EE346 NLO

1/20/21, #4 slide 1

4_Phasematching and Dispersion

EE 346 Nonlinear Optics

M.M. Fejer
fejer@stanford.edu
01/20/21
 EE346 NLO
1/20/21, #4 slide 2
Previously
Lecture 2 Lecture 3
−5
How to obtain n2ω − nω ~ 10 ?
Birefringent media

Anisotropic dielectric tensor:

Phase velocity mismatch ∆k
a key parameter in the efficiency: ε XX 0 0
4π ε= ε YY
∆k = k2ω − 2kω = ( n2ω − nω ) 0 0
λω 0 0 ε ZZ
I 2ω ( L ) ∆k L Propagation constants depend on direction.
η≡ = η PM ,0 sinc 2
Iω (0) 2 Uniaxial media (εXX = εYY ) simplest case:
−1/2
η 1 sin 2 θ cos 2 θ
ηPM ne (θ ) = +
ne2 no2
0.5

0
-10 -5 0 5 10
∆kL / 2
 EE346 NLO
1/20/21, #4 slide 3
Today
• Phasematching in birefringent crystals

• Type I and Type II phasematching

• Tolerances

• Reading
– Harris 5.4.2
– Yariv 16.5
– Yariv and Yeh 12.4.1

• Description of linear dispersion

– classical electron oscillator model
– reading
Harris (third appendix after Ch. 2 (not in T.of C.))
Boyd, Ch. 1.4
A. Siegman, Lasers, Ch. 2.1
 EE346 NLO
1/20/21, #4 slide 4
Birefringent Phasematching

• Assume n2ω >nω (true over most of spectrum): no ,ω < no ,2ω , ne ,ω < ne ,2ω
• Consider a uniaxial medium [like KDP (42m ), LiNbO3 (3m), BBO (3m) …]
– take negative uniaxial case: ne < no
Z Z n
o ,2ω
no ,ω

ne ,ω ne ,2ω
X X

– phasematching possible if positive uniaxial case: no < ne

curves intersect, i.e. if: ne ,2ω ≤ no ,ω ≤ no ,2ω PM possible if:
no ,2ω Z k Z no ,ω ≤ no ,2ω ≤ ne,ω
ne ,2ω (θ PM ) = no ,ω no ,2ω k
no ,ω θ PM ne ,ω (θ PM ) = no ,2ω
θ PM
ne ,2ω ne ,ω
X X
no ,ω
 EE346 NLO
1/20/21, #4 slide 5
Type I vs Type II Phasematching
Experimental configuration for SHG in negative uniaxial medium:
Eω ,o Type I phasematching
ko ,ω E2ω ,e
ayin Z χ (2)
Heptane θ PM ke ,2ω
X⊗ k
e ray − i 2 ko ,ω z return to χ eff shortly
(2)

Nonlinear polarization: P2ω ,e = ε 0 χ Eω ,o ∝ e

(2) 2
eff
ko ,ω ko ,ω
Phasematching condition: ke,2ω = 2ko,ω ⇒ n2ω ,e (θ PM ) = nω ,o
ke ,2ω
Positive uniaxial same, except role of e and o reversed

What if input polarized at 45°? Type II phasematching

Eω ,e u
kω E2ω ,e
Z χ (2) maybemore
θ PM k 2ω useful insome
Eω ,o X⊗ k
cases
later
edmiscussmore
i ( k +k ) z
Nonlinear polarization: P2ω ,e = ε 0 χ eff
(2)
Eω ,o Eω ,e ∝ e o ,ω e ,ω ke ,ω ko ,ω
Phasematching condition: ke,2ω = ko,ω + ke,ω
ke ,2ω
⇒ n2ω ,e (θ PM ) = [nω ,o + nω ,e (θ PM )] / 2
 EE346 NLO
1/20/21, #4 slide 6
Summary of Uniaxial Phasematching Conditions

Type I Type II
1
positive birefringent no ,2ω = ne ,ω (θ ) no ,2ω =
2
( ne,ω (θ ) + no,ω )
1
negative birefringent ne ,2ω (θ ) = no ,ω ne ,2ω (θ ) =
2
( ne,ω (θ ) + no,ω )

extension to biaxial cases straightforward, but too tedious to enumerate

 EE346 NLO
1/20/21, #4 slide 7
Acceptance Bandwidths
E
kω ,PM ω ,o E2ω ,e
Z χ (2) k 2ω
δθ θ PM k
kω X⊗

• What is the acceptance for a change δθ in the propagation angle?

– sets alignment tolerances
– more fundamentally, sets limit on focusing of Gaussian beam input
• Efficiency falls to half of peak when
∆kL / 2 = ±0.443π η ∆k L
= sinc 2
– take tolerance as ∆k < π / L η PM ,0 2
• Expand ∆k in Taylor series: 1
η
d ∆k 1 d ∆k 2
2
ηPM
∆k = ∆k0 + δθ + δθ
dθ 2 dθ 2
−1 0.5 0.443π
π d ∆k π /2
⇒ δθ <
L dθ π
∆kL / 2
– tolerance ∝ 1/L 0
-10 -5 0 5 10
 EE346 NLO
1/20/21, #4 slide 8
Angular Acceptance I

• Form of ∆k given in #2.8: ∆k = (4π / λ )(n2ω − nω )

−1 no ,2ω Z
d ∆k 4π d k
= ( n2ω − nω ) ⇒ δθ < λ d ( n2ω − nω )
dθ λ dθ 4 L dθ θ PM
#4.7 π d ∆k
−1
no ,ω
δθ < ne ,2ω
L dθ
• To simplify algebra: X
−1/2
sin θ cos θ
2 2
ne (θ ) = + ≈ no + ( ne − no ) sin 2 θ
ne2 no2
to first order in ne – no (#3.14)

dne (θ )
⇒ ≈ 2 ( ne − no ) sin θ cosθ = ( ne − no ) sin 2θ
dθ

• For negative uniaxial media (#4.4)

λ 1
δθ <
4 L ( ne ,2ω − no ,2ω ) sin 2θ PM

– typical case (KDP at λ = 1 µm): ne − no = 0.04 θ PM = 45°

for L = 1 cm: δθ int = 0.6 mrad very strict tolerance
note: Snell’s law ⇒ δθ int ≈ n −1δθ ext
 EE346 NLO
1/20/21, #4 slide 9
Angular Acceptance II

• Angle phasematching for θPM ≠ 0 known as “critical phasematching”

– because of tight angular tolerance
• Angular acceptance much looser for θPM = 90° (“noncritical phasematching”)
negative uniaxial: Z
ne ,2ω (θ ) dne ,2ω
=0
dθ θ PM =90°
no ,ω
k ne,2ω (θ PM = 90°) = no,ω d 2 ne ,2ω
ne ,2ω X ≈ 2 ( ne ,2ω − no ,2ω ) cos 2θ
dθ 2 θ PM =90° #4.8

4π
⇒ ∆k ≈
λ
(n e ,2ω − no ,2ω ) δθ 2
λ Taylor series, #4.7
⇒ δθ <
4 L ( ne ,2ω − no ,2ω )
with tolerance from #4.7

• With parameters of example from #4.8: δθ < 25mrad

– 40x larger than critical PM example (#4.8)
– much preferable to critical PM
 EE346 NLO
1/20/21, #4 slide 10
Temperature Tuning

• Noncritical PM requires additional degree of freedom

– temperature most widely used
– microstructures for quasi-PM an alternative (discuss later)

• Temperature PM
– makes use of difference between thermo-optic coeff’s at ω and 2ω
– e.g. take negative uniaxial with dne ,2ω / dT > dno ,ω / dT (commonly true)
Z Z Z
ne ,2ω (θ ) ne ,2ω (θ )
ne ,2ω (θ )
k
no ,ω no ,ω no ,ω
k
ne ,2ω X X X
ne ,2ω ne ,2ω

T > TPM T = TPM T < TPM

no phasematching
 EE346 NLO
1/20/21, #4 slide 11
Temperature Acceptance

• Same idea as angular acceptance from #4.7:

−1
π d ∆k
⇒ δT <
L dT
– for negative uniaxial example:
d ∆k 4π dne ,2ω dno ,ω
= −
dT λ dT dT
−1
λ dne,2ω dno ,ω
⇒ δT = −
4L dT dT

– e.g. for d (ne ,2ω − no ,ω ) / dT ~ 3 × 10−5 K −1, λ = 1 µm, L = 1 cm: δ T ≈ 1°C

Convenient tool for phasematching calculations:

freeware, SNLO downloadable at http://www.as-photonics.com/SNLO.html
 EE346 NLO
1/20/21, #4 slide 12
Phasematching in Biaxial Media
don'tworry too Z
• Concepts similar, enumeration tedious much notas
– must specify two angles; common k
θ
– use conventional polar angles (θ, ϕ)
• Consider case where nZ > nY > nX and nZ ,ω > ( nY ,2ω , nX ,2ω ) ϕ

Z Y
X
Z
nY ,2ω
k
nX ,ω k nY ,ω θ
θ PM,1
PM,2

nZ ,ω nX ,2ω nZ ,ω 1.06 µm SHG in LBO

X
Y

(θ PM ,ϕPM ) = (θ PM,1 ,0) (θ PM ,ϕPM ) = (θ PM,2 , π / 2)

ˆ
E2ω Y ˆ
Eω ⊥ Y ˆ ˆ
E2ω X Eω ⊥ X
Velsko, IEEE J.Q.E. 27, 2182 (1991)

continuous family of solutions will exist:θ PM = θ PM (ϕ PM )

 EE346 NLO
1/20/21, #4 slide 13
Linear Susceptibility

• Evidently detailed knowledge of linear dispersion essential for NLO

• First principles microscopic calculation rarely adequate

– difficult to get 10-4 – 10-5 accuracy
– efficient parameterization of empirical results very useful

• Qualitative microscopic discussion useful

– sets scale for expected behavior
– motivates common empirical representations
 EE346 NLO
1/20/21, #4 slide 14
Lorentz Classical Electron Oscillator Model

• Simple model
– reveals essential features of EM field
interacting with bound electron
F = −k q q
– considers electron bound to massive −e
nucleus with a linear spring E
– driven by a sinusoidal electric field

m q = −k q − e E
Newton ⇒ accel not adequate to describe resonance behavior
spring field need damping term − mγ x

q + γ q + ( k / m) q = −(e / m) E forced oscillator

phenomenological k / m ≡ ω 02
damping term resonant frequency

iωt
Assume sinusoidal driving field: E (t ) = E eiω t ⇒ sinusoidal response: q (t ) = q (ω ) e
−ω 2 q + i γ ω q + ω02 q = −(e / m) E ⇒ q = (−eE / m) / (ω02 − ω 2 + i γ ω )
Im[q ]
Resonant response of electron
when driven by field near natural
frequency
Re[q ]
How connected to EM phenomena?
 EE346 NLO
1/20/21, #4 slide 15
Dipole Response and EM Phenomena

• Use CEO result to obtain expression for linear susceptibility χ(ω)

for each atom: µ = −e q P = N µ = −N e q
atomic dipole moment number density of atoms

– with q from #4.14:

χ = P / (ε 0 E )
= −( Ne / ε 0 ) q / E
= ( Ne 2 / ε 0 m)[(ω02 − ω 2 ) + i γ ω ]−1 note that χ is a complex quantity
with CEO result so ε and thus n must be as well

• Dielectric constant: ε = 1 + χ ⇒ ε = 1 + ( Ne / ε 0 m)[(ω0 − ω ) + i γ ω ]

2 2 2 −1
3

2.5
nRe
• Connection to EM wave propagation:
2 ~ (ω − ω0 ) −1
1.5

n 2 = 1 + ( Ne 2 / ε 0 m)[(ω02 − ω 2 ) + i γ ω ]−1
1

0.5

− i 2π nre z / λ 2π nim z / λ
E( z) ∝ e e -0.5
~ (ω − ω0 ) −2 nIm
phase attenuation -1

-1.5

-2

-2.5
ω / ω0
0 0.5 1 1.5
 EE346 NLO
1/20/21, #4 slide 16
CEO vs Reality

…
• Classical result: n 2 = 1 + ( Ne 2 / ε 0 m)[(ω02 − ω 2 ) + i γ ω ]−1 3
2
– quantum result:
1
n = 1 + ( Ne / ε 0 m)∑ f gp [(ω − ω ) + i γ gp ω ]
2 2 2
gp
2 −1

p
oscillator strength
g
– close correspondence between classical and quantum results
• Motivates common parameterization of empirical results:
– “Sellmeier” form
)

– applied away from absorptive regions

n 2 single neff nA + nB
simple: A A′λ 2
n2 = 1 + 2 n 2
= 1 + polepretty
or
ω0 − ω 2 λ 2 − λ02 goodapprox
multiple pole: nA
A A
n 2 = 1 + 2 1 2 + 2 2 2 + ... nB
ω1 − ω ω2 − ω
many variants:
A1 A2
e.g. n = nb + +
2 2

ω12 − ω 2 ω22 − ω 2 ω / ω0
temperature dependent coefficients, … x
 EE346 NLO
1/20/21, #4 slide 17
Material Dispersion
A
• Single resonance: n 2 = 1 + 2
ω0 − ω 2
– key features:
except in absorbing region:
nre monotone increasing with frequency
n(ω )
slope decreases away from resonance
d 2n d 2n
<0 >0
dω 2 dω 2

• More realistic (especially for IR interactions)

– two resonances ω ph < ω < ω e
e.g. electronic (UV) and phonon (IR) poles
avoid near-resonant Lorentzian form
ω
A B
n(ω ) − 1 = χ ′(ω ) ≈ 2
2
+ 2
ω e − ω ω ph − ω 2
2
ωe
ω ph
note that in transparency range dn
ng = n + ω
terms contribute to GVD with opposite sign dω
n monotonic, ng positive, but not GVD d 2k
GVD =
will see has important implications for pulse propagation dω 2
 EE346 NLO
1/20/21, #4 slide 18
CEO in Anisotropic Media
qZ
• Can follow same pattern for anisotropic case

F = − k Z qZ • Obtain different eqs of motion for qX and qZ

−e F = −k X q X m q X = −k q X − e E X
qX
m qZ = − k qZ − e EZ
E
• Two different susceptibilities
– similar to #4.15:
χ X = PX / (ε 0 E X )
= −( Ne / ε 0 ) q X / E X
= ( Ne 2 / ε 0 m)[(ω X2 − ω 2 ) + i γ X ω ]−1
ω X2 = k X / m
similarly:
χ Z = ( Ne 2 / ε 0 m)[(ωZ2 − ω 2 ) + i γ Z ω ]−1
ωZ2 = kZ / m