Measurement of the effective nonlinear and dispersion coefficients in optical fibers by the induced grating autocorrelation technique

doi:10.1364/OE.19.001755

« Show journal navigation

Measurement of the effective nonlinear and dispersion coefficients in optical fibers by the induced grating autocorrelation technique

Robinson Kuis, Anthony Johnson, and Sudhir Trivedi »View Author Affiliations

Optics Express, Vol. 19, Issue 3, pp. 1755-1766 (2011)
http://dx.doi.org/10.1364/OE.19.001755

View Full Text Article

Acrobat PDF (1328 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Article Navigation

▸ Article Outline
– Abstract
1. Introduction
2. IGA technique description
3. Experimental discussion and results
4. Conclusion
– References and links

Article Tools
Full Text
Article Info
References
Cited By
Figures
Metrics
Download PDF

Viewing Equations in the
HTML Article

Optimal Viewing Recommendations

Full Text
Article Info
References (25)
Cited By
Figures (12)
Metrics

Abstract

The induced grating autocorrelation technique, a technique based on temporally resolved two-beam coupling in a photorefractive crystal, was used to measure the nonlinear coefficient γ of three photonic crystal fibers (PCFs): a 30-cm long highly nonlinear PCF, and two large mode area PCFs of 4.5-m and 4.9-m lengths. The measurement used intense 2-ps, 800-nm (850-nm in one case) pulses from a Ti: sapphire laser that experienced self-phase modulation and group velocity dispersion as it travels inside the fibers. This technique was also expanded to measure γ and the dispersion coefficient β₂ simultaneously.

1. Introduction

Photonic crystal fibers (PCFs) have revolutionized the field of nonlinear optics since their introduction in the mid 1990’s [1

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “Pure silica single-mode fiber with hexagonal photonic crystal cladding,” Conf. Optical Fiber Commun. (OFC) San Jose, CA, (1996).

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996). [CrossRef] [PubMed]

]. Unlike conventional fibers where different materials for core and cladding are required to have guidance, PCFs can be drawn from a single material. Light in PCFs is guided by the index difference [3

J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A 15(3), 748–752 (1998). [CrossRef]

K. Saitoh and M. Koshiba, “Numerical modeling of photonic crystal fibers,” J. Lightwave Technol. 23(11), 3580–3590 (2005). [CrossRef]

] created between the solid core and surrounding air-hole silica structure that behaves as an effective cladding. The cladding structure can be designed to optimize the dispersion properties of the fiber in a wavelength region of interest. The fiber can also operate in single mode for long ranges of wavelengths by properly choosing air-hole diameter (d) and air-hole separation or pitch (Λ) [5

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22(13), 961–963 (1997). [CrossRef] [PubMed]

]. PCFs can have higher modal confinement leading to larger nonlinear coefficients than conventional silica fibers. High nonlinear coefficients allows for observation of nonlinear phenomena {self-phase modulation (SPM), Raman effect, supercontinuum generation, etc.} using shorter lengths than conventional fibers. There are commercially available PCFs that have theoretical values for the effective nonlinear coefficient γ of 70 W⁻¹km⁻¹ (NKT Photonics’ NL-2.4-800) at 800 nm. Broderick et al. [6

N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, “Nonlinearity in holey optical fibers: measurement and future opportunities,” Opt. Lett. 24(20), 1395–1397 (1999). [CrossRef]

] first measured γ in PCFs at 1550 nm using a method devised by Boskovic et al [7

A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, ““Direct continuous-wave measurement of n₂ in various types of telecommunication fiber at 1.55 µm,” Opt. Lett. 21, 1966–1968 (1996). For an additional measurement approach in single-mode fiber see for example, C. Vinegone, M. Wegmuller, and N. Gisin, “Measurements of the Nonlinear Coefficient of Standard SMF, DSF, and DCF Fibers Using a Self-Aligned Interferometer and a Faraday Mirror,” IEEE Photon. Technol. Lett. 13, 1337–1339 (2001). [CrossRef] [PubMed]

]. A high-power dual-frequency beat signal is propagated in the fiber under test. Sidebands are generated due to SPM and the ratio between the signal and first sideband gives the nonlinear phase shift. Effective nonlinear index measurements were also conducted by Hensley et al [8

C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, “Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers,” Opt. Express 15(6), 3507–3512 (2007). [CrossRef] [PubMed]

]. These experiments determined the contribution from the silica glass to the nonlinear coefficient of several photonic bandgap fibers. These nonlinear coefficient measurements required complicated setups or multiple sources. We propose using a less complicated technique, the induced grating autocorrelation (IGA), to measure γ in PCFs.

The IGA technique [9

H. Garcia, A. M. Johnson, F. A. Oguama, and S. Trivedi, “New approach to the measurement of the nonlinear refractive index of short (<25 m) lengths of silica and erbium-doped fibers,” Opt. Lett. 28(19), 1796–1798 (2003). [CrossRef] [PubMed]

,10

F. A. Oguama, H. Garcia, and A. M. Johnson, “Simultaneous measurement of the Raman gain coefficient and the nonlinear refractive index of optical fibers: theory and experiment,” J. Opt. Soc. Am. B 22(2), 426–436 (2005). [CrossRef]

] has been shown to be very accurate in measuring the nonlinear refractive index n ₂ in conventional single mode fibers requiring a single laser source. Our results will show that the IGA technique can be applied to measurements of γ in PCFs. The technique was used to measure γ in three PCFs. These measured γs had excellent agreement with calculated theoretical values. The measurements demonstrated that γ could be determined with fiber lengths as short as 95 cm.

2. IGA technique description

The IGA technique is based on time-resolved two-beam coupling in a photorefractive crystal [11

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, and D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting, Vol. 11 of 1988 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 128.

–15

X. S. Yao, V. Dominic, and J. Feinberg, “Theory of beam coupling and pulse shaping of mode-locked laser pulses in a photorefractive crystal,” J. Opt. Soc. Am. B 7(12), 2347–2355 (1990). [CrossRef]

]. Mutually coherent pulses (Beam 1 and Beam 2) intersect and produce an interference pattern (bright and dark regions) in a photorefractive crystal (Fig. 1 ).

Fig. 1 Description of the photorefractive two-beam coupling in CdMnTe: V.

Electrons are released from donor sites (e.g., vanadium mid-gap states in CdMnTe: V crystals) in the bright regions due to photoionization and migrate to the dark regions by diffusion or drift. The motion of the free carriers reduces the spatial amplitude of the electron density when compared to the spatial amplitude of the ionized donors. The amplitude difference gives rise to a space-charge distribution modulated in phase with the interference pattern. This space-charge distribution generates an electric field (E_SC ) [15

E_{S C} = i \frac{b}{a} \frac{〈 E_{1} (r, t) E_{2}^{*} (r, t - τ_{d}) 〉}{I_{o}},

(1)

where a and b are functions that depend on material parameters and beam geometry [16

F. P. Strohkendl, J. M. C. Jonathan, and R. W. Hellwarth, “Hole - electron competition in photorefractive gratings,” Opt. Lett. 11(5), 312–314 (1986). [CrossRef] [PubMed]

] but not on the optical intensities. E ₁(r,t) and E* ₂(r,t) are the slowly varying complex electric field envelopes, 〈〉 denotes an average over time T_p (

\int_{- T_{P} / 2}^{T_{P} / 2} F (τ) d τ

), τ_d is a time delay between Beam 1 and Beam 2, and I_o = |E ₁(r,t)|² + |E ₂(r,t)|² is proportional to the total intensity. This field modulates the refractive index through the electro-optic effect [15

Δ {\overset{\leftrightarrow}{ε}}_{r} = - {\overset{\leftrightarrow}{ε}}_{r} \cdot (\overset{\leftrightarrow}{R} \cdot {\hat{k}}_{g}) \cdot {\overset{\leftrightarrow}{ε}}_{r} E_{S C},

(2)

where

Δ \overset{\leftrightarrow}{ε}

is the induced change to the dielectric tensor by the space-charge field,

{\overset{\leftrightarrow}{ε}}_{r}

is the relative dielectric tensor of the medium,

\overset{\leftrightarrow}{R}

is the electro-optic tensor of the crystal, and

{\hat{k}}_{g}

is the unit vector along the grating direction. The space-charge induced grating can cause gain (or loss) of intensity between Beam 1 and Beam 2 (or Beam 2 to Beam 1). The intensity change of Beam 1 at the expense of Beam 2 is described by:

〈 {| E_{1} (z, t) |}^{2} 〉 - 〈 {| E_{1} (0, t) |}^{2} 〉 = \frac{4 η_{12}}{η_{12} - η_{21}} {| 〈 E_{1} (z, t) E_{2}^{*} (z, t - τ_{d}) 〉 |}^{2} [\exp (\frac{1}{2} {η_{12} - η_{21}} z) - 1] .

(3)

To arrive at Eq. (3), the coupled-wave equations were obtained from Maxwell’s equations and a solution was found assuming the intensities of Beam 1 and Beam 2 were the same (〈|E ₁|²〉 = 〈|E ₂|²〉). The quantities η ₁₂ and η ₂₁ are the coupling constants between Beam1 and Beam 2 [15

]. This result shows that the change in intensity of Beam 1 due to the induced grating is proportional to the squared electric field correlation function of the input fields.

The result of Eq. (3) was used to develop an experimental technique (IGA technique) for measuring n ₂ in conventional fibers [9

,10

,17

H. Garcia, “Time domain measurement of the nonlinear refractive index in optical fibers and semiconductors films,” Ph.D. dissertation (New Jersey Institute of Technology, Newark, NJ, 2000).

,18

F. A. Oguama, A. M. Johnson, and W. A. Reed, “Measurement of the nonlinear coefficient of telecommunication fibers as a function of Er, Al, and Ge doping profiles by using the photorefractive beam-coupling technique,” J. Opt. Soc. Am. B 22(8), 1600–1604 (2005). [CrossRef]

]. An IGA measurement is performed as follows. Pulses transmitted through a fiber enter a modified Michelson interferometer (illustrated in Fig. 2 ). The light beam is separated into two (Beam 1 and Beam 2) by 50/50 beam splitter producing the arms of the interferometer.

Fig. 2 IGA experimental setup.

Both arms of the interferometer are stepper motor driven, however, one arm is fixed resulting in a relative delay, τ between beams. The optical pulses then interfere in a photorefractive crystal (CdMnTe: V in this work) and the diffracted intensity is plotted versus the delay (τ) between pulses producing an IGA trace:

I G A (τ_{d}) = {| \int_{- \infty}^{\infty} U (t) U^{*} (t + τ_{d}) d t |}^{2} .

(4)

In Eq. (4), U(t) is the normalized complex electric field envelope and at τ_d = 0, IGA(0) = 1. The limits of the integral are set to infinity since the pulse period is much larger than the pulsewidth (T _p >> T ₀). To determine U(t), one must solve the propagation equation that best describes ps pulses traveling in an optical fiber [19

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).

\frac{\partial U}{\partial z} + i \frac{α}{2} U - \frac{β_{2}}{2 T_{_{0}}^{2}} \frac{\partial^{2} U}{\partial T^{2}} + γ P_{0} {| U |}^{2} A = 0,

(5)

where T = t/T ₀, α is the absorption coefficient, β ₂ is the dispersion (GVD) coefficient, T ₀ is the 1/e pulsewidth, P ₀ is the peak power inside the fiber and γ is the nonlinear coefficient and is defined as:

γ = \frac{n_{2} ω_{0}}{c A_{e f f}},

(6)

where n ₂ is the nonlinear refractive index, ω ₀ is the central frequency of the pulse, c is the speed of light in vacuum and A_eff is the effective area of the fiber [19

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).

]. The length where the nonlinearities become important is called the nonlinear length L_NL = 1/ (P ₀ γ). The fiber length where dispersion becomes relevant is termed the dispersion length L_D = T₀ ²/β ₂. The validity of Eq. (5) to describe propagation of picosecond pulses was confirmed by performing theoretical simulations using the split-step method [19

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).

] at three pulsewidths: 30 ps, 10 ps, and 2 ps. A freely available Matlab routine [20

The ssprop code is available from the Photonics Research Laboratory at University of Maryland, College Park at http://www.photonics.umd.edu/software/ssprop

] that implements the split-step method was used to deternine U(t) and the routine was modified to include stimulated Raman scattering (SRS). The simulations included self-steepening and SRS alongside SPM. In addition to GVD, third-order dispersion was also included. The results showed that propagation of these pulses, in fiber lengths less than 5 m of single-mode silica glass fiber, was not significantly affected by self-steepening, SRS and third-order dispersion [21

R. Kuis, “Theoretical and Experimental Study of the Nonlinear Optical and Dispersive Properties of Conventional and Photonic Crystal Fibers,” Ph.D. dissertation (University of Maryland, Baltimore County, Baltimore, MD 2009).

2.1 IGA model with SPM only

Previous IGA experiments used large 1064 nm pulses [50 ps (30 ps 1/e width)] and fiber lengths ≤ 15 m long. The typical value for L_D in single-mode silica fiber is ~53 km at this pulsewidth. Therefore, nonlinearities dominate at this fiber length and effects of dispersion can be ignored. To examine the effects of GVD, simulations of Eq. (5) were performed using both 50 ps and 2 ps pulses. There was no observable broadening in 15-m long silica fiber with 50 ps pulses as expected. Simulations at 2 ps showed significant GVD-induced broadening with fiber lengths as short as 35 cm. However, the original IGA model is useful with short pulses (< 50 ps) for fibers that have small or zero GVD coefficients. Thus, the model can be applied to measurements of γ in highly nonlinear fibers such as NL-2.4-800, where β ₂ ≅ 0. Without GVD, Eq. (5) can be analytically solved:

U (L_{e f f}, T) = U (0, T) \exp [i ϕ_{N L} (L_{e f f}, T)],

(7)

ϕ_{N L} (L_{e f f}, T) = Δ ϕ_{N L} {| U (0, T) |}^{2},

(8)

Δ ϕ_{N L} = \frac{L_{e f f}}{L_{N L}} = L_{e f f} P_{0} γ,

(9)

where ϕ_NL is the time-dependent phase generated through SPM, Δϕ_NL is the nonlinear phase shift experienced by the pulse, and L_eff = (1-exp(-αL))/α (L is the fiber length) is the effective length of the optical fiber and L_NL is the fiber nonlinear length.

The normalized complex pulse envelope U(0,T) for the Ti: Sapphire laser system must be obtained prior to IGA measurements. Second-harmonic generation frequency resolved optical gating (SHG-FROG) [22

K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11(11), 2206–2215 (1994). [CrossRef]

] was used to determine both the intensity and phase of the laser pulse.

Figure 3(a) shows the experimental SHG-FROG trace of laser and Fig. 3(b) shows the intensity and phase retrieved from the FROG trace. The intensity profile was fitted using

I_{n} (\frac{t}{τ_{p}}) = \sec h {(1.763 \frac{t}{τ_{p}})}^{2},

(10)

where I_n is the normalized intensity profile, τ_p ( = 1.763T ₀) is the pulsewidth at full width at half maximum (FWHM). The fit of the intensity profile is shown in Fig. 4 . The data was approximated well by the fit. However, the secant function did not fit along the wings of the intensity profile. Improved fits were obtained using asymmetric hyperbolic secant functions. IGA data fitted using either Eq. (10) or the asymmetric secant function did not have a significant difference in Δϕ_NL values.

Fig. 3 (a) experimental SHG-FROG trace for the 2 ps Ti: sapphire laser system and (b) the intensity and phase derived from FROG trace.

Fig. 4 Intensity profile and hyperbolic secant squared fit.

The results from the above analysis allows U to be approximated by sech(1.763t/τ_p ) and a value for τ _p of 1.8 ps was determined from the fit. Theoretical IGA traces are shown in Fig. 5 for values of Δϕ_NL = 2, 4 and 6. The central peak of IGA traces becomes narrower as the Δϕ_NL increases. Oscillations develop around the central peak as the peak power P _o increases inside the fiber. The oscillations are due to the time-dependent phase shift experienced by the pulse. As the relative delay τ_d between the pulses changes, so does the phase of each of the pulses [Eq. (4)]. Therefore, at specific delay times the total phase of the IGA trace will add or cancel giving rise to peaks and valleys in the trace.

Fig. 5 Simulation of IGA traces (SPM only model) at Δϕ_NL = 2, 4 and 6.

The total number of peaks in an IGA trace depends on the Δϕ_NL . Thus the narrowing and oscillation in an IGA trace are clear signatures of SPM. Once the Δϕ_NL is obtained from the IGA trace, it can be related to γ using

γ = \frac{Δ ϕ_{N L}}{P_{o} \cdot L_{e f f}} .

(11)

The Δϕ_NL is obtained by integrating Eq. (4) and fitting it to the normalized (normalized to one at zero delay) experimental IGA data.

2.2 IGA model with SPM and GVD

Initial IGA experiments with 2 ps pulses were first performed using 15 m long silica fibers as in previous IGA measurements. Figure 6(a) shows an IGA trace for this fiber fitted with the SPM only model. The fit suggests that the IGA trace should have acquired lobes around the central peak. In Fig. 6(b), a new model that includes both SPM and GVD was used to fit the same IGA trace, which produced a much better fit.

Fig. 6 IGA trace of 15 m long silica fiber fitted using (a) SPM only model and (b) SPM and GVD model.

To determine γ using the using this new IGA, Eq. (5) must be solved numerically using the split-step method. Equation (5) is now normalized into a more useful form:

i \frac{\partial U}{\partial ξ} + sgn (β_{2}) \frac{1}{2} \frac{\partial^{2} A}{\partial {τ^{'}}^{2}} + N^{2} {| U |}^{2} U = 0,

(12)

where ξ = z/L_D {L_D = τ_p ²/(1.763² β ₂)}, sgn(β ₂) is a function giving the sign (- or + ) of β ₂, and τ' = 1.763t /τ_p are normalized distance and time variables respectively and the parameter N is given by:

N^{2} = \frac{L_{D}}{L_{N L}} = \frac{γ P_{o} τ_{p}^{2}}{{1.763}^{2} β_{2}} .

(13)

The relative strength of SPM and dispersion on the pulse evolution is related to the value of N. Dispersive effects dominate for values of N ˂˂ 1 and for N ˃˃ 1 SPM dominates. While for values of N ~1 both effects are equally important on the pulse evolution. The effect of dispersion on the IGA measurement is shown in Fig. 7 for a fiber with L_eff equal to 6L_NL . GVD causes the pulse to stretch reducing the P_o inside the fiber. The Δϕ_NL acquired by the pulse will be lower than the case where there is no dispersion. Thus, GVD must be included for pulses of 2 ps in length.

Fig. 7 Theoretical IGA traces with and without dispersion.

To obtain γ using this new IGA model, the parameters N ² and ξ _max must be first determined by fitting IGA data. The parameter ξ _max is related to the effective length of the fiber:

ξ_{\max} = \frac{L_{e f f}}{L_{D}} .

(14)

By substituting for L_D in Eq. (14), a relationship for γ is found and is given by:

γ = \frac{N^{2} ξ_{\max}}{P_{o} \cdot L_{e f f}} .

(15)

This relationship is similar to Eq. (11) and Δϕ_NL is, in this case, equal to N ² ξ _max. The GVD parameter can be determined by solving for β₂ in Eq. (14) to obtain:

β_{2} = \frac{τ_{p}^{2}}{{1.763}^{2}} \frac{ξ_{\max}}{L_{e f f}} .

(16)

3. Experimental discussion and results

Prior to using the IGA setup, the system was calibrated using a fiber with a known γ value. For this task, γ was measured in a single-mode silica fiber (Newport F-SF). Initial IGA measurements were conducted using 15 m, 10 m, 5m, 0.95 m lengths. The parameters of this fiber are located in Table 1 . Figure 8(a) shows an experimental IGA trace for the 0.95 m long fiber. The trace was fitted using the Eq. (4), where U(t) was obtained by numerically solving Eq. (12) using split-step method routine [20

The ssprop code is available from the Photonics Research Laboratory at University of Maryland, College Park at http://www.photonics.umd.edu/software/ssprop

]. This routine was used to develop the new fitting program for IGA measurements that required the addition of GVD. However, GVD for the 0.95 m fiber is minimal. The fitting routine determines values of N ² and ξ _max for each experimental IGA trace. Using the new fitting program, it was deternined that the measured Δϕ_NL ( = N ² ξ_max ) experienced by the pulse was 2.36 for a peak power of 309 Watts. Figure 8(b) shows a plot of the Δϕ_NL vs. peak power for each IGA trace taken for this fiber with a linear regression fit.

Table 1 Fiber parameters

Fiber	F-SF	NL-2.4-800	LMA-20	LMA-25
Hole Diameter (μm)	N/A	2.4	6.11	8.2
Pitch (μm)	N/A	2.9	13	16.4
Mode Field Diameter (μm)	5.6 (@ λ = 830 nm)	1.5 (@ λ = 800nm)	15 ± 1.5 (@ λ = 780 nm)	19.8 ± 2 (@ λ = 850 nm)
Cladding Diameter (μm)	125 ± 12	105 ± 1	230 ± 5	268 ± 5
Numerical Aperture	0.10-0.14	0.19	0.04-0.05	0.032
Cut-off Wavelength (nm)	660-800		None	None
Effective Area (μm²)	24.63	2.8	177	307
Dispersion Coefficient (ps²/km)	~44 [21 R. Kuis, “Theoretical and Experimental Study of the Nonlinear Optical and Dispersive Properties of Conventional and Photonic Crystal Fibers,” Ph.D. dissertation (University of Maryland, Baltimore County, Baltimore, MD 2009). ]	~0	~38 [25 Obtained from www.nktphotonics.com. ]	~29 [25 Obtained from www.nktphotonics.com. ]
Absorption Loss (dB/km)	5 (@λ = 830 nm)	80 (@λ = 800 nm)	<7 (@λ = 800 nm)	8 (@λ = 850 nm)

Fig. 8 (a) Experimental IGA trace for 0.95 m long F-SF fiber using 309 W peak power and (b) plot of the measured N ² ξ_max = Δϕ_NL vs. peak power.

The slope of this line gives a value for the ratio N ² ξ_max/P_o and placing this value in Eq. (15) a value for γ of 7.64 ± 0.5 W⁻¹km⁻¹ was determined. The value n ₂ of bulk silica [23

D. Milam, “Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica,” Appl. Opt. 37(3), 546–550 (1998). [CrossRef]

] was used to obtain a γ (7.6 ± 0.8 W⁻¹km⁻¹) value to compare to the measured F-SF fiber. The measured value was in excellent agreement with the value of bulk glass, where little deviation is expected.

Once the IGA setup was calibrated, the system was used to measure γ in three fibers: a 30-cm length of highly nonlinear PCF (Crystal Fibre NL-2.4-800 [Fig. 9(a) ], and two large mode area PCFs of lengths 4.5-m and 4.9-m (Crystal Fibre LMA-20 [Λ = 13.0 and d/Λ = 0.47] and Crystal Fibre LMA-25 [Λ = 16.4, d/Λ = 0.50] [Fig. 9(b)]). The highly nonlinear fiber parameters are located in Table 1. Due to the central wavelength of the laser (800 nm) coinciding with the zero-dispersion wavelength of the fiber, modulation instability (MI) [24

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59(8), 880–883 (1987). [CrossRef] [PubMed]

] can manifest itself. To reduce MI effects, IGA measurements were conducted at peak power levels that produced less than 1% of power conversion to MI induced Stokes and anti-Stokes fields.

Fig. 9 SEM Images of a) SC-PCF NL-2.4-800 and b) LMA-25 from data sheets.

A typical IGA trace is shown in Fig. 10(a) . The trace was fitted using Eq. (4), where U(t) was obtained from Eqs. (7), (8), and (9). The measured Δϕ_NL ( = L_eff /L_NL ) for this trace was 1.77 for a peak power of 17 Watts. A plot of the Δϕ_NL vs. peak power for this fiber was fitted using linear regression and is shown in Fig. 10(b). The measured γ was 74 ± 2.9 W⁻¹km⁻¹ and is within 2% of the theoretical value of 70 W⁻¹km⁻¹ [25

Obtained from www.nktphotonics.com.

Fig. 10 (a) Experimental IGA trace with fit for a 30-cm long NL-2.4-800 fiber with 17 Watts of peak power and (b) plot of

Δ φ_{N L}

vs. P_o for each IGA trace with linear regression fit, where the slope of the line is ∝ γ.

IGA traces in the large mode area PCFs where fitted using Eq. (13) since GVD can influence pulse propagation at the input wavelengths used (800 nm for LMA-20 and 850 nm for LMA-25). Their respective β ₂ values (38 ps²/km for LMA-20 and 29 ps²/km for LMA-25) are shown in Table 1. Figure 11(a) shows an IGA trace obtained for a peak power of 259 W and the fit to the trace determined a value of 1.23 for the Δϕ_NL . A value of 1 ± 0.1W⁻¹km⁻¹ was measured for γ [Fig. 11(b)]. Since both fibers are made from pure silica glass, a similar comparison as with the F-SF fiber can be applied here. Using the A_eff value of LMA-20 and the n ₂ of bulk silica a value of 0.98 ± 0.1 1W⁻¹km⁻¹ is obtained for γ. IGA measurements in fiber LMA-25 required tuning the wavelength of the laser system to 850 nm from 800 nm due to the high confinement loss at 800 nm. Figure 11(c) shows an IGA trace for this fiber and Fig. 11(d) shows the plot of the nonlinear phase shift vs. peak power for all the IGA measurements. At this wavelength γ was found to be 0.66 ± 0.08 W⁻¹km⁻¹. The estimated γ value using the n₂ of silica glass is 0.59 ± 0.6 W⁻¹km⁻¹.

Fig. 11 (a) Experimental IGA trace with fit for a 4.5-m long LMA-20 fiber with 259Watts of peak power, (b) plot of

Δ φ_{N L}

vs. P_o for each IGA trace with linear regression fit, (c) experimental IGA trace with fit for a 4.9-m long LMA-25 fiber with 600 Watts of peak power, and (d) plot of

Δ φ_{N L}

vs. P_o for each IGA trace with linear regression fit.

Due to the sensitivity of the IGA measurements to dispersion when 2 ps pulses are involved, attempts were made to determine both γ and β ₂ simultaneously from the IGA trace. Experiments for this purpose began with the F-SF fiber. IGA data obtained during the IGA setup calibration for fiber lengths of 15 m, 10 m, and 5 m were used. The IGA traces were then fitted to determine both N ² and ξ _max as in the case of the 0.95-m long fiber. However, ξ _max was not a fitting parameter in the 0.95-m long fiber case since a value for β ₂ was used from Table 1. Since the value ξ _max is dependent on the values of L_eff , T_o and β ₂, ξ _max should remain constant for IGA traces taken for a particular fiber length. The fiber length of 15 m was the fiber length with values of ξ _max with small differences (less than 13% from the highest to the lowest ξ _max value) from IGA trace to IGA trace. Values of ξ _max varied randomly (more than 100% in some cases) for the fibers of length 5 and 10 m. The GVD induced changes to the pulse propagation at these fibers lengths are in the range where the fitting routine cannot extract a value for ξ _max without large errors as in the case of 15 m. Therefore, with the present experimental IGA setup and fitting program, it is only possible to measure γ and β ₂ with fiber lengths greater than 15 m. It may, still be possible to measure γ and β ₂ in shorter lengths of fiber if the value of β ₂ is 2 to 3 times larger than the typical value for silica fibers; however, the limits on the measurement are yet to be determined. From the IGA fits obtained at 15-m [Fig. 12(a) ], values were obtained for both γ and β ₂ of 7.99 ± 0.4 W⁻¹km⁻¹ [Fig. 12(b)] and 46.5 ± 1 ps²/km, respectively.

Fig. 12 (a) Experimental IGA trace with fit for a 15-m long F-SF fiber with 27 Watts of peak power and (b) plot of

Δ φ_{N L}

vs. P_o for each IGA trace with linear regression fit, where the slope of the line is ∝ γ .

4. Conclusion

Measurements of γ conducted in a highly nonlinear PCF (NL-2.4-800) and two large mode area PCFs (LMA-20 and LMA-25) were conducted using the IGA technique. This technique is based on time resolved two-beam coupling in a photorefractive crystal. The measured γ values were compared to theoretical values calculated using the n ₂ from bulk silica glass. The measured values deviated less than 5% from the calculated values, except in the case of the LMA-25 fiber, where its measured value deviated less than 12%. Due to the sensitivity of the IGA measurements to GVD, attempts were made to simultaneously determine both γ and β ₂ from an IGA trace. Results for the 15-m long single-mode silica glass fiber deviated less than 6% from typical values for both γ and β ₂. Table 2 summarizes the results obtained for all the fibers measured.

Table 2 Summary of results

Fiber	Fiber Length (m)	A_eff (μm²)	Measured γ (W⁻¹km⁻¹)	Calculated γ (W¹km⁻¹)	Measured β ₂(ps²/km)	Accepted β ₂(ps²/km)
F-SF	0.95	24.63	7.64 ± 0.5	7.6 ± 0.8	—	~44 [ 21 R. Kuis, “Theoretical and Experimental Study of the Nonlinear Optical and Dispersive Properties of Conventional and Photonic Crystal Fibers,” Ph.D. dissertation (University of Maryland, Baltimore County, Baltimore, MD 2009). ]
F-SF	15	24.63	7.5 ± 0.18	7.6 ± 0.8	46.5 ± 1	~44 [ 21 R. Kuis, “Theoretical and Experimental Study of the Nonlinear Optical and Dispersive Properties of Conventional and Photonic Crystal Fibers,” Ph.D. dissertation (University of Maryland, Baltimore County, Baltimore, MD 2009). ]
NL-2.4-800	0.30	2.8	74 ± 2.9	70 ± 0.7	—	~0 [ 25 Obtained from www.nktphotonics.com. ]
LMA-20	4.5	177	1 ± 0.1	0.98 ± 0.1	—	38 [ 25 Obtained from www.nktphotonics.com. ]
LMA-25	4.9	307	0.66 ± 0.08	0.59 ± 0.06	—	29 [ 25 Obtained from www.nktphotonics.com. ]

References and links

1.	J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “Pure silica single-mode fiber with hexagonal photonic crystal cladding,” Conf. Optical Fiber Commun. (OFC) San Jose, CA, (1996).
2.	J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996). [CrossRef] [PubMed]
3.	J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A 15(3), 748–752 (1998). [CrossRef]
4.	K. Saitoh and M. Koshiba, “Numerical modeling of photonic crystal fibers,” J. Lightwave Technol. 23(11), 3580–3590 (2005). [CrossRef]
5.	T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22(13), 961–963 (1997). [CrossRef] [PubMed]
6.	N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, “Nonlinearity in holey optical fibers: measurement and future opportunities,” Opt. Lett. 24(20), 1395–1397 (1999). [CrossRef]
7.	A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, ““Direct continuous-wave measurement of n₂ in various types of telecommunication fiber at 1.55 µm,” Opt. Lett. 21, 1966–1968 (1996). For an additional measurement approach in single-mode fiber see for example, C. Vinegone, M. Wegmuller, and N. Gisin, “Measurements of the Nonlinear Coefficient of Standard SMF, DSF, and DCF Fibers Using a Self-Aligned Interferometer and a Faraday Mirror,” IEEE Photon. Technol. Lett. 13, 1337–1339 (2001). [CrossRef] [PubMed]
8.	C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, “Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers,” Opt. Express 15(6), 3507–3512 (2007). [CrossRef] [PubMed]
9.	H. Garcia, A. M. Johnson, F. A. Oguama, and S. Trivedi, “New approach to the measurement of the nonlinear refractive index of short (<25 m) lengths of silica and erbium-doped fibers,” Opt. Lett. 28(19), 1796–1798 (2003). [CrossRef] [PubMed]
10.	F. A. Oguama, H. Garcia, and A. M. Johnson, “Simultaneous measurement of the Raman gain coefficient and the nonlinear refractive index of optical fibers: theory and experiment,” J. Opt. Soc. Am. B 22(2), 426–436 (2005). [CrossRef]
11.	A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, and D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting, Vol. 11 of 1988 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 128.
12.	A. M. Johnson, A. M. Glass, W. M. Simpson, and D. H. Olson, “Infrared picosecond pulse diagnostics using photorefractive beam coupling,” in Conference on Lasers and Electro-Optics, Vol. II of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 226.
13.	A. M. Johnson, W. M. Simpson, A. M. Glass, M. B. Klein, D. Rytz, and R. Trebino, “Infrared picosecond pulse correlation measurements using photorefractive beam coupling and harmonic generation in KNbO3 and BaTiO3,” in OSA Annual Meeting, Vol. 18 of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 53.
14.	V. Dominic, X. S. Yao, R. M. Pierce, and J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56(6), 521–523 (1990). [CrossRef]
15.	X. S. Yao, V. Dominic, and J. Feinberg, “Theory of beam coupling and pulse shaping of mode-locked laser pulses in a photorefractive crystal,” J. Opt. Soc. Am. B 7(12), 2347–2355 (1990). [CrossRef]
16.	F. P. Strohkendl, J. M. C. Jonathan, and R. W. Hellwarth, “Hole - electron competition in photorefractive gratings,” Opt. Lett. 11(5), 312–314 (1986). [CrossRef] [PubMed]
17.	H. Garcia, “Time domain measurement of the nonlinear refractive index in optical fibers and semiconductors films,” Ph.D. dissertation (New Jersey Institute of Technology, Newark, NJ, 2000).
18.	F. A. Oguama, A. M. Johnson, and W. A. Reed, “Measurement of the nonlinear coefficient of telecommunication fibers as a function of Er, Al, and Ge doping profiles by using the photorefractive beam-coupling technique,” J. Opt. Soc. Am. B 22(8), 1600–1604 (2005). [CrossRef]
19.	G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).
20.	The ssprop code is available from the Photonics Research Laboratory at University of Maryland, College Park at http://www.photonics.umd.edu/software/ssprop
21.	R. Kuis, “Theoretical and Experimental Study of the Nonlinear Optical and Dispersive Properties of Conventional and Photonic Crystal Fibers,” Ph.D. dissertation (University of Maryland, Baltimore County, Baltimore, MD 2009).
22.	K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11(11), 2206–2215 (1994). [CrossRef]
23.	D. Milam, “Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica,” Appl. Opt. 37(3), 546–550 (1998). [CrossRef]
24.	G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59(8), 880–883 (1987). [CrossRef] [PubMed]
25.	Obtained from www.nktphotonics.com.

OCIS Codes
(060.2300) Fiber optics and optical communications : Fiber measurements
(060.2310) Fiber optics and optical communications : Fiber optics
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: November 18, 2010
Revised Manuscript: January 12, 2011
Manuscript Accepted: January 13, 2011
Published: January 14, 2011

Citation
Robinson Kuis, Anthony Johnson, and Sudhir Trivedi, "Measurement of the effective nonlinear and dispersion coefficients in optical fibers by the induced grating autocorrelation technique," Opt. Express 19, 1755-1766 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-1755

Sort: Author | Year | Journal | Reset

References

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “Pure silica single-mode fiber with hexagonal photonic crystal cladding,” Conf. Optical Fiber Commun. (OFC) San Jose, CA, (1996).
J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996). [CrossRef] [PubMed]
J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A 15(3), 748–752 (1998). [CrossRef]
K. Saitoh and M. Koshiba, “Numerical modeling of photonic crystal fibers,” J. Lightwave Technol. 23(11), 3580–3590 (2005). [CrossRef]
T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22(13), 961–963 (1997). [CrossRef] [PubMed]
N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, “Nonlinearity in holey optical fibers: measurement and future opportunities,” Opt. Lett. 24(20), 1395–1397 (1999). [CrossRef]
A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, ““Direct continuous-wave measurement of n2 in various types of telecommunication fiber at 1.55 µm,” Opt. Lett. 21, 1966–1968 (1996). For an additional measurement approach in single-mode fiber see for example, C. Vinegone, M. Wegmuller, and N. Gisin, “Measurements of the Nonlinear Coefficient of Standard SMF, DSF, and DCF Fibers Using a Self-Aligned Interferometer and a Faraday Mirror,” IEEE Photon. Technol. Lett. 13, 1337–1339 (2001). [CrossRef] [PubMed]
C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, “Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers,” Opt. Express 15(6), 3507–3512 (2007). [CrossRef] [PubMed]
H. Garcia, A. M. Johnson, F. A. Oguama, and S. Trivedi, “New approach to the measurement of the nonlinear refractive index of short (<25 m) lengths of silica and erbium-doped fibers,” Opt. Lett. 28(19), 1796–1798 (2003). [CrossRef] [PubMed]
F. A. Oguama, H. Garcia, and A. M. Johnson, “Simultaneous measurement of the Raman gain coefficient and the nonlinear refractive index of optical fibers: theory and experiment,” J. Opt. Soc. Am. B 22(2), 426–436 (2005). [CrossRef]
A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, and D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting, Vol. 11 of 1988 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 128.
A. M. Johnson, A. M. Glass, W. M. Simpson, and D. H. Olson, “Infrared picosecond pulse diagnostics using photorefractive beam coupling,” in Conference on Lasers and Electro-Optics, Vol. II of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 226.
A. M. Johnson, W. M. Simpson, A. M. Glass, M. B. Klein, D. Rytz, and R. Trebino, “Infrared picosecond pulse correlation measurements using photorefractive beam coupling and harmonic generation in KNbO3 and BaTiO3,” in OSA Annual Meeting, Vol. 18 of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 53.
V. Dominic, X. S. Yao, R. M. Pierce, and J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56(6), 521–523 (1990). [CrossRef]
X. S. Yao, V. Dominic, and J. Feinberg, “Theory of beam coupling and pulse shaping of mode-locked laser pulses in a photorefractive crystal,” J. Opt. Soc. Am. B 7(12), 2347–2355 (1990). [CrossRef]
F. P. Strohkendl, J. M. C. Jonathan, and R. W. Hellwarth, “Hole - electron competition in photorefractive gratings,” Opt. Lett. 11(5), 312–314 (1986). [CrossRef] [PubMed]
H. Garcia, “Time domain measurement of the nonlinear refractive index in optical fibers and semiconductors films,” Ph.D. dissertation (New Jersey Institute of Technology, Newark, NJ, 2000).
F. A. Oguama, A. M. Johnson, and W. A. Reed, “Measurement of the nonlinear coefficient of telecommunication fibers as a function of Er, Al, and Ge doping profiles by using the photorefractive beam-coupling technique,” J. Opt. Soc. Am. B 22(8), 1600–1604 (2005). [CrossRef]
G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).
The ssprop code is available from the Photonics Research Laboratory at University of Maryland, College Park at http://www.photonics.umd.edu/software/ssprop
R. Kuis, “Theoretical and Experimental Study of the Nonlinear Optical and Dispersive Properties of Conventional and Photonic Crystal Fibers,” Ph.D. dissertation (University of Maryland, Baltimore County, Baltimore, MD 2009).
K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11(11), 2206–2215 (1994). [CrossRef]
D. Milam, “Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica,” Appl. Opt. 37(3), 546–550 (1998). [CrossRef]
G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59(8), 880–883 (1987). [CrossRef] [PubMed]
Obtained from www.nktphotonics.com .

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper