mode here will have a different effective polarization. If we now solve the light-cone problem shifting the terminations in the opposite direction we also see the increase of the Q. Though experimentally it looks like it is more difficult to realize. We’ve seen the increase from 300 to Maybe it is more sensitive to disorder.
K.F Karlsson, K.A. Atlasov unpublished

18. 1 2 SUMMARY high Q small V Strong coupling
So, in summary it is many interesting things that can be done with these photonic-crystal microcavities, especially with 2D ones, owing to their high Q and small V.
Sources on Purcell effect, lasers at low threshold, single photon sources with high degree of indistinguishability, strong light-matter coupling regime. Optimization is possible via solving the light-cone problem.

19. end

20. additional stuff

21. For non-lasing devices β>0.9 have been estimated (0.97)
(Kress et al, PRB, 71, , (2005)