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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 β 2 simultaneously.
©2011 Optical Society of America
1. Introduction
Photonic crystal fibers (PCFs) have revolutionized the field of nonlinear optics since their introduction in the mid 1990’s [1,2]. 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 [3,4] 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]. 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] first measured γ in PCFs at 1550 nm using a method devised by Boskovic et al [7]. 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]. 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 [9,10] 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 [11–15]. Mutually coherent pulses (Beam 1 and Beam 2) intersect and produce an interference pattern (bright and dark regions) in a photorefractive crystal (Fig. 1 ).
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 (ESC) [15]:
where a and b are functions that depend on material parameters and beam geometry [16] 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 Tp (), τd is a time delay between Beam 1 and Beam 2, and Io = |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]:where is the induced change to the dielectric tensor by the space-charge field, is the relative dielectric tensor of the medium, is the electro-optic tensor of the crystal, and 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: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 [9,10,17,18]. 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.
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:
In Eq. (4), U(t) is the normalized complex electric field envelope and at τd = 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]: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: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 Aeff is the effective area of the fiber [19]. The length where the nonlinearities become important is called the nonlinear length LNL = 1/ (P 0 γ). The fiber length where dispersion becomes relevant is termed the dispersion length LD = 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] at three pulsewidths: 30 ps, 10 ps, and 2 ps. A freely available Matlab routine [20] 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].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 LD 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:
where ϕNL is the time-dependent phase generated through SPM, ΔϕNL is the nonlinear phase shift experienced by the pulse, and Leff = (1-exp(-αL))/α (L is the fiber length) is the effective length of the optical fiber and LNL 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] 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
where In is the normalized intensity profile, τp ( = 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 ΔϕNL 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.
The results from the above analysis allows U to be approximated by sech(1.763t/τp) 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 ΔϕNL = 2, 4 and 6. The central peak of IGA traces becomes narrower as the ΔϕNL 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 τd 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.
The total number of peaks in an IGA trace depends on the ΔϕNL. Thus the narrowing and oscillation in an IGA trace are clear signatures of SPM. Once the ΔϕNL is obtained from the IGA trace, it can be related to γ using
The ΔϕNL 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:
where ξ = z/LD {LD = τp2/(1.7632 β 2)}, sgn(β 2) is a function giving the sign (- or + ) of β 2, and τ' = 1.763t /τp are normalized distance and time variables respectively and the parameter N is given by: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 Leff equal to 6LNL. GVD causes the pulse to stretch reducing the Po inside the fiber. The ΔϕNL 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.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:
By substituting for LD in Eq. (14), a relationship for γ is found and is given by:This relationship is similar to Eq. (11) and ΔϕNL 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: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]. 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 ΔϕNL ( = N 2 ξmax) experienced by the pulse was 2.36 for a peak power of 309 Watts. Figure 8(b) shows a plot of the ΔϕNL vs. peak power for each IGA trace taken for this fiber with a linear regression fit.
Table 1. Fiber parameters
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 ξmax = ΔϕNL vs. peak power.
The slope of this line gives a value for the ratio N 2 ξmax/Po 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] 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] 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.
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 ΔϕNL ( = Leff/LNL) for this trace was 1.77 for a peak power of 17 Watts. A plot of the ΔϕNL 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].
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 vs. Po 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 ΔϕNL. 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 Aeff 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 vs. Po 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 vs. Po 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 Leff, To 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 vs. Po 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
References and links
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