Lecture 9 - CREOL€¦ · Lecture 9 1 2 Phase matching: spatial walkoffeffect Walk-off between...

1

1

Quasi-phase matching (QPM) via domain inversion in

ferroelectrics and other media; phase mismatch &

coherence length.

Lecture 9

1

2

Phase matching: spatial walkoff effect

Walk-off between ordinary and extraordinary beams in a birefringent medium

If the angle θ between the propagation direction and the optic axis is not 0 or 90 degrees, the Poynting vector (S) and the propagation vector kare not parallel for extraordinary rays. The ordinary and extraordinary rays with parallel propagation vectors diverge from one another as they propagate through the crystal.

k-vector 𝐷×𝐵

Poynting vector 𝑺 = 𝐸×𝐻

Poynting vector S; energy flows along the normal to the k-vector surface

k-vector

z- axis

o-wave

e-wave

tan 𝜃, = (./.0)2tan 𝜃 →

𝜌 = 𝜃, − 𝜃 ≈𝑛8 − 𝑛9𝑛9

sin 2𝜃

spatial walkoff:

S-vector

𝜃

k-vector surface

k𝜌

In anisotropic crystals, Pointing vector (the power flow) and k-vector generally do not coincide.

This effect limits the spatial overlap of the two waves and decreases the efficiency of any nonlinear mixing process.

2

2

3

Lithium niobateLiNbO3 point group 3m uniaxial crystal

from lecture 7: 𝑑𝑖𝑗 =

𝑑22 = 2.1𝑝𝑚𝑉

𝑑31 = −4.35𝑝𝑚𝑉

𝑑33 = 27.2𝑝𝑚𝑉

Dmitriev et al., Handbook of NLO crystals (1997), p.125

𝑛8 < 𝑛9

Two scenarios for SFG: 1) Type-I SFG 𝜔1 + 𝜔2 = 𝜔3

2) Type-II SFG 𝜔1 + 𝜔2 = 𝜔3

𝑜 + 𝑜 = 𝑒

𝑒 + 𝑜 = 𝑒

In traditional phase matching approach using birefringence, one can not have access to the highest tensor elements like d33 in lithium niobate

3

4

Quasi-phase matching (QPM)

The other method satisfying momentum

conservation is quasi-phase-matching.

A nonlinear material with spatially modulated nonlinear properties is used; the crystal axis is flipped at a regular interval Λ, typically 10–100 µm in length.

example: lithium niobate

4

3

5

Quasi-phase matching (QPM): theory

Use the example of second harmonic generation (SHG).

A plane wave field of amplitude 𝐸N at frequency 𝜔N and wavevector 𝑘N =.PQPR

, passes through a medium with quadratic nonlinear susceptibility 𝑑, generating a nonlinear polarization wave proportional to 𝑑𝐸N2 at frequency 𝜔2=2𝜔N and wavevector 2𝑘N.

𝐸N 𝐸2𝑃TU

The polarization wave radiates a free second harmonic wave with wave vector 𝑘2 =.VQVR

.

Δ𝑘 = 𝑘2 − 2𝑘N

The forced and free waves accumulate a phase shift of 𝜋 over a distance known as the coherence length

𝑙R Δ𝑘 = 𝜋 where

where 𝜆 is the vacuum wavelength of the fundamental wave (at 𝜔N). or

The direction of power flow between the fundamental and harmonic depends on the relative phase of the forced and free waves, and hence changes sign every coherence length.

𝑙R =𝜋

𝑘2 − 2𝑘N=

𝜆4(𝑛2 − 𝑛N)

5

6

Quasi-phase matching (QPM): theory

[\V[]

=−𝑖 ^2𝐴N2 𝑒`∆b]=−𝑖𝛾 𝑒`∆b]

g = [R

QPVQV.PV.V

NL coupling coefficient

𝐴𝑖 =𝑛`𝜔`𝐸𝑖

Δ𝑘 = 𝑘2 − 2𝑘N

Second harmonic generation (SHG)

The slowly-varying envelope equation (low conversion limit) from Lecture 6:

𝐴2 =−𝑖𝛾 ∫fU 𝑒`∆b]𝑑𝑧 = −𝑖𝛾 N

`∆b(𝑒`∆bU − 1) =

= −𝑖𝛾 N`∆b

𝑒h∆ijV 2𝑖 sin ∆bU

2= −𝑖𝑒`(

∆ijV ) 𝛾𝐿 sin ∆bU

2/ ∆bU

2=−𝑖𝑒`(

∆ijV ) 𝛾𝐿 𝑠𝑖𝑛𝑐 ∆bU

2

Now integrate:

where 𝛾 = ^2𝐴N2 = [

2RQPVQV

.PV.V

𝐴N2

self-test : ∆𝑘=0 à get familiar formula (6.10) for 𝐴22

use this one for sinc

also: recall uncertainty relation ∆𝑘𝐿~ 1

(9.2)

(9.1)

phase termΦo = ΦNf (

^2)2|𝐴N|2𝐿2 𝑠𝑖𝑛𝑐2

∆bU2

flow of photons at 2𝜔flow of photons at 𝜔

6

4

7

Quasi-phase matching (QPM): theoryNow imagine we have a sign-reversal grating of the nonlinear coefficient d(z):

dmax

-dmax

𝑙R

QPM period Λ = 2𝑙R

𝑙R

beamFor a medium with a periodic modulation of the nonlinear susceptibility with period Λ= 2𝑙R and hence fundamental spatial frequency Kg = 2π/Λ, we can write the nonlinear susceptibility as a Fourier series

𝑑 𝑧 = rstf,±N,±2…

𝑑s𝑒x`syz]

where the m-th spatial harmonic is mKg, and dm is the corresponding Fourier coefficient.

(9.3)

Now plug (9.4) into the integral (9.2)

𝐴2 =−𝑖 ∫fU ∑ 𝛾s𝑒x`syz] 𝑒`∆b]𝑑𝑧 = − 𝑖 ∫f

U ∑ 𝛾s 𝑒`(∆bxsyz)]𝑑𝑧 = −𝑖 ∫fU ∑ 𝛾s 𝑒`∆b(Nxs)]𝑑𝑧 = ⋯

(the modified NL coefficient 𝛾(𝑧) will have the same sign reversal )

The modified NL coefficient 𝛾 𝑧– will have the same form as 𝑑s :

𝛾 𝑧 = rstf,±N,±2…

𝛾s𝑒x`syz] (9.4)

only 𝑚 = 1 term of the sum will ‘survive’

since 𝑙R Δ𝑘 = 𝜋

Δ𝑘 =𝜋𝑙R

=2𝜋(2𝑙R )

=2𝜋Λ = 𝐾

𝐾^ = Δ𝑘

with Fourier coefficients 𝛾s=1Λ�f

�

𝛾(𝑧)𝑒`syz]𝑑𝑧

𝛾 = ^2 𝐴N

2 = [��2R

QPVQV.PV.V

𝐴N2

7

8

Quasi-phase matching (QPM): theoryNow we take into account that Δ𝑘 = 𝐾

𝐴2 = −𝑖 �f

U𝛾(stN) 𝑒`∆b]x`yz]𝑑𝑧 = − 𝑖 �

f

U𝑖2𝜋 𝛾 𝑑𝑧 = (

2𝜋 )𝛾𝐿

We see that the largest QPM nonlinear coefficient is reduced below that of a truly phase-matched uniform medium by a factor 2/𝜋, and for m-th order QPM a further reduction by a factor 1/𝑚 appears.

While a sign reversal grating is the most effective, other forms, such as an “on-off” grating can also work, though with a smaller Fourier component and lower efficiency.

|𝐴2|2 = (2�)2 𝛾2𝐿2

the price you pay: the effective NL

coefficient 𝛾 is reduced by a factor 2�QPM photon flux

output (9.6)

𝑚 = 1 Fourier component

𝛾stN=1Λ�f

�

𝛾(𝑧)𝑒`yz]𝑑𝑧 = 𝑖2𝜋 𝛾and also

We get:

(9.5)

𝐾^ = Δ𝑘

8

5

9


By changing the sign of the nonlinear susceptibility every coherence length, the phase of the polarization wave is shifted by 𝜋, effectively rephasing the interaction and leading to monotonic power flow into the harmonic wave.

Changing the direction of z or the direction of the domain alignment, changes the sign of the nonlinear coupling and reverses the direction of energy flow.

9

10


SH field SH intensity

= Λ

10

6

11


Periodically-poled lithium niobate (PPLN) is a domain-engineered lithium niobate crystal, used mainly for achieving quasi-phase-matching in nonlinear optics. The ferroelectric domains point alternatively to the +c and the −c direction, with a period of typically between 5 and 35 µm. The shorter periods of this range are used for second harmonic generation, while the longer ones for optical parametric oscillation. Periodic poling can be achieved by electrical poling .

Lithium niobate –a ferroelectric material

11

12


Schematic diagram of the structure of lithium niobateshowing the effects of poling.

oxygen

lithium

oxygen

oxygen

oxygen

oxygen

oxygen

niobiumThe lithium, niobium, and oxygen atoms lie in layers.

Lithium niobate: ferroelectric domain reversal

The positive z direction, is determined by the position of the layer of lithium atoms in the structure. Poling causes this layer of lithium atoms to move through the oxygen layer into the adjacent space, reversing the direction of the crystal structure.

lithium

niobium

12

7

13

13

14


14

8

15


Using photolithography allows patterning of multi-grating devices. Here the +z surface of a 0.5-mm-thick PPLN chip is revealed by etching with hydrofluoric acid. Each grating is 500-μm wide. The QPM periods are 29–30.5 μm

These 15-μm-period domains propagate through 0.5-mm-thick substrates; aspect ratios greater than 100:1 are observed.

Domain patterns expand underneath the electrode region when electric fields larger than the coercive field are applied.

High Voltage~ 20 kV

Lithium niobate (Stanford Univ.)

15

16


The first, and still the most widely exploited of the PP ferroelectrics are the lithium niobate (LiNbO3, LN) and lithium tantalate (LiTaO3, LT) family.

Both are readily available in three- and four-inch diameter substrates, convenient for lithographic patterning and processing, and have established waveguide technologies compatible with periodic domain structures.

The d33 coefficient in LN, 28 pm/V, is the largest among commonly used ferroelectrics, while thatof LT is somewhat smaller, 16 pm/V.

Both materials offer transparency from the near-UV (~ 300 nm) to the IR 4–5 μm

16

9

17


17

18


Benefits of QPM:

• utilize a large nonlinear coefficient (d33 in LN)• can work with crystals with weak or no birefringence at all (GaAs)• no spatial walk-off• all beams can be co-polarized

Limitations of QPM:

• possible with only certain crystal materials (LN, LT, KTP and familty, GaAs, GaP)

• limited thickness – excludes high power levels• parasitic higher-order processes can generate light at additional

frequencies

18

10

19


The QPM crystal itself provides the additional wavevector k = 2π/Λ (and hence momentum) to satisfy the phase-matching condition.

𝒌𝟏 + 𝒌𝟏 + 𝑮 = 𝒌𝟑

SHG

19

20


𝑮

𝒌𝟏 + 𝒌𝟐 + 𝑮 = 𝒌𝟑

𝒌𝟏

𝒌𝟐

𝒌𝟑

Producing backward waves

DFG

20

11

21


𝑮

𝒌𝟏 + 𝒌𝟏 + 𝑮 = 𝒌𝟑

𝒌𝟏

𝒌𝟑

Producing backward SH waves

𝒌𝟏SHG

21

22


Even more exotic scenarios: 2D QPM

22

12

23

23

24


x

y

z

24

13

25

+ + +---

(100)

(011)

(011)

1) GaAs wafer 2) MBE growth of inverted GaAs on nonpolar Ge layer

+ (100)

(011)

4) MBE regrowth of orientation-patterned film (2-3µm thick)

+ + +---

(011)

(100)5) HVPE growth of thick-

film device (>1000µm)

+

- (100)

(011) 3) Photolithography& Etching

+ + +---

(100)

(011)

(011)

(100)

First Practical QPM Semiconductor: Orientation-Patterned GaAs (OP-GaAs)All-epitaxial growth pioneered at Stanford University1,2/University of Tokyo3,4

[1] C.B. Ebert, L.A. Eyres, M.M. Fejer, and J.S. Harris, J. Crystal Growth 201/202, 187 (1999).[2] T. J. Pinguet et al., in OSA Trends in Optics and Photonics Vol. 56, (Optical Society of America, Washington DC, 1998) pp. 226-229.[3] S. Koh, T. Kondo, T. Ishiwada, C. Iwamoto, H. Ichinose, H. Yamaguchi, T. Usami, Y. Shiraki, and R. Ito, Jpn. J. Appl. Phys. 37 (1998) L1493.[4] S. Koh, T. Kondo, M. Ebihara, T. Ishiwada, H. Sawada, H. Ichinose, I. Shoji, and R. Ito, Jpn. J. Appl. Phys. 38 (1999) L508.

25

26

QPM GaAs: MBE and HVPE Growth

Stain-etched cross-sections of OP-GaAs with 80-μm period

0.5–1mm

26

14

27

27

Why use waveguides?• Offers compatibility with fiber pump laser or

QCL pump laser[no free-space coupling]

• Beam is tightly confined resulting in high intensity – thus high nonlinear gain

• high nonlinear gain over long lengths• Thick material fabrication not needed

M.B. Oron, S. Shusterman and P. Blau, "Periodically oriented GaAs templates and waveguide structures for frequency conversion", SPIE Proceedings, Vol. 6875, 68750F-1 (2008).

Waveguides in QPM Materials based on GaAs

MOCVD & Chemical Polish

27

28

28

QPM was actually suggested [Bloembergen] as a phasematching technique before BPM.

The first demonstration* of QPM was done by cutting up crystals into coherence lengths and stacking them. But there were disadvantages:

• thicknesses involved are a few microns so the slabs were difficult to fabricate

• added surfaces added significant loss making nonlinear conversion efficiencies poor [even a loss of 0.1% in 1000 interfaces yields a total loss of 1-.9991000 = 63% !!]

• assembling a QPM device was labor intensive

Attempts were made to grow crystals with alternating domains. There was some success with LiNbO3 but it was difficult to maintain the exact same periodicity for hundreds of domain thicknesses.

The breakthrough came with the discovery ~1995 that ferroelectric crystals can have their domains flipped by applying an electric field.*Szilagyi, Hordvik and Schlossberg, “A quasi-phase-matching technique for efficient optical mixing and frequency doubling,” J of Appl Phys, 47 (5) 2025-2032 (1976).

QPM History

28

15

29


29

Lecture 9 - CREOL€¦ · Lecture 9 1 2 Phase matching: spatial walkoffeffect Walk-off between...

Documents

Transcript of Lecture 9 - CREOL€¦ · Lecture 9 1 2 Phase matching: spatial walkoffeffect Walk-off between...