One stage pulse compression at 1554nm through highly anomalous dispersive photonic crystal fiber

doi:10.1364/OE.19.021809

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One stage pulse compression at 1554nm through highly anomalous dispersive photonic crystal fiber

Maggie Yihong Chen, Harish Subbaraman, and Ray T. Chen »View Author Affiliations

Optics Express, Vol. 19, Issue 22, pp. 21809-21817 (2011)
http://dx.doi.org/10.1364/OE.19.021809

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▸ Article Outline
– Abstract
1. Introduction
2. Highly anomalous dispersive photonic crystal fiber design and characterization
3. Simulation of pulse propagation
4. Experimental results of pulse compression
4. Summary
– References and links

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Abstract

We demonstrate the pulse compression at 1554 nm using one stage of highly anomalous dispersive photonic crystal fibers with a dispersion value of 600 ps/nm∙km. A 1.64 ps pulse is compressed down to 0.357 ps with a compression factor of 4.6, which agrees reasonably well with the simulation value of 6.1. The compressor is better suited for high energy ultra-short pulse compression than conventional low dispersive single mode fibers.

1. Introduction

Over the past two decades, much effort has been made in generating ultrashort pulses through higher-order soliton compression [1

N. Akhmediev, N. V. Mitzkevich, and F. V. Lukin, “Extremely high degree of N-soliton pulse compression in an optical fiber,” IEEE J. Quantum Electron. 27(3), 849–857 (1991). [CrossRef]

] after the first observation of this phenomenon by Mollenauer et al. in 1980 [2

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental-observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980). [CrossRef]

]. It is believed that in this method, intrinsic spectral broadening and simultaneous temporal compression can be exploited once a proper length of the medium is chosen, thus obviating the need of post-compression devices [3

G. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

]. In recent years, especially with the development of waveguides with special dispersion and nonlinear properties, this one step straight forward compression method draws a lot of attention. Foster demonstrated soliton-effect compression in photonic nanowires [4

M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express 13(18), 6848–6855 (2005). [CrossRef] [PubMed]

]. In 2007, a 30 fs pulse near 1.55 µm was obtained using a 7 cm highly nonlinear fiber [5

B. Kibler, R. Fischer, R. A. Lacourt, E. Courvoisier, R. Ferriere, L. Larger, D. N. Neshev, and J. M. Dudley, “Optimized one-step compression of femtosecond fibre laser soliton pulses around 1550 nm to below 30 fs in highly nonlinear fibre,” Electron. Lett. 43(17), 915–916 (2007). [CrossRef]

]. In another experiment, a 50 fs pulse was achieved with Xe filled photonic crystal fiber [6

D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, “Soliton pulse compression in photonic band-gap fibers,” Opt. Express 13(16), 6153–6159 (2005). [CrossRef] [PubMed]

]. Recently, a sub-two-cycle pulse was observed using a 4 mm long highly nonlinear photonic crystal fiber [7

A. A. Amorim, M. V. Tognetti, P. Oliveira, J. L. Silva, L. M. Bernardo, F. X. Kärtner, and H. M. Crespo, “Sub-two-cycle pulses by soliton self-compression in highly nonlinear photonic crystal fibers,” Opt. Lett. 34(24), 3851–3853 (2009). [CrossRef] [PubMed]

]. However, current state of the art photonic crystal fiber technology used for compression of ultrashort pulses is typically limited by nonlinearities to nanojoule energy levels. To maintain the same number of solitons for higher energy level pulse, we need to design and fabricate new optical fibers or waveguides with reasonable nonlinearities and adequate dispersion values.

In this paper, we present a single-stage compression with highly anomalous dispersive photonic crystal fiber (PCF). The dispersion parameter, D which is commonly used in fiber optics, is related to the group velocity dispersion (GVD) parameter β₂ given by:

D = \frac{- 2 π c}{λ^{2}} β_{2}

(1)

Depending on the sign of β₂ parameter, the nonlinear effects in a fiber can be used to control the behavior of a pulse propagating through the fiber.

On the other hand, the refractive index of the fiber depends on the intensity of light inside the fiber. The general expression for the refractive index of the fiber is given by the relation:

n (ω, {| E |}^{2}) = n (ω) + n_{2} {| E |}^{2}

(2)

where the first term on the right hand side of the equation gives the linear term, and the second term on the right hand side of the equation gives the intensity dependence of refractive index.

n_{2}

is the non-linear refractive index coefficient (~2.2x10⁻²⁰ m²W⁻¹ in silica glass). Depending on the initial pulse power (P₀), and initial width (T₀) of the pulse, the evolution of the pulse through the fiber based on the interplay between the dispersive and nonlinear effects can be studied. The two important terms are the dispersion length (L_D), and the nonlinear length (L_NL).

L_{D} = \frac{T_{0}_{}^{2}}{| β_{2} |}

(3)

L_{N L} = \frac{1}{γ P_{0}}

(4)

where γ is the effective nonlinearity. γ is calculated as γ=2πn₂/λA_eff, where A_eff is the effective mode area. A_eff is expressed as

A_{e f f} = \frac{{[\int \int_{- \infty}^{\infty} {| F (x, y) |}^{2} d x d y]}^{2}}{\int \int_{- \infty}^{\infty} {| F (x, y) |}^{4} d x d y}

(5)

where F(x,y) is the fundamental mode distribution.

To generate solitons in an optical fiber, the chirp induced by non-linear effect should cancel out as much as possible with the chirp induced via dispersion. For this reason, solitons can be achieved only in the anomalous dispersion region as the signs of chirps are opposite for the two cases, which means the dispersion parameter D is positive. To generate N-th order soliton, the following equation should be satisfied:

N^{2} = \frac{L_{D}}{L_{N L}}

(6)

The optimum length at which to extract the compressed pulse is predicted in following equation [3

G. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

\frac{z_{o p t}}{L_{D}} = \frac{π}{2} (\frac{0.32}{N} + \frac{1.1}{N^{2}})

(7)

The compression factor F_c and the quality factor Q_c are defined to describe the efficiency [3

G. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

F_{c} = \frac{T_{0}}{T_{c o m p}}

(8)

Q_{c} = \frac{P_{c o m p}}{F_{c}}

(9)

where T_comp is the width of the compressed pulse, and P_comp is the peak power of the compressed pulse normalized to the input pulse.

In our design, through maintaining the similar effective mode field diameter, the effective nonlinearity is maintained in the similar order as silicon single mode fiber. For a given input pulse width, if the dispersion of the photonic crystal fiber is 60 times larger, to support same order of soliton, the peak power of input pulse are 60 times those supported by conventional low dispersive single mode fibers.

2. Highly anomalous dispersive photonic crystal fiber design and characterization

Photonic crystal fibers have generated a lot of interest due to their unusual and attractive properties [8

L. P. Shen, W. P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15(4), 540–542 (2003). [CrossRef]

–11

J. Broeng, S. E. Barkou, T. Søndergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25(2), 96–98 (2000). [CrossRef] [PubMed]

]. The dispersion of the PCFs is tuned by changing the pitch (Λ) of the periodic array, the hole diameter (d) and the doping concentration (n) of the core, as shown in Fig. 1 [8

J. A. West, N. Venkataramam, C. M. Smith, and M. T. Gallagher, “Photonic crystal fibers,” in Proc. 27th Eur. Conf. on Opt. Comm. (2001), Vol. 4, pp. 582 –585.

Fig. 1 Transverse section of a model highly dispersive PCF. The box with dimensions D x D corresponds to the supercell used to implement boundary conditions.

We used a two-core PCF design to achieve high dispersion. The inner core is a doped silica rod, and the outer core is 12 concentric doped silica rods, as shown in Fig. 1. Both cores are doped to have higher refractive index than pure silica, but the refractive index of the inner core is greater than that of the outer core. This two-core PCF can support two supermodes, which are analogous to the two supermodes of a directional coupler [10

K. Thyagarajan, R. K. Varshney, P. Palai, A. K. Ghatak, and I. C. Goyal, “A novel design of a dispersion compensating fiber,” IEEE Photon. Technol. Lett. 8(11), 1510–1512 (1996). [CrossRef]

]. These modes are nearly phase matched at 1550 nm. Close to the phase matching wavelength, the mode index of the PCF changes rapidly due to strong coupling between the two individual modes of the inner core and outer core. Due to strong refractive index asymmetry between the two cores, there is a rapid change in the slope of the wavelength variation of the fundamental mode index. This leads to a large dispersion around 1550 nm. The air hole structure helps not only to guide the mode, but also to increase the dispersion value.

The dispersion of PCFs can be calculated using the full vectorial plane-wave expansion (PWE) method, which is fast and accurate compared to other methods [11

J. Broeng, S. E. Barkou, T. Søndergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25(2), 96–98 (2000). [CrossRef] [PubMed]

]. We simulated the PCFs by using Bandsolve^TM software that is based on full vectorial PWE. Since our PCF design is not a perfect crystal without defects, we need to use a supercell having a size of 8 × 8 instead of a natural unit cell to implement the periodic boundary conditions [11

J. Broeng, S. E. Barkou, T. Søndergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25(2), 96–98 (2000). [CrossRef] [PubMed]

The group velocity dispersion or simply the dispersion parameter D(λ) of the guided mode of the PCF can be directly calculated from the modal effective index n_eff(λ) of the fundamental mode over a range of wavelengths [12

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. 24(5), 276–278 (1999). [CrossRef] [PubMed]

]

D(λ) = - \frac{λ}{c} \frac{d^{2} n_{e f f} (λ)}{d λ^{2}}

(10)

where the effective refractive index of the mode is given by n_eff = β[λ,n_m(λ)]/k₀, where β is the propagation constant and k₀ is the free-space wave number. The dispersion parameter, D(λ), of PCF is strongly related to the structure and refractive index perturbation, and can thus be changed to achieve the desired characteristics. In our design, we used different doping concentrations, period Λ, and hole diameters d₀, d₁, d₂, d₃, to tune the dispersion. Here d₀ and d₂ are the diameters of the inner and outer doped silica rods, respectively. And d₁ and d₃ are the diameters of the air holes. Figure 2 shows the theoretical simulation results of highly dispersive PCFs with differing parameters using the full vectorial PWE method. Design #1 has the flattest and highest dispersion of about D = 620 ps/nm∙km at 1554 nm. The measured dispersion curve of the fabricated PCF is compared with simulation in Fig. 3 .

Fig. 2 Dispersion parameter D for dispersion enhanced PCF with various microstructure parameters (refractive index of pure silica glass n = 1.444) #1: Λ = 3.5 μm, d₀ = 1.72 μm, d₁ = 1.45 μm, d₂ = 1.08 μm, d₃ = 0.86 μm, Δn₁/n = 1.9%, Δn₂/n = 1.3% #2: Λ = 3.5 μm, d₀ = 1.74 μm, d₁ = 2.00 μm, d₂ = 0.70 μm, d₃ = 0.87 μm, Δn₁/n = 2.0%, Δn₂/n = 1.7% #3: Λ = 3.5 μm, d₀ = 1.74 μm, d₁ = 1.80 μm, d₂ = 0.87 μm, d₃ = 0.87 μm, Δn₁/n = 2.0%, Δn₂/n = 1.7% #4: Λ = 3.5 μm, d₀ = 1.72 μm, d₁ = 1.45 μm, d₂ = 1.08 μm, d₃ = 0.86 μm, Δn₁/n = 1.9%, Δn₂/n = 1.2%.

Fig. 3 The simulated and measured dispersion parameter D for the highly dispersive PCF.

3. Simulation of pulse propagation

The effective mode area A_eff is calculated to be about 44µm². The effective nonlinearity γ is about 2 km^-1W^-1. Given an input pulse of 1.64 ps and peak power of 1000 W with an 1554 nm center wavelength, the nonlinear length L_NL is 0.5 m. To avoid the effects of the mismatch between simulated and measured values of dispersion, we use the measured dispersion value at 1554 nm, which is about 600ps/nm/km, for the calculation of β₂. According to Eq. (1), for 1554 nm center wavelength, β₂ is calculated to be -770 ps²/km. Dispersion length L_D is estimated to be around 4.4 m. According to Eq. (6), the given pulse would excite a soliton of order N≈3. Using Eq. (7), an optimum PCF length is predicted to be 1.58 m.

We simulate the pulse propagation through solving the following nonlinear Schrödinger equation to incorporate the fiber propagation loss and third-order dispersion

\frac{\partial A (t, z)}{\partial z} = - \frac{i β_{2}}{2} \frac{\partial^{2} A (t, z)}{\partial t^{2}} + \frac{β_{3}}{6} \frac{\partial^{3} A (t, z)}{\partial t^{3}} - \frac{α}{2} A (t, z) + i γ [{| A (t, z) |}^{2} A (t, z) + \frac{i}{ω_{0}} \frac{\partial ({| A |}^{2} A)}{\partial t}]

(11)

where A is the magnitude of the pulse envelope, ɷ₀ is the center angular frequency, and β₃ is the fiber’s third-order dispersion.

We solve the above equation using split-step method. The propagation loss of the PCF is estimated to be 40dB/km, and counted in the simulation. Again, we use the measured dispersion value of 600ps/nm/km to avoid the effects of mismatch between simulated and measured values. The results of the simulation are shown in Figs. 4 , 5 and 6 . In Fig. 4, output profiles for different propagation length z are compared. At the optimum fiber length z_opt of 1.7 m, the compressed pulse has the greatest power and shortest duration. The initial 1.64 pm pulse compresses down to a duration of 269 fs with a compression factor F_c of 6.1 and a quality factor of compression Q_c of 0.79. The optimum fiber length 1.58m predicted by Eq. (7) agrees reasonably well with the simulated 1.7 m. The discrepancy comes from the non-uniformity of dispersion within the entire simulation bandwidth. Figure 5 illustrates the 3D waterfall plot of the evolution of the field during propagation with the characteristic of a typical third order soliton [3

G. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

]. This agrees with the result calculated from Eq. (6). Figure 6 shows the input and output spectral intensity at optimum fiber length of 1.7 m, which agrees well with the typical spectral of third order soliton at optimum propagation distance [3

G. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

Fig. 4 Simulated temporal evolution of a 1000 W peak power 1.64 ps hyperbolic-secant pulse at (a) 1.9 m, (b) 1.7 m, (c) 1.5 m, and (d) 1.3 m. T₀ is the initial pulse width 1.64 ps.

Fig. 5 Simulated 3D waterfall plot of the evolution of the field during propagation.

Fig. 6 Simulated input and output spectral intensity at optimum fiber length of 1.7m. v is the frequency, v₀ is the center frequency and T₀ is the initial pulse width.

Higher degree of pulse compression is possible with higher number of solitons [1

N. Akhmediev, N. V. Mitzkevich, and F. V. Lukin, “Extremely high degree of N-soliton pulse compression in an optical fiber,” IEEE J. Quantum Electron. 27(3), 849–857 (1991). [CrossRef]

]. According to Eq. (6), higher order soliton can be generated through increasing the peak power of the input pulse. Further experiment will be carried out with available high power pulse laser to increase the quality factor and compression factor.

4. Experimental results of pulse compression

Experiments are conducted to demonstrate the compression around 1554 nm using the setup in Fig. 7 . As in the figure, short pulse laser source and autocorrelator are required equipments for the measurement.

Fig. 7 Setup for PCF compressor demonstration.

An Erbium doped fiber laser is used, producing optical pulses in the vicinity of 1550 nm with a 1000 W peak level and a repetition rate of 19 MHz. The launched pulses have hyperbolic-secant temporal profiles and are free of any frequency chirp.

Figure 8 depicts the optical spectrum and time domain feature of the pulse laser as measured by an optical spectrum analyzer and a digital communication analyzer. Figure 8(a) shows the broad optical spectrum, which is typical for a short pulse. Figure 8(b) shows a series of optical pulses in the time domain with a repetition rate of 19 MHz. The oscillation after the main peak comes from the response of the high-speed photodetector.

Fig. 8 Measured (a) optical spectrum and (b) time domain feature of the pulse laser.

We carry out pulse compression experiment using the designed photonic crystal fiber with 600 ps/nm/km measured dispersion value. The length we used is 1.7 m as simulated in section 3. The initial pulse fed into the PCF is measured and shown in Fig. 9(a) . By launching the short light pulses in the photonic crystal fiber, we are able to achieve pulse compression. Figure 9(b) shows the pulse coming out of the PCF. It can be seen that the pulse is compressed by a factor of 4.6. With 3 dB coupling loss estimated, the quality factor is measured to be 0.71. The performance can be further improved through adjusting the input peak power, pulse quality, and fine tuning the fiber length.

Fig. 9 The measured (a) initial pulse launched into PCF, (b) output pulse compressed by a factor of 4.6. FWHM is full width half maximum.

4. Summary

In conclusion, a 1.64 ps pulse generated by mode-locking fiber laser centered at 1554 nm is compressed down to a 0.357 ps pulse, through a one-stage 1.7 m highly dispersive PCF compressor. With very high peak power pulse, higher numbers of soliton can be generated easily with satisfying compression factor, despite the large dispersion value. The highly dispersive PCF is better suited for solitons with high peak powers than conventional low dispersive single mode fibers. The pulse compression factor and the quality factor can be improved through the adjustment of fiber length, dispersion value, input pulse power and pulse quality. According to the author’s best knowledge, this is the first report about one-stage highly anomalous dispersive photonic crystal fiber based compressor around 1550nm.

References and links

1.	N. Akhmediev, N. V. Mitzkevich, and F. V. Lukin, “Extremely high degree of N-soliton pulse compression in an optical fiber,” IEEE J. Quantum Electron. 27(3), 849–857 (1991). [CrossRef]
2.	L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental-observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980). [CrossRef]
3.	G. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
4.	M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express 13(18), 6848–6855 (2005). [CrossRef] [PubMed]
5.	B. Kibler, R. Fischer, R. A. Lacourt, E. Courvoisier, R. Ferriere, L. Larger, D. N. Neshev, and J. M. Dudley, “Optimized one-step compression of femtosecond fibre laser soliton pulses around 1550 nm to below 30 fs in highly nonlinear fibre,” Electron. Lett. 43(17), 915–916 (2007). [CrossRef]
6.	D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, “Soliton pulse compression in photonic band-gap fibers,” Opt. Express 13(16), 6153–6159 (2005). [CrossRef] [PubMed]
7.	A. A. Amorim, M. V. Tognetti, P. Oliveira, J. L. Silva, L. M. Bernardo, F. X. Kärtner, and H. M. Crespo, “Sub-two-cycle pulses by soliton self-compression in highly nonlinear photonic crystal fibers,” Opt. Lett. 34(24), 3851–3853 (2009). [CrossRef] [PubMed]
8.	L. P. Shen, W. P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15(4), 540–542 (2003). [CrossRef]
9.	J. A. West, N. Venkataramam, C. M. Smith, and M. T. Gallagher, “Photonic crystal fibers,” in Proc. 27th Eur. Conf. on Opt. Comm. (2001), Vol. 4, pp. 582 –585.
10.	K. Thyagarajan, R. K. Varshney, P. Palai, A. K. Ghatak, and I. C. Goyal, “A novel design of a dispersion compensating fiber,” IEEE Photon. Technol. Lett. 8(11), 1510–1512 (1996). [CrossRef]
11.	J. Broeng, S. E. Barkou, T. Søndergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25(2), 96–98 (2000). [CrossRef] [PubMed]
12.	A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. 24(5), 276–278 (1999). [CrossRef] [PubMed]

OCIS Codes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(320.5520) Ultrafast optics : Pulse compression
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Ultrafast Optics

History
Original Manuscript: August 11, 2011
Revised Manuscript: September 27, 2011
Manuscript Accepted: September 27, 2011
Published: October 20, 2011

Citation
Maggie Yihong Chen, Harish Subbaraman, and Ray T. Chen, "One stage pulse compression at 1554nm through highly anomalous dispersive photonic crystal fiber," Opt. Express 19, 21809-21817 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21809

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References

N. Akhmediev, N. V. Mitzkevich, and F. V. Lukin, “Extremely high degree of N-soliton pulse compression in an optical fiber,” IEEE J. Quantum Electron.27(3), 849–857 (1991). [CrossRef]
L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental-observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett.45(13), 1095–1098 (1980). [CrossRef]
G. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express13(18), 6848–6855 (2005). [CrossRef] [PubMed]
B. Kibler, R. Fischer, R. A. Lacourt, E. Courvoisier, R. Ferriere, L. Larger, D. N. Neshev, and J. M. Dudley, “Optimized one-step compression of femtosecond fibre laser soliton pulses around 1550 nm to below 30 fs in highly nonlinear fibre,” Electron. Lett.43(17), 915–916 (2007). [CrossRef]
D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, “Soliton pulse compression in photonic band-gap fibers,” Opt. Express13(16), 6153–6159 (2005). [CrossRef] [PubMed]
A. A. Amorim, M. V. Tognetti, P. Oliveira, J. L. Silva, L. M. Bernardo, F. X. Kärtner, and H. M. Crespo, “Sub-two-cycle pulses by soliton self-compression in highly nonlinear photonic crystal fibers,” Opt. Lett.34(24), 3851–3853 (2009). [CrossRef] [PubMed]
L. P. Shen, W. P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett.15(4), 540–542 (2003). [CrossRef]
J. A. West, N. Venkataramam, C. M. Smith, and M. T. Gallagher, “Photonic crystal fibers,” in Proc. 27th Eur. Conf. on Opt. Comm. (2001), Vol. 4, pp. 582 –585.
K. Thyagarajan, R. K. Varshney, P. Palai, A. K. Ghatak, and I. C. Goyal, “A novel design of a dispersion compensating fiber,” IEEE Photon. Technol. Lett.8(11), 1510–1512 (1996). [CrossRef]
J. Broeng, S. E. Barkou, T. Søndergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett.25(2), 96–98 (2000). [CrossRef] [PubMed]
A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett.24(5), 276–278 (1999). [CrossRef] [PubMed]

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