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Nonlinear effects in Silicon Waveguides
Govind P. Agrawal
Institute of Optics
University of Rochester
Rochester, NY 14627
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2007
G. P. Agrawal
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Outline
• Why use silicon for photonics?
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• Silicon-on-Insulator (SOI) Waveguides
• Two-Photon Absorption and Free-Carrier Effects
• Self-Phase Modulation and Soliton Formation
• Higher-Order Solitons and Supercontinuum Generation
• Cross-Phase Modulation and Optical Switching
• Four-Wave Mixing and Parametric Amplification
• Raman Amplification and Silicon Raman Lasers
• Concluding Remarks
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Silicon Photonics
• Silicon dominates microelectronics industry totally.
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• Silicon photonics is a new research area trying to capitalize on the
huge investment by the microelectronics industry.
• It has the potential for providing a monolithically integrated
optoelectronic platform on a silicon chip.
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Credit: Intel and IBM Websites
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Silicon-on-Insulator Technology
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• Core waveguide layer is made of silicon (n1 = 3.45).
• A silica layer under the core layer is used for lower cladding.
• Silica layer formed by implanting oxygen, followed with annealing.
• A rib or ridge structure used for two-dimensional confinement.
• Air on top confines mode tightly (large index difference).
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SOI Waveguide Geometry
[0 0 1]
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[1 1 0]
• SEM image of an SOI device;
Izhaky et al., JSTQE, (2006).
• Typically W ∼ 1 µm,
H ∼ 1 µm, h ∼ H/2.
• Dispersion normal at telecom
wavelengths (λ0 ∼ 2.2 µm).
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Waveguide Dispersion
• Mode index n̄(ω) = n1(ω) − δ nW (ω).
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• Material dispersion results from n1(ω) (index of silicon).
• Waveguide dispersion results from δ nW (ω).
• Total dispersion D = DM + DW can be controlled.
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Major Nonlinear Effects
• Self-Phase Modulation (SPM)
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• Cross-Phase Modulation (XPM)
• Four-Wave Mixing (FWM)
• Stimulated Brillouin Scattering (SBS)
• Stimulated Raman Scattering (SRS)
Origin of Nonlinear Effects in Silicon
• Third-order nonlinear susceptibility χ (3).
• Real part leads to SPM, XPM, and FWM.
• Imaginary part leads to two-photon absorption (TPA).
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Third-order Nonlinear Response
3ε0
(3)
e
e∗
e
=
χ
i jkl (−ωi ; ω j , −ωk , ωl )E j (ω j )Ek (ωk )El (ωl )dω j dωk
2
16π
• Two distinct contributions from electronic and Raman responses:
(3)
Pei (ωi)
ZZ
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(3)
χi jkl = χiejkl + χiRjkl .
• Electronic response for silicon is governed by
e
e
e
χiejkl = χ1122
δi j δkl + χ1212
δik δ jl + χ1221
δil δ jk + χdeδi jkl .
• Raman response for silicon is governed by
eR(ωl − ωk )(δik δ jl + δil δ jk − 2δi jkl ) + a similar term.
χiRjkl = χRH
• A relatively narrow Lorentzian Raman-gain spectrum:
2
Ω
Ω
Γ
R
R
R
eR(Ω) =
H
=
15.6
THz,
= 105 GHz .
π
Ω2R − Ω2 − 2iΓRΩ 2π
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Third-order Nonlinear Susceptibility
• The tensorial nature of χ (3) makes theory quite complicated.
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• It can be simplified considerably when a single optical beam excites
the fundamental TE or TM mode of the Si waveguide.
e
• Only the component χ1111
(−ω; ω, −ω, ω) is relevant in this case.
• Its real and imaginary part the Kerr coefficient n2 and the TPA
coefficient βT as
i
3ω
ω
e
n2(ω) + βTPA(ω) =
(−ω; ω, −ω, ω).
χ1111
2
2
c
2
4ε0c n0
• A recent review provides more details:
Q. Lin, O. J. Painter, G. P. Agrawal, Opt. Express 15, 16604 (2007).
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Measured Nonlinear Parameters
4
0.7
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0.6
3
0.5
0.4
2
0.3
0.2
1
0.1
0.0
0
1200
1400
1600
1800
2000
2200
2400
1200
1400
1600
1800
2000
2200
2400
Lin et al, Appl. Phys. Lett. 91, 021111 (2007)
• We measured n2 and βTPA for silicon in the spectral range of 1.2 to
2.4 µm using the z scan technique.
• n2 larger for silicon by >100 compared with silica fibers.
• As expected, βTPA vanishes for λ > 2.2 µm
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Nonlinear Parameters
• Refractive index depends on intensity as (Kerr effect):
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n(ω, I) = n̄(ω) + n2(1 + ir)I(t).
• Material parameter n2 = 3 × 10−18 m2/W is larger for silicon by a
factor of 100 compared with silica fibers.
• Dimensionless parameter r = βTPA/(2k0n2) is related to two-photon
absorption (TPA) occurring when hν exceeds Eg/2.
• TPA parameter: βTPA = 5×10−12 m/W at wavelengths near 1550 nm.
• Dimensionless parameter r ≈ 0.1 for silicon near 1550 nm.
• TPA is a major limiting factor for SOI waveguides because it creates
free carriers (in addition to nonlinear losses).
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Free-Carrier Generation
• TPA creates free carriers inside a silicon waveguide according to
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∂ Nc βTPA 2
Nc
=
I (z,t) − .
∂t
2hν0
τc
• Carrier lifetime is relatively large for silicon (τc > 10 ns).
• It limits the device response time if carriers cannot be removed
quickly enough.
• Free carriers also introduce loss and change the refractive index.
• Pulse propagation inside silicon waveguides is governed by
∂ A iβ2 ∂ 2A
σ
αl
2
+
=
ik
n
(1
+
ir)|A|
A
−
(1
+
iµ)N
A
−
A.
0 2
c
∂z
2 ∂t 2
2
2
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Free-Carrier Absorption (FCA)
• Loss induced by FCA: α f = σ Nc with σ = 1.45 × 10−21 m2.
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• Free carriers also change the refractive index by ∆n = −(µ/2k0)σ Nc
(free-carrier dispersion).
• This change is opposite to the index change n2I resulting from the
Kerr effect.
• Parameter µ is known as the “linewidth enhancement factor” in the
context of semiconductor lasers.
• Its value for silicon is close to 7.5 in the spectral region near 1550 nm.
• Absorption and index changes resulting from free carriers affect the
performance of silicon waveguides.
• Quick removal of carriers helps (e.g., by applying a dc electric field).
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Removal of Free Carriers
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Jones et al, Opt. Exp. 13, 519 (2005)
• A reversed-biased p-n junction is used for this purpose.
• Electric field across the waveguide removes electrons and holes.
• Drift time becomes shorter for larger applied voltages.
• Effective carrier lifetime can be shortened from >20 to <1 ns.
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Self-Phase Modulation (SPM)
• Refractive index depends on intensity as n0 = n̄ + n2I(t).
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• Propagation constant also becomes intensity-dependent:
β 0 = β + k0n2(P/Aeff) ≡ β + γP.
• Nonlinear parameter γ = k0n2/Aeff can be larger by a factor of
10,000 compared with silica fibers.
• Nonlinear Phase shift:
φNL =
Z
0
L
(β − β ) dz =
0
Z
L
γP(z) dz = γPinLeff.
0
Here, P(z) = Pine−αz and Leff = (1 − e−αL )/α.
• Optical field modifies its own phase (SPM).
• Phase shift varies with time for pulses (chirping).
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Chirping and Spectral Broadening
1
2
(a)
(b)
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0.8
Chirp, δωT0
Phase, φNL
1
0.6
0.4
0
−1
0.2
−2
0
−2
−1
0
1
Time, T/T0
2
−2
−1
0
1
Time, T/T0
2
• In the case of optical pulses, φNL(t) = γP(t)Leff.
• Chirp is related to the phase derivative dφNL/dt.
• Phase and chirp profiles for super-Gaussian pulses are shown
using P(t) = P0 exp[−(t/T )2m] with m = 1 and m = 3.
• SPM creates new frequencies and leads to spectral broadening.
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Self-Phase Modulation and TPA
• Preceding analysis neglected two-photon absorption.
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• Its impact on SPM can be studied by solving:
∂A
αl
= iγ(1 + ir)|A|2A − A.
∂z
2
• This equation ignores dispersive and free-carrier effects.
√
• Using A = P exp(iφNL), we obtain the following analytic solution:
P(L,t) =
P(0,t) exp(−αl L)
,
1 + 2rγP(0,t)Leff
1
ln[1 + 2rγP(0,t)Leff].
2r
• TPA converts linear dependence of φNL on power to a logarithmic
one.
φNL(L,t) =
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Impact of Two-Photon Absorption
• In the absence of TPA,
φ0 = φmax = γP0Leff.
• Inset shows the reduction
using r = 0.1.
φmax= 3π
φmax= 2π
2
φmax= π
8
φ0 / π
2.5
0
φ0 = ln(1 + 2rφmax)/(2r).
10
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nonlinear phase shift φ / π
• TPA reduces the maximum phase shift:
3
6
4
2
0
0
1.5
2
4
φmax/ π
6
8
10
1
0.5
0
0.2
0.4
0.6
βTPA/(k0n2)
• TPA-induced reduction becomes severe at high powers.
• When φmax = 100, φ0 is limited to a value of 15.
0.8
1
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Impact of Free-Carrier Generation
Spectral density J/(cm2⋅nm)
φmax= 1.5π, I0= 1.2 GW/cm2
0.015
TPA=0
0.01
max
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0.025
TPA
only
0.02
Full
2
0.015
0.01
0.005
0.005
0
1.549 1.5495
1.55 1.5505 1.551
λ (µm)
0
φmax= 15.5π, I0= 12.5 GW/cm2
4
0.015
TPA
0.01 only
TPA
+FCA
0.005
Full
1.548
φ
1.55
λ (µm)
1.552
= 15.5π, I = 12.5 GW/cm2
max
3
Phase shift φ / π
Spectral density J/(cm2⋅nm)
= 7.5π, I0= 6.0 GW/cm2
φ
0
TPA
only
2
1
0
−1
−2
TPA
+FCA
P(t) = P0e−t /T
T0 = 10 ps
L = 2 cm
τc = 1 ns
αl = 1 dB/cm
2
Yin and Agrawal,
Opt. Lett. (2007)
Full
−3
0
1.548
1.549
1.55 1.551
λ (µm)
1.552
−5 −4 −3 −2 −1 0 1 2 3 4 5
t / T0
Free carriers produce a nonlinear phase shift in the opposite direction.
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Experimental Results
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Boyraz et al., Opt. Exp. 12, 829 (2004).
• First observation of SPM-induced spectral broadening in 2004.
• 4-ps pulses launched inside a 2-cm-long SOI waveguide.
• The 3-peak output spectrum broadened by a factor of 2 when peak
intensity was 2.2 GW/cm2 (P0 ≈ 100 W).
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mitted intensity but not in spectral shape and position. The spectral full width at half
m (FWHM) is identical to pulses measured without sample. Hence, the applied
f the two low-power spectra belong to the regime where the waveguide is responding
linear. However, by increasing the peak power to P = 1.55 W (I ≈ 2.6 GW/cm2), the
hape of the pulses broadens dramatically; now exhibiting several spectral side wings.
dening continues for higher powers which can be seen in the spectrum with P = 6.85
.6 GW/cm2).
Experimental Results
-70
Transmission (dBm)
100
-75
-80
-85
6.85 W
1.55 W
100
200
300
400
8 mW
1.8 mW
200 300
400
53.2
49.9
43.6
37.4
31.2
24.9
18.7
12.5
6.23
0
500
600
-90
• Larger broadening by 2006.
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• 1.8-ps pulses launched
inside a 4-mm-long
waveguide.
• Width 470 nm
height 226 nm.
-95
-100
-105
1480
1500
1520
Wavelength (nm)
• Spectral asymmetry is due
to free-carrier effects.
Fig. 1. Transmission spectra of a 4 mm long SOI waveguide as a function of coupled peak
power. Linear optical response is observed for low powers (black and red) whereas SPMinduced spectral broadening occurs for high powers (green and blue). Inset: Time- and
spectrally resolved FROG trace of input pulses at 1500 nm.
Dulkeith et al., Opt. Exp. 14, 5524
(2006).
• Inset shows the FROG
trace.
measured spectral pulse distortions can be explained by self-phase-modulation,
known to cause an intensity dependent phase shift of the pulse carrier frequency.
s own duration, the pulse experiences an intensity- and thus time-dependent
index. New frequency components are generated and the initial pulse spectrum
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re broadened and asymmetric. The significant difference is that SPM-induced
ng becomes more efficient for longer wavelengths. The spectral distance between the
velength SPM oscillation and the originally injected laser wavelength increases from
m when the OPA is tuned from 1400 to 1650 nm. As will be discussed in the next
his enhancement for longer wavelengths is probably due to an increase of group
nd nonlinear refractive index n2.
Wavelength Dependence of SPM
Transmission (dB)
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• SPM-broadened spectra at
three different wavelengths.
-80
-85
-90
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• 1.8-ps pulses tunable from
1400 to 1650 nm.
-95
-100
-105
1400 1450 1500 1550 1600 1650
Wavelength (nm)
• Larger broadening at longer
wavelengths.
• Consistent with a larger n2
near 1550 nm
Fig. 4. SPM-induced spectral broadening of optical pulses for the same coupled
peak power (6.85 W) injected into the SOI waveguide at 1400, 1500 and 1600 nm.
The spectral broadening increases with the wavelength.
Dulkeith et al., Opt. Exp. 14, 5524
(2006).
Experimental
Results agree with theoretical predictions.
arison
with theory and discussion
l description
ze the experimental data we have used a recently developed theoretical model to
he pulse dynamics in Si photonic wires. It accounts for GVD, parametric and nonc nonlinear effects such as SPM, XPM, TPA, or Raman interaction; free carrier-
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Formation of Optical Solitons
• SPM-induced phase shift φNL > 1 is easily realized.
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• φNL = 1 occurs at z = LNL = 1/(γP0).
• Nonlinear length LNL ∼1 cm at moderate peak powers <10 W.
• Dispersion length (LD = T02/|β2) can also be made ∼1 cm for pulses
100-fs wide or less.
• Pulses propagate as fundamental solitons (N = 1) when
LD
γP0T02
N =
=
= 1.
LNL
|β2
2
• Formation of optical solitons possible if waveguide is designed to
provide anomalous dispersion at the operating wavelength.
• Numerical simulations confirm this expectation.
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Formation of Optical Solitons
Normalized P ower
P uls e S hape
1
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0.8
P ath averaged
0.6
0.4
TPA
Input
0.2
0
500
400
300
200
100
0
100
time (fs )
P uls e S pectrum
200
300
400
500
S pectral Dens ity
1
0.8
0.6
0.4
P ath averaged
TPA
Input
α
0.2
0
1.51
1.52
1.53
1.54 1.55 1.56
Wavelength (µm)
1.57
1.58
1.59
Yin, Lin, and Agrawal, Opt. Lett. 31, 1295 (2006)
130-fs pulses launched inside a 5-mm-long waveguide (N = 1).
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Path-Averaged Solitons
• Perfect solitons do not for inside silicon waveguides.
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• Linear and nonlinear (TPA) losses broaden the pulse as N = 1 is
not maintained along the whole device.
• Concept of path-averaged solitons can be used to reduce
pulse broadening.
• Input peak power increased such that N = 1 on average.
• Required peak power is estimated using
1
P̄0 =
L
Z
0
L
P0(z) dz =
1
.
γLD
• Solion formation has been observed in a recent experiment.
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Observation of Optical Solitons
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• In a 2007 experiment, Zhang et al. [Opt. Exp. 15, 7682 (2007)]
launched 110-fs Gaussian pulses inside a 5-mm-long waveguide.
• Gaussian spectrum broadened at 1250 nm because of β2 > 0.
• It narrowed and acquired a “sech” shape at 1484 nm where β2 < 0.
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0
0
−5
−5
−10
−10
Normalized Spectrum (dB)
Normalized Intensity (dB)
Numerical Simulations
−15
−20
−25
−15
−20
−25
−30
−30
−35
−35
−40
−8
−6
−4
−2
0
Time τ/T0
2
4
6
8
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−40
1420
1440
1460
1480
1500
1520
1540
Wavelength (nm)
• Numerical results under experimental conditions.
• Input: blue curves; Output: red curves; low power case (green).
• Experimental data in agreement with numerical predictions.
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Supercontinuum Generation
• Ultrashort pulses are affected by a multitude of nonlinear effects,
such as SPM, XPM, FWM, and SRS, together with dispersion.
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• All of these nonlinear processes are capable of generating new
frequencies outside the input pulse spectrum.
• For sufficiently intense pulses, the pulse spectrum can become so
broad that it extends over a frequency range exceeding 100 THz.
• Such extreme spectral broadening is referred to as supercontinuum
generation.
• This phenomenon was first observed in solid and gases more than
35 years ago (late 1960s.)
• Since 2000, microstructure fibers have been used for supercontinuum generation.
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SC Generation in a microstructured fiber
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Ranka et al., Opt. Lett. 25, 25 (2000)
• Output spectrum generated in a 75-cm section of microstructured
fiber using 100-fs pules with 0.8 pJ energy.
• Even for such a short fiber, supercontinuum extends from
400 to 1600 nm.
• Supercontinuum is also relatively flat over the entire bandwidth.
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SC Generation through Soliton Fission
• Input pulses correspond to a higher-order soliton of large order;
typically N = (γP0T02/|β2|)1/2 exceeds 5.
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• Higher-order effects lead to their fission into much narrower
fundamental solitons: Tk = T0/(2N + 1 − 2k).
• Each of these solitons is affected by intrapulse Raman scattering.
• Spectrum of each soliton is shifted toward longer and longer wavelengths with propagation inside the fiber.
• At the same time, each soliton emits nonsolitonic radiation at a
different wavelength on the blue side.
• XPM and FWM generate additional bandwidth and lead to a broad
supercontinuum.
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SC Generation in Silicon Waveguides
• SOI waveguides also support higher-order solitons when
N = (γP0T02/|β2|)1/2 exceeds 1.
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• Higher-order dispersion leads to their fission into much
shorter fundamental solitons: Tk = T0/(2N + 1 − 2k).
• Spectrum of each soliton shifts toward longer wavelengths by a
smaller amount because of a narrower Raman-gain spectrum of silicon.
• Similar to the case of optical fibers, each soliton is expected to emit
dispersive waves with frequencies on the blue side.
• Numerical simulations confirm the potential of SOI waveguides for
SC generation.
• Spectral broadening over 350 nm is predicted for femtoseond pulses.
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Normalized Power
Spectral and Temporal Evolution
0.1
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0.05
0
−0.2
Solitons
0
0.2
Dispersive
wave
0.4
0.6
0.8
Normalized Power (dB)
time (ps)
0
−10
−20
−30
−40
1.2
Cherenkov
radiation
1.3
1.4
Solitons
1.5
1.6
λ (µm)
1.7
1.8
1.9
Yin, Lin, and Agrawal, Opt. Lett. 32, 391 (2007)
50-fs pulses launched with 25-W peak power into a 1.2-cm waveguide.
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Normalized Power (dB)
Impact of TPA on SC Generation
0
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L=3 mm
TPA=0
−10
−20
−30
−40
1.2
L=3 mm
with TPA
1.3
1.4
1.5
1.6
λ (µm)
1.7
1.8
1.9
Yin, Lin, and Agrawal, Opt. Lett. 32, 391 (2007)
• TPA reduces SC bandwidth but is not detrimental.
• Nearly 400-nm-wide supercontinuum created within
a 3-mm-long waveguide.
• Required pulse energies are relatively modest (∼1 pJ).
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Optical Switching
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• A Mach-Zehnder interferometer can be made using two Si waveguides. Such a device exhibits SPM-induced optical switching.
• In each arm, optical field accumulates linear and nonlinear
phase shifts.
• Transmission through the bar port of theinterferometer:
T = sin2(φL + φNL);
φNL = (γP0/4)(L1 − L2).
• T changes with input power P0 in a nonlinear fashion.
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Cross-Phase Modulation
• Consider two optical fields propagating simultaneously.
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• Nonlinear refractive index seen by one wave depends on the
intensity of the other wave as
∆nNL = n2(|A1|2 + b|A2|2).
• Total nonlinear phase shift in a fiber of length L:
φNL = (2πL/λ )n2[I1(t) + bI2(t)].
• An optical beam modifies not only its own phase but also of other
copropagating beams (XPM).
• XPM induces nonlinear coupling among overlapping optical pulses.
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XPM-Induced Spectral Changes
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Hsieh et al., Opt. Exp. 15, 1135 (2007)
• 200-fs pump and probe pulses (at 1527 and 1590 nm) launched into
a 4.7-mm-long SOI waveguide (w = 445 nm, h = 220 nm).
• Pump and probe pulses travel at different speeds (walk-off effect).
• XPM-induced phase shifts occurs as long as pulses overlap.
• Asymmetric XPM-induced spectral broadening depends on pump
power (blue curve); Probe spectra without pump (red curve).
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XPM-Induced Switching
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• A Mach–Zehnder interferometer is often used for optical switching.
• Output switched to a different port using a control signal that shifts
the phase through XPM.
• If control signal is in the form of a pulse train, a CW signal can be
converted into a pulse train.
• Turn-on time quite fast but the generation of free carriers widens
the switching window (depends on the carrier lifetime).
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Experimental Demonstration
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Boyraz et al., Opt. Exp. 12, 4094 (2004)
• A Mach–Zehnder interferometer used for optical switching.
• Short pump pulses (<1 ps) at 1560 nm pass through the arm
containing a 2.5-cm-long SOI waveguide.
• CW probe experiences XPM-induced phase shift in that arm.
• Temporal slice of the probe overlapping with the pump is optically
switched.
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XPM-Induced Switching
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Boyraz al., Opt. Exp. 12, 4094 (2004)
• Instantaneous switching on the leading edge, as expected, with high
on–off contrast.
• Long trailing edge results from the free-carrier effects.
• Free carriers provide an additional contribution to the probe phase
by changing the refractive index.
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Four-Wave Mixing (FWM)
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• FWM is a nonlinear process that transfers energy from pumps
to signal and idler waves.
• FWM requires conservation of
? Energy
ω1 + ω2 = ω3 + ω4
? Momentum
β1 + β2 = β3 + β4
• Degenerate FWM:
Single pump (ω1 = ω2).
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Parametric Amplifiers
• FWM can be used to amplify a weak signal.
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• Pump power is transferred to signal through FWM.
• The idler (generated as a byproduct) acts as a copy of the signal at
a new wavelength (useful for wavelength conversion).
• Parametric amplifiers can provide gain at any wavelength using
suitable pumps.
• They are also useful for all-optical signal processing.
• Optical fibers are often used, but the use of SOI waveguides would
result in a much more compact device.
• Impact of two-photon and free-carrier absorption requires further
study.
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FWM Theory for Silicon Waveguides
• Theory should include TPA and free-carrier effects fully.
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• Polarization and Raman effects should also be included.
• Full vector theory by Lin et al., Opt. Exp. 14, 4786 (2006).
• Free-carrier absorption limits the gain for a CW pump.
• β2 < 0 (red); β2 = 0 (blue); β2 > 0 (green).
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FWM with Short Pump Pulses
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Lin et al., Opt. Exp. 14, 4786 (2006)
• FCA is reduced significantly for pump pulses much shorter than
carrier lifetime τc.
• Figure shows the case of 10-ps pump pulses with τc = 1 ns.
• Phase-matching condition is satisfied even for signal that is shifted
by 70 nm from the pump wavelength.
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Singe and Dual-Pump Configurations
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Lin et al., Opt. Exp. 14, 4786 (2006)
• Parametric amplifiers with a large bandwidth can be realized by
pumping an SOI waveguide with two pumps.
• This is possible because of a relatively short device length.
• Recent experiments with SOI waveguides are encouraging.
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Experimental Results (Single Pump)
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Foster et al., Nature 441, 760 (2006)
• 3.5-ps pump pulses at 1550 nm; signal tunable over 80 nm.
• Up to 5 dB of gain observed in devices <2 cm long.
• 150-nm bandwidth realized in 2007 [Opt. Exp. 15, 12949 (2007)].
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Photon-Pair Generation
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Lin et al., Opt. Lett. 31, 3140 (2006)
• Spontaneous FWM in fibers creates entangled photon pairs but
suffers from the noise induced by Raman scattering.
• The use of SOI waveguides avoids this problem because Raman
scattering does not occur when TM mode is excited.
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Experimental Results
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Sharping et al., Opt. Exp. 14, 12388 (2006)
• 5-ps pump pulses launched into a 9-mm-long SOI waveguide.
• Total (red) and accidental (blue) coincidence rates measured.
• Their ratio exceeded 10 at low pump powers.
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Stimulated Raman Scattering
• Scattering of a pump beam from vibrating molecules creates a
Stokes beam down-shifted in frequency by a specific amount.
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• Frequency shift is set by a vibrational mode (phonons).
• Raman gain spectrum exhibits a dominant peak at 15.6 THz with
a 105-GHz bandwidth (≈1 nm wide near 1550 nm).
• Peak gain for silicon >1000 larger compared with silica.
Claps et al., Opt. Exp.
13, 2459 (2006)
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Raman Amplifiers
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Jalali et al., IEEE JSTQE 12, 412 (2006)
• CW pumping leads to accumulation of free carriers through TPA.
• Free-carrier absorption introduces losses for pump and signal.
• No signal gain occurs for τeff > 10 ns.
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Pulsed Raman Amplifiers
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Jalali et al., IEEE JSTQE 12, 412 (2006)
• Pulsed pumping can provide >20-dB gain if spacing among pulses
is much larger than τeff (R pτeff 1).
• Free carriers can then decay before the next pulse arrives.
• Pump pulses (∼30 ps) at 1540 used to amplify a 1673-nm signal.
• 20-dB net gain realized at 37-W peak power of pump pulses.
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Raman Lasers
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Boyraz and Jalali, Opt. Exp. 12, 5269 (2004)
• Pumped with 30-ps pulses at 1540 nm at 25-MHz repetition rate.
• Produced 18 ps pulses at 1675 nm at the same repetition rate.
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CW Raman Amplifiers
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Jones et al, Opt. Exp. 13, 519 (2005)
• CW pumping can be used if free carriers are removed quickly.
• A reversed-biased p-n junction is used for this purpose.
• Electric field across the waveguide removes electrons and holes.
• Drift time of carriers is shorter for larger applied voltages.
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CW Raman Amplifiers
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Jones et al, Opt. Exp. 13, 519 (2005)
• A 4.8-cm-long waveguide CW pumped at 1458 nm (signal at 1684 nm).
• Output pump and signal powers increase with applied voltage.
• Effective carrier lifetime decreases from 16 to 1 ns.
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Theory behind Raman Amplifiers and Lasers
• Pump and signal powers satisfy the set of two coupled equations:
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∂ Pp
= −(αlp + αfp)Pp − βTppPp2 − 2βTpsPsPp − gRPsPp
∂z
∂ Ps
= −(αls + αfs)Ps − βTssPs2 − 2βTspPpPs + gRPpPs.
∂z
• Signal loss αfs by free carriers (generated from the pump-induced
TPA) limit the performance severely.
• For net amplification to occur, the carrier lifetime should satisfy
h̄ω pnsā2pp(gR − 2βTsp)2
τ0 < τth ≡
.
2αlsσasn0sβTppā2sp
• A Raman laser cannot function if this condition does not hold.
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Concluding Remarks
• Nonlinear effects in silicon waveguides can be used to make many
active and passive components.
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• SPM is useful for soliton formation and supercontinuum generation.
• SPM and XPM can also be used for optical switching, wavelength
conversion, and all-optical regeneration.
• Four-wave mixing converts silicon waveguides into parametric
amplifiers.
• It can also be used for quantum applications requiring entangled
photon pairs.
• Stimulated Raman scattering converts silicon waveguides into
Raman amplifiers.
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Further Reading
• L. Pavesi and D. J. Lockwood, Silicon Photonics (Springer, 2004).
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• G. T. Reed, Silicon Photonics: State of the Art (Wiley, 2007).
• IEEE J. Sel. Topics Quantum Electron., Special issue on Silicon
Photonics, 12 (2006).
• B. Jalali and S. Fathpour, “Silicon Photonics,” J. Lightwave Technol. 24, 4600 (2006).
• B. Jalali, “Teaching silicon new tricks,” Nature Photonics 1, 193
(2007).
• Q. Lin, O. J. Painter, G. P. Agrawal, “Nonlinear optical phenomena
in silicon waveguides,” Opt. Express 15, 16604 (2007).
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