## Abstract

The effect of dispersion fluctuations on the conversion efficiency of large frequency shift parametric sidebands is studied by numerical simulation and experiment. Numerical results based on periodic and random dispersion models are used to fit the experimental results. The fitting parameters provide a measure of the uniformity of the photonic crystal fiber used in the experiment. This allows us to place limits on the required uniformity of a photonic crystal fiber for strong frequency conversion.

©2006 Optical Society of America

## 1. Introduction

It has long been known that phase-matching around the zero-dispersion wavelength (ZDW) of an optical fiber is a simple way to generate large frequency shift optical parametric sidebands [1]. More recently this effect was shown to be a manifestation (in the normal dispersion regime), of what is called modulation instability when it occurs in the anomalous dispersion regime [2] and it has been used to demonstrate widely tunable narrowband optical parametric amplification when a strong pump propagates near the ZDW of a highly nonlinear photonic crystal fiber (PCF) [3]. Similar results have also been obtained around the ZDW of a standard dispersion-shifted fiber [4, 5]. This wide tuning range is made possible by the dispersion profile around the fiber’s ZDW. Experimentally this effect has been exploited to demonstrate a widely tunable pulsed optical source with over 250 THz of continuous wavelength tunability centered around 650 nm. The theoretical limit for the tuning range of the source is over 500 THz which, if realizable, would be of considerable interest. Two effects limit the experimentally observed conversion efficiency and tuning range, namely the walk-off between the pump pulse and the optical parametric sidebands due to their different group velocities, and the dispersion fluctuations of the fiber along its length. The first of these effects can be overcome by using lower power CW (or long pulse quasi-CW) pumps [6] and hence requires the use of longer lengths of PCF; the second is more fundamental and is the subject of this paper.

The effect of dispersion fluctuations on optical parametric amplification has been widely studied in the literature [7–15]. Early papers investigated the effect of periodic fluctuations and were able to derive analytic expressions for the small signal parametric gain [7–12]. Later studies investigated the effect of randomly varying dispersion fluctuations and used numerical simulations to obtain values for the mean and variance of the parametric gain [12–15]. In this paper we analyze the effect of both periodic and random dispersion fluctuations on the conversion efficiency of large frequency shift parametric sidebands. We are able to demonstrate that the sensitivity of the sidebands to dispersion fluctuations increases with increasing sideband frequency shift. Comparing the results of these simulations to the experimentally measured sideband conversion efficiency enables us to estimate the uniformity of the fiber used. The simulations also allow us to determine the fiber uniformity necessary to exploit the full tuning range.

## 2. Parametric gain in fibers with fluctuating dispersion

The highly nonlinear, large air filling fraction PCFs used to generate large frequency shift parametric sidebands are typically strongly birefringent (Δ*n*~10^{-4}). As a result when light is

launched polarized parallel to one of the fiber’s principal axes, the PCF is well modeled as a single polarization optical fiber with an arbitrary dispersion and a standard Kerr nonlinearity. Typically these PCFs also have a non-negligible loss which must be accounted for. These considerations allow us to write the coupled mode equations describing the nonlinear interaction between the pump wave (amplitude *A*
* _{p}*) and the two sidebands (amplitudes

*A*

*and*

_{a}*A*

*) symmetrically detuned by a frequency Ω from the pump as [16]*

_{s}where *γ* is the nonlinear interaction coefficient and α is the fiber attenuation coefficient. For simplicity we assume the same attenuation coefficient for each of the three waves. *ϕ* is the phase of the parametric process given by

where *ϕ*
_{a,s,p} are the phases of the anti-Stokes, Stokes and pump waves, respectively. Δ*β*
* _{L}* is the linear wavevector mismatch and varies along the length of the fiber as the dispersion varies,

A Taylor expansion of the wavevector, *β*, around the pump frequency allows us to rewrite Eq. (6) as a sum of the even powers of dispersion,

Close to the zero dispersion wavelength the first two terms of this expansion (*β*
_{2} and *β*
_{4}) are sufficient to accurately predict the sideband’s frequency shift [4, 5]. However to correctly predict the large frequency shifts dealt with in this paper we find it necessary to include higher orders of dispersion. This is most easily done by computing *β* directly from a model (discussed in the next section) and then calculating the linear wavevector mismatch using Eq. (6).

We wish to solve the above set of equations to obtain an expression for the small signal gain of equal-amplitude sidebands. Under these conditions Eqs. (1–4) reduce to

where *A* is the sideband amplitude and *P* is the pump power at the beginning of the fiber. Equation (8) can be directly integrated to obtain the sideband’s small signal intensity gain per unit length for a fiber of length *L*,

This expression depends upon the evolution of *ϕ* which for an arbitrary dispersion has no analytic solution. Nonetheless Eqs. (9) and (10) give us a simple method to numerically calculate the parametric gain for an optical fiber with an arbitrary dispersion fluctuation. From Eq. (10) it is then simple to calculate the relative small signal conversion efficiency of a sideband with frequency shift Ω compared to a zero frequency shift sideband as

This calculated value can then be compared to the experimentally measured conversion efficiency and used to estimate the uniformity of the fiber.

Although the full problem as stated above requires a numerical solution, considerable insight can be obtained if we assume that the attenuation of the fiber may be neglected, and that the magnitude of the wavevector mismatch |Δ*β _{L}*(Ω,

*z*)| is much larger than

*γP*. Under these assumptions the evolution of the parametric phase becomes linear,

and the small signal gain of the sidebands can be written as

For a perfectly uniform fiber, the maximum sideband gain occurs at the frequency shift corresponding to Δ*β _{L}*=0, with an initial phase of

*φ*(0)=

*π*/2. Under these conditions, the phase does not evolve and the parametric sidebands experience a constant small signal gain of 2

*γP*. For a fiber with fluctuating dispersion, the maximum sideband gain occurs at the frequency shift where the integral of Δ

*β*is zero [9]. In this case the phase does not remain constant at π/2 and so the presence of the dispersion fluctuation lowers the sideband gain. Furthermore, for a given dispersion fluctuation, increasing the frequency shift of the sidebands results in an increase of |Δ

_{L}*β*(Ω,

_{L}*z*)|, and hence a decrease in parametric gain. This can be seen by considering the first term of the expansion in Eq. (7). Equations (12) and (13) also allow us to identify a figure of merit for the parametric gain for a fiber with a periodic dispersion fluctuation, namely the product of the fluctuation amplitude and the fluctuation length. This confirms the previously identified effect that parametric gain is more sensitive to dispersion fluctuations with long rather than short fluctuation periods [14].

## 3. Results

The PCF discussed in this paper was drawn using a commercial draw tower. The fluctuations in the dispersion of the PCF along its length are a result of the small fluctuations in the fiber’s core diameter, hole size, and hole spacing [17]. In the analysis which follows we consider only the effects of core diameter fluctuations on the dispersion as these are the simplest for us to model. The fiber’s core diameter can be estimated using a scanning electron microscope (SEM) to view its end face, but the accuracy of this measurement is only ±2%. In addition, the profiling of the core diameter fluctuations along the length of a fiber using an SEM is a destructive measurement. Several non-destructive measurements of fiber dispersion fluctuations exist, however none has the required combination of resolution and accuracy (better than 1 m and ±0.1%, respectively) [18]. For this reason we must resort to indirect measurements. By modeling the fiber’s dispersion fluctuations as either periodic or random, we are able to estimate the uniformity of the PCF by comparing the experimentally measured and numerically calculated sideband conversion efficiency as a function of frequency shift. This technique is similar to previous methods which use a tunable pump and seed to map the parametric gain of the fiber about its zero dispersion wavelength and infer from this the dispersion profile of fiber [19].

The experimental results we wish to analyze for the effect of dispersion fluctuations were previously presented in Ref. [20]. The fiber used was a 43 m long, highly nonlinear, large air filling fraction PCF. An SEM image of this fiber is shown in Fig. 1(a) of Ref. [20]. The loss of the fiber was measured by the cut-back technique and found to be 90 dB/km at 647 nm, which corresponds to 40% transmission for a 43 m fiber. The nonlinear interaction coefficient of the fiber was estimated to be 140 W^{-1}km^{-1} at 650 nm. The dispersion of the PCF’s high group-index mode has been calculated from an analysis of the sideband frequency shifts and is shown in the inset to Fig. 1 [20]. The small core diameter of approximately 1.6 *µ*m, and the large index step (this PCF has a cladding air filling fraction of 88%) produce a large waveguide dispersion which shifts the ZDW to 671 nm. We find that this dispersion is well modeled by that of a step-index fiber with a core diameter set to the effective core diameter of the PCF, and a cladding index set to the mean refractive index of the PCF’s cladding [20]. This simple model can be used to calculate the dispersion of the fiber analytically and allows us to relate fluctuations in the core diameter to fluctuations in the dispersion.

The pump laser used in this experiment was a cavity-dumped DCM dye laser tunable between 630 and 690 nm. The peak power of the pump coupled into the fiber was 3.2 W, with a pulse width of 13 ns. This pulse width is long enough so that the walk-off between the parametric sidebands and the pump is negligible [6], and the pulses can be considered as quasi-CW. In Fig. 1 we plot the theoretical phase-matching curve of the high group-index mode (solid line), superimposed with the experimentally measured sideband frequency shifts (circles). Figure 1 clearly shows the large potential tuning range of the process. Tuning the pump wave-length from 670 to 640 nm results in a theoretical sideband tuning range of 420 to 1350 nm (490 THz). Experimentally we find the sideband tuning range is limited to between 550 and 845 nm (190 THz). With the pump polarized parallel to the low rather than the high groupindex mode we measure an almost identical tuning curve but with the pump wavelength displaced 2 nm to the red due to the slightly different dispersion of the two axes.

In Fig. 2 we plot experimentally measured spectra at four different pump wavelengths when the pump is polarized parallel to the high group-index mode. These results show that the sideband conversion efficiency and the bandwidth of the sidebands decrease with increasing sideband frequency shift. In the analyses which follow we will show that the reduced conversion efficiency can be explained by the effect of dispersion fluctuations on the parametric gain, while the reduced bandwidth of the sidebands has been discussed previously [6].

## 3.1. Periodic dispersion fluctuations

We restrict our analysis to a simple stepwise periodic fluctuation in the core diameter with period *dL* and percentage amplitude fluctuation±*da*. For the PCF discussed above, a ±1% fluctuation in the core diameter corresponds to a ±0.7 ps^{2}/km fluctuation in the dispersion (±2 nm in the zero dispersion wavelength). In Fig. 3 we plot the experimentally observed relative conversion efficiency of the anti-Stokes (frequency up-shifted) and Stokes (frequency down-shifted) sidebands as a function of frequency shift. The data obtained with the pump polarized parallel to the high group-index mode, and those obtained with the pump polarized parallel to the low group-index mode, show the same characteristic fall-off in conversion efficiency as the sideband frequency shift increases. This suggests that the core diameter fluctuations of the two axes are of the same order of magnitude, as might be expected. Fig. 3 also shows the effect of stimulated Raman scattering (SRS) on the sidebands. Below 20 THz SRS is responsible for an amplification of the Stokes sideband and an attenuation of the anti-Stokes sideband. Above 20 THz the Raman gain drops rapidly and the effect of SRS is negligible. This means our simple theory presented in Section 2 should be valid for sideband frequency shifts above 20 THz.

Using Eq. (11) and the full numerical theory presented in Section 2, we can calculate the theoretical relative conversion efficiency of the sidebands. The fiber parameters used for this calculation were those given at the start of this section. For these parameters the magnitude of the linear wavevector mismatch fluctuations at the phase-matched frequency, |Δ*β _{L}*(Ω

_{pm},

*z*)|, is indeed much larger than the nonlinear term,

*γP*, so the effect of the periodic fluctuations is well described by the proposed figure of merit, i.e. the product of the fluctuation magnitude and the fluctuation period. The solid line superimposed on Fig. 3 is the calculated relative conversion efficiency for a stepwise periodic fluctuation with

*da*·

*dL*=±0.022%·m. The agreement between the modeled and measured conversion efficiencies is very good, except at low frequencies where the effects of Raman gain are visible on the experimentally measured points. For the product

*da*·

*dL*=±0.022%·m, we find the numerically calculated relative conversion efficiency changes negligibly as the fluctuation period

*dL*is varied from 1 cm to 5 m. This confirms the validity of our figure of merit. For periods longer than 5 m, the partial periods in the 43 m fiber start to have a noticeable effect on the calculated conversion efficiency. For periods shorter than 1 cm, the magnitude of the core diameter fluctuation becomes so large that the relationship between the core diameter fluctuation and the resultant dispersion fluctuation is no longer linear. To demonstrate the sensitivity of the conversion efficiency to the product

*da*·

*dL*, we superimpose on Fig. 3 the calculated relative conversion efficiencies obtained with

*da*·

*dL*=±0.011%·m and

*da*·

*dL*=±0.044%·m. These superimposed traces show that the fall off in parametric gain as a function of frequency shift is a strong function of the product

*da*·

*dL*.

In Fig. 4 we plot Ω_{3dB}, the sideband frequency shift at which the relative conversion efficiency falls 3 dB below that of the zero frequency shift sideband, as a function of *da*·*dL*. For the PCF used in this paper the fitted fluctuation product is *da*·*dL*=±0.022%·mwhich gives an Ω_{3dB} bandwidth of approximately 47 THz. Figure 4 shows the Ω_{3dB} bandwidth decreases as the magnitude of the dispersion fluctuations increases, and allows us to place an upper bound on the magnitude of the dispersion fluctuations that can be tolerated for a desired Ω_{3dB} bandwidth.

## 3.2. Random dispersion fluctuations

We now consider the effect of random fluctuations in the fiber’s dispersion on the conversion efficiency of large frequency shift parametric sidebands. We choose to study the piecewise constant stochastic model (model I) of Ref. [14]. In this model the fiber is made up of piecewise constant sections of fiber. Each section has a core diameter that is a normally distributed random variable with mean *α*
_{0}, standard deviation σ* _{a}*, and a length given by

where *U* is a uniformly distributed random variable with a range from 0 to 1, and *L*
* _{c}* is the correlation length, the typical length scale of the fluctuations. This gives the autocorrelation function of the core diameter fluctuations the physically reasonable form of

In Fig. 5 we plot an example of the core diameter fluctuations given by this model for a 43 m fiber with *L*
* _{c}*=1 m.

For a random fluctuation, the figure of merit which we were able to define for a periodic fluctuation no longer applies. Nonetheless, it is still true that long length scale fluctuations will have a stronger effect on the parametric gain than short ones. For a random fluctuation the largest length scale fluctuation will typically be of the order of the length of the fiber. This implies that the longer the fiber length, the more sensitive the parametric gain is to dispersion fluctuations. In Fig. 6 we plot the results of the random core diameter fluctuation simulation on top of the experimentally measured relative conversion efficiencies. This is done for an ensemble of 400 fibers with the mean and standard deviation of the conversion efficiencies shown as squares and error bars, respectively. The correlation length was set to 1 m, which requires a value of 0.0031% for the standard deviation of the core diameter random variable to give the best fit to the experimental conversion efficiency. Figure 6 shows the same general trend previously observed for periodic fluctuations, with increasing sensitivity to core diameter fluctuations as the frequency shift of the sidebands increases. The simulations also show, as might be expected, that increasing either *L _{c}* or σ

_{a}results in decreased conversion efficiency for the large frequency shift sidebands. While Fig. 6 shows the best fit for a correlation length of 1 m, it is possible to adjust σ

*to obtain an equally good fit to the data for other correlation lengths. In Fig. 7 we plot this ‘best fit’ curve in the (*

_{a}*L*,σ

_{c}*) parameter space. It shows that there is a clear relationship between the correlation length and the standard deviation of the core diameter fluctuation, with an increase in*

_{a}*L*requiring a corresponding decrease in σ

_{c}*to maintain a good fit to the experimentally measured conversion efficiency versus frequency shift curve.*

_{a}## 4. Conclusion

In this paper we have studied the effects of dispersion fluctuations on widely tunable optical parametric amplification in a PCF. To model the parametric gain in these fibers more accurately, we have extended the standard analysis of the problem to include the effects of fiber loss and higher orders of dispersion. We show that it is the evolution of the parametric phase which is the key parameter that governs the parametric gain in fibers with non-uniform dispersion, and demonstrate that the characteristic fall-off of the sideband conversion efficiency at large frequency shifts is a result of dispersion fluctuations in the fiber. Good agreement is obtained between the experimentally measured relative conversion efficiency and the two fluctuation models considered. Both models suggest that the core diameter of the PCF used is very uniform, with *da*·*dL*=±0.022%·mfor the periodic model, and σ * _{a}*~0.001-0.005% of the mean core diameter for the random model. As the largest fluctuation period of the random model is in the order of the fiber length, one can see that the fluctuation amplitudes predicted by the two models are of the same order of magnitude, and conclude that the models presented are indeed consistent. This work suggests that efficient large frequency shift (~100 THz) parametric amplification in long (>50 m) fibers would require a fiber uniformity two to three times higher than the above values. Further progress in fabrication technology will be needed to achieve this goal.

## Acknowledgements

The fibers used in these experiments were fabricated at and supplied by the University of Bath in the U.K. We would like to thank Nicolas Joly, William Wadsworth and Jonathan Knight for helpful discussions concerning these fibers.

## References and links

**1. **C. Lin, W. A. Reed, A. D. Pearson, and H. T. Sbang, “Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing,” Opt. Lett. **6**, 493–5 (1981). [CrossRef] [PubMed]

**2. **J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. **28**, 2225–7 (2003). [CrossRef] [PubMed]

**3. **A. Y. H. Chen, G. K. L. Wong, S. G. Murdoch, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Widely tunable optical parametric generation in a photonic crystal fiber,” Opt. Lett. **30**, 762–764 (2005). [CrossRef] [PubMed]

**4. **S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. **226**, 415–22 (2003). [CrossRef]

**5. **M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Wideband tuning of the gain spectra of one-pump fiber optical parametric amplifiers,” J. Sel. Top. Quantum. Electron. **10**, 1133–1141 (2004). [CrossRef]

**6. **G. K. L. Wong, A. Y. H. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Continuous-wave tunable optical parametric generation in a photonic-crystal fiber,” J. Opt. Soc. Am. B **22**, 2505–2511 (2005). [CrossRef]

**7. **N. J. Smith and N. J. Doran, “Modulational instabilities in fibers with periodic dispersion management,” Opt. Lett. **21**, 570–2 (1996). [CrossRef] [PubMed]

**8. **J. C. Bronski and J. N. Kutz, “Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management,” Opt. Lett. **21**, 937–9 (1996). [CrossRef] [PubMed]

**9. **S. G. Murdoch, R. Leonhardt, J. D. Harvey, and T. A. B. Kennedy, “Quasi-phase matching in an optical fiber with periodic birefringence,” J. Opt. Soc. Am. B **14**, 1816–1822 (1997). [CrossRef]

**10. **E. Ciaramella and M. Tamburrini, “Modulation instability in long amplified links with strong dispersion compensation,” IEEE Photon. Technol. Lett. **11**, 1608–1610 (1999). [CrossRef]

**11. **A. Kumar, A. Labruyere, and P. Tchofo Dinda, “Modulational instability in fiber systems with periodic loss compensation and dispersion management,” Opt. Commun. **219**, 221–239 (2003). [CrossRef]

**12. **F. K. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A **220**, 213–218 (1996). [CrossRef]

**13. **M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B **15**, 2269–2275 (1998). [CrossRef]

**14. **M. Farahmand and M. de Sterke, “Parametric amplification in presence of dispersion fluctuations,” Opt. Express **12**, 136–142 (2004). [CrossRef] [PubMed]

**15. **B. Kibler, C. Billet, J. M. Dudley, R. S. Windeler, and G. Millot, “Effects of structural irregularities on modulational instability phase matching in photonic crystal fibers,” Opt. Lett. **29**, 1903–1905 (2004). [CrossRef] [PubMed]

**16. **G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B **8**, 824–838 (1991). [CrossRef]

**17. **K. L. Reichenbach and C. Xu, “The effects of randomly occurring nonuniformities on propagation in photonic crystal fibers,” Opt. Express **13**, 2799–2807 (2005). [CrossRef] [PubMed]

**18. **D. Derickson, *Fiber Optic Test and Measurement*, 1st ed. (Prentice Hall, Upper Saddle River, NJ, 1998).

**19. **A. Mussot, E. Lantz, A. Durecu-Legrand, C. Simonneau, D. Bayart, T. Sylvestre, and H. Maillotte, “Zero-dispersion wavelength mapping in short single-mode optical fibers using parametric amplification,” IEEE Photon. Technol. Lett. **18**, 22–4 (2006). [CrossRef]

**20. **G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Y. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express **13**, 8662–8670 (2005). [CrossRef] [PubMed]