The generation of dispersive waves from a photonic crystal fiber by higher-order mode excitation

doi:10.1364/OE.18.005338

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The generation of dispersive waves from a photonic crystal fiber by higher-order mode excitation

Naoki Karasawa and Kazuhiro Tada »View Author Affiliations

Optics Express, Vol. 18, Issue 5, pp. 5338-5343 (2010)
http://dx.doi.org/10.1364/OE.18.005338

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– Abstract
1. Introduction
2. Methods
3. Results and discussions
– References and links

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Abstract

Dispersive waves were generated from a photonic crystal fiber by higher-order mode excitation and the dependence of their wavelengths on polarization was measured. The dispersion properties of various spatial modes with different symmetry numbers were calculated theoretically and four combinations of linearly-polarizing higher-order modes were identified. The phase-matching conditions of dispersive waves for higher-order modes were calculated and it was found that the wavelengths of dispersive waves with identical spatial modes depended on polarization directions. The dependence measured experimentally agreed well with results obtained by theoretical calculations.

1. Introduction

The generation of a broadband light pulse in a photonic crystal fiber (PCF) using an optical pulse from a Ti:sapphire laser was reported in 2000 [1

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

] and it has been studied extensively [2

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

]. In a PCF, the group velocity dispersion (GVD) can be made negative at the wavelength of a Ti:sapphire laser (~ 810 nm). When a femtosecond pulse is propagated inside the PCF, soliton pulses are created and dispersive waves are emitted from these soliton pulses [3

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]

]. When a Ti:sapphire laser is used, the wavelengths of dispersive waves are in the visible range and these can be used as ultrashort light pulses in various applications including nonlinear spectroscopy and microscopy. The radiation of a dispersive wave can be regarded as Cherenkov radiation emitted by a soliton pulse [4

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

]. The generation of intense dispersive waves has been demonstrated using a fundamental mode of a PCF [5

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express 12, 124–135 (2004). [CrossRef] [PubMed]

]. However, the core size of a PCF had to be smaller than about 2 μm when the pump wavelength was about 810 nm. In this study, a PCF with a relatively large core size of about 3 μm is used and the dispersive waves emitted by higher-order mode excitation are studied. In particular, the dependence of the wavelengths of dispersive waves on polarization is studied when LP₁₁ modes are excited. Dispersion properties and the phase matching conditions of dispersive waves are calculated theoretically and experimental results are compared with calculations. Previously, supercontinuum generation by higher-order mode excitation in a PCF was studied [6

J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton, and S. Coen, “Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,” J. Opt. Soc. Am. B 19, 765–771 (2002). [CrossRef]

, 7

R. Cherif, M. Zghal, L. Tartara, and V. Degiorgio, “Supercontinuum generation by higher-order mode excitation in a photonic crystal fiber,” Opt. Express 16, 2147–2152 (2008). [CrossRef] [PubMed]

], however the dependence of spectral broadening on polarization was not mentioned. Recently, the radiation of the Cherenkov peaks belonging to different linear modes in arrays of silicon-on-insulator photonic wires was reported [8

C. J. Benton, A. V. Gorbach, and D. V. Skryabin, “Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires,” Phys. Rev. A 78, 033818 (2008). [CrossRef]

2. Methods

The PCF used in this study was made of fused silica and it contained five rings of air holes in a regular hexagonal lattice arrangement. The diameter and the pitch of air holes were measured to be 1.57 μm and 2.27 μm respectively. The spatial modes and dispersion properties of this PCF were calculated using two rings of air holes as functions of wavelength by a multipole method [9

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]

], where electric and magnetic fields were treated as vector quantities. In experiment, a linearly-polarized output beam from a Ti:sapphire oscillator (center wavelength 810 nm, repetition rate 78 MHz, pulse width 50 fs, and average power 80 mW) was propagated in a 28-cm-long PCF using a 40 × objective for coupling the beam to the PCF. The polarization direction of an input pulse was controlled by a half-wave plate. A fundamental LP₀₁ mode was propagated in a PCF first and higher-order LP₁₁ modes were propagated by slightly offsetting the fiber position in vertical and horizontal directions. The offsetting distances were 1.7 μm for a vertical direction and 1.0 μm for a horizontal direction. Spectra and polarization directions were measured for these higher-order modes using a spectrometer and a polarizer. Four combinations of spatial modes and polarization directions of LP₁₁ modes were measured. The orientation of a PCF and its relation to spatial modes and polarization directions were measured by a CCD camera.

3. Results and discussions

In Fig. 1(a) and 1(b), calculated electric field vectors and spatial modes classified by symmetry numbers (p1, p2, p3, p5 and p6) [9

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]

] at 800 nm are shown. In this figure, p3 mode is a fundamental LP₀₁ mode and other modes are higher-order modes with the lowest loss for each symmetry number. The GVD curves of these modes were calculated from the dependences of effective refractive indices of modes on wavelength and are shown in Fig. 2. Here, GVD at angular frequency ω is calculated by

β_{2} (ω) = \frac{d^{2} β (ω)}{d ω^{2}} = \frac{λ^{3}}{2 π c^{2}} \frac{d^{2} n_{eff, r} (λ)}{d λ^{2}},

where β(ω) = n _eff,r(ω)(ω/c is the propagation constant of a PCF, n _eff,r(λ) is the real part of an effective index of refraction at wavelength λ for each mode, and c is the speed of light. As Fig. 2, the dispersion properties of higher-order modes at dispersive wave wavelengths (about 600 nm) are very similar (In this figure, only GVD values are shown, but propagation constants and group velocities are also very similar). Thus, the linear combinations of these modes may be considered as approximate propagation modes. In Fig. 1(c), the four combined modes with approximately linear polarization are shown, which are similar to the higher-order modes of a standard optical fiber [10

D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971). [CrossRef] [PubMed]

]. As can be seen from this figure, modes p1 and p5 (p2 and p6) have vertical (horizontal) nodal lines when the polarization direction is horizontal. Similarly, modes p1 and p5 (p2 and p6) have horizontal (vertical) nodal lines when the polarization direction is vertical. The wavelengths of dispersive waves were calculated by considering the phase-matching conditions using the dispersion relations of higher-order modes using the following equation [3

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express 12, 124–135 (2004). [CrossRef] [PubMed]

Fig. 1. (a) Electric field vectors of fundamental (p3) mode. (b) Electric field vectors of higher-order modes (p1, p2, p5, and p6). (c) Electric field vectors of four LP₁₁ higher-order modes.

Fig. 2. Group velocity dispersion versus wavelength for various spatial modes.

Fig. 3. Dispersive wave wavelength versus pump wavelength for higher-order modes.

n_{eff, r} (ω) ω / c - (ω - ω_{0}) / ν_{g} (ω_{0}) - n_{eff, r} (ω_{0}) ω_{0} / c = 0 .

where ω ₀ is the angular frequency of a pump soliton, and ν_g (ω ₀) = (dβ(ω ₀)/dω)^-1 is the group velocity of the soliton. In this equation, the nonlinear phase shift is neglected. The dispersive wave wavelengths were calculated as functions of pump wavelength using this equation and the results are shown in Fig. 3 for different higher-order modes, where p1, p2, p5 and p6 show the symmetry numbers of spatial modes. As shown in this figure, the wavelengths of dispersive waves are calculated to be about 600 nm, 580 nm and 570 nm for p2, p1, and p5/p6 modes respectively when a pump pulse at 810 nm is propagated. From the calculations, the wavelength of the dispersive wave for mode 2 (mode 4) in Fig. 1(c) is expected to be longer than that of mode 1 (mode 3) in the same figure, since mode 2 (mode 4) contains p2 mode instead of p1 mode in mode 1 (mode 3).

Fig. 4. The spectra of an input pulse (black curve) and an output pulse (red curve) from a PCF for a fundamental mode. Insets show the cross section of the PCF (right) and the picture of the spatial pattern of a fundamental mode (left).

In Fig. 4, the experimental spectrum for the fundamental mode of the PCF is shown. Also in this figure, the picture of the spatial mode and the cross section of the PCF used in experiment are shown. As shown in this figure, soliton pulses and dispersive waves were not observed in this mode since the zero-dispersion wavelength of this mode (~ 950 nm) was much longer than 810 nm. In Fig. 5, the experimental spectra and the pictures of spatial modes for the four LP₁₁ dispersive waves are shown. In these cases, soliton pulses can be propagated and the spectral peaks corresponding to fundamental soliton pulses (about 830 nm for modes 1 and 2 and about 840 nm for modes 3 and 4) were observed as shown in Fig. 6. The spatial modes of these soliton pulses were identical to the spatial modes of corresponding dispersive waves. The coupling efficiency of LP₀₁ mode and LP₁₁ mode were 10 % and 15 % respectively, after considering the loss of objectives. As shown in Fig. 5, dispersive waves were generated at wavelength from 570 to 610 nm. The estimated dispersive wave wavelengths from Fig. 3 using the soliton peak wavelengths (830 – 840 nm) in Fig. 6 were about 530 – 550 nm and somewhat shorter than experimental wavelengths. The difference is presumably due to the imperfection of the size and the position of air holes in the PCF used in experiment. The spatial patterns of mode 1 and mode 2 were identical, but the polarization directions were perpendicular. Similarly, mode 3 and mode 4 had the identical spatial patterns with perpendicular polarizations. The wavelength of dispersive wave of mode 2 (mode 4) was longer than mode 1 (mode 3) by about 20 nm in experiment and it agreed well with theoretical calculations. The intensity of dispersive wave of mode 4 was much larger than other modes. In Table 1, the normalized effective area [9

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]

] Aeff for various modes calculated from the following equation are shown

Fig. 5. The spectra of four different LP₁₁ higher-order dispersive waves. Insets show the pictures of spatial modes and the combinations of modes with different symmetry numbers. Arrows indicate the polarization directions of various modes.

Fig. 6. The spectra of soliton pulses corresponding to four different LP₁₁ higher-order dispersive waves shown in Fig. 5.

A_{eff} = \frac{{(\int {∣ E_{z} ∣}^{2} dS)}^{2}}{\int {∣ E_{z} ∣}^{4} dS},

where E_z is the electric field component along a propagation direction (z), and the integral is calculated in the plane perpendicular to z direction. As shown in this table, the normalized effective areas of higher-order modes are smaller than that of a fundamental mode (p3). Also, the normalized effective area of mode 4 is the smallest, which is consistent with the experimental spectra shown in Fig. 4, since the nonlinear effect is enhanced as the normalized effective area is reduced.

In summary, we have generated the dispersive waves of four LP₁₁ higher-order modes. The wavelengths and the polarization directions of these dispersive waves were measured and it was found that the wavelengths of modes containing p2 mode (mode 2 and mode 4) were longer than modes containing p1 mode (mode 1 and mode 3) with the identical spatial mode patterns, which agreed well with results obtained by theoretical calculations. Since the spatial mode pattern of mode 2 (4) is identical to mode 1 (3) and only the polarization directions between mode 2 (4) and mode 1 (3) are different, it is suggested that the wavelength of a dispersive wave can be controlled by changing only the polarization direction of an input pulse when higher-order mode excitation is used, which may be useful in applications such as optical switching and modulation.

Table 1. The normalized effective area A _eff of each mode in μm² at 800 nm.

p3	p1	p2	p5	p6	mode1	mode2	mode3	mode 4
5.47	3.25	3.84	5.05	5.05	4.23	4.99	3.83	3.78

Acknowledgements

This work was supported in part by a Grant-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science.

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 800 nm,” Opt. Lett. 25, 25–27 (2000). [CrossRef]
2.	J. M. Dudley and G. Genty, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]
3.	A. V. 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.	N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]
5.	I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express 12, 124–135 (2004). [CrossRef] [PubMed]
6.	J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton, and S. Coen, “Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,” J. Opt. Soc. Am. B 19, 765–771 (2002). [CrossRef]
7.	R. Cherif, M. Zghal, L. Tartara, and V. Degiorgio, “Supercontinuum generation by higher-order mode excitation in a photonic crystal fiber,” Opt. Express 16, 2147–2152 (2008). [CrossRef] [PubMed]
8.	C. J. Benton, A. V. Gorbach, and D. V. Skryabin, “Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires,” Phys. Rev. A 78, 033818 (2008). [CrossRef]
9.	F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]
10.	D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971). [CrossRef] [PubMed]

OCIS Codes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: January 20, 2010
Revised Manuscript: February 23, 2010
Manuscript Accepted: February 24, 2010
Published: February 26, 2010

Citation
Naoki Karasawa and Kazuhiro Tada, "The generation of dispersive waves from a photonic crystal fiber by higher-order mode excitation," Opt. Express 18, 5338-5343 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-5338

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References

J. K. Ranka, R. S. Windeler, and A. J. Stentz, "Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm," Opt. Lett. 25,25-27 (2000). [CrossRef]
J. M. Dudley and G. Genty, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78,1135-1184 (2006). [CrossRef]
A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87,203901 (2001). [CrossRef] [PubMed]
N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51,2602-2607 (1995). [CrossRef] [PubMed]
I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, "Dispersive wave generation by solitons in microstructured optical fibers," Opt. Express 12,124-135 (2004). [CrossRef] [PubMed]
J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton, and S. Coen, "Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping," J. Opt. Soc. Am. B 19,765-771 (2002). [CrossRef]
R. Cherif, M. Zghal, L. Tartara, and V. Degiorgio, "Supercontinuum generation by higher-order mode excitation in a photonic crystal fiber," Opt. Express 16,2147-2152 (2008). [CrossRef] [PubMed]
C. J. Benton, A. V. Gorbach, and D. V. Skryabin, "Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires," Phys. Rev. A 78,033818 (2008). [CrossRef]
F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005). [CrossRef]
D. Gloge, "Weakly guiding fibers," Appl. Opt. 10,2252-2258 (1971). [CrossRef] [PubMed]

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