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 (β ₂) 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 (β ₂≲0). In addition, it is well-known that for optimal performance positive fourth-order dispersion (β ₄) can be used to counter β ₂-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 β ₄ [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 β ₄ [1–3, 6, 8], which reduces the flatness and bandwidth of the FWM gain; or, at much smaller transverse dimensions, a positive β ₄ 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 β ₂, but also β ₄, to enhance the FWM performance of high-index waveguides.

This paper has two aims. The first of these is to illustrate that a negative β ₄ 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-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 β ₂=0) in relation to the sign and magnitude of β ₄. Using an order-of-magnitude estimation of the maximum anomalous waveguide dispersion, we propose a method of tailoring not only β ₂, but β ₄ as well through control of the core-cladding index difference. We demonstrate its effectiveness by comparing the examples of a β ₂-engineered chalcogenide fiber taper suspended in air and a β ₂/β ₄-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 β ₄ on degenerate FWM are discussed. In Section 3, we present a generic model of the mode-propagation constant and discuss its implications for conventional β ₂-engineering. The concept of multi-order dispersion engineering is introduced in Section 4, and an example of β ₂/β ₄-engineering is presented. In Section 5, we go through a specific example, comparing the FWM performance of an As₂Se₃ fiber taper with β ₂-engineering versus the same with β ₂/β ₄-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 ω_p, to both the signal, at ω_s, and into the creation of an idler, at ω_i, such that ω_i=2ω_p-ω_s. Considering up to fifth-order dispersion, the dispersive phase-mismatch is [10, 11, 17]

The top equation uses the full dispersion relation, where β_s, β_i and β_p are the modepropagation constant evaluated at ω_s, ω_i and ω_p, and the more commonly used bottom equation uses a Taylor expansion where β ₂ and β ₄ are evaluated at ω_p and Δω=|ω_s-ω_p|. 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

where the exponential gain coefficient, g, is defined as

Although the effect of β ₄ is typically much smaller than β ₂, FWM operates best near the ZDP where β ₂~0, making the β ₄ term in Eq. (1) significant for large frequency offsets. It is well known that the dispersive phase-mismatch can be tailored through engineering β ₂ [1, 18]; however, it has been shown that with further refinement the gain bandwidth can be increased if β ₄ is also controlled [11–13]. The following analysis of Eqs (1) to (3) indicates the possibilities this enables and the constraints imposed on β ₄.

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, Δω_c, which is the frequency offset at 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 β ₂ dominates Eq. (1), therefore β ₂ must be negative (anomalous). With this in mind, we look at maximizing the bandwidth defined by Δω_c.

 

Fig. 1. (a) The dispersive phase-mismatch, Δβ, for the case when Δβ becomes less than -2γP. The dashed line has β ₂<0 and β ₄=0; the solid line has β₂<0 and β ₄>0, which results in a broader bandwidth. (b) Δβ when β₂<0 and β ₄>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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First considering a cutoff at Δβ=-2γP, we can write

Although this is not an explicit expression, it confirms that if β ₄ has the opposite sign to β ₂, 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 β ₄=0 (Δω_c0, dashed lines) to the case with β ₂<0 and β ₄>0 (Δω _c+, solid lines). Similarly, if β ₄ has the same sign as β ₂, Δω_c is further reduced from the case with β ₄=0.

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

Thus, this solution only exists if β ₂ and β ₄ have opposite signs, and Δω_c increases with a smaller β ₄. According to Eq. (1), since β ₂ is negative, Δβ is negative at small Δω and at higher Δω, a positive β ₄ 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 β ₄ can improve the flatness of the gain spectrum. This is explained explicitly in Appendix 1.

Although many groups have used dispersion engineering to achieve anomalous dispersion at a target wavelength, in all cases β ₄ 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 β ₄ can be achieved, we must take a closer look at how waveguide dispersion affects β ₄ in relation to β ₂.

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, β_i, using the standard equations

 

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 n_core=2 and n_clad=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.

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At very high frequencies (short wavelengths) the effective index, n_eff, approaches the core index, n_core, since the mode is fully confined; and at very low frequencies (long wavelengths) n_eff approaches the cladding index, n_clad, since the mode expands greatly outside the core. Although we initially consider ω as the independent variable and calculate β at a single transverse waveguide dimension, D, these can be exchanged (D considered variable and calculate β at a single frequency) due to the scale invariance of the Maxwell equations [19]. Fig. 2(a) shows a generic schematic representation of this model which emphasizes the key features. The slope of β transitions between the lower slope (n_clad/c) at low ω to a higher slope (n_core/c) at high ω, which necessitates a region with a slope greater than n_core/c, as shown in the plot of β ₁. Differentiation for each higher-order term results in a zero-crossing at each maximum or minimum, creating an additional zero and a new maximum or minimum. Because of this, all odd-order dispersion terms approach zero from above as ω→∞, and even terms approach zero from below.

As an example, Fig. 2(b) shows simulation results for a square waveguide exhibiting the same features as Fig. 2(a), although the magnitude of the peaks in the higher derivatives make the behavior at high frequencies difficult to see. The same features are found for fiber taper, ridge and rib planar waveguide and other geometries as well.

Conventional β ₂-engineering [1–4, 6–8, 18] uses anomalous waveguide dispersion (β _2,wg), caused by strong confinement, to offset normal material dispersion (β _2,mat) and control the total dispersion at a given frequency. Typically,

where min β _2,wg is indicated in Fig. 2(a). Appendix 2 shows that an order-of-magnitude estimate yields

where ω ₀ is the frequency of the minimum and Δn=n_core-n_clad. For Δn~2 and an operating wavelength of 1550 nm, Eq. (8) yields min β _2,wg≈-11 ps²/m. At λ=1550 nm, β _2,mat=1.0, 1.2, 0.8 ps²/m for silicon, AlGaAs and As₂Se₃, respectively, thus Eq. (7) holds true.

Following the same argument, Fig. 3 illustrates how the total β ₂ and β ₄ at a specific wavelength vary with D for a generic high-index waveguide. The total β ₂ is the sum of waveguide and material dispersions. Since we have targeted a single wavelength, β _2,mat can be considered a constant. This results in two values of D giving a ZDP at the specified wavelength. Usually, ZDP₁ in Fig. 3 is used because it occurs at a larger dimension, but this always results in a negative β ₄. ZDP₂ occurs at a much smaller dimension and results in a positive β ₄, but with such a large magnitude that the FWM gain bandwidth is smaller than that at ZDP₁. Thus, the typical dispersion engineered waveguide has a negative β ₄.

 

Fig. 3. Illustration of conventional β ₂-engineered for a high-index waveguide. Total β ₂ and β ₄ are shown as a function of the transverse dimension of the waveguide. Typical normal material dispersion (β _2,mat) is indicated by the horizontal dotted line. Markers show the two ZDPs and their corresponding β ₄ values. β _4,mat is typically insignificant compared to β _4,wg for high-index waveguides and would be indistinguishable from the horizontal origin line.

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4. Methodology of multi-order dispersion engineering

In Section 2, we found that a positive β ₄ enhances FWM performance and, from Fig. 2(a), that operating at a dimension near the minimum in β ₂ results in a positive β ₄. However, a ZDP occurring in this region requires that

which is inconsistent with Eq. (7). Since β _2,mat is fixed, β _2,wg must be reduced, and, from Eq. (8), this is possible if Δn is decreased. This concept has been demonstrated previously, both theoretically [9] and experimentally [5], with silica fiber tapers to achieve a target ZDP by reducing the waveguide dispersion. Although the target ZDP could be reached by further reduction of the taper width, this may have been impractical. No mention was made of the higher-order dispersion terms in this work.

Fig. 4 demonstrates this principle of multi-order dispersion engineering. By decreasing the waveguide dispersion with respect to the material dispersion, a positive β ₄ occurs at the ZDP. In fact, if the ZDP is targeted at a dimension just below the inflection point in β ₂, then β ₄ is just past its zero-crossing such that β ₂ is negative, β ₄ is positive, and both have a small magnitude. From Eq. (5), this results in a very wide FWM gain bandwidth.

Fig. 5 shows simulation results which demonstrate this principle using an As₂Se₃ chalcogenide fiber taper of varying width (refractive index n=2.76, β _2,mat=+0.81 ps²/m, β _4,mat=+0.37×10^-6 ps⁴/m) suspended in air (n=1) and immersed in a high-index liquid (n=1.65). When Δn is changed from 1.76 to 1.11, the minimum total β ₂ is reduced from -4.8 to -0.04 ps²/m, maintaining just enough waveguide dispersion to offset the normal material dispersion and resulting in the two very close ZDPs shown in Fig. 5(b). As well, β ₄ at the first ZDP (shown as the vertical dotted lines in Fig. 5) has shifted from -6.9×10^-6 to +0.5×10^-6 ps⁴/m, which is optimal for FWM. However, by optimizing both β ₂ and β ₄, the width necessary to attain the first ZDP at 1550 nm has become smaller, from 1.19 to 0.86 µm, which could make fabrication more difficult. This confirms that if the core-cladding index difference can be adjusted, in addition to adjusting the waveguide dimension, multi-order dispersion engineering is possible.

 

Fig. 4. Illustration of high-order dispersion engineering applied to β ₂ and β ₄ for a high-index waveguide. Total β ₂ and β ₄ are shown as a function of the transverse dimension of the waveguide. The magnitude of β _2,wg is now only slightly larger than β _2,mat. Markers show the targeted ZDP and its corresponding β ₄ value. β _4,mat is typically small compared to β _4,wg for high-index waveguides and has not been included.

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Fig. 5. Simulated β ₂ and β ₄ at 1550 nm of an As2Se3 fibre taper, n=2.76, (a) suspended in air, and (b) immersed in a liquid of n=1.65. The vertical dotted line indicates the taper width at which the ZDP is at the target wavelength of 1550 nm.

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5. FWM performance of an immersed As₂Se₃ fiber taper

As an example to illustrate the benefits of multi-order dispersion engineering, we consider the FWM performance of two As₂Se₃ tapers, taper A having a width of 1.19 µm suspended in air, and taper B having a width of 0.86 µm and suspending in a liquid with n=1.65, whose widths are optimized to have a ZDP at 1550 nm, as indicated by the vertical dotted lines in Fig. 5(a) and (b) respectively. The parameters of the D-FWM calculation are shown in the left-hand side of Table 1. The effective area of taper B is much smaller than that of taper A; so as to maintain a fair comparison, the pump power is adjusted to equalize the γP product between calculations. This would also result in similar pump intensities in both tapers.

Figure 6(a) shows Δβ for taper A and B calculated from Eq. (1), which requires the full dispersion relation since the bandwidth is so large. In practice this means that dispersion up to

Table 1. Degenerate and non-degenerate FWM parameters

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β ₆ needs to be included. The values used in our calculations are given in Table 1. The pump wavelength for both tapers is tuned to maximize the gain bandwidth while maintaining gain across the entire range, ±Δω_c. For taper A this implies the pump operates at the ZDP, while for taper B the pump wavelength is offset from the ZDP to introduce anomalous β ₂, which is then compensated by β ₄ such that the minimum Δβ occurs at the optimum value of -γP. The resulting signal gain, from Eq. (2), for a 5 cm-long taper is shown in Fig. 6(b). As predicted, taper B, for which β ₂ and β ₄ have opposite signs, achieves a much broader, flatter bandwidth than taper A, for which β ₂ and β ₄ have the same sign. Although both tapers achieve a maximum gain of more than 15 dB, the β ₂-engineered taper has a bandwidth of 350 nm with >8.6 dB of gain, while the β ₂/β ₄-engineered taper achieves 650 nm, almost double.

 

Fig. 6. (a) D-FWM dispersive phase-mismatch for taper A, 1.19 µm As₂Se₃ taper in air (blue, dashed line), and taper B, 0.86 µm immersed in a high-index fluid with n=1.65 (red, solid line). The left-hand side of Table 1 contains the D-FWM parameters for both tapers. (b) Signal gain for the same tapers with a taper length of 5 cm. The vertical lines indicate the position of the pumps, and the grey area denotes the gain region.

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The above results are for one-pump D-FWM, however using two-pump non-degenerate FWM (ND-FWM) allows more flexibility in the gain spectrum [20]. The equations governing ND-FWM are similar to those of D-FWM, and was discussed by McKinstrie et al. [11]; however rather than depending on β ₂ and β ₄ at ω_p, as in D-FWM, ND-FWM depends on β ₂ and β ₄ at the average frequency between the pumps, ω_a. Using ND-FWM it is possible to achieve a flat, continuous gain region between the pumps. Again, β ₂ must be negative and a positive β ₄ enhances the performance. The right-hand side of Table 1 contains the parameters for the ND-FWM calculation, for the same tapers A and B, where the subscripts ‘1’ and ‘2’ indicate the two pumps. The γP products are equalized as in the previous calculation, but the power is halved between the two pumps so that the total intensity within the taper is the same as for D-FWM.

 

Fig. 7. (a) ND-FWM dispersive phase-mismatch for taper A, 1.19 µm As₂Se₃ taper in air (blue, dashed line), and taper B, 0.84 µm immersed in a high-index fluid with n=1.66 (red, solid line). The right-hand side of Table 1 contains the ND-FWM parameters for both tapers. (b) Signal gain for the same tapers with a taper length of 5 cm. The vertical lines indicate the position of the pumps, and the grey area denotes the gain region.

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As indicated in Fig. 7(a), the gain region for ND-FWM extends to -3γP<Δβ<+γP, and the maximum gain remains at Δβ=-γP. The pump wavelengths have been optimized to achieve perfectly flat gain at ω_a while maximizing the bandwidth. For taper A, ω_a must be offset from the ZDP to introduce normal dispersion and achieve Δβ=-γP between the pumps. For taper B, the pumps must be placed symmetrically about the ZDP. Again, both tapers achieve a maximum gain of more than 15 dB, shown in Fig. 7(b). In this example, the 3-dB bandwidth of taper B is not greatly improved, but this was not the goal since we have optimized primarily for gain flatness. Comparing the 0.5-dB bandwidth shows that the β ₂/β ₄-engineered taper has a range of 390 nm versus 140 nm for the β ₂-engineered taper. For ND-FWM, multi-order dispersion engineering has almost tripled the targeted FWM bandwidth.

6. Discussion

We have shown that by varying the cladding refractive index, it is possible to engineer both β ₂ and β ₄ simultaneously. This method is not specific to any geometry — it is valid for any highly-confining waveguide, including fiber tapers and ridge or rib planar waveguides. Using an As₂Se₃ fiber taper as an example, we demonstrated the effectiveness of this technique by comparing the FWM performance of a β ₂/β ₄-engineered taper to a β ₂-engineered one, the former achieving double the bandwidth of the latter. One could also imagine applying this principle to fluid filled micro-structured optical fibers, or to the design of planar waveguides by choosing a cover material with a targeted refractive index or even a liquid crystal cover providing tunable dispersive characteristics.

Taking as an example the taper from Section 5, we presented an optimized case for operation at 1550 nm; with Δn=1.11, the ZDP is very near the inflection point in β ₂ where β ₄=0. However, the tolerances on Δn are very tight. The range over which β ₄ is positive is only 1.08<Δn<1.11, and Δn<1.08 does not result in enough waveguide dispersion at 1550 nm to compensate the material dispersion fully, i.e. |min β _2,wg|<|β _2,mat|, thus setting an lower limit on Δn. But expanding the required ZDP wavelength to a range 1500<ZDP<1600 nm increases the range over which β ₄>0. With Δn=1.14 and ZDP=1500 nm, β ₄ is both very small and positive. With a ZDP=1600 nm, Δn=1.06 provides just enough waveguide dispersion to compensate the material contribution. So, if a single wavelength is targeted for the ZDP, the tolerance on Δn is ±0.01, however by loosening the ZDP to ±50 nm, the tolerance on Δn increases to ±0.04. Depending on the geometry and materials of the waveguide, it may be possible to tune Δn after fabrication. A photosensitive core or cladding can be utilized, or temperature tuning can be used if the thermal index change varies between the core and cladding.

Although attaining a positive β ₄ may not be possible due to limitations in available materials for the core and cladding, the FWM gain spectrum can still be improved by lowering the magnitude of a negative β ₄, which is a much easier task. For instance, immersing an As₂Se₃ fiber taper in an epoxy with n=1.55 (Δn=1.20) results in β ₄=-3.2×10^-6 ps⁴/m, compared to -6.9×10^-6 ps⁴/m in air, giving a 20% increase in the FWM gain bandwidth even though β ₄ remains negative. In practice, fabrication may also be a limiting factor, as β ₂/β ₄-engineering demands smaller transverse waveguide dimensions. When targeting a ZDP at a particular wavelength, a larger Δn results in a larger necessary cross-section.

Here we have focused on the use of Δn to control the minimum β _2,wg, however another option available in some geometries is the use of a gradient in the waveguide index profile. In deriving Eq. (17) in Appendix 2, we assumed that β ₁ drops from its maximum value to its asymptotic value over a range of ΔV≈1. If we leave this as a variable, Eq. (17) becomes

By introducing a spatial index gradient between the core and cladding, ΔV increases and the magnitude of min β _2,wg decreases.

Some geometries, such as planar waveguides, allow control over the shape of the core. Although this introduces polarization mode dispersion, this can also be used to tailor the magnitude of the β _2,wg minimum. For instance, the square waveguide in Fig. 2(b) confines the mode from all four sides resulting in a minimum β _2,wg=-1.6 ps²/m, where a flatter waveguide (same 1 µm2 core area with a 5:1 width to height ratio) has weaker mode confinement, since the mode is affected predominantly from the top and bottom interfaces, resulting in a minimum β _2,wg=-0.8 ps²/m for the TM mode and -0.1 ps²/m for the TE mode. As well, if the cover index differs from the substrate index, the core shape can alter the “effective” Δn by emphasizing or de-emphasizing the role of the substrate in confining the mode. A tall, thin core emphasizes the role of the sidewalls; and a short, wide core emphasizes the bottom substrate interface. If the planar waveguide is not fully etched, the magnitude of the β _2,wg minimum can be controlled through the etch depth; a smaller etch depth results in less waveguide dispersion.

In Sections 4 and 5, we have focused on using multi-order dispersion engineering to optimize FWM performance; however, the same principle can be used to target other dispersive orders or characteristics for other processes. For supercontinuum generation, it may be beneficial to target a low β ₂ while minimizing β ₃ [5]. Or for cross-phase modulation, minimizing both β ₂ and β ₃ would result in a greatly broadened wavelength conversion bandwidth [21]. While low β ₃ has been realized in photonic crystal fibers [22, 23], it has also been shown in immersed tapered silica fibers [5] despite the greatly reduced flexibility of the waveguide geometry. The process of multi-order dispersion engineering demonstrates how this can be achieved in high-index materials as well. Since high-index materials tend to have a positive β _3,mat, a negative β _3,wg is needed to compensate. As a consequence, the dimension of the waveguide must be made smaller than min β _2,wg, where the slope of β _2,wg is negative, as shown in Fig. 2(a). Thus the transverse waveguide dimensions necessary to achieve low β ₃ is smaller than necessary for low β ₄.

7. Conclusion

Dispersion engineering is necessary for high-index waveguides, such as silicon, AlGaAs and chalcogenides, to achieve the low anomalous β ₂ needed for four-wave mixing at telecom wavelengths. However, the FWM performance depends strongly on β ₄; if a positive β ₄ can be achieved, as is found in silica fiber, the exponential gain bandwidth can be made much broader and flatter than the gain for a device with a negative β ₄. However, with the high-index waveguide materials currently in use, β ₂-engineered waveguides result in β ₄<0.

Conventional dispersion engineering uses control over the transverse waveguide dimensions as a means to achieve a target β ₂. In this paper, we proposed a method of multi-order dispersion engineering that varies the core-cladding index difference as well, to provide the additional degree of freedom required to simultaneously control β ₂ and β ₄. Using an As₂Se₃ fiber taper as an example, we demonstrated the effectiveness of this technique by achieving a much broader and flatter FWM gain spectrum with β ₂/β ₄-engineering than with conventional β ₂-engineering.

Appendix 1: Bandwidth optimization

The analytic arguments presented in Section 2 show that a positive β ₄ provides a broader exponential gain bandwidth; however a positive β ₄ also results in a flatter gain peak, which can be seen in Fig. 1(d).

From Eq. (3), the maximum gain occurs at the frequency when Δβ=-γP. If we define this frequency of maximum gain as ω_m, then flatness of the gain near its maximum is related to the rate of change in Δβ with respect to ω, evaluated at ω_m, or

(11)$$\frac{d}{d\omega }\Delta \beta {|}_{\omega ={\omega }_m}=\left[{\beta }_2+\frac{1}{6}{\beta }_4{\left({\omega }_m-{\omega }_p\right)}^2\right]\left({\omega }_m-{\omega }_p\right).$$

Therefore, at ω_m it is possible to achieve an optimized gain spectrum if the rate of change goes to zero, which occurs when

(12)$${\beta }_4=\frac{-6{\beta }_2}{{\left({\omega }_m-{\omega }_p\right)}^2}.$$

Thus, β ₄ must have the opposite sign to β ₂, and for exponential gain at small frequency offsets, β ₂ must be negative. If β ₂ and β ₄ have the same sign, the right-hand side of Eq. (11) increases with the magnitude of β ₄, resulting in narrower peaks in the gain spectrum.

Appendix 2: Estimate of the anomalous waveguide dispersion minimum

In Section 3, we showed that for a step-index waveguide, the waveguide contribution to β ₂ has one normal maximum and one anomalous minimum, as shown in Fig. 8(a) at V=1 and 2, respectively. In Section 4, we argued that this anomalous minimum can be used to compensate a normal material dispersion; thus the amplitude of this minimum determines where the ZDP occurs with respect to the higher-order waveguide dispersion curves. The following is an order-of-magnitude estimation of this minimum value in waveguide β ₂ with the goal of understanding which variables may be used to control the ZDP position.

A series of assumptions are used, shown in Fig. 8, to build this final estimation. These assumptions are centered on the typical mode behavior expected at a specific V-number, a dimensionless variable defined as

(13)$$V=\frac{\omega D}{c}{\left(n_{\mathrm{core}}^2-n_{\mathrm{clad}}^2\right)}^{\frac{1}{2}},$$

 

Fig. 8. Illustrations of the estimates used (solid green lines), to the generic waveguide dispersion curves (dashed red lines). (a) β ₂ as a function of V-number. (b) Estimating the β-curve as three lines. (c) β ₁ resulting from the estimate of β, indicating β _1,max. (d) Estimate of the β ₁, falling from its maximum value to its asymptotic value over a range of ΔV.

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where D is the typical transverse dimension of the waveguide [19]. For the duration of this appendix, β_i refers to waveguide dispersion only (i.e. material dispersion is not included).

As shown in Fig. 8(b), β can be estimated by three linear segments, transitioning from the two asymptotic limits at V=1 and V=2. This would result in the step-wise constant function for β ₁ in Fig. 8(c), thus the slope of the linear section between 1<V<2 is an estimate of the maximum value of β ₁.

To solve for the slope, we define ω, β and V at V=1, 2 as

(14)$$V_1=1,{\omega }_1=\frac{d\omega }{dV}{V}_1,\beta \left({\omega }_1\right)=\frac{n_{\mathrm{clad}}}{c}{\omega }_1=\frac{n_{\mathrm{clad}}}{c}\frac{d\omega }{dV}{V}_1$$
$$V_2=2,{\omega }_2=\frac{d\omega }{dV}{V}_2,\beta \left({\omega }_2\right)=\frac{n_{\mathrm{core}}}{c}{\omega }_2=\frac{n_{\mathrm{core}}}{c}\frac{d\omega }{dV}{V}_2.$$

Thus the maximum slope in β is

(15)$${\beta }_{l,max}\approx \frac{\beta \left({\omega }_2\right)-\beta \left({\omega }_1\right)}{{\omega }_2-{\omega }_1}=\frac{2n_{\mathrm{core}}-n_{\mathrm{clad}}}{c}.$$

From this maximum β _1,max, we assume that β ₁ drops to its asymptotic value over a range ΔV~1, as shown in Fig. 8(d), giving an estimate of the anomalous dispersion minimum:

(16)$${\beta }_{2,min}\approx \frac{{\Delta \beta }_1}{\Delta V}\frac{\mathrm{dV}}{d\omega }=-\frac{D}{c^2}{(\Delta n)}^{\frac{3}{2}}{\left(n_{\mathrm{core}}+n_{\mathrm{clad}}\right)}^{\frac{1}{2}}.$$

We take this anomalous dispersion minimum, occurring near V~2, to correspond to a ZDP at ω ₀, thus we can use Eq. (13) to substitute for D in Eq. (16) to get

(17)$${\beta }_{2,min}\approx -\frac{2}{c}\frac{\Delta n}{{\omega }_0}.$$

The following table compares the estimates above to calculated values for two cases: a 1 µm² square silicon waveguide embedded in silica, and a 1 µm wide As₂Se₃ fiber taper suspended in air.

Table 2. Comparison of estimated values to simulation at 1550 nm

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Overall, the estimates from Eq. (15) and (17) are within an order of magnitude, demonstrating the validity of the arguments used. However, it is the dependence of β _2,min on Δn that is of real importance to the method of multi-order dispersion engineering described in this paper.

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