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Multi-order dispersion engineering for optimal

four-wave mixing

Michael R.E. Lamont,* Boris T. Kuhlmey, and C. Martijn de Sterke

Centre for Ultra-high-bandwidth Devices for Optical Systems

School of Physics, University of Sydney, Sydney, NSW 2006, Australia

*Corresponding author: m.lamont@physics.usyd.edu.au

Abstract: Four-wave mixing in high refractive index materials, such as

chalcogenide glass or semiconductors, is promising because of their large

cubic nonlinearity. However, these materials tend to have normal

dispersion at telecom wavelengths, preventing phase matched operation.

Recent work has shown that the waveguide dispersion in strongly confining

guided-wave structures can lead to anomalous dispersion, but the resulting

four-wave mixing has limited bandwidth because of negative quartic

dispersion. Here we first show that the negative quartic dispersion is an

inevitable consequence of this dispersion engineering procedure. However,

we also demonstrate that a slight change in the procedure leads to positive

quartic dispersion, resulting in a superior bandwidth. We give an example

in which the four-wave mixing bandwidth is doubled in this way.

©2008 Optical Society of America

OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (130.4310) Integrated optics,

nonlinear; (230.7370) Waveguides.

References and links

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Received 14 Mar 2008; revised 24 Apr 2008; accepted 28 Apr 2008; published 9 May 2008

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14. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell,

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1. Introduction

The fabrication of sub-micron dimensioned step-index planar waveguides is now possible

using high-index materials such as silicon [1, 2] and AlGaAs [3]. These structures have

enhanced nonlinear effects due to a reduced effective mode area and the large cubic

nonlinearity intrinsic to these materials. As well, fiber tapers utilizing these same

enhancements have been demonstrated in a variety of materials including, recently, highly

nonlinear chalcogenide glass [4]. At this reduced dimension, the dispersion due to the

waveguide geometry can be larger than the material dispersion. This has lead to an interest in

tailoring the group-velocity dispersion (β2) for processes that depend strongly on dispersive

characteristics such as supercontinuum generation [5-7] and four-wave mixing (FWM) [1, 2,

8, 9].

Degenerate FWM is a nonlinear process utilizing a single pump with a phase-matching

condition that depends only on the even ordered dispersion parameters evaluated at the pump

frequency [10, 11]. Specifically, for continuous gain on either side of the pump frequency

with a useful bandwidth, the pump must experience low anomalous dispersion (β2 ≲ 0). In

addition, it is well-known that for optimal performance positive fourth-order dispersion (β4)

can be used to counter β2-induced phase mismatch, thus increasing the bandwidth over which

the device is phase matched [11-13]. Measurements of dispersion-shifted silica fiber, which

has a material dispersion that is anomalous at telecom wavelengths, resulted in both negative

β4 [14, 15] and positive β4 [10, 16]. However this is not the case for high-index materials,

which have large normal material dispersion at 1550 nm. Dispersion engineering high-index

waveguides has resulted in either a negative β4 [1-3, 6, 8], which reduces the flatness and

bandwidth of the FWM gain; or, at much smaller transverse dimensions, a positive β4 that is

an order of magnitude larger than the material value [2, 3, 8], resulting in an even lower

bandwidth. It is therefore important to understand how waveguide dispersion affects not only

β2, but also β4, to enhance the FWM performance of high-index waveguides.

This paper has two aims. The first of these is to illustrate that a negative β4 is a generic

property of the standard waveguide engineering process. The second is to propose a modified

process that corrects this. We present a simple, generic model for the waveguide mode-

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Received 14 Mar 2008; revised 24 Apr 2008; accepted 28 Apr 2008; published 9 May 2008

12 May 2008 / Vol. 16, No. 10 / OPTICS EXPRESS 7552

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propagation constant based on effective indices. This model, which is consistent with

numerical simulation, demonstrates many standard characteristics of the various dispersive

orders, specifically the placement of the zero dispersion points (ZDP, frequency when β2 = 0)

in relation to the sign and magnitude of β4. Using an order-of-magnitude estimation of the

maximum anomalous waveguide dispersion, we propose a method of tailoring not only β2, but

β4 as well through control of the core-cladding index difference. We demonstrate its

effectiveness by comparing the examples of a β2-engineered chalcogenide fiber taper

suspended in air and a β2/β4-engineered taper immersed in a high-index fluid, and show that

multi-order dispersion engineering more than doubles the FWM gain bandwidth as well as

providing a much flatter gain spectrum.

In Section 2, the effects of β4 on degenerate FWM are discussed. In Section 3, we

present a generic model of the mode-propagation constant and discuss its implications for

conventional β2-engineering. The concept of multi-order dispersion engineering is introduced

in Section 4, and an example of β2/β4-engineering is presented. In Section 5, we go through a

specific example, comparing the FWM performance of an As2Se3 fiber taper with β2-

engineering versus the same with β2/β4-engineering. Section 6 contains a discussion of the

benefits and difficulties of implementing multi-order dispersion engineering and the paper is

concluded in Section 7.

2. Optimal dispersion characteristics for FWM

FWM is a nonlinear parametric process governed by a phase-matching condition. In the

degenerate case involving a single pump (D-FWM), if the dispersive phase-mismatch is

compensated by the nonlinear phase, then energy is transferred from the pump, at an angular

frequency ωp, to both the signal, at ωs, and into the creation of an idler, at ωi, such that

ωi = 2ωp – ωs. Considering up to fifth-order dispersion, the dispersive phase-mismatch is [10,

11, 17]

(

()

2

2

ωβΔ=

)

() ,

2

4

4

24

1

2

1

2

1

ωΔβ

ββββ

+

−+=

Δ

pis

(1)

The top equation uses the full dispersion relation, where βs, βi and βp are the mode-

propagation constant evaluated at ωs, ωi and ωp, and the more commonly used bottom

equation uses a Taylor expansion where β2 and β4 are evaluated at ωp and Δω = |ωs – ωp|. The

nonlinear phase is given by γP, where P is the pump power, and γ is the nonlinear coefficient

of the waveguide, resulting in a net phase of κ = Δβ + γP. The signal gain is then

(

1

PGs

g

γ+=

)(),sinh

2

2

zg

(2)

where the exponential gain coefficient, g, is defined as

( )

P

γ

( )

P

γ

() .

22

2

2

P

γβκ+Δ−=−=

g

(3)

Although the effect of β4 is typically much smaller than β2, FWM operates best near the

ZDP where β2 ~ 0, making the β4 term in Eq. (1) significant for large frequency offsets. It is

well known that the dispersive phase-mismatch can be tailored through engineering β2 [1, 18];

however, it has been shown that with further refinement the gain bandwidth can be increased

if β4 is also controlled [11-13]. The following analysis of Eqs (1) to (3) indicates the

possibilities this enables and the constraints imposed on β4.

From Eq. (2), exponential gain only occurs while g remains real and, from Eq. (3), this is

true for -2γP < Δβ < 0, with maximum gain occurring when Δβ = -γP. The gain bandwidth

can be characterized by the cutoff frequency difference, Δωc, which is the frequency offset at

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(C) 2008 OSA

Received 14 Mar 2008; revised 24 Apr 2008; accepted 28 Apr 2008; published 9 May 2008

12 May 2008 / Vol. 16, No. 10 / OPTICS EXPRESS 7553

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which g becomes imaginary, or equivalently when Δβ = 0 or -2γP. These two cases are

illustrated in Fig. 1. To achieve exponential gain at frequencies close to the pump, Δβ must be

negative at very small Δω where β2 dominates Eq. (1), therefore β2 must be negative

(anomalous). With this in mind, we look at maximizing the bandwidth defined by Δωc.

First considering a cutoff at Δβ = -2γP, we can write

()

.

4

β

2

4

12

1

2

c

c

P

ωΔβ

γ

ωΔ

+

−=

(4)

Although this is not an explicit expression, it confirms that if β4 has the opposite sign to β2, it

lowers the denominator in Eq. (4), thus increasing the gain bandwidth. This is illustrated in

Fig. 1(a) and (c) by comparing the case with β4 = 0 (Δωc0, dashed lines) to the case with β2 < 0

and β4 > 0 (Δωc+, solid lines). Similarly, if β4 has the same sign as β2, Δωc is further reduced

from the case with β4 = 0.

However, if the cutoff occurs when Δβ = 0, the cutoff frequency is

.

12

β

4

2

β

ω

Δ−=

c

(5)

Thus, this solution only exists if β2 and β4 have opposite signs, and Δωc increases with a

smaller β4. According to Eq. (1), since β2 is negative, Δβ is negative at small Δω and at

higher Δω, a positive β4 causes Δβ to increase. As demonstrated in Fig. 1(b), this system can

be optimized to have a zero slope at Δβ = -γP where g is maximized, resulting in the broad

gain bandwidth shown in Fig. 1(d). As well as increasing the gain bandwidth, a positive β4

can improve the flatness of the gain spectrum. This is explained explicitly in Appendix 1.

Δβ

Δω

-2γP

-γP

0

Δωc

0

Δβ

Δω

-2γP

-γP

0

Δωc+

Δωc0

0

Δω

0

γP

Re(g)

Δωc

0

Δω

0

Δωc+

γP

Re(g)

Δωc0

0

(a)

(b)

(c)

(d)

Fig. 1. (a) The dispersive phase-mismatch, Δβ, for the case when Δβ becomes less than -2γP.

The dashed line has β2 < 0 and β4 = 0; the solid line has β2 < 0 and β4 > 0, which results in a

broader bandwidth. (b) Δβ when β2 < 0 and β4 > 0, optimized to have zero-slope at -γP, giving

it a broad and flat gain spectrum. The grey-shaded areas indicate the gain region. The real

part of the exponential gain coefficient, g, is calculated from Eq. (3) and shown in (c) and (d)

for Δβ curves shown in (a) and (b), respectively.

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Although many groups have used dispersion engineering to achieve anomalous dispersion

at a target wavelength, in all cases β4 was either negative or positive but very large [1-3, 6],

neither of which result in a broad and flat gain spectrum. To understand how general these

dispersion engineering results are and when a positive β4 can be achieved, we must take a

closer look at how waveguide dispersion affects β4 in relation to β2.

3. Modeling the mode-propagation constant

In this section, we consider only the waveguide contribution to the mode-propagation

constant. Starting from a simple, generic model of the effective index of the fundamental

mode for a step-index waveguide, general trends can be seen in the mode-propagation

constant, β, and its derivatives with respect to angular frequency, βi, using the standard

equations

.;

β

ω

d

β

ω

c

β

i

i

i eff

d

n

==

(6)

(a) (b)

Fig. 2. (a) Schematic of the effective index model of the mode-propagation constant and its

consequences on the various orders of dispersion. (b) RSoft FemSIM results of a 1×1 μm

waveguide with ncore = 2 and nclad = 1, which match the predicted features in (a). Although ω

is the independent variable, similar results are obtained if ω is kept constant and the

waveguide size varied; a low frequency is equivalent to a small size, high frequency to a large

size.

β βwg

ncoreω ω/c

ncladω ω/c

β β1,wg

ncore/c

nclad/c

β β3,wg

0

β β4,wg

0

ω

β β2,wg

0

min β2,wg

-2000

-1000

0

1000

×10-6

0 5001000

ω ω [THz]

1500 2000

β β4[ps4/m]

0

8

5

10

15

×106

β β0[1/m]

-120

0

120

×10-3

β

3[ps3/m]

2

4

6

β β1[ps/m]

×103

-10

0

10

20

β β2[ps2/m]

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