## Abstract

We measure the degree of coherence of supercontinua generated in tapered fibers by subsequent fs pulses. By means of interference experiments we study its dependence on the input pulse duration and power. We also present numerical simulations that allow us to explain the experimental observations which show a decreasing degree of coherence with increasing input power. We attribute this loss of coherence to phase noise due to the cross-phase modulation by several solitons with randomly varying parameters due to quantum noise.

© 2007 Optical Society of America

## 1. Introduction

Supercontinua (SC) generated in microstructure fibers (MFs) as well in tapered fibers with nJ-pulses [1, 2] have encouraged tremendous research activities to investigate their origin and their characteristics in different ranges of fiber and pump parameters. As a new type of coherent white-light source they have found their way as a tool into many applications such as frequency metrology, optical coherence tomography, or absorption spectroscopy. The SC generated in MFs or tapered fibers in the anomalous dispersion range are connected with the splitting of the input pulse into several fundamental solitons, which emit phase-matched non-solitonic radiation [3, 4]. The threshold for this highly efficient mechanism is significantly lower than for self-phase modulation or any other known spectral broadening process. Therefore this process has attracted significant interest of many groups (for a recent review see [5]). For many applications the noise and coherence properties of this octave-spanning white light is of crucial importance. In particular, the spectral width of more than one octave in the visible and near-IR could be used for the generation of extremely short pulses and is able to support sub-cycle fs pulses. However, in order to achieve the required flat phase, pulse compression based on a pulse shaper is needed to adjust the phases of the different spectral components with respect to each other [6]. Therefore the pulse-to-pulse phase noise of the spectral components is a very critical issue. Supercontinua with small excess noise are also essential for optical frequency metrology [7], optical coherence tomography, and other applications. Recently, experimental studies and theoretical simulations have shown that the SC coherence is very sensitive to both the fundamental quantum noise [8, 9] and the shot-to-shot pump intensity fluctuations (technical noise) [10, 11, 12]. SC coherence has been measured by using radio frequency noise analysis [9, 10], by Young’s interference [13] between independently generated SC from two separated MFs [14], or by means of the delayed-pulse method [15]. These studies have shown that highly coherent SC can be obtained using pulses with durations of about 50 fs or less [14, 16], or for input wavelengths tuned deeper into the anomalous dispersion region [15]. In this work, we present experimental und numerical results for the phase noise characteristics of SC generated under various conditions. The *phase noise* characteristic of the SC are measured using Young interference between the SC of two delayed *subsequent pulses*. The main emphasis is put on the comparison of SC generated by pulses with different input durations in the fs regime and different input power. The experimental results and the numerical simulations suggest that the origin of the phase noise and its dependence on the fiber and pump parameters is closely related with
the fission of the input pulse into several fundamental solitons and its emission of non-solitonic radiation under the influence of the superposition of several solitons. The increased phase noise for longer pulses or high power and hence for the loss of coherence is interpreted as a result of cross-phase modulation by the superposition of several solitons with quantum-noise-induced irregular variations of the soliton parameters.

## 2. Experimental results

In our experiments, we have used flame-drawn tapered fibers [17] to provide a nonlinear medium that exhibits anomalous dispersion in the spectral region of our pump pulses. The drawn fibers consist of a taper region where the outer diameter decreases from 125 micrometers to a few micrometers over a distance of about 15 millimeters. It is followed by a waist region with a diameter adjustable from less than one to about three micrometers, followed by another taper region. The zero dispersion wavelength can be adjusted by choosing the appropriate waist diameter [2]. For the experiment pump pulses from a Ti:sapphire laser with a repetition rate of 80 MHz and a pulse duration of 148 fs at a center wavelength of 800 nm are coupled into a tapered fiber. A typical output spectrum is shown in Fig. 2(a). Longer pump pulses of about 400 fs are created by limiting the spectral width with a bandpass filter of 3 nm bandwidth.

To measure the pulse-to-pulse phase noise, the generated SC trains are sent into an asymmetric Michelson interferometer [15, 18], consisting of two unequal arms. One arm is longer by the exact distance between two subsequent pulses (see Fig. 1). At the output of the interferometer, the spectrally resolved interference pattern has been measured [see Fig. 2(b)]. The contrast of the interference patterns allows direct access to the visibility *V*. As the degradation of the contrast is not only affected by the phase noise but also by the amplitude noise, the latter has to be measured independently for different frequency bands by monitoring a train of pulses using a fast photodiode. A simple simulation of the dependence of the visibility on the amplitude noise allows to calculate the phase noise part separately as follows: We assume a perfectly stable phase for a certain frequency component ω, but we allow a Gaussian distributed amplitude noise of the e-field:

where 〈*P*〉 denotes the average power and Δ*P* the deviation from 〈*P*〉. We choose a Gaussian distribution due to the fact that it serves in the case of high photon numbers as an approximation for a Poisson distribution, which describes the photon statistic of coherent laser light. Since the detector integrates over a large number of subsequent pulses with individual amplitudes, the measured power reads as

Herer ρσ(Δ*P*) describes the Gaussian distribution with the standard deviation σ of the amplitude noise. σ is determined by evaluating the standard deviation of the normalized amplitudes of the electric field recorded by the photodiode.

For the visibility we have to consider the interference of infinite combinations of two pulses with a relative time delay τ [19]:

$$\phantom{\rule{1.3em}{0ex}}=\frac{{\left(\underset{-1}{\overset{\infty}{\int}}{\rho}_{\sigma}\left(x\right)\sqrt{1+x\mathit{dx}}\right)}^{2}}{\underset{-1}{\overset{\infty}{\int}}{\rho}_{\sigma}\left(x\right)\left(1+x\right)\mathit{dx}\cdot \underset{-1}{\overset{\infty}{\int}}{\rho}_{\sigma}\left(\mathit{x}\right)\mathit{dx}}\mathrm{cos}\mathrm{\omega \tau}\phantom{\rule{1.3em}{0ex}}$$

$$\phantom{\rule{1.3em}{0ex}}=V\left(\sigma \right)\mathrm{cos}\mathrm{\omega \tau}\phantom{\rule{.2em}{0ex}},$$

where *V*(σ) denotes the fraction in the previous expression, with *x*=Δ*P* for better readability. The numerically calculated decrease of the amplitude *V*(σ) of the visibility fringes given by Eq. 3 [see also Fig. 2(b)] in dependence on the width of the Gaussian distribution is shown in Fig. 3. It is remarkable that even a completely random amplitude of the electric field (rectangu-lar distribution with σ=1) leads to a decrease of the visibility of only 12 %.

With a measured average amplitude noise standard deviation of around 35 % we find a drop of the visibility of only 3.5 %, so that we conclude that the main contribution to the decrease of the visibility is actually *phase noise*.

Figure 4 shows the experimental results of the spectrally resolved phase noise for input pulse durations of 148 fs and 410 fs. The visibility is presented for different input peak powers. The results show that pumping with 148 fs pulses leads to a very good overall spectral coherence up to a peak power of about 6 kW. With further increase of the input power and hence further broadening of the spectrum, the overall coherence collapses until there is only some phase stability left in the vicinity of the pump wavelength. In contrast, the results for the 410 fs pumping show a different behavior. Interference fringes and hence a stable phase relationship of subsequent pulses can only be observed in the vicinity of the pump wavelengths. Despite a significant broadening of the spectrum, nearly no interference can be detected in all other parts of the spectrum. This behavior is in agreement with the results shown in [20] and will be explained by the numerical model and the interpretation presented in the chapters 3 and 4.

## 3. Numerical model and results

In order to achieve a deeper understanding of the experimental results and the coherence properties of the generated supercontinua, we have performed a numerical analysis of the spectral, temporal, and noise properties of the pulse propagating through the tapered fiber. The nonlinear propagation is governed by the effects of Kerr nonlinearity, Raman scattering, linear dispersion including higher-order terms, and higher-order nonlinear effects like self-steepening. In the limit of high photon number the quantum fluctuations of the field can be described by the Wigner quasi-probability representation (see e.g. [21] and references therein), where the evolution equations for quantum field operators are mapped onto stochastic partial differential equations which has up to an additional noise term the same form as the classical nonlinear evolution equations. In our description we employ the forward Maxwell equation as classical evolution equation which takes into account all of the above effects and does not rely on the slowly-varying-envelope approximation [3]:

In this equation, *E*(*z*,ω) is the field strength characterizing the evolution of the pulse during propagation along the *z* coordinate. The transverse distribution of the field is determined by the spatial structure of the fundamental mode, which satisfies a corresponding Helmholtz eigenvalue equation with the wavenumber β(ω), and *n _{g}* is the characteristic group refractive index. For tapered fibers, the transverse mode distribution and the wavenumber including both bulk and waveguide dispersion can be found by an analytical solution of the Helmholtz equation; the difference to standard fibers is here only in the smaller core radius and the large index contrast between the fused silica core and the air surrounding. The quantity

*P*(

_{NL}*z*,ω) is the Fourier transform of the nonlinear polarization

$$\phantom{\rule{2.9em}{0ex}}\text{}+f{\epsilon}_{0}{\chi}^{\left(3\right)}\frac{2\pi {T}_{R}{\tau}_{R}^{2}}{{T}_{R}^{2}+4{\pi}^{2}{\tau}_{R}^{2}}E\left(z,\tau \right){\int}_{-\infty}^{t}{E\left(z,t\right)}^{2}$$

$$\phantom{\rule{2.9em}{0ex}}\times \mathrm{exp}(-\left(t-\tau \right)/{\tau}_{R})\mathrm{sin}\left(2\pi \left(t-\tau \right)/{T}_{R}\right)\mathrm{d\tau}\phantom{\rule{.2em}{0ex}},$$

where *f* is the fraction of the Raman contribution to the instantaneous nonlinear polarization, χ^{(3)} is the nonlinear third-order susceptibility which is related to the nonlinear refractive index *n*
^{2} by *n*
^{2}=(3/4)χ^{(3)}/(*c*ε_{0}
*n*
^{2}) with *n* being the linear refractive index, τ* ^{R}* and

*T*are the Raman decay time and Raman period, correspondingly. The initial quantum state in the Wigner presentation described by Eq. 4 is a coherent state (Glauber state) and therefore quantum effects here arise only from the initial vacuum fluctuations. In addition both fiber loss and Raman gain leads to an additional noise term that contribute to quantum noise. Since the fiber loss is small and the Raman effect influences only the red-shifted part of the spectrum by the Raman self-frequency shift, the inhomogeneous noise term can be neglected in our analysis. Besides shot-to-shot noise, input pulse parameter variations (technical noise) can influence the coherence properties of the SC, but for the condition of our experiment this contribution is negligible. The effect of initial vacuum fluctuations is included in our simulations by adding to the input pulse shape

_{R}*E*(

_{in}*t*) a perturbation Δ

*E*(

*t*) which is determined by

where δ(∙) is the delta function, < ∙ > denotes the average over initial noise realizations, and *n*¯ = *E*
_{0}/(*h*¯ω_{0}) is the number of photons in the pulse, *E*
_{0} being the pulse energy. The quantity which characterizes the coherence of the supercontinuum is defined as

where the average in the numerator is taken over all non-identical pairs of noise realization, while in the denominator the average is taken over all noise realizations, and *L* is the length of the fiber. The quantity *g*(ω) directly corresponds to the visibility *g*=*V*=(*I*
_{max} - *I*
_{min})/(*I*
_{max} + *I*
_{min}) measured in the interference experiment. To characterize the coherence of the radiation, we used also the average coherence *g*¯ calculated by using the spectral power as weight. The propagation equation (4) was solved by the split-step Fourier method with nonlinear steps performed by the fourth-order Runge-Kutta method. To model the SC coherence of a pulse train as observed in the experiment, the numerical results were smoothed as functions of ω to reproduce experimental conditions. The values of the material parameters for fused-silica fibers are *n*
^{2}=3×10^{-16} cm^{2}/W, *f* = 0.18, *T _{R}* = 70 fs, and τ

_{R}=30 fs. For the calculation of wavenumber β(ω) we have used the Sellmeyer coefficients of fused silica [22], and a fiber diameter of 2.1 μm. These values yield a zero-dispersion wavelength of 730 nm and a GVD-parameter at the input wavelength of 775 nm of -11.2 fs

^{2}/mm.

In Fig. 5 the results for the spectral broadening and the coherence properties are presented for a 148-fs input pulses for two different input peak power. As can be seen in Fig. 5(a), for the pulse with input peak power of 2.5 kW, the spectrum has the width of 470 nm with a coherence close to unity, and an average coherence *g*¯ of 0.93. For the higher power of 5.9 kW and the same pulse duration [Fig. 5(b)], the average coherence drops to 0.15 which is within the uncertainty due to the finite number of the noise realizations. The coherence has a peak around the input wavelength but is quite low in other parts of the spectrum. Note that coherence is observed for 2.5 kW and disappears for 5.9 kW input power except for the peak at the input wavelength, in good agreement with the experimental results in Fig. 4. The widths of the spectrum correspond also to the experimental values in both cases. Note that we simulated pulse propagation over a shorter distance than the experimental length of the fiber of around 30 cm, however this is justified by the saturation of the spectral broadening and coherence after about 4-cm propagation.

In Fig. 6 the coherence properties of the SC and the spectrum of the longer 410-fs input pulse are illustrated. The supercontinuum starts to arise for input peak powers around 1 kW [Fig. 6(b)], and reaches the width of one octave at 5.9 kW [ Fig. 6(c)]. The behavior of the coherence, however, differs from the case of shorter 148-fs pulses. For all input powers with relatively broad spectra the calculated coherence was low; in contrast to the coherence behavior in Fig. 5(a) for the 148 fs pulse with 2.5 kW. For the case of longer input pulses, the deterioration of coherence starts for the same powers as spectral broadening, yielding low g of only 0.4 for 0.9 kW input power as shown in Fig. 6(b). Also note that the spectral width is larger than for the shorter 148-fs pulse with the same intensity. The comparison of these numerical findings with the experimental data reveals reasonable general agreement, especially in the coherence curves, despite some disagreement in the threshold power for SC generation.

In order to achieve clearer understanding of the coherence degradation mechanism, we have additionally studied the case of 15-fs input pulses which is not included in the experimental study. The corresponding spectral and coherence properties are illustrated in Fig. 7, where one can see very high (close to unity) average coherence and spectrum of approximately 300 nm width. Compared to the long pulse with the same input peak power [Fig. 6(c)] which yields incoherent SC, the short pulse leads to a completely coherent broad spectrum.

## 4. Discussion

The main noise source responsible for the destruction of the coherence in the SC for the current conditions is the fundamental quantum noise arising in the Wigner representation as input quantum shot noise. Besides, technical noise due to random variations of the input pulse parameters and the noise induced by the Raman gain could influence the coherence properties of the SC. In our experiments, the random variations of the input power in the train were kept below 1%. We have performed simulations with random 1% variations of the peak power instead of the input quantum shot noise and obtained coherence degradations smaller than those shown in Fig. 5 and 6. The spontaneous Raman noise term has been previously shown to yield noise two levels of magnitude below the contribution from quantum shot noise [9]. Thus these two noise sources play only a minor role for the current conditions. However, the physical mechanism of such strong coherence decrease for certain conditions still needs a qualitative explanation. In principle one should expect a connection between the mechanism for SC generation due to soliton emission and the degree of coherence of the SC. The strong dependence of the coherence on the pulse duration demonstrating increasing coherence with decreasing pulse duration suggest such correlation between the coherence and the temporal shape of the pulse during propagation. The broadening mechanism here is mainly determined by the splitting of the input pulse into several fundamental solitons which arise due to pumping in the anomalous-dispersion region, and emission of non-solitonic radiation (NSR) by these solitons [3]. The number of solitons grows with the input pulse duration, as illustrated by Fig. 8, demonstrating exactly one soliton for a 15-fs input pulse and several solitons for a 410-fs input pulse with the same input peak power. The emission of radiation *f _{i}*(

*z*,

*t*) by a certain soliton with amplitude

*A*after the fission of the input pulse can be described by the soliton perturbation theory. Using the ansatz

_{i}*E*(

*z*,

*t*)=∑

*(*

_{i}A_{i}*z*,

*t*)

*e*

*+∑*

^{iωit-ikiz}*i fi*(

*z*,

*t*)e

^{i$\stackrel{\xb4}{\omega}$it-iḱiz}the non-solitonic component

*fi*(

*z*,

*t*) is described by the following linear propagation equation:

$$\phantom{\rule{2.2em}{0ex}}+\mathit{i\gamma}{f}_{i}\left(z,t\right){\mid \underset{j}{\Sigma}{A}_{j}\left(z,t\right){e}^{i{\omega}_{i}t-i{k}_{i}z}\mid}^{2}$$

$$\phantom{\rule{2.2em}{0ex}}+({\mathit{i\beta}}^{\left(3\right)}{\partial}^{3}/{{\partial t}^{3}-\beta}^{\left(4\right)}{\partial}^{4}/{\partial t}^{4}+...){A}_{i}\mathit{\left(}\mathit{z},t\right)\phantom{\rule{.2em}{0ex}},$$

where the operator Δβ∼ in the first term on the RHS corresponds in the frequency representation to the phase mismatch between the non-solitonic radiation and the solitons (see Ref. [4]). The generation of SC is described in Eq. 8 by the inhomogeneous third term on the RHS as a source term arising from the soliton with number i due to third-order dispersion and higher-order effects. Since each soliton with the amplitude *A _{i}*(

*z*,

*t*) has different central frequency, the phase-matched non-solitonic radiation

*f*(

_{i}*z*,

*t*) is emitted at different frequency intervals. The second term on the RHS in Eq. 8 describes the effect of cross-phase modulation of the non-solitonic radiation by the solitons. Here the superposition and interference of all solitons

*A*(

_{i}*z*,

*t*) play a crucial role.

*The input vacuum fluctuations lead to variations of the soliton parameters sensitively influencing the mutual phase of the solitons and their interference.*For longer input pulses with the same intensity, the number of solitons is higher and therefore this interference has a stronger effect due to a larger number of terms in the sum, leading to a stronger coherence degradation. The findings of Ref. [15], which show improving coherence for increasing anomalous dispersion, can be explained by the same mechanism: for fixed pulse parameters but larger anomalous GVD the coherence degradation is reduced due to a lower number of solitons. Note that other authors have given an explanation for the coherence degradation different from that given here. Dudley

*et al*. [5] carefully compared a numerically simulated evolution of the spectrum and the coherence for a shorter and a longer pulse. The interpretation of reduced coherence for longer pulses given in [5] is that the modulation-instability-amplified noise background becomes a dominant feature in propagation dynamics. However, for parameters of Fig. 6(c), at a position where the pulse is split, the incoherent sidebands of modulation instability are much lower than the highly-coherent central peak, even for the 410-fs pulses. Therefore coherent solitons are generated from the split pulse with high coherence, and an explanation is necessary how the emission of dispersive waves by these solitons becomes incoherent. A further investigation of the problem of coherence degradation is subject of current research and will be published elsewhere [23].

## 5. Conclusion

In summary, interference experiments of subsequent pulses yield spectra and spectral coherence of the supercontinua generated in tapered fibers in dependence on the pump pulse duration and peak power. We demonstrated improved coherence properties of the supercontinua when using shorter input pulses. We have numerically studied the coherence and spectral properties of the SC due to soliton emission and found good agreement with the experiments. In our explanation, the coherence degradation is related to the noise-sensitive soliton dynamics as follows: Phase-sensitive superposition of several fundamental solitons induces cross-phase modulation of the SC. Therefore longer input pulses split to a higher number of solitons, which enhance the fluctuations of the cross-phase modulation term and reduces the coherence.

The authors would like to thank J. Kuhl for invaluable advice. We acknowledge support from DFG (FOR 557), BMBF (13N8340) and DFG Focus program ”Photonic crystals” (SPP 1113).

## References and links

**1. **J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm, ” Opt. Lett . **25**, 25–27(2000). [CrossRef]

**2. **T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers, ” Opt. Lett . **25**, 1415–1417 (2000). [CrossRef]

**3. **A. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers, ” Phys. Rev. Lett . **87**, 203901 (2001). [CrossRef] [PubMed]

**4. **J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. S. J. Russell, and G. Korn, “Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers, ” Phys. Rev. Lett . **88**, 173901 (2002). [CrossRef] [PubMed]

**5. **J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fibers, ” Rev. Mod. Phys . **78**, 1135–1184 (2006). [CrossRef]

**6. **B. Schenkel, R. Paschotta, and U. Keller, “Pulse compression with supercontinuum generation in microstructure fibers, ” J. Opt. Soc. Am. B **22**, 687–693 (2005). [CrossRef]

**7. **R. Holzwarth, Th. Udem, T. W. Haänsch, J. C. Knight, W. J. Wadsworth, and p. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy, ” Phys. Rev. Lett . **85**, 2264–2267 (2000). [CrossRef] [PubMed]

**8. **J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers, ” Opt. Lett . **27**, 1180–1182 (2002). [CrossRef]

**9. **K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber, ” Phys. Rev. Lett . **90**, 113904 (2003). [CrossRef] [PubMed]

**10. **T. M. Fortier, Ye J., S. T. Cundiff, and R. S. Windeler, “Nonlinear phase noise generated in air-silica microstructure fiber and its effect on carrier-envelope phase, ” Opt. Lett . **27**, 445–447 (2002). [CrossRef]

**11. **N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, “Noise amplification during supercontinuum generation in microstructure fiber, ” Opt. Lett . **28**, 944–946 (2003). [CrossRef] [PubMed]

**12. **A. L. Gaeta, “Nonlinear propagation and continuum generation in microstructured optical fibers, ” Opt. Lett . **27**, 924–926 (2002). [CrossRef]

**13. **M. Bellini and T.W. Haänsch, “Phase-locked white-light continuum pulses: toward a universal optical frequency-comb synthesizer, ” Opt. Lett . **25**, 1049–1051 (2000). [CrossRef]

**14. **X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, and R. S. Windeler, “Experimental studies of the coherence of microstructure-fiber supercontinuum,” Opt. Express **11**2697–2703 (2003). [CrossRef] [PubMed]

**15. **F. Lu and W. H. Knox, “Generation of a broadband continuum with high spectral coherence in tapered single-mode optical fibers, ” Opt. Express **12**, 347–353 (2004). [CrossRef] [PubMed]

**16. **J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, and S. T. Cundiff, “Excess noise generation during spectral broadening in a microstructured fiber, ” Appl. Phys. B **77**, 279–284 (2003). [CrossRef]

**17. **J. Teipel, K. Franke, D. Türke, F. Warken, D. Meiser, M. Leuschner, and H. Giessen, “Characteristics of supercontin-uum generationin tapered fibers using femtosecond laser pulses, ” Appl. Phys. B **77**, 245–251 (2003). [CrossRef]

**18. **J. Stenger and H. R. Telle, “Kerr-lens mode-locked lasers for optical frequency measurements, ” in Laser Frequency Stabilization, Standards, Measurement, and Applications, John L. Hall and Jun Ye, eds., Proc. SPIE **4269**, 72–76 (2001). [CrossRef]

**19. **L. Mandel and E. Wolf, *Optical coherence and quantum optics* (Cambridge University Press, Cambridge, 1995).

**20. **J. W. Nicholson and M. F. Yan, “Cross-coherence measurements of supercontinua generated in highly-nonlinear, dispersion shifted fiber at 1550 nm, ” Opt. Express **12**, 679–688 (2004). [CrossRef] [PubMed]

**21. **P. D. Drummond and J. F. Corney, “Quantum noise in optical fibers. I. Stochastic equations, ” J. Opt. Soc. Am. B **18**, 139–152 (2001). [CrossRef]

**22. **M. Bass (ed.), *Handbook of optics* (McGraw-Hill, New York, 1995).

**23. **A. Husakou and J. Herrmann, in preparation.