Second-harmonic generation, Phase-matching bandwidth, and Group velocity mismatch (GVM)

Phase-matching in SHG

Phase-matching bandwidth

Group-velocity mismatch

Nonlinear-optical crystals

Practical numbers for SHG

Electro-optics

Phase-matching in Second-Harmonic Generation

How does phase-matching affect SHG? It’s a major effect, one of the most important reasons why you just don’t see SHG every day.

First Demonstration of Second-Harmonic Generation

P.A. Franken, et al, Physical Review Letters 7, p. 118 (1961)

The second-harmonic beam was very weak because the process was not phase-matched.

First demonstration of SHG: The DataThe actual published results…

Input beamThe second harmonic

Note that the very weak spot due to the second harmonic is missing. It was removed by an overzealous Physical Review Letters editor,who thought it was a speck of dirt.

Sinusoidal Dependence of SHG Intensity on Length

Large Δk Small Δk

The SHG intensity is sharply maximized if Δk = 0.

which will only be satisfied when:

Unfortunately, dispersion prevents this from ever happening!

Phase-matching Second-Harmonic Generation

So we’re creating light at ωsig = 2ω.

0

2 2 ( )polk k ncω ω= =

0 0

(2 )( ) (2 )sigsig sigk n n

c cω ωω ω= =

sig polk k=

(2 ) ( )n nω ω=

ω 2ωFrequency

Ref

ract

ive

inde

x

And the k-vector of the polarization is:

The phase-matching condition is:

The k-vector of the second-harmonic is:

We can now satisfy the phase-matching condition.

Use the extraordinary polarizationfor ω and the ordinary for 2ω.

Phase-matching Second-Harmonic Generation using BirefringenceBirefringent materials have different refractive indices for different polarizations. Ordinary and extraordinary refractive indicescan be different by up to ~0.1 for SHG crystals.

(2 ) ( )o en nω ω= ω 2ωFrequencyR

efra

ctiv

e in

dex

ne

on

ne depends on propagation angle, so we can tune for a given ω.Some crystals have ne < no, so the opposite polarizations work.

Noncollinear SHG phase-matching

0

ˆ2 cos

2 ( )cos

pol

pol

k k k k z

k nc

θ

ω ω θ

′= + =

⇒ =

2 (2 )sigo

k ncω ω=

(2 ) ( ) cosn nω ω θ=

ˆˆcos sink k z k xθ θ= −

ˆˆcos sink k z k xθ θ′ = +

z

x

But: So the phase-matching condition becomes:

θθ

I

Δk

Phase-Matching Bandwidth

Phase-matching only works exactly for one wavelength, say λ0. Since ultrashort pulses have lots of bandwidth, achieving approximate phase-matching for all frequencies is a big issue.

The range of wavelengths (or frequencies) that achieve approximate phase-matching is the phase-matching bandwidth.

[ ]4( ) ( ) ( / 2)k n nπλ λ λλ

Δ = −

0λ 0

2λWavelength

Ref

ract

ive

inde

x

ne

no

2 2( ) ( / ) sinc ( / 2)sigI L L k Lλ∝ Δ

Recall that the intensity out of an SHG crystal of length L is:

where:

( ) )/ 2 (n nλ λ≠

2λ

λ

Phase-matching efficiency vs. wavelength for BBO

These curves also take into account the (L/λ)2 factor.

While the curves are scaled in arbitrary units, the relative magnitudes can be compared among the three plots. (These curves don’t, however, include the nonlinear susceptibility, χ(2)).

Phase-matching efficiency vs. wavelength for the nonlinear-optical crystal, beta-barium borate (BBO), for different crystal thicknesses:

10 μm 100 μm 1000 μm

Note the huge differences in phase-matching bandwidth and efficiency with crystal thickness.

Phase-matching efficiency vs. wavelength for KDP

The curves for the thin crystals don’t fall to zero at long wavelengths because KDP simultaneously phase-matches for two wavelengths, that shown and a longer (IR) wavelength, whose phase-matching ranges begin to overlap when the crystal is thin.

Phase-matching efficiency vs. wavelength for the nonlinear-optical crystal, potassium dihydrogen phosphate (KDP), for different crystal thicknesses:

10 μm 100 μm 1000 μm

The huge differences in phase-matching bandwidth and efficiency with crystal thickness occur for all crystals.

Calculation of Phase-Matching Bandwidth

Δk (λ) =4πλ

n(λ)− n(λ / 2)[ ]

0 0 0 00 0

4( ) 1 ( ) ( ) ( / 2) ( / 2)2

k n n n nπ δλ δλλ λ δλ λ λ λλ λ

⎡ ⎤ ⎡ ⎤′ ′Δ = − + − −⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

Assuming the process is phase-matched at λ0, let’s see what the phase-mismatch will be at λ = λ0 + δλ

x xBut the process is phase-matched at λ0

because, when the input wavelength changes by δλ, the second-harmonic wavelength changes by only δλ/2.

The phase-mismatch is:

0 00

4 1( ) ( ) ( / 2)2

k n nπ δλλ λ λλ

⎡ ⎤′ ′Δ = −⎢ ⎥⎣ ⎦to first order in δλ

The sinc2 curve will decrease by a factor of 2 when Δk L/2 = ± 1.39.

So solving for the wavelength range that yields |Δk | < 2.78/L

yields the phase-matching bandwidth.

01

0 02

0.44 /( ) ( / 2)FWHM

Ln n

λδλλ λ

=′ ′−

0 00

4 12.78 / ( ) ( / 2) 2.78 /2

L n n Lπ δλ λ λλ

⎡ ⎤′ ′− < − <⎢ ⎥⎣ ⎦

I

Δk

FWHM

2.78/L-2.78/L

Calculation of Phase-Matching Bandwidth (cont’d)

sinc2(ΔkL/2)

Phase-Matching Bandwidth

BBO KDP

The phase-matching bandwidth is usually too small, but it increases as the crystal gets thinner or the dispersion decreases (i.e., the wavelength approaches ~1.5 microns).

The theory breaks down, however, when the bandwidth approaches the wavelength.

Group-Velocity MismatchInside the crystal the two different wavelengths have different group velocities.

Define the Group-Velocity Mismatch (GVM):

0 0

1 1v ( / 2) v ( )g g

GVMλ λ

≡ −

Crystal

As the pulse enters the crystal:

As the pulseleaves the crystal:

Second harmonic createdjust as pulse enters crystal(overlaps the input pulse)

Second harmonic pulse lagsbehind input pulse due to GVM

Group-Velocity Mismatch

0 / ( )v ( )1 ( )

( )

gc n

nn

λλ λ λλ

=′−

0 0 0 00 0

0 0 0 0

( / 2) / 2 ( )1 ( / 2) 1 ( )( / 2) ( )

n nn nc n c n

λ λ λ λλ λλ λ

⎡ ⎤ ⎡ ⎤′ ′= − − −⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

00 0

0

1( ) ( / 2)2

GVM n ncλ λ λ⎡ ⎤′ ′= −⎢ ⎥⎣ ⎦

Calculating GVM:

0

1 ( ) 1 ( )v ( ) ( )g

n nc nλ λ λ

λ λ⎡ ⎤′= −⎢ ⎥⎣ ⎦

So:

0 0

1 1v ( / 2) v ( )g g

GVMλ λ

≡ −

x xx x x xBut we only care about GVM when n(λ0/2) = n(λ0)

Group-velocity mismatch lengthens the SH pulse.

Assuming that a very short pulse enters the crystal, the length of the , SH pulse, δt, will be determined by the difference in light-travel times through the crystal:

δ t =L

v g(λ0 / 2)−

Lv g(λ0 )

= L GVM

Crystal

L GVM << τ pWe always try to satisfy:

L /LD

Group-velocity mismatch pulse lengtheningSecond-harmonic pulse shape for different crystal lengths:

It’s best to use a very thin crystal. Sub-100-micron crystals are common.

LD ≡τ p

GVM

Inputpulseshape

LD is the crystal length that doubles the pulse length.

Group-velocity mismatch numbers

Group-velocity mismatch limits bandwidth.Let’s compute the second-harmonic bandwidth due to GVM.

Take the SH pulse to have a Gaussian intensity, for which δt δν = 0.44. Rewriting in terms of the wavelength,

δt δλ = δt δν [dν/dλ]–1 = 0.44 [dν/dλ]–1 = 0.44 λ2/c0

where we’ve neglected the minus sign since we’re computing the bandwidth, which is inherently positive. So the bandwidth is:

01

0 02

0.44 /( ) ( / 2)FWHM

Ln n

λδλλ λ

≈′ ′−

Calculating the bandwidth by considering the GVM yields the sameresult as the phase-matching bandwidth!

2 20 0 0 00.44 / 0.44 /

FWHMc c

t L GVMλ λδλ

δ≈ = 0

0 00

1( ) ( / 2)2

GVM n ncλ λ λ⎡ ⎤′ ′= −⎢ ⎥⎣ ⎦

SHG efficiency

22 0( , ) exp( 2)sinc( 2)

2LE L t i P i kL kL

kω μ ω

= − Δ Δ

2 (2) 2 2 22 20

2 30

( ) ( ) sinc ( 2)2

I LI kLc n

ωω η ω χ

= Δ

I 2ω

Iω =2η0ω 2d 2I ω L2

c02n3

28 2[5 10 / ]I W I L

I

ωω

ω−≈ ×

The second-harmonic field is given by:

The irradiance will be:

Dividing by the input irradiance to obtain the SHG efficiency:

Substituting in typical numbers:

Take Δk = 0

d ∝ χ(2), and includes crystal additional parameters.

Amazing second-harmonic generation

Frequency-doublingKDP crystals atLawrence LivermoreNational Laboratory

These crystals convertas much as 80% of theinput light to its secondharmonic. Then additionalcrystals produce thethird harmonic withsimilar efficiency!

These guys are serious!

ω1

ω1

ω3

ω2 = ω3 − ω1

Parametric Down-Conversion(Difference-frequency generation)

Optical Parametric Oscillation (OPO)

ω3

ω2

"signal"

"idler"

By convention:ωsignal > ωidler

Difference-Frequency Generation: Optical Parametric Generation, Amplification, Oscillation

ω1

ω3 ω2

Optical Parametric Amplification (OPA)

ω1

ω1

ω3ω2

Optical Parametric Generation (OPG)

Difference-frequency generation takes many useful forms.

mirror mirror

Optical Parametric GenerationEquations are just about identical to those for SHG:

2(2) *1

1 2 321 1

2(2) *2

2 1 322 2

2(2) 3

3 1 223 3

1v 2

1v 2

1v 2

i k z

g

i k z

g

i k z

g

E i E E ez t c k

E i E E ez t c k

E i E E ez t c k

ωχ

ωχ

ωχ

Δ ⋅

Δ ⋅

− Δ ⋅

⎛ ⎞∂ ∂+ = −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

⎛ ⎞∂ ∂+ = −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

⎛ ⎞∂ ∂+ = −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

where:ki = wave vector of ith waveΔk = k1 + k2 - k3vgi = group velocity of ith wave

The solutions for E1 and E2involve exponential gain!

OPA’s etc. are ideal uses of ultrashort pulses, whose intensities are high.

Phase-matching applies.We can vary the crystal angle in the usual manner, or we can vary the crystal temperature (since n depends on T).

Free code to perform OPO, OPA, and OPG calculations

Public domain software maintained by Arlee Smith at Sandia National Labs. Just web-search ‘SNLO’.

You can use it to select the best nonlinear crystal for your particular application or perform detailed simulations of nonlinear mixing processes in crystals.

Functions in SNLO:

1. Crystal properties2. Modeling of nonlinear crystals in various applications.3. Designing of stable cavities, computing Gaussian focus parameters

and displaying the help file.

Optical Parametric Generation

Recent results using the nonlinear medium,periodically poled RbTiOAsO4

Sibbett, et al., Opt. Lett., 22, 1397 (1997).

signal:

idler:

An Ultrafast Noncol-linear OPA (NOPA)Continuum generates an arbitrary-color seed pulse.

NOPA Specs

Crystals for far-IR generation

With unusual crystals, such as AgGaS2, AgGaSe2 or GaSe, one can obtain radiation to wavelengths as long as 20 μm.

These long wavelengths are useful for vibrationalspectroscopy.

Gavin D. Reid, University of Leeds, and Klaas Wynne, University of Strathclyde

10 μm 1 μm

Wavelength

Elsaesser, et al., Opt. Lett., 23, 861 (1998)

Difference-frequency generation in GaSe

Angle-tuned wavelength

Another 2nd-order process: Electro-opticsApplying a voltage to a crystal changes its refractive indices and introduces birefringence. In a sense, this is sum-frequency generation with a beam of zero frequency (but not zero field!).

A few kV can turn a crystal into a half- or quarter-wave plate.

VIf V = 0, the pulse polarization doesn’t change.

If V = Vp, the pulse polarization switches to its orthogonal state.

“Pockels cell”

(voltage may be transverse or longitudinal)

Abruptly switching a Pockels cell allows us to switch a pulse into or out of a laser.

Polarizer

Before switching After switching

Pockels’ cell as wave plate w/ axes at ±45°

0° Polarizer Mirror

Pockels’ cell as wave plate w/

axes at 0° or 90°

0° Polarizer Mirror

The Pockels’ Cell (Q-Switch)

The Pockels effect involves the simple second-order process:

0sigω ω= +dc field

The Pockels effect is a type of second-order nonlinear-optical effect.

The signal field has the orthogonal polarization, however.