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NonlinearNonlinear OpticalOptical EffectsEffects in in PhcPhc and and MetamaterialsMetamaterials
Concita Sibilia
Dipartimento di Energetica
Universita’ di Roma La Sapienza
Introduction
:Nl
OpticsNonlinear 1-D Photonic Crystals: enhancement
of
quadratic
interaction. Fields localization & Phase matchingRelevance of fields’overlap on frequency
conversion
efficiency.
Generation of Entangled two-photon state
Conclusions
Outline
Metallo-Dielectrics
Nanostructured metals
Introduction
Question:Is it possible to change the color of a monochromatic light?
Answer:Not without a laser light
output
NLO
sample
input
2. The Essence of Nonlinear Optics
When the intensity of the incident light to a material system increases the response of medium is no longer linear
Input intensity
Output
How does optical nonlinearity appear
The strength of the electric field of the light wave should be in the range of atomic fields
N
a0
e
νh20/ aeEat =
220 / mea =
esu102 7=×≈atE
Response of an optical Medium
The response of an optical medium to the incident electro magnetic field is the induced dipole moments inside the medium
μ
νh
νhνh
νh
μ μ
μ
μ
)(ˆ)( tEtP α=
Lasers made available highly coherent radiation that could be concentrated and focused to give extremely high local intensities that can reach 1018 W/cm2.Discovery of second harmonic generation by Franken et al. in 1961 is considered the beginning of the field of nonlinear optics. A rich stream of new phenomena soon followed. Nonlinear optics plays an important role in telecommunications and future computer technologies. The relatively long interaction lengths and small cross sections available in waveguides and fibers means that low energy optical pulses can achieve sufficiently high peak intensities to put in evidence non linear effects also in many transparent optical materials with weak nonlinearities.
When the strength of the applied field is comparable to the atomic field strength, the linear approximation is no longer valid. By performing a power expansion of P it is possible to consider the nonlinear terms in the polarization function:
...)()()(...)()()()( )3()2()1(32 +++=+++= tPtPtPtEtEtEtP γβα
Introduction to non linear optics
Nonlinear Polarization
• Permanent Polarization
• First order polarization:
• Second order Polarization
• Third Order Polarization
0iP
jiji EP)1(1 χ=
kjijki EEP)2(2 χ=
lkjijkli EEEP)3(3 χ=
Nonlinear Optical Interactions• The E-field of a laser beam
• 2nd order nonlinear polarization
C.C.)(~ += − tiEetE ω
)C.C.(2)(~ 22)2(*)2()2( ++= − tieEEEtP ωχχ
ωω
ω2)2(χ
2nd Order Nonlinearities • The incident optical field
• Nonlinear polarization contains the following terms
..)(~ 21 21 CCeEeEtEtiti ++= −− ωω
(OR) )(2)0(
(DFG) 2)(
(SFG) 2)(
(SHG) )2(
(SHG) )2(
*22
*11
)2(
*21
)2(21
21)2(
21
22
)2(2
21
)2(1
EEEEP
EEP
EEP
EP
EP
+=
=−
=+
=
=
χ
χωω
χωω
χω
χω
1ω
2ω)2(χ
1ω
2ω213 ωωω +=
Sum Frequency Generation
1ω3ω
2ωApplication:Tunable radiation in the UV Spectral region.
Application:Tunable radiation in the UV Spectral region.
Application:The low frequency photon, amplifies in the presence of high frequency beam . This is known as parametric amplification
Application:The low frequency photon, amplifies in the presence of high frequency beam . This is known as parametric amplification.
2ω
1ω
1ω
2ω)2(χ
2ω
1ω213 ωωω −=
Difference Frequency Generation
1ω3ω
2ω
Linear susceptibility tensor:
EP )1(0)1( χ̂ε=
For anisotropic media is a second rank tensor with 9 components (3x3 matrix).
For isotropic media
)1(χ̂
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
100010001
ˆ )1()1( χχ EP )1(0)1( χε=
Nonlinear susceptibility tensor:
zyxi EEPzyxkj
kjijki ,,;ˆ,,,
)2(0
)2( == ∑=
χε)2(χ̂ Third rank tensor (27 components)
zyxi EEEPzyxlkj
lkjijkli ,,;ˆ,,,,
)3(0
)3( == ∑=
χε
The Non linear Optical Susceptibility Tensor
[m/V]
)3(χ̂ Fourth rank tensor (81 components) [m2/V2]
Only non centrosymmetric crystals can posses a non-vanishing second order nonlinear susceptibility tensor. In a centrosymmetric crystal, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Thus a reversal of the sign of Ej and Ek must cause a reversal in the sign of Pi(2):
( ) ( )kjizyxkj
kjijkkji EEPEEEEP ,))((ˆ,)2(
,,,
)2(0
)2( −=−−=−− ∑=
χε
Possible only if all the elements of the nonlinear susceptibility tensor are zero.
The Non linear Optical Susceptibility Tensor
Second order nonlinear effects
( ),:ˆ )2(22
002
2
0002 EE
tE
tE
tE r χεμεεμσμ ∂
∂+
∂∂
+∂∂
=∇
In our one dimensional model we have:
( ),),(),(),(),( )()()(22
2321
tzEtzEtzEz
tzE ωωω ++∂∂
=∇
Performing the second order derivative and assuming that the variation of the complex field envelopes with z are small enough so that:
),()( 122
11 zEdzdkzE
dzd
ii >>Slow varying envelope approximation
SVEA
( )[ ]( )[ ]( )[ ]..)(
21),(
..)(21),(
..)(21),(
333
222
111
3)(
2)(
1)(
ccezEtzE
ccezEtzE
ccezEtzE
zktijj
zktikk
zktiii
+=
+=
+=
−
−
−
ωω
ωω
ωω
Being i,j,k the Cartesian coordinates and can each take on values x and y.
[ ]
( )
( )
( )
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
⎥⎦
⎤⎢⎣
⎡++
+⎥⎦⎤
⎢⎣⎡ ++
+⎥⎦⎤
⎢⎣⎡ +
−≅++=∇
−
−
−
zktijj
zktikk
zktiii
edz
zdEikzEk
edz
zdEikzEk
edz
zdEikzEk
tzEtzEtzEdzdE
33
22
11
321
)(2)(
)(2)(
)(2)(
21),(),(),(
333
23
222
22
111
21
)()()(2
22
ω
ω
ω
ωωω
Second order nonlinear effects
The valuation of the Laplacian within the SVEAT approximation gives:
Finally, with some algebra:
zkkki
yxkikijik
rj
r
j
zkkki
yxjijikij
rk
r
k
zkkki
yxkjkjijk
ri
r
i
eEEiEdz
dE
eEEiEdz
dE
eEEiEdz
dE
)(
,,21
)2(
3
0033
30
033
)(
,,
*31
)2(
2
002*2
20
02*2
)(
,,
*23
)2(
1
0011
10
011
321
231
123
ˆ22
ˆ22
ˆ22
−+−
=
+−−
=
−−−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛′−−=
⎟⎟⎠
⎞⎜⎜⎝
⎛′+−=
⎟⎟⎠
⎞⎜⎜⎝
⎛′−−=
∑
∑
∑
χε
εμωεε
μσ
χε
εμωεε
μσ
χε
εμωεε
μσ
Phase Matching
ω)2(χ
ω2
•Since the optical (NLO) media are dispersive,The fundamental and the harmonic signals havedifferent propagation speeds inside the media.
•The harmonic signals generated at different points interfere destructively with each other.
•Since the optical (NLO) media are dispersive,The fundamental and the harmonic signals havedifferent propagation speeds inside the media.
•The harmonic signals generated at different points interfere destructively with each other.
frequency
Ref
ract
ive
inde
x
2ω
ω
NOTE: In order to have efficient energy transfer among the fields, a long interaction length is required, the phase mismatch should be as close as possible to zero. In other words:
k3 =k2 +k1
Second order nonlinear effects
In a more general case, the condition to be fulfilled is:
213 kkk +=
It is called phase matching condition and it can be interpreted as a momentum conservation requirement. Example: Two pump fields at frequencies w1 and w2 can generate a sum frequency (ω3 ) field. The wavevector of the generated field will fulfill:
2k1k
3k
To obtain an expression for the second harmonic power output, consideringA linearly polarized pump with non vanishing electric field component E1x we use the relation:
*3330
)2(
21
jjr EEcAreaP εε
ω
=
If we consider the SH generated linearly polarized with non vanishing field componentE3y we can write an expression for the conversion efficiency:
( );
2/)2/sin(
21 2)(
22
30
2)2(22
)(
)2(
⎥⎦⎤
⎢⎣⎡
ΔΔ′
==kL
kLAreaP
nncL
PP yxx ω
ωωω
ω
εχω
η
The conversion efficiency is linearly growing with the pump intensity. This means that the generated second harmonic intensity with respect to the pump intensity follows a quadratic law. The conversion efficiency increases with the squared length of the nonlinear medium
Optical Second harmonic Generation
Optical Second harmonic Generation
First experimental report on second harmonic generation was performed by Franken, Hill, Peters and Weinreich in 1961. (Fig: A. Yariv, Quantum electronics)
SHG can be studied as the limiting case of the three frequency interaction where two of the frequencies ω1 and ω2 are equal and ω3 =2ω1 .We assume as first approximation that the amount of power lost by the input beam is negligible so that dE1i /dz=0 and the medium is transparent at ω3 ( σ=0)
Maximum conversion efficiency is achieved for ΔkL/2=0,Δk=k2ω
-2kω
=0 is called phase matching condition and it is fulfilled as long asThe refractive indices at FF and SH frequencies are the same: n2ω
=nω
.If such condition is fulfilled, the FF and SH fields propagate with the same phase velocities
We note that the conversion efficiency is crucially determined by the sinc function:
Phase matching
In common materials the refractive index is an increasing function of the frequency thus we have nω
0.During the non linear process there is an exchange of energy between FF and SH field and the SH production is zero for any propagation length that satisfy the law:
ΔkL/2=mπ, where m is an integer.
For a mismatched interaction, the length of non linearmaterial that produce the maximum generatedsecond harmonic field can be calculated by:
( )ωωλπ
π
nnkl
kl
c
c
−=
Δ=
=Δ
2
0
4
;22
Typically in nonlinear materials Δn is of the order of 10-1to 10-2. Coherence lengths are only a few numbers of wavelenghts.
Phase matching
Undepleted pump approximation
Phase matching techniques
A widely used technique takes advantage of the natural birefringence of anisotropic crystals.Under certain circumstances it is possible to use the different refractive indices for the ordinary wave and the extraordinary wave. For example, a typical behaviour of dispersion of the refractive indices of a negative (ne
Quasi Phase matching
Phase matching techniques
Periodic modulation of the non linear coefficients tensor elements responsible for the interaction. It can be shown that the phase matching condition becomes:
Δk=2πm/ΛWhere m is an integer and Λ is the period of the nonlinearity.
EXAMPLE: If the sign of the non linear interaction is reversed at every coherence length d(z) is a periodic function of period 2lc =2π/Δk. QPM is achieved for m=1.
deff =2dbulk /π
Periodically poled LiNbO3Periodical reversal of electric field in order to induce a permanent periodically modulated electric polarization
Phase matching techniques
Considering dielectric waveguides, the modal dispersion can be used to achieve phase matching. The phase matching relation for second harmonic generation becomes:
ωω ββ~~
2 =
This condition can be easily fulfilled by considering modes for the pump and for the SH of different order or polarization.
Nevertheless the non linear process is governed by how the fields overlap: usually overlaps between fields belonging to different orders are not efficient.
( ) ( ) 2*22)2( )()(:ˆ ⎥⎦⎤
⎢⎣⎡ ⋅∝ ∫
∞+
∞−dxxExE nm
ωωχη
Phase matching techniques
Photonic CrystalsIn a photonic crystal both the linear and the nonlinear susceptibility functions can be periodically modulated. Modulation of the linear susceptibility is responsible for peculiar properties of the linear dispersion curves of these structures. Fields are characterized by a Bloch wave vector and an periodic function over the unit cell. Thus the quasi momentum conservation for second harmonic generation is:
02 =−− Gkk ωωWhere G is a reciprocal lattice vector. Usually unit cells are of the order of the wavelength or less unlike in the QPM regime where domains are inverted every coherence length.
Three wave mixing
The basic equations govern the interaction of three fields due to second order nonlinearity:
kzi
r
kzi
r
kzi
r
eEEidz
dE
eEEidz
dE
eEEidzdE
Δ−
Δ
Δ−
′−=
′+=
′−=
21)2(
3
0033
*31
)2(
2
002*2
*23
)2(
1
0011
ˆ2
ˆ2
ˆ2
χε
εμω
χε
εμω
χε
εμω
Different processes can happen depending on the initial condition: ω3ω1
ω2Sum frequency generationFrequency up-conversion
Parametric amplificationDifference frequency generationStimulated parametric down conversion
Spontaneous parametric down conversionSpontaneous parametric fluorescence
Summary
Third order optical nonlinearities
Third order nonlinear polarization allow coupling of fields at different frequencies, Here we present an sketched overview of the most relevant effects that will be analyzed later:
ω1
ω3ω2
ω4
Four wave mixing
ω
3ω
Third harmonic generation
ωω
ω
ω
Optical Kerr effect
ω−ω
ω
ωs
Raman effect
−ωωs
Intensity dependence of the refractive index
The refractive index of many materials depends on the intensity of the incident optical field. If we consider an optical field of the form:
( ) 2202 )()()(,~ ωωωω EnnEn +=
[ ]xcceEt ti ˆ.)(21)(E += − ωω
Time averaged intensity is defined:
200 )(2
1 ωε EncI =
The nonlinear refractive index is defined:
For instantaneous response:
(t)E(t)E(t)Eˆ)(P )3(0χε=tNL
Intensity dependence of the refractive index
Intensity dependence of the refractive index has drastic effects both on the spatial properties of optical beams and on temporal (spectral) properties of ultrashort pulses.
Self phase modulation (SPM)Phase shift experienced by short pulse propagating through a nonlinear refractive index- spectral broadening; - Optical solitons in anomalous dispersion regimes of fibres.
Cross phase modulation (XPM)Phase shift of a field induced by a co-propagating intense field at different wavelength.
Self focusing and self defocusing of optical beamsDepending on the sign on the nonlinearity, the central (more intense) portion of aBeam experiences higher (lower) index of refraction with respect to the outer edges.An effective lens is created.
Photonic
Crystals
1D PCh
-
Band Edge
EffectsFinite size 1D photonic crystals transmission bands exhibit resonance peaks
due to the boundary conditions with the external ( homogenous ) world.In particular, resonances are sharper in proximity of the band edge.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Frequency0.00.10.20.30.40.50.60.70.80.91.0
Tran
smis
sion
Pass bands
Band gaps
0
0.2
0.4
0.6
0.8
1.0
0.56 0.57 0.58 0.59 0.60 0.61
ΔωII ΔωI
0
4
8
12
0 4 8 12
z (μm)
| Φ ω|2
No Hermitian
Eigenproblems
3
5
7
(in u
nits
of 1
/c)
tieTyixtφ
ωωω =+= )()()(
πφ mxyt ±−= )/(1tan
Dnc
Dk efft )()( ωωωφ ==
Density of modes :
dϕ/dw
= (1/D)dk/d w
DOM = dk/dw=
(1/D)(y’x-x’y)/(x^2+y^2)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Frequency
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Tran
smis
sion
Pass bands
Band gaps
:Transmission function for the structure depicted in Fig.(1). Note the frequency pass bands and band gaps.
No Hermitian
Eigenproblems
tieTyixtφ
ωωω =+= )()()(
πφ mxyt ±−= )/(1tan
Dnc
Dk efft )()( ωωωφ ==
Vg= dw/dk=1/DOM
0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
0 . 5 1 . 0 1 . 5
ω / ω 0
|t|2
1D 1D PChPCh
--
Band Band EdgeEdge
EffectsEffects
-- GroupGroup
velocityvelocity
( ) 022
2
2
=+ ωωω εω E
cz
dzEd
The boundary conditions at the input (z=0) and output (z=L) surfaces are: EIw+Erw=Ew(0); Etw=Ew
(L)exp[-i(w/c)L)]; i(w/c)(EIw-Erw)=dEw
(0)/dz; i(w/c)Etw=(dEw
(L)/dz)exp[-i(w/c)L).
Φω
(z) =Eω
(z)/EIω
; tω
=(Etω
exp[i(ω/c)L]/EIω
;
( )
( )∫
∫
Φ
⎥⎥⎦
⎤
⎢⎢⎣
⎡ ΦΦ−
=
∗
L
L
E
dzz
dzdz
dicV
0
2
0
2Re
ωω
ωω
ω
εω
No Hermitian
EigenproblemsThe energy velocity is defined as the ratio of the Poynting vector integrated over the volume to the average total energy stored in the same volume
1D 1D PChPCh
--
Band Band EdgeEdge
EffectsEffects
-- EnergyEnergy
velocityvelocity
( ) ( )ωω
ωgE VtV
2=
0 .4
0 .6
0 .8
1 .0
units
of c
) ;
|t|2
(a.
u)
0
0.2
0.4
0.6
0.55 0.57 0.59 0.61
N→ ∞N=20N=200
N=200N=20
ω/ω0
V E
(in
units
of
c)
G.D’Aguanno et al
-χ(2)
–
Nonlinear frequency Conversion-
SHG
ωω
2ωPnl(2ω) χ(2) E2(ω)
-Down Conversion
ω
ω
Nonlinear Interaction
2ωIn Bulk:
PM and χ(2)
Nonlinear quadratic Interaction in PhC-χ(2)
, interface , bulk (in each layer)
-P.M. (equal PHASE velocity),
For SH 2K1 = K2n(w) = n(2 w)
thanks to the geometrical dispersion-Field enhancement due to field localization
( band edge and when T=1)
-
Large conversion efficiencyM.Scalora et al 1997
Effective refractive index
πϕ mxytg ±⎟
⎠⎞
⎜⎝⎛= −1t
*Centini et al, Phys. Rev. E 60, 4891-4898 (1999)
frequency
Ref
ract
ive
inde
x 2ω
ω
Φ≡≡+= ii eeTiyxt tϕ
Dnc
DkTi effefft ˆˆlnωϕ ≡≡−=Φ
⎥⎦⎤
⎢⎣⎡ +−= )ln(
2ˆ 22 yxi
Dcn teff ϕω
Bulk Medium
ωF 2ωF
Re(
n eff)
frequency
Multilayer Structure
Phase Matching
But
Field
overlap
more important
than
P.M !!!!
The expression of the conversion efficiency (η=ΙSH /Ιpump )in the non-depleted pump regime is similar to the bulk case
22 ( , ) 2( , )
2 ( , ) ( , ) (2 , )0
8 eff pumpp s p
eff eff eff
d L Ic n n nω ω ωπ
ηε λ
+ −+ − =
( , ) (2) 2( ) *( , )2
0
1 ( ) ( ) ( )D
effd z z z dzL ω ωχ+ − + + −= Φ Φ∫
deff contains both information on PM conditions and fields overlap
*G. D’Aguanno, et al. J. Opt. Soc. Am. B 19, 2111 (2002)
Conversion Efficiency U-Pho
NL layers =λ0
/4Linear layers = λ0
/2
24
68
1030
210
60
240
90
270
120
300
150
330
180 0
n1=1; n2(ωF )=1.428; n2(2ωF )=1.616.
0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 20
0 . 2
0 . 4
0 . 6
0 . 8
1
w / w 0
T
n1=1; n2(ωF )=1.428; n2(2ωF )=1.676.
0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 20
0 . 2
0 . 4
0 . 6
0 . 8
1
w / w 0
T
Phase matched interaction
Non Phase matched interaction
Fields’ Overlap
* Centini et al. Opt. Letters (2004)
( , ) (2) 2( ) *( , )2
0
1 ( ) ( ) ( )D
effd z z z dzL ω ωχ+ − + + −= Φ Φ∫
22 ( , ) 2( , )
2 ( , ) ( , ) (2 , )0
8 eff pumpp s p
eff eff eff
d L Ic n n nω ω ωπ
ηε λ
+ −+ − =
For periodic finite structures, tuning the pump at the band edge resonance, PM is always achieved if the second harmonic is tuned to the second resonance*
* Centini et al Phys Rev E 60 4891 (199
Phase Matching
Huge conversion for random sequences
0.85 0.9 0.95 1 1.05 1.1 1.15 1.210-3
10-2
10-1
100
101
102
103
104
105
106
12 μm etalonof nl material
Perfectly phase matched12 micron etalon
ω/ ω
0
SH energy(a.u.)
random
Coll. with D.Wiersma (Lens)
D’Aguanno et al. “Energy Exchange Properties during SHG in finite,1-D ,PBG structures with deep gratings”
Fig.2
η+SHη-SH
(GW/cm2)
0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10
RFF
TFF
η SH
+ &
ηSH
−
IFF(MW/cm2)
0
0 .0 2
0 .0 4
0 .0 6
1 4 7 1 0
)(inputFFI Fig.2
η+SHη-SH
(GW/cm2)
0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10
RFF
TFF
η SH
+ &
ηSH
−
IFF(MW/cm2)
0
0 .0 2
0 .0 4
0 .0 6
1 4 7 1 0
)(inputFFI
Undepleted pumpregime
SHG Conversion SaturationSHG Conversion Saturation
FF Anom
alous Transm
ission/Reflection
N^6
0
10
20
30
0 4 8 12
b)
z(μm)
|E|2
(arb
. uni
ts)
0
0.25
0.50
0.75
1.00
-500 0 500
Δϕ=0 Δϕ=0
Δϕ=πΔϕ=π a)
z (μm)
|E|2
(arb
. uni
ts)
δφω (units of π)
Con
vers
i on
E ffic
i enc
y
Q=0.5
Fig.(6a)
0
0.005
0.010
0.015
0.020
0 0.5 1.0 1.5 2.0
Control of SHG process
0
0.2
0.4
0.6
0.8
1.0
0.5 0.7 0.9 1.1 1.3 1.5
Tran
smitt
ance
ω /ω0 Fig.1(a)
FF SH
Q=Ipump2 /Ipump1
Second Harmonic Generation for counterSecond Harmonic Generation for counter--propagating propagating pumpspumps
62 NIT ≈
Y. Dumeige et al APL 2001, JOSAB 2002
Experiments on SHG
AlGaAs/Al2 O3
Simultaneous perfect phase matching for second and third harmonic generations in ZnS/YF3 photonic crystal for visible emissions
Weixin Lu, Ping Xie, Zhao-Qing Zhang, George K. L. Wong, and Kam S. Wong
Optics Express, Vol. 14, Issue 25, pp. 12353-12358
M. Centini, et al,"Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E 64, 046606 (2001)
Experiment on Type II SHG :AlGaAs/Al2 O3λp=1.55 microns
ω-p polarized
ω-s polarized
2ω-p polarized
Non collinear Type II second harmonic generation
kω kω
k2ω
Momentum Conservation
p,θ2ω p,θωs,θω
p,kωs,kωp,k2ω
p,θ2ω p,θωs,θω
p,kωs,kωp,k2ω
k y[μ
m-1
]
kx [μm-1]0 5 10
5
10
GAP
( , )sk ω⎡ +⎣( , )pk ω ⎤⎦
12
(2 , )12
pk ω= (2 , )12 pk ω=
U-Pho
Seco
nd H
arm
onic
Sig
nal [
a.u.
]
θ ω ,p [deg]
0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
20 22 24 26 28 30 32 34 36
Seco
nd H
arm
onic
Sig
nal [
a.u.
]
θ ω ,p [deg]
0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
20 22 24 26 28 30 32 34 360
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
20 22 24 26 28 30 32 34 3620 22 24 26 28 30 32 34 36
P spectrum
S spectrum
P spectrum around SH
Ref
lect
ion
wavelength
1.51 μm
1.51 μm
755 nm
1.51 μm
1.51 μm
755 nm
1.51 μm
1.51 μm
755 nm
1.51 μm
1.51 μm
755 nm
Angular measurements:
PM interaction: FF fields tuned at the band edge resonance, SH tuned at the second resonance near the band edge
Slight Phase mismatch but the SH is tuned at the band edge resonance (higher field localization)
Fields at the Pass BandsNo Overlap inside the Structure
Almost no energy inside the photonic crystal
27 degsk ,ω
pk ,2ωp,ωθ pk ,ω
*Bosco et al APL 05
U-Pho
Nonlinearities in metals• In metals bulk production of second harmonic
radiation is usually inhibited due to symmetry reasons and only an electric quadrupole and a magnetic dipole terms may be active
• These reasons do not exhist at surface due to the lack of symmetry in a sheet region of the order of a few Å’s thickness between vacuum and the metal. In this case the discontinuity of the electric field is important
• Therefore, although the production of radiation has a very low efficiency (~10-10) it is interesting to examine what happens
the term E x B originating from the Lorentz force acts on the current produced at the first order by B (it is nonlinear in B). It is a bulk current in the metal and does not irradiate except if there is a boundary.
The other term is a surface electric quadrupole term. It is present in nonlinear reflection from a metal and originates from the discontinuity of the normal component of E at the interface which generates nonlinear currents. The surface currents are a tangential component and a bulk one directed along z.
The SH polarization from the perpendicular and parallel surface currents can be written as
(2 ) ( ) ( ) ( )surfacez z zaω ∝ ω ω ωP E E (2 ) ( ) ( ) ( )surfacex x zbω ∝ ω ω ωP E E
Dependence on fundamental beam polarization direction
• A typical apparatus is shown in the figure
• Since the surface nonlinear sources are in the simplest theory proportional to (∇·E)E, then incident fields polarized only in the plane of the surface (normal to the plane of incidence) do not contribute to the harmonic signal.
• However the bulk terms proportional to E×H lead to harmonic generation for all incidence polarizations
• Neglecting bulk terms, a [ cos(φ)
]4 dependence on the polarization is predicted, where φ
is the electric field angle
relative to the plane of incidence.
• Deviations from this relation are indicative of contributions from the bulk term, and in particular the ratio
• M = ISH (φ=90°) / ISH (φ=0°)
• yields information about the relative strengths of the surface versus bulk terms
• F.Brown et al., PRL 14 (1965) 1029• F.Brown and R.E.Parks, PRL 16 (1966) 507• N. Bloembergen et al., PR 174 (1968) 813
Dependence on fundamental beam polarization direction
High Pass band filter
800 nm150 fs1 kHz
Neutral density filters
A
PM
VA
Sample
Dichroic filters
BS
Prism
L2
L1
800nm
400nm
LaserSystem
HP-F
λ/2
α
A
PM
VA
Sample
Dichroic filters
BS
Prism
L2
L1
800nm
400nm
LaserSystem
HP-F
λ/2
α
PM
VA
Sample
Dichroic filters
BS
Prism
L2
L1
800nm
400nm
LaserSystem
HP-F
λ/2
α
p (TM)
s (TE)
φ
φ=0°
φ=90°Experimental Setup
Nonlinearities in metals
p (TM)
s (TE)
φ
φ=0°
φ=90°
When the fundamental wave is s polarized, only the bulk contribution is present
0153045607590
105120135150
-100 -50 0 50 100
Sample6 Ag/Ta2O5 Noise ≈ 1 mVSample5 Ag/Ta2O5
s polarized (TE)
p polarized (TM)
φ= angle between E and the incindent plane [ ° ]
SH
Sig
nal [
mV
]
Metallo-dielectric multilayer SH production in reflection
0153045607590
105120135150165180
-100 -50 0 50 100Noise ≈ 1mV
Fitting curve: Smax⋅ cos(φ)
4
Fitting curve: Smax⋅ cos(φ)
4- a ⋅ cos(φ)2 ⋅ sin(φ)2 + Smin⋅ sin(φ)4
s polarized (TE)p polarized (TM)
φ= angle between E and the incindent plane [ ° ]
SH
Sig
nal [
mV
]
0
10
20
30
40
50
200 400 600 800 1000
Wavelength [ nm ]
Tran
smitt
ance
[ %
]
Ag (20 nm)
Ta2 O5 (124 nm)
BK7 glass
The surface contribution is TM when the fundamental is TM(P), if the fundamental is TE there is no discontinuity of the normal component of E and the SH vanishes.
The bulk is always present, but only at the interface.
If fundamental is TM surface and bulk are present; if fundamental is TE only bulk is present.
600nm
0
2x10-12
4x10-12
6x10-12
8x10-12
-50 -25 0 25 50
pump ⊥ to the ripplespump = to the ripples
incidence angle (deg)
SH
con
vers
ion
effic
ienc
y
gold nanowires 5nm
S.H. signal
Sample AGold 5 nm/glass
ω 2ω
Second
Harmonic
Generation from nanostructured metals
Ti: Sapphireλ=800 nmλ nm SHG
λ=400 nmλ=400 nmλ=800 nmλ=800 nm
λ=400 nm+ λ=800 nm
λ=400 nm+ λ=800 nm
Dichroic filter
PM
Second Harmonic Generation Set-up (Reflection)
Laser p polarized
Ti:SapphireLambda=800nmPulse duration =130fsRepetition rate=1kHzAverage Power=78mW
Spot diameter on the sample=1.6mm
SHG
λ=400 nmλ=400 nm
λ=800 nmλ=800 nm
λ=400 nm+ λ=800 nm
λ=400 nm+ λ=800 nm
Dichroic filter
Ti: Sapphireλ=800 nmλ nm
PM
Second Harmonic Generation Set-up (Transmission)
pω )2(χiω
sω
Low efficiencySizeSpatial and frequency filtering needed
BULK MATERIAL
ENTANGLED PHOTON GENERATION
STATE of the ART: 10^6 twin photons/s at low temperature in a
Photonic Crystal Fiber
Optics Express, Vol. 13, Issue 2, pp. 534-544
1+1 D Ph Crystal
is ωω ,
si ωω ,
SizeHigh brightness per modeNarrow linewidthGuided entangled photon source
ENTANGLED PHOTON GENERATION In PhC
SizeHigh brightness per modeNarrow linewidth
1D Ph Crystal
pω
pω
( ) ( ) )()()()( )()( zzAzzAE nnnnn−−++ Θ+Θ=
Conclusions
Structures, suitable for degenerate and nondegenerate
generations of entangled photon pairs, were suggested using GaN/ALN layers
and AlGaAs/Al2O3.
We showed that it is possible to enhance or suppress the nonlinear effect by properly choosing the parameters of the structure.
Despite of what happens in bulk materials, the key feature for high efficiency of the nonlinear process is localization and overlap of
interacting fields in nonlinear layers.
- INTERlink Project -Miur.Italy-Phoremost NoE. EU
COST P11-ESF
ERO-US
Slide Number 1Slide Number 2Introduction2. The Essence of Nonlinear Optics How does optical nonlinearity appear Response of an optical MediumSlide Number 7Nonlinear PolarizationNonlinear Optical Interactions2nd Order Nonlinearities Sum Frequency GenerationSlide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Phase Matching Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 311D PCh - Band Edge EffectsSlide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Second Harmonic Generation for counter-propagating pumps�Slide Number 47Slide Number 48Slide Number 49Slide Number 50Slide Number 51Slide Number 52Slide Number 53Nonlinearities in metals Slide Number 55Dependence on fundamental beam polarization direction Dependence on fundamental beam polarization direction Slide Number 58Slide Number 59Slide Number 60Slide Number 61Slide Number 62Slide Number 63Slide Number 64Slide Number 65Slide Number 66