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GUIDED WAVES AND INTEGRATED OPTICS - PHENOMENOLOGICAL APPROACH, TECHNOLOGIES AND APPLICATIONS Part 2 - MASTER 2 Optique Image Vision Spécialité Optique, Photonique et Hyperfréquences ISTASE 3 Nathalie DESTOUCHES 2005- 2006 SUMMARY VI – PASSIVE COMPONENTS 1- Total reflection mirrors 2- Fiber Bragg grating 3- Phase zones 4- Waveguide Fresnel lens 5- Dual-channel directional coupler 6- Y-junction 7- 1xN and NxN couplers 8- Array waveguide grating 9- Mach-Zehnder interferometer 10- Optical add/drop multiplexer/demultiplexer 11- Polarization elements 12- Example of application

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GUIDED WAVESAND INTEGRATED OPTICS

-PHENOMENOLOGICAL APPROACH,

TECHNOLOGIESAND APPLICATIONS

Part 2-

MASTER 2 Optique Image VisionSpécialité Optique, Photonique et Hyperfréquences

ISTASE 3

Nathalie DESTOUCHES2005- 2006

SUMMARY

VI – PASSIVE COMPONENTS1- Total reflection mirrors2- Fiber Bragg grating 3- Phase zones4- Waveguide Fresnel lens5- Dual-channel directional coupler6- Y-junction7- 1xN and NxN couplers8- Array waveguide grating9- Mach-Zehnder interferometer10- Optical add/drop multiplexer/demultiplexer11- Polarization elements12- Example of application

SUMMARY

VII – ACTIVE MODULATORS, DEFLECTORS AND SWITCHES1- Introduction2- Basics definitions 3- Electro-optic effect and applications4- Single waveguide electro-optic modulator5- Directional couplers as modulators6- Directional couplers as switches7- Mach-Zehnder type electro-optic modulator8- Electro-absorption modulators9- Basic principle of acousto-optic effect10- Raman-Nath type modulator11- Bragg type modulator12- Bragg type deflectors and switches13- Magneto-optic effects14- Exercice

SUMMARY

VIII – SOURCES, AMPLIFIERS AND DETECTORS1- Laser diode2- Heterojunction laser structures3- Vertical cavity lasers4- Distributed feed-back lasers5- Amplifiers : introduction6- Optical fiber amplifiers7- Semi-conductor optical amplifiers8- Comparison ion-doped FA and SOA9- Photodiode10- Exercice

IX – APPLICATIONS1- RF spectrum analyzer2- Monolithic wavelength-multiplexed optical source3- IO Doppler velocimeter4- IO optical disk readhead5- IO temperature sensor6- IO surface plasmon resonance biosensor

X – INTRODUCTION TO PHOTONIC CRYSTALS(Excerpt from a S. Johnson course, MIT)

VI – PASSIVE COMPONENTS

VI-1 Total reflection mirrors

The in-plane junction between two differing slab waveguides supporting a guided wave of effective index nei and neo represents a dioptre at which Snell’s laws apply. In particular there is total reflection of a wave impinging on the junction at an incident angle θi if

ei

eoi n

narccos<θ

nei neo

z

Interrupting the coating along a line

overlay, e.g. SiO2waveguide, e.g. Si3N4

The wavefield propagating under the overlay sees a higher average index than that which could propagate in the non-coated slab. The critical angle θc can be rather large if the basis slab is a highly guiding waveguide

Digging a trench and filling it with low index material

ne

z

t

trench with low indexwaveguidesubstrate

Also partially reflecting mirror with frustrated total reflection depending on t

VI-2 Fiber Bragg gratingWavelength filter

Index n1 Index n2

λΒ

λ1, λ2, λ4, ...

Λ

λ1, λ2, λΒ, λ4, ...

Periodic variation of the refractive index

Effect similar to that of multilayer coating with alternating high and low refractive index

Constructive interference for the successive reflected waves at the Bragg wavelength λB or phase matching condition for the first diffraction order

2ΛΛΛΛneff = λλλλB

Λ : grating period, neff : effective index

VI-3 Phase zonesWaveguide section inducing a phase retardation

One may want to induce a distortion of the wave front in a slab waveguide in order to correct or to generate some kind of aberrations.

Phase retardation element : dielectric overlay of length L on top of a slab waveguide.

The presence of an overlay slows down the wave propagating underneath (nep > nep ). Therefore the phase retardation ∆φ relative to the wave progressing outside the dielectric patch is : ∆φ = k0L(nep - nep)

Radiation loss due to the non perfect field matching at the dielectric patch edges

L

incident wave front

modified wave front neo nep

field matching region => transition loss

VI-4 Waveguide Fresnel lensSpecial phase zones

neo

nep

condition of constructive interference at F :

Fyo

y1

y2

yn

y

eon n

mf2y λ=

Shape and length of phase zones :

( )( )221n

eoep

eo yynnf2

n)y(L −−

= +

yn

yn+1

y

neo

nep

L(y)

VI-5 dual-channel directional couplerPrinciple

n1

n2

n1>n2

Coupling region

substrate

Parallel channel optical waveguides sufficiently closely spaced so that energy is transferred from one to the other by optical tunneling

Principle

• Coupling: Mixing of two adjacent modes, exchanging power as they propagate along adjacent paths.

• Energy transfer in a coherent fashion → Direction of propagation maintained.

VI-5 dual-channel directional coupler

Coupled-mode theory for synchronous coupling(Both waveguides are identical)

β: propagation constant of the mode in each of the guides

κ: coupling coefficient

VI-5 dual-channel directional coupler

Power transfer in synchronous coupling

VI-5 dual-channel directional coupler

Coupled-mode theory for asynchronous coupling(Both waveguides are not identical)

VI-5 dual-channel directional coupler

Power transfer in asynchronous coupling

Assuming lossless for the figures

Ψ = g K = κ

Power flow:

Incomplete power transfer Complete power transfer

VI-5 dual-channel directional coupler

Coupling coefficient

Fraction of the power coupled per unit length determined by overlaps of the modes

see part “coupled mode theory” (A. Trouillet) for the detailed calculation

guide 0

guide 1

Coupling coefficient

κ01

VI-5 dual-channel directional coupler

Applications : splitter, switch

Determine the amount of transmitted power by bending away the secondary channel at proper point.

Coupling length

Guide 0

LC=π/2 κ100 % coupler for beam switching

Guide 1Intensity

50/50 coupling→ 3dB coupler for beam splitting

90/10 coupling→ 10dB coupler for

measurement padding

50%

50%LC/2

0%

100%LC

VI-5 dual-channel directional coupler

Applications : splitter, switch

100 % directional coupler for beam switching

3dB directional coupler for beam splitting

VI-5 dual-channel directional coupler

Applications : wavelength division multiplexer/demultiplexer (WDM)

Wave vector k and propagation constants βstrongly depend on the wavelength

Field distribution of the guided modes and refractive index of the material slightly depend on the wavelength

Coupling coefficient κκκκvary with wavelength

Wavelength λ

Pow

er fl

ow

λ1 λ2

A to D A to B

Lossless waveguides

Design of the coupling length so that : ( ) mLc1 π=λκ

( )2

nLc2π+π=λκ

where n and m are integers

Wavelength separation Demultiplexer

VI-5 dual-channel directional coupler

Applications : wavelength division multiplexer/demultiplexer (WDM)

λ1, λ2

λ2λ1

A C

B D

Demultiplexer

When the light is delivered in the reverse direction, it will function as a multiplexer

VI-5 dual-channel directional coupler

VI-6 Y junctionPrinciple

The simplest coupler to divide guided light into two waveguides

Drawbacks : - not convenient for splitting ratios different from 1:1- loss due to large θ : transmission rapidly deteriorates as θ > 1° (as θ²)

θ

Analogous to a beam-splitting mirror in volume optics

Causes of loss

2- Change of the modal field in the transition section and imperfect field overlap in amplitude

For high integration density a Y-junction must be shorter, therefore the branching angle must be large (larger than 1°). However the loss must be kept low. In order to have low junction loss, the phase front must be tilted somehow.

Two causes of loss :1- Tilted coupled wave fronts ; imperfect field overlap

VI-6 Y junction

Solutions to reduce this loss

Input waveguide is expanded with a tapered region :

The guiding mode gradually expands to a larger region, making the mode better at splitting into the two latter waveguides

Input waveguide

Output waveguides

Taper region

VI-6 Y junction

Solutions to reduce this loss

Placing a refractive low index zone within the Y-junction :

When the guiding mode in the input waveguide propagates into this region, the mode expands due to a lack of guiding effect of the reduced refractive index. This makes the wave front of the guiding mode curved so it can be perpendicular to the latter waveguides for outputs.

n1

n2

n2

n1

n2 n1

n3

Input waveguide

Output waveguide

Output waveguiden3 < n2 < n1

Rays based on a geometrical optics point of view

A usual Y-junction without special treatment typically experiences loss of no less than 3 dB.This kind of solution can reduce the loss to only 1 dB.

VI-6 Y junction

Behaviour of non-symmetrical singlemode Y-junction

Absence of symmetry => the second order mode plays an important role

ng2< ng1

ng1

ng

ns

ns

ns

3 possible practical cases All 3 cases can be basically described qualitatively by a sigle structure

input

slow, β1> β2

fast, β2

VI-6 Y junction

Behaviour of non-symmetrical singlemode Y-junction

Shape of the eigenfield in the branching section :

This is the mode splitter behaviour which occurs if : - the taper of the Y is shallow enough- the birefringence β1-β2 is large enough

If β1-β2 is small and for θ large, then the device acts as a power divider. Its control is difficult as a large amount of coupling between local normal modes takes place.

Fast

Slow

VI-6 Y junction

VI-7 1xN and NxN couplers1xN coupler

Input

Output

A 1 x 4 coupler formed by three 3dB dual-channel directional couplers

NxN coupler

A 8 x 8 coupler using 12 3dB dual-channel directional couplersin

pu t

3dBcoupler

3dBcoupler

3dBcoupler

3dBcoupler

3dBcoupler

3dBcoupler

3dBcoupler

3dBcoupler

3dBcoupler

3dBcoupler

3dBcoupler

3dBcoupler

outp

ut

( )Nlog2N

2 dual-channel directional couplers are required to form an NxN couplers

For a 1xN or an NxN coupler, the input signals travel through log2(N) dual-channel couplers

Ideally each output share a fraction 1/N of the input power.

If we assume that each 3dB coupler has the excess loss of α, the power at each output is α log2(N)/N times of the input power

VI-7 1xN and NxN couplers

NxN star coupler

The approach consisting of a cascade of 2-waveguide 3dB directional couplers or Y branching devices is not viable with large N since the losses and the size of the device increase with N.

Alternative : single mode slab waveguide located between two arrays of N laterally confined single mode waveguides.

The N inputs and the N outputs are located on an arc of circle. The center of each circle is located in the other array.

The waves exiting the input waveguides expand freely and arrive at the arrayed output waveguides.

Losses originate in spill-over and in the poor field matching between the far field and the modal field of the output ports.

This device is practically wavelength independent

Port 1Arrayed waveguide Output waveguide

VI-7 1xN and NxN couplers

VI-8 Array Waveguide Grating

Array Waveguide Grating (AWG) are also named

Phased Arrays (PHASAR) and

Waveguide Grating Routers (WGR)

Application : Wavelength Division Multiplexing (WDM)

λ1, λ2, λ3, λ4, ...λ1

λ2λ3

λ4

M arrayed waveguides

Free expansionregion

First star coupler (N X M)

second star coupler (M X N)

To have constructive interference at the port p of the output waveguides, the condition is :

where ∆L is the length difference of two adjacent waveguides of the arrayed waveguides

and ∆l is the path difference for the light propagating from two successive waveguides to output port p.

Application : Wavelength Division Multiplexing (WDM)

path1path2pathM

Port 1

Secondstar coupler

Arrayed waveguide Output waveguide

( ) π=∆+∆λ

π m2lL2p

Channel spacing of 0.4 nm (50 GHz) can be achieved with the WDM based on the AWG

The number of channel can exceed 48.

The propagation loss in the AWg can be less than 0.1 dB/cm

The most severe loss is the the junction loss between the waveguides and the free propagation region of the first and the second star couplers, 1-1.5 dB for each junction and 2-3 dB for the total device.

VI-8 Array Waveguide Grating

VI-9 Mach-Zehnder interferometer

path13dB

coupler13dB

coupler2

∆Lpath2

λ1, λ2, . . . λ1

Port1

Port2

The 3 dB coupler splits the input light equally to path 1 and path 2.

Path 2 is longer than path 1 for a length of ∆L such that : neff ∆L = m λ1, while λ2 has

neff ∆L = (n+1/2) λ2

When the waves combines again at the second 3dB coupler, the light at wavelength λ1 experiences constructive interference at port 1, while other wavelength experience destructive interference

When the light in the mach-Zehnder interferometer experiences destructive interference at one output, it should experiences constructive interference at the other output.

It then works as a wavelength division multiplexer

Wavelength filter, wavelength division multiplexer

VI-10 Optical Add/Drop Multiplexer/ DemultiplexerOADM based on fiber Bragg grating and Mach-Zehnder interferometer

path1

3dBcoupler

λ1, λ2, λ3, λ4, ...

λn

Fiber Bragg grating

Path2

3dBcoupler

λ1, λ2, λ3, λ4, ...

λn

λnλn

λn λn

1

2

1

2

Principle : All wavelength are sent to the same input port of the first 3 dB coupler. Their power are split equally but their phase are different. The wave at λn is reflected by the fiber Bragg grating for both path 1 and path 2. The reflected wave travels back to the first 3 dB coupler where they combine. the phase difference between the waves in path 1 and path 2 causes the reflected wave at λn to go to the input port 2. => function of signal dropping for λn

The other wavelength continue to travel through the two paths and then combine at the second 3 dB coupler. with proper adjustment of the phase difference due t the path difference, they exit at outpu port 2

In a similar way another signal at λn can be added to the same outport for the other wavelengths

VI-11 Polarization elements

In most waveguide technologies the guidance conditions for TE modes imply the existence of TM modes also. In particular a waveguide said to be single mode is actually a dual-mode waveguide propagating one TE and one TM mode.

neTE > neTM : the TE mode is the slow wave

neTE and neTM differ for two reasons :

Maxwell’s equations differ for TE and TM polarisation

The photoelastic effect which originates in the in-plane stress that is always associated with vacuum or chemical vapor depositions as well as in diffusion and ion exchange techniques.

In most cases of practical interest the stress is compressive and leads generally to a larger increase for neTM than for neTE

The coexistence of two non-degenerate polarization modes calls for elements that are polarization selective...

Plasmon polarizer

Idea : Filter out the TM modePrinciple : To couple the guided TM mode to the highly lossy TM plasmon mode propagating along the surface of a metal film deposited on top of the waveguide

ETMETE

ETMETE ETE

ETE

waveguide

substrate

Low index buffer, thickness wb

Metal film, length L

VI-11 Polarization elements

Plasmon polarizer

Only two modes of the same polarization can exchange power in such isotropic structure.The TE mode will therefore only experience some attenuation due to the dissipation of the metal electrons under the driving form of its small evanescent field.The TM slab mode can transfer its full guided power to the plasmon mode where it will be dissipated after a few micron propagation provided these two waves propagate in synchronism, i.e. at the same spatial frequency.As the field shape of the plasmonsuggests, its effective index nep is very sensitive in the average index seen by the field tail of the dielectric, therefore in the low index buffer width wb, whereas the slab mode index remains practically constant => there exists a synchronism condition

Modal fields of the 3 modes that may propagate in the polarizing waveguide section

waveguide

substrate

Low index buffer

Metal film

wb

TMTE

Waveguide mode βTM

Plasmon mode βplasmon

wb

nb

ns

ngne ne waveguide mode

ne plasmon mode

Synchronism condition for the coupling of the TM waveguide mode to the highly lossy TM plasmon mode : βTM = βplasmon

VI-11 Polarization elements

VI-12 Example of application

Objective : to design a multiplexer for a amplifier working in the band C of telecommunications ( λ1 =[ 1530- 1560] nm) and pumped at λ2 = 980 nm.

We want to make the following directional coupler

with a technology such that : nc=1.515 and ng=1.51

The free parameters are the following :

the distance between the waveguides : s

the waveguide width : w

The coupling length : Lc

Perturbative method + weak coupling (s not too small)

VI-12 Example of application

Determination of w :

- single mode waveguide at both wavelengths λ1 and λ2

- curved waveguides => greatest possible refractive index

The waveguide becomes multimode for V>1.57=> we choose V=1.537 => w=3.9 µm for λ2 =0.98 µmFor w=4 µm, V=1.577 and the waveguide supports two modes

VI-12 Example of application

Influence of parameter s on the multiplexer response :

s = 6 µm => Lc ~ 20 mms = 12 µm => Lc ~ 14 mm

VI-12 Example of application

Wavelength response of the multiplexer :

Best tolerance for s = 12 µm => Lc = 13500 µm

VI-12 Example of application

VI-12 Example of application

VII – ACTIVE MODULATORS,

DEFLECTORS AND

SWITCHES

VII-1 Introduction

Modulator

Deflector

switches

VII-2 Basic definitions

Modulation depth

Bandwidth

m

0mI

II −=η

0

0I

II −=η

Intensity modulation by interference :

φ∆=η

2²sin

Bandwidth ∆f = fmax - fmin

Normally fmin ≈ 0

Switching time T = 2π / ∆f

Insertion loss

Power consumption

For modulators : driving power per unit bandwidth P / ∆f (mW / MHz)For switches : holding power is critical

Isolation [dB] = I1 : optical intensity in the driving portI2 : optical intensity in the driven port when switch is off

Modulator

It

I0 with no applied signalIm with max signal applied

=

0

ti I

Ilog10L

=

m

ti I

Ilog10L

Isolation (cross talk)

1

2IIlog10

VII-2 Basic definitions

VII-3 Electro-optic effect

Change in index of refraction produced by the application of an electric field

r : Pockels coefficient or linear electro-optic coeff

r ~ 10-12-10-10 m/V typically

crystals that do not possess inversion symmetry

Exemple : NH4H2PO4 (ADP), KH2PO4 (KDP), LiNbO3, LiTaO3, CdTe, GaAs, GaP

Pockels effect

...ESn21Ern

21n...Ea

21Ean)E(n 2332

21 +−−=+++= 32

31

naS,

na2r −=−=

Kerr effect

Ern21n)E(n 3−= 23ESn

21n)E(n −=

S : Kerr coefficient

S ~ 10-18-10-14 m/V in crystals

S ~ 10-19-10-22 m/V in liquids

Basic principle of electro-optic effect

VII-3 Electro-optic effectElectro-optic effect in anisotropic media

VII-3 Electro-optic effectPockels effect in anisotropic media

VII-3 Electro-optic effectPockels coefficients

VII-4 Single waveguide electro-optic modulatorPhase modulation

Polarization modulation

Substrate ns

x

y z

V

waveguide ngLinearly polarized incident beam at 45° to the x and y axes.

Phase change only for waves polarized in the y direction

Change of polarization which can be detected with polarization sensitive detector

tg

nc

Intensity modulation

nc << ns ~ ng

Threshold for 0th order guided mode2

g

0

gsg tn32

1nnn

λ>−=∆

Applying E field change ng : Guided mode ↔ unguided mode or vice versa

VII-4 Single waveguide electro-optic modulator

VII-5 Directional couplers as modulators

VI-6 Directional couplers as switches

Synchronous coupling => Cross state.When the effective indices, and therefore propagation constants, in the two waveguides aresufficiently different by applying bias, no coupling will occur => Bar-state

VII-7 Mach-Zehnder type electro-optic modulator

Linear intensity modulator: Modulating around BOn/Off switch: Modulating between A and C

VII-8 Electro-absorption modulators

Franz-Keldysh effect

VII-9 Basic principle of acousto-optic effect

Change in the refractive index caused by mechanical strain introduced by an acoustic wave

Periodic index variation, with a wavelength equal to that of the acoustic wave

Photoelastic effect => photoelastic tensor that relates the strain tensor to the optical indicatrixanalogy with the electro-optics tensor

Change in refractive index :

- n : index of refraction in the unstrained medium- p : appropriate element of the photoelastic tensor- Pa : total acoustic power (W)- ρ : mass density- νa : acoustic velocity- A : cross sectional area through which the waves travel- M2 : acousto-optics figure of merit

∆n is small ~ 10-4

)A2/(P10M)A2/(P10pnn a7

23aa

726 =ρν=∆

Travelling acoustic waves => grating in motion with respect the incident optical beam

=> frequency shift

c/λ0+c/Λ

Λ c/λ0-c/Λ

c/λ0c/λ0

λ0 << Λ

Effect generally negligible in acousto-optic phase or intensity modulators and beam deflectors

Can be used to produce optical frequency division multiplexing of signals

VII-9 Basic principle of acousto-optic effect

VII-10 Raman-Nath type modulator

no multiple diffraction : l << Λ² / λ (optical wavelength within the modulator material)

Diffraction by a simple phase grating diffraction :

Intensity

piezo-electric transducer

incident light I0 0th order

-1st order

1st order

2nd order

-2nd orderl

Λλ=θ 0msin

( )[ ]( )[ ]

=φ∆

>φ∆=

0m,J

0m,2/JII

20

2m

0

(Bessel functions) )a2/(lP10M2nl2a

72

00 λπ=

λ∆π=φ∆

a

Small index modulation in the 0th order mode; not convenient for optical switch

VII-11 Bragg type modulator

Multiple diffraction occurs: l >> Λ² / λ

Input angle optimally equals the Bragg angle :

Modulation depth : I0 transmitted without acoustic beam, I in 0th order

piezo-electric transducer

incident light I 0

0th order Output

-1st order

l

Λλ=θ

2sin B

a

θB

2θB

φ∆=−

2²sin

III

0

0

)a2/(lP10M2a

72

0λπ=φ∆ gives approximate results <= optical and acoustic fields not uniform

other higher orders have negligible intensity

Switch : constant frequency of the acoustic wave, variation of the acoustic power

=> 100 % diffraction from the 0th order beam to the 1st order beam

Deflector : variation of the acoustic frequency (wavelength)

=> change in the Bragg diffraction angle

piezo-electric transducer

incident light I 0

0th order Output

-1st order

la

θB

2θB

VII-12 Bragg type deflectors and switches

If b >> Λ, ∆θ1 = λ / b and ∆θ2 = λ / Λ

If the frequency of the acoustic wave is varied from that for which the Bragg condition is exactly satisfied, the diffracted beam is angularly scanned, but its intensity is reduced => bell-shaped pattern : ∆θ3 = 2 Λ / l

Number of resolvable spots : lb2N

1

3λΛ=

θ∆θ∆=

VII-12 Bragg type deflectors and switches

Maximum bandwidth of the acoustic signal :

νa : acoustic velocity

Number of resolvable spots :

t : transit time of the acoustic wave across the optical beam width :

Overall response time of an acoustic beam deflector :

∆fa : bandwidth of the acoustic transducer

If N is desired to be >> 1, then 1/ ∆fa must be << t =>

In order to obtain maximum speed of operation, τ must be minimized by reducing t and increasing ∆fa.

Trade-off between number of resolvable spots and speed => transit time can be reduced by making b smaller, but that results in fewer resolvable spots

∆fa may be the limiting factor

l2f a

0 λΛυ≅∆

0f.tN ∆=

a

btυ

=

tf1

f1

a0+

∆+

∆=τ

aaa

bf1t

f1

υ+

∆=+

∆≅τ

VII-12 Bragg type deflectors and switches

VII-13 Magneto-optic effects

( )0

nnlλ

−π=θ−+

Faraday effect : rotation of the direction of polarization of linearly^^ polarized light caused by an applied magnetic field

magnetization dependent difference in the index of refraction for right circularly polarized light (n+) as compared to left circularly polarized light (n-) propagating in the direction of polarization

Rotation angle : , l : length of travel

Magneto-optic materials not transparent in the visible range and strong absorption in the near IR and IR regions

In integrated devices this effect provides a couplign between TE and TM modes

VII-14 Exercice

A dual-channel coupler type of modulator has been designed so that κL = π/2 + mπ, m = 0, 1, 2, ... where κ is the coupling coefficient and L is the length. Thus complete transfer of light will occur from channel 0 to channel 1. 1- If we now apply a voltage to produce a ∆β = β0 - β1 , give the condition satisfied by g for complete cancellation of the transfer, where g² = κ² + (∆β/2)²2- Derive an expression for ∆β required to produce this complete cancellation in terms of length L.3- What change in refractive index in one waveguide (∆ng)would be required for such a cancellation ? Give your answer in terms of an expression in which the parameters are wavevectors (k) and coupling length (L)4- What would be the change of index required for the case of light with vacuum wavelength λ0 = 900 nm and a coupling length L = 1 cm ?5- A dual-channel directional coupler has κ = 4 cm-1, α = 0.6 cm-1, and ∆β=0. What length should it be to produce a 3 dB power divider? If that length is doubled, what fraction of the input power is in each channel at the output?

Solution : 1- gL = π+ mπ, m = 0, 1, 2, ... 2- ∆β L = 30.5 π3- ∆ng= 30.5 π /kL4- ∆ng= 7.8.10-5

5- L= π/(4 κ) = 1.963 mmat 2L, I0=0 and I1=exp(- 2αL) = 79 %

VII-14 Exercice

A Bragg modulator is formed in a LiNbO3 planar waveguide which is capableof propagating only the lowest order mode for light of 6328 Å (vacuum) wavelength.This mode has β=2.085 ×105 cm- 1. The bulk index of refraction for LiNbO3 atthis wavelength is 2.295. The acoustic wavelength in the waveguide is 2.5 µm, theoptical beam width 4.0 mm, and the interaction length 2.0 mm. Obtain the following:a) The angle (in degrees, with respect to the acoustic wave propagation direction)at which the optical beam must be introduced so as to obtain maximum diffractionefficiency.b) If the input optical beam is a uniform plane wave, what is the angular divergence(in degrees) between the half-power points of the diffracted beam, and at whatangle does it leave the modulator?c) Make a sketch of the modulator, labeling the angles found in a) and b). (Donot attempt to draw the angles to scale.)

VII-14 Exercice

VII-14 Exercice

An acousto-optic, Bragg-deflection-type RF spectrum analyzer is made ina planar waveguide in LiNbO3 with an index n =2.0. Signals with modulationfrequencies ranging from 50 to 500 MHz are analyzed by using them to drive theacousto-optic transducer which produces a travelling acoustic wave in the LiNbO3with a velocity νa=3000 m/s. The optical beam has a wavelength in air of λ0=6328 Å and a width b =1 mm. The interaction length l =5mm.a) What is the angular deflection (in degrees) when the modulation frequency is500 MHz?b) What is the number of resolvable spots (the ratio of the envelope width to thespot width) at a frequency of 500 MHz?

VII-14 Exercice

VII-14 Exercice

VIII – SOURCES,

AMPLIFIERS AND

DETECTORS

VIII-1 Laser diodeBasic structure : p-n junction

0

+V

L

Wn

p

Small size, relatively simple construction, high reliability

Usual materials : GaAs, Ga(1-x)AlxAs, GaxIn(1-x)As(1-y)Py

pumping energy => inverted population

mirrors => optical feedback

Optical modes

Fabry-Perrot cavity =>

Mode spacing :

Optical power

λ

spontaneous emission below threshold

lasing mode structure above threshold

m = 2Ln/ λ0

00200 d

dnL2Ln2ddm

λλ+

λ−=

λSC laser are always operated near the bandgap wavelength, where n is a strong function of λ

For dm=-1, the mode spacing is given by :

λλ−

λ=λ

00

20

0

ddnnL2

d

Usually several longitudinal modes will coexist, with λ near the peak wavelength of spontaneous emission.Typical mode spacing for a AsGa laser : 0.3 nm

20 nm

0.1 nm

VIII-1 Laser diode

Lasing threshold conditions

Corresponds to the point at which the increase in the number of lasing mode photons (per second) due to stimulated emission just equals the number of photons lost (per sec) because of scattering, absorption or emission from the laser

Optical Power

Pump current density

Threshold current density

Distance normal to

junction plane

D

p

n

d

D

p

n

d

Density of photons

Photon distribution spreads into inactive regions (diffraction)

=> Only d/D photons in the lasing mode can generate new photons by stimulated emission

=> Reduced gain

VIII-1 Laser diode

Lasing threshold conditions

Threshold calculation :

0.5 Pout

0.5 Pout R Ps

R Ps Ps

PsPs : optical power incident internally on each end face0.5 Pout = (1-R)Ps : emitted power

( )[ ]LDdgexpRPP ss α−= with gain

ν∆πλη

=den8J

g 2

20q ηq : internal quantum efficiency

λ0 : vacuum wavelength emission : refractive index at λ0∆ν : linewidth of spontaneous emissione : electronic chargeJ : inhected current densityα : all kinds of optical loss

( )R1ln

L1Den8J 2

0q

2

TH +αλη

ν∆π=

spreading of photons => increase of threshold current density

D=d for optimum performances => confined field lasers

VIII-1 Laser diode

Output power and efficiency

Losses over a small distance ∆z: ( )( )( ) zPz11PP

e1PPePP

loss

zzloss

∆α=∆α−−≈−=−= ∆α−∆α−

or PdzdPloss α=

Power absorbed over length L: ∫α=∫=L

0

L

0lossloss PdzdPP

Similarly, power generated per pass: ∫=∫=L

0

L

0gengen Pdz

DdgdPP

Internal power efficiency:

R1lnL

1R1ln

L1

Ddg

Ddg

PPP

gen

lossgen

+α=

α−=

−=η

Power out: ( )[ ]νηη=η= hLxWeJPP qinout

Increasing T => larger α, reduced ηq increased JTH, decreased power out preferably pulsed laser diodes

VIII-1 Laser diode

VIII-2 Heterojunction laser structuresCharacteristics

Waveguiding structure => optical confinement => reduction of thresholdincrease in efficiency

Higher carrier injection efficiency at the p-n junction

Confine the carriers to the junction region

p+ GaAs

p+ Ga0.94Al0.06As

p Ga0.98Al0.02As

n Ga0.94Al0.06As

n+ GaAs substrate

5 µm

5 µm

0.8 µm

refractive index

3.58

3.58

3.56

3.53

3.53

double heterostructure laser diode

VIII-3 Vertical cavity lasersVertical-cavity surface-emitting lasers (VCSEL)

Advantages over fabry-Perrot endface diode lasers : - can be made in a circular shape with a diameter matched to the core diameter of a fiber to optimize coupling- relatively large emitting area => beam remains well collimated and doe not diverge due to diffraction- can be conveniently arranged into a surface array to couple to a fiber bundle- surface orientation of emission facilitates integration with planar electronic circuitry

VIII-4 Distributed feed-back lasersBasic principle

End-face mirrors planar diffraction grating

GaAs substrate

Ga0.7Al0.3AsGaAs

Λ

higher order Bragg diffraction gratings

Grating formula :Λλ+θ=θ msinnsinn igmg

θi θm

ng

Bragg condition : mth order is reflected (θi = - θm)mg sinn

m2θ

λ=Λ

Let θi equals -90° : p = 0, ..., mgn

m2 λ=Λ Λλ+−=θg

p nm1sin

p = 0p = 1

1st order Bragg grating 2nd order 3rd order 4th order

loss for the optical waveguide

VIII-4 Distributed feed-back lasers

Distributed Bragg reflection structure

Two Bragg gratings located at both ends of the laser and outside the electrically-pumped active region

VIII-4 Distributed feed-back lasers

Performance characteritics

Wavelength selectability :

cleaved-end-face lasers : gain curve and Fabry-Perrot resonator=> number of longitudinal modes lase simultaneously

DFB or DBR : gain curve and grating period=> single-longitudinal-mode operation

Optical emission linewidth :

grating much more wavelength selective than a cleave end-face=> narrower emission linewidth for a DFB or DBR laser

Stability :

DFB lasers offer improved wavelength stability as compared to cleaved-end-face lasers, because the grating tends to lock the laser to a given wavelength

Lower threshold current density and high output power for DFB lasers

VIII-4 Distributed feed-back lasers

VIII-5 Amplifiers : Introduction

In lightwave communication systemsneed for repeaters at regular intervals to amplify the signal to compensate for losses

Early systemsphotodetector : lightwave signal electrical current waveformelectronic amplifierlaser or high speed LED (ligth emitting diode) : electrical signal optical signal

Drawbacks : Use of additional components => reduced reliabilityElectronic amplification => limited bandwidth

Optical amplifiers : directly amplifies the optical signal

Optical fiber amplifiersSemi-conductor optical amplifiers

More efficient and reliable=> Predominant type of amplifier used in long-distance lightwave telecom and datacom systems

VIII-6 Optical fiber amplifiers

Optical fiber amplifiers = glass optical fiber doped with optically active atoms such as Erbium

Active atoms enter the structure of the host glass as ionscharacteristic splitting of ion energy levels

stimulated emission of photon when optically pumped at an appropriate wavelengthstimulated by a lightwave signal

The number of photons produced by stimulated emission

=

The number of photons present in the stimulating lightwave signal

The amplitude modulation of the amplified lightwaves is preserved

Principle

Erbium doped fiber amplifiers (EDFA)

Absorption sprectrum of erbium-doped silica fiber

Energy levels and transitions in an EDFA

Emission central wavelength in the third telecommunication windowBand energy levels => gain in the range 1520 nm – 1600 nm

Semi-conductor pumping laser sources at 0.81, 0.98, 1.48 µm

VIII-6 Optical fiber amplifiers

Physical configuration of an EDFAVIII-6 Optical fiber amplifiers

Characteristics of an EDFA

Optical gain : stim abs aseopt

sig

P P PGP

− −=

stim aseopt

sig

P PGP−= for a strongly inverted population N2>>N1

Psig : input signal to be amplifiedPstim : power due to stimulated emissionPase : power due to amplified spontaneous emission

stim aseopt

sig

P PG 10logP

−=

In decibels (dB)

A gain of 30 dB (103) is easily attainable with only few mW of pump power, Ppump at 980 nm

Pumping at 980 nm is more efficient than pumping at 1480 nm

A gain efficiency (Gopt/Ppump) of 8-10 dB/mW is typical for pumping at 980 nm

EDFAs are very stable with respect to ambient environnemental conditions (temperature, eletric field (Stark effect negligeable), magnetic field)

But, the range of wavelength available is limited (1525-1575 nm with a max efficiency)

VIII-6 Optical fiber amplifiers

Absorption characteristics of glass optical fibers

Modern low OH- content silica fibers have less than 0.35 dB/km loss over the wavelength range from 1300 nm to 1600 nm, permitting the distribution of DWDM channels over the entire range as

long as suitable amplifiers can be found

VIII-6 Optical fiber amplifiers

Raman optical fiber amplifiers

Can extend the range of useful wavelengths into the S band

VIII-6 Optical fiber amplifiers

VIII-7 Semi-conductor optical amplifiers

Electrical pumping

Use of p-n junction below the lasing threshold so that oscillation does not occur• The travelling-wave type : the facets are antireflection coated => one pass through the inverted population region of the diode

• Fabry-Perrot type : the end-facets are left uncoated => the incident light is amplified during successive passes before being emitted at a higher power level

The injection-locked type : use of p-n junction above threshold => operate as a laserthe injection of an intensity modulated light signal at one facet causes the light waves produced by the lasing action to « lock » with the input light waves so as to follow the same intensity and phase variations, but at a higher optical power level.

Advantages : fast-switchingsmallwavelength range from 800nm to 1600 nm

Advantages of travelling-wave laser amplifiers : high gain, wide spectral bandwidth, low sensitivity to changes in either temperature or drive current level

VIII-8 Comparison ion-doped FA and SOAWavelength range

(1570 to 1620nm) SOARaman

L-band

(1525 ta 1565 nm) SOAEDFARaman

C-band

(1480 to 1520nm) SOATDFARaman

Third Window S-band

(approximately 1300 ta 1360 nm) SOAPDFA (Praseadymium)NDFA (Neadymium)

Second Window

(approximately 800 to 900 nm) SOATDFA (Thulium)

First Window

Performance characteristicsVIII-8 Comparison ion-doped FA and SOA

VIII-9 PhotodiodePrinciple

p-n junction stateEr

Photo electric effect : If hν > Eg, the electrons are pulled up into the conduction band, leaving holes in their place in the valence band.

Electron-hole pairs occur throughout the P-layer, depletion layer and N-layer materials.

In the depletion layer the electric field accelerates these electrons toward the N-layer and the holes toward the P-layer. Cross-section

a positive charge in the P-layer and a negative charge in the N-layer

Photoconductor mode

VVseuil

Increasing flux

RmIr=Vd+Es

RmEs

Vd

Ir

I=-IrLinear behaviour

PN

A bias voltage is applied

e-

VIII-9 Photodiode

Photovoltaic mode

VVth

Increasing flux

RmIr=Vd

I=-Ir

Vco

No bias voltage is applied

Diode ≡ electrical generator

or Rm

I

VIII-9 Photodiode

Features 1.Excellent linearity with respect to incident light 2.Low noise 3.Wide spectral response 4.Mechanically rugged 5.Compact and lightweight 6.Long life

See course “capteurs photoniques”, master 1

VIII-9 Photodiode

Waveguide photodiodes

Light is incident transversally on the active volume => 100% quantum efficiency can be obtained

-V

0

n - substrate

n

p

W

L

( )L0J e 1 e−α= φ − e : electronic charge

: incident photon flux (in photons.cm-2)0φV can be chosen so that the depletion layer thickness = WL can be chosen so that αL >> 1

Low capacitance => improved response time

VIII-9 Photodiode

VIII-10 Exercice

Ex1 : We would like to design a p-n junction laser for use as the transmitter ina range finder. The output is to be pulsed with a peak power out of each end of10 W (only the output from one end will be used) and a pulse duration of 100 ns.Wavelength is to be 9000 Å. Room temperature operation is desired and some of thepertinent parameters have been either measured or established so that:1) Half-power points of the emission peak for spontaneous emission have beenmeasured for this material at room temperature to be at 9200 and 8800 Å.2) Index of refraction: 3.3.3) Thickness of light emitting layer: 10 µm.4) Thickness of active (inverted pop.) layer: 1 µm.5) Internal quantum efficiency: 0.7.6) Average absorption coefficient: 30 cm-1 .7) W =300 µm.8) Reflectivity of the Fabry-Perot surfaces: 0.4.a) What must be the separation between Fabry-Perot surfaces if we wish to havea peak pulse current density of 3 ×104 A.cm-2 ?b) What is the threshold current density?

h=6.62 10-34 J.se=1.6 10-19 C

Ex1 :

VIII-10 Exercice

Ex2 :What is the grating spacing required for a DFB laser to operate at a wavelength λ0=8950 Å in GaAs (n =3.58) if a first-order grating is desired?

Ex3 : a) If a surface grating is formed on a waveguide, what is the grating spacing Λ required for fourth-order Bragg diffraction, in terms of λ and ng?b) Sketch the reflected wave directions, and indicate the angles between them.c) In general, can a scattered wave perpendicular to the direction of the waveguidebe observed for odd orders, such as the first and third order, of Bragg diffraction?

VIII-10 Exercice

Ex2 :

Ex3 :

VIII-10 Exercice

VIII-10 Exercice

IX – APPLICATIONS

IX-1 RF spectrum analyzer

Originally : to enable the pilot of a military aircraft to obtain an instantaneous spectral analysis of an incoming radar beam, in order to determine if his plane is being tracked by a ground station, air-to-air missile,…

Laser source

Each detector element represents a particular frequency channel.The output signal at each frequency is proportional to the RF power at that frequency

RF signal to be spectrally analysed

Acoustic transducer

Hamilton et al., Opt. Eng. 16, 475 (1977)

IX-2 Monolithic wavelength-multiplexed optical source

DFB laser coupled to a GaAlAs waveguide by direct transmission

High index layer

Aiki et al., IEEE J. QE-13, 597 (1977)

Schematic drawing of wavelength multiplexed light source

Aiki et al., IEEE J. QE-13, 220 (1977)

IX-2 Monolithic wavelength-multiplexed optical source

IX-3 IO Doppler velocimeter

H. Toda et al., IEEE J. LT-5, 901 (1987)

IX-4 IO optical disk readhead

Ura et al., IEEE J. LT-4, 913 (1986)

IX-5 IO temperature sensor

No electrical connection⇒useful in explosive or flammable environments

High resolutionWide range

( )

out0

in

eff eff

P 21 mcos b LTP z

d Ldn nbdT L dT

γ π = + ∆ + ∆φ λ ∆

= +∆

Johnson et al., Appl. Phys. Lett. 41, 134 (1982)

IX-6 IO surface plasmon resonance biosensor

Spectral interrogation of surfaceplasmon resonance

Measure bulk refractive index changes smaller than 1.2 10-6

In conjunction with specific biomolecular recognition elements

=> Biological sensor

X- Introduction to photonic crystals

Excerpt

from a S. Johnson course online, MIT

To Begin: A Cartoon in 2d

planewave

r E ,

r H ~ ei(

r k ⋅ r x −ωt )

r k = ω / c = 2π

λ

r k

scattering

To Begin: A Cartoon in 2d

planewave

r E ,

r H ~ ei(

r k ⋅ r x −ωt )

r k = ω / c = 2π

λ

r k

• • •

• • •

• • •

• • •

• • •

• • •

• • •

• • •

• • •

• • •

••

••

••

••

••

••

••

••

••

••

••

••

••

••

for most λ, beam(s) propagatethrough crystal without scattering(scattering cancels coherently)

...but for some λ (~ 2a), no light can propagate: a photonic band gap

a

A mystery from the 19th century

e–

e–

r E

+

+

+

+

+

r J = σ

r E current:

conductivity (measured)

mean free path (distance) of electrons

conductive material

A mystery from the 19th century

e–

e–

r E

+

r J = σ

r E current:

conductivity (measured)

mean free path (distance) of electrons

+ + + + + + +

+ + + + + + + +

+ + + + + + + +

+ + + + + + + +

crystalline conductor (e.g. copper)

10’sof

periods!

A mystery solved…

electrons are waves (quantum mechanics)1

waves in a periodic medium can propagate without scattering:

Bloch’s Theorem (1d: Floquet’s)

2

The foundations do not depend on the specific wave equation.

Electronic and Photonic Crystalsatoms in diamond structure

wavevector

elec

tron

ener

gy

Peri

odic

Med

ium

Blo

ch w

aves

:B

and

Dia

gram

dielectric spheres, diamond lattice

wavevectorph

oton

freq

uenc

y

interacting: hard problem non-interacting: “easy” problem

1887 1987

Photonic Crystalsperiodic electromagnetic media

with photonic band gaps: “optical insulators”

2-D�

periodic in�two directions�

3-D�

periodic in�three directions�

1-D�

periodic in�one direction�

(need a more

complex topology)

Photonic Crystals in Nature

wing scale:

Morpho butterfly

6.21µm

[ L. P. Biró et al., PRE 67, 021907

(2003) ]

Peacock feather

[J. Zi et al, Proc. Nat. Acad. Sci. USA,100, 12576 (2003) ]

[figs: Blau, Physics Today 57, 18 (2004)]

Layer-by-Layer Lithography

• Fabrication of 2d patterns in Si or GaAs is very advanced(think: Pentium IV, 50 million transistors)

So, make 3d structure one layer at a time

…inter-layer alignment techniques are only slightly more exotic

Need a 3d crystal with constant cross-section layers

A Schematic

[ M. Qi, H. Smith, MIT ]

7-layer E-Beam Fabrication

5 µm

[ M. Qi, et al., Nature 429, 538 (2004) ]

The Woodpile Crystal

[ S. Y. Lin et al., Nature 394, 251 (1998) ]

gap

(4 “log” layers = 1 period)

http://www.sandia.gov/media/photonic.htm

Si

[ K. Ho et al., Solid State Comm. 89, 413 (1994) ] [ H. S. Sözüer et al., J. Mod. Opt. 41, 231 (1994) ]

an earlier design:(& currently more popular)

Holographic Lithography[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]

absorbing material

Four beams make 3d-periodic interference pattern

(1.4µm)

k-vector differences give reciprocal lattice vectors (i.e. periodicity)

beam polarizations + amplitudes (8 parameters) give unit cell

One-PhotonHolographic Lithography

[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]

huge volumes, long-range periodic, fcc lattice…backfill for high contrast

10µm

Mass-production II: Colloids

microspheres (diameter < 1µm)silica (SiO2)

sediment by gravity intoclose-packed fcc lattice!

(evaporate)

Mass-production II: Colloids

http://www.icmm.csic.es/cefe/

Inverse Opals

fcc solid spheres do not have a gap… …but fcc spherical holes in Si do have a gap

Infiltration

sub-micron colloidal spheres

Template(synthetic opal)3D

Remove Template

“Inverted Opal”

complete band gap

~ 10% gap between 8th & 9th bandssmall gap, upper bands: sensitive to disorder

[ figs courtesyD. Norris, UMN ]

[ H. S. Sözüer, PRB 45, 13962 (1992) ]

Inverse-Opal Photonic Crystal[ fig courtesy

D. Norris, UMN ]

[ Y. A. Vlasov et al., Nature 414, 289 (2001). ]

Properties of Bulk Crystalsby Bloch’s theorem

(cartoon)

cons

erve

d fr

eque

ncy

ω

conserved wavevector k

photonic band gap

band diagram (dispersion relation)

dω/dk → 0: slow light(e.g. DFB lasers)

backwards slope:negative refraction

strong curvature:super-prisms, …

(+ negative refraction)

synthetic mediumfor propagation

Superprisms[Kosaka, PRB 58, R10096 (1998).]

from divergent dispersion (band curvature)

Negative-refractiveall-dielectric photonic crystals

[ M. Notomi, PRB 62, 10696 (2000). ]

negative refraction focussing

(2d rods in air, TE)

Superlensing with Photonic Crystals

Here, using positive effective index but negative “effective mass”…

[ Luo et al, PRB 68, 045115 (2003). ]

2/3 diffraction limit

the magic of periodicity:unusual dispersion without scattering

Intentional “defects” are good

microcavities waveguides (“wires”)

Review: Why no scattering?

forbidden by gap(except for finite-crystal tunneling)

forbidden by Bloch(k conserved)

Benefits of a complete gap…

broken symmetry –> reflections only

effectively one-dimensional

“1d” Waveguides + Cavities = Devices

splitters�

Lossless Bends

symmetry + single-mode + “1d” = resonances of 100% transmission

[ A. Mekis et al.,Phys. Rev. Lett. 77, 3787 (1996) ]

protectivepolymersheath

Optical Fibers Today(not to scale)

silica claddingn ~ 1.45

more complex profilesto tune dispersion

“high” indexdoped-silica core

n ~ 1.46

“LP01”confined modefield diameter ~ 8µm

losses ~ 0.2 dB/kmat λ=1.55µm

(amplifiers every50–100km)

but this is~ as good as

it gets…[ R. Ramaswami & K. N. Sivarajan, Optical Networks: A Practical Perspective ]

The Glass Ceiling: Limits of Silica

Long DistancesHigh Bit-Rates

Dense Wavelength Multiplexing (DWDM)

Loss: amplifiers every 50–100km…limited by Rayleigh scattering (molecular entropy)

…cannot use “exotic” wavelengths like 10.6µm

Nonlinearities: after ~100km, cause dispersion, crosstalk, power limits(limited by mode area ~ single-mode, bending loss)

also cannot be made (very) large for compact nonlinear devices

Compact Devices

Radical modifications to dispersion, polarization effects?…tunability is limited by low index contrast

Breaking the Glass Ceiling:Hollow-core Bandgap Fibers

1000x betterloss/nonlinear limits

(from density)

Photonic Crystal

1dcrystal

Bragg fiber[ Yeh et al., 1978 ]

+ omnidirectional

= OmniGuides

2dcrystal

PCF[ Knight et al., 1998 ](You can also

put stuff in here …)

Breaking the Glass Ceiling: Hollow-core Bandgap Fibers

Bragg fiber[ Yeh et al., 1978 ]

+ omnidirectional

= OmniGuidefibers

PCF[ Knight et al., 1998 ]

white/grey= chalco/polymer

5µm[ R. F. Cregan et al.,

Science 285, 1537 (1999) ]

[ figs courtesy Y. Fink et al., MIT ]

silica

Breaking the Glass Ceiling II:Solid-core Holey Fibers

solid core

holey cladding formseffective

low-index material

[ J. C. Knight et al., Opt. Lett. 21, 1547 (1996) ]

Can have much higher contrastthan doped silica…

strong confinement = enhancednonlinearities, birefringence, …

Bibliography

• R.G. Hunsperger, “Integrated Optics- Theory and technology”, fifth edition, Springer (535.14 HUN)

• Ching-Fuh Lin, “Optical components for communications”, Kluwer Academic Publishers (621.391.8 LIN)

• T. Tamir, “Integrated Optics”, second edition, Springer-Verlag(535.374 INT)

• T. Tamir, “Guided-wave optoelectronics”, second edition, Springer-Verlag(535.13 GUI)

• “Introduction to integrated optics”, edited by M. Barnoski(Sacha)

• J.P.Meunier, “Télécoms optiques”, Hermès Lavoisier(621.391.8 MEU)

• H. Haus, “Waves and fields in optoelectronics”, (621.371 HAU)

• Steven G. Johnson website (MIT) on photonic crystals : http://ab-initio.mit.edu/photons/tutorial/• Lih Y. Lin website (University of Washigton) on intergarted optics and optical MEMS : http://www.ee.washington.edu/people/faculty/lin_lih/EE539/index.html• O. Parriaux, cours Optique guidée