Abstract

We measured, within 6% accuracy, the nonlinear refractive index (n2) of 10 meter long multimode silica fiber of 17μm core diameter, using a modified induced grating autocorrelation technique (IGA). This measurement technique, based on time-delayed two beam coupling in a photorefractive crystal has been used to accurately measure n2 in short lengths of single mode fibers. For the first time to our knowledge, IGA is used to measure n2 of multimode fiber with a passively modelocked Nd:YVO4 laser operating with a dual-line near 1342 nm.

© 2015 Optical Society of America

1. Introduction

In the past decades, nonlinear effects in silica based single mode fibers have been intensively investigated. Currently, these nonlinearities can be measured with several measurement techniques and they can be managed in fiber telecommunication industries. Despite the impressive progress made on single mode fibers, the study of ultrashort pulse propagation and nonlinear phenomena in multimode fibers has been neglected, mainly because of the complex interactions between the modes during laser pulse propagation.

As laser power continues to grow, multimode fibers have become the ideal carrier for delivering high optical power in industrial, medical and research fields. They have great potential for information transmission by exploiting the spectral and the two spatial degrees of freedom to increase the number of transmission channels [1].

Recently, the interest in generating supercontinuum light utilizing highly nonlinear materials such as photonic crystal fibers has encouraged the study of mode interactions and the nonlinear dynamics occurring in multiple mode light carriers such as multimode optical fibers. A generalized nonlinear Schrodinger equation (GNLSE) has been developed to investigate Kerr and Raman nonlinearities, high order dispersion, and mode coupling in multimode fibers [2]. Furthermore, it has been demonstrated that during propagation in multimode fibers the pulse energy is distributed only among the lowest modes [3].

In this paper, we numerically derived the total electric field of a passively modelocked dual-line Nd:YVO4 laser pulse subjected to self-phase modulation during propagation through 10 meters of 17 μm core diameter multimode fiber, and we determine the nonlinear refractive index of the fiber by using the IGA technique [4–10].

2. Theory

The electric field of laser pulses traveling through a multimode fiber can be represented by:

E(r,z,φ,t)=jFj(r,φ)Aj(z,t)exp(iβ0j)
where β0j are the propagation constants, the transverse mode functions Fj(r,φ) are written in cylindrical coordinates as Fj(r,φ)=Fj(r)exp(iljφ) where lj are integers and φ the azimuthal coordinate. Fj(r) are solutions of Maxwell’s equations for the weakly guiding approximation for step index fibers [11–13].
r22F/r2+rF/r+(U2r2/a2l2)F=0for0<r<a(core)
r22F/r2+rF/r+(W2r2/a2l2)F=0forr>a(cladding)
where a is the radius of the fiber core.
U=a(k02n12β2)1/2andW=a(β2k02n22)1/2
The solutions of Eq. (2) are the transverse mode functions given by:
F(r,φ)=AJl(U)Jl(Ura)[cos(lφ)sin(lφ)]0<r<a(core)
F(r,φ)=AKl(W)Kl(Wra)[cos(lφ)sin(lφ)]r>a(cladding)
where A is the fiber core area and Jl are Bessel functions and Kl the modified Bessel functions. The modal envelope function terms Aj in Eq. (1) are solutions of the GNLSE for multimode fiber (MM-GNLSE) [2].
Ap(z,t)/z=i(β0(p)β0)Ap(z,t)(β1(p)β1)Ap(z,t)/tin2βn(p)n!(i/t)nAp(z,t)+(in2ω0/c)(1+iτ0/t)l,m,n{Qplmn(ω0)[2(1fR)Al(z,t)Am(z,t)An(z,t)+3fRAl(z,t)dτh(τ)Am(z,tτ)An(z,tτ)]}
where ω0, β0 and β1 are parameters of the fundamental mode, βn(p)=nβ(p)/ωn|ω=ω0 ƒR is the functional contribution of the Raman response, h(τ) the delayed Raman response function and n2 is the nonlinear index of refraction of the fiber. Qplmn are the overlap integrals. To measure the nonlinear refractive index of optical fibers with the IGA technique, moderate power level laser light is coupled into the multimode fiber under investigation. Under these conditions of moderate power level, the electric field of the input optical pulse is only subjected to phase modulation. The total self-phase modulation strength (SPMS) can be expressed as a sum of the self-phase modulation strengths (spms) of the individual electric field modes [3].
SPMS=j|Aj|2(spms)j/j|Aj|2
This total self-phase modulation strength is determined by comparing the experimental IGA data to the theoretical expression of the IGA signal given by [4–10]:
Wdet|K(τ)|2=|+E(r,z,φ,t)E(r,z,φ,t+τ)dt|2
where τ is the time delay, the electric field E(r,z,φ ,t) is expressed in terms of the solutions of Eqs. (5), (6), (7). The nonlinear refractive index n2 is determined from the relation [5–10]:
n2/Aeff=(nτpcλ/tRL)[32π2ln2/π107]1(SPMS/Pavg)
where Aeff is the effective area of the fiber, n the linear refractive index, τp is the laser pulsewidth, tR the laser repetition period, L the fiber length and Pavg the average power in the fiber.

3. Experiment

The IGA experiment is performed by coupling the light of a well characterized laser source into a relatively short length of fiber sample (10 meters or less). Figure 1 shows the layout of the experiment. The laser light coupling is achieved with a 10X microscope objective mounted on a XYZ stage. The bare fiber input is held in a fiber chuck mounted in front of the microscope objective. The laser beam from the output end of the fiber is collected and collimated with another microscope objective lens. The beam is then split in a modified Michelson interferometer, with one arm (pump) of fixed optical delay and the other arm (probe) is delayed by a stepper motor-retroreflector combination. The probe beam is also mechanically chopped at 1.3 KHz. The two beams are then focused at an appropriate intersection angle (2θ) onto a photorefractive crystal (CdMnTe:V) with a pair of AR-coated fused silica lenses where they interfere and generate an interference pattern. The interference pattern causes an optically induced redistribution of charges which gives rise to an internal electric field. The electric field modulates the refractive index of the crystal by virtue of the linear electro-optic effect. The index modulation creates a photorefractive grating from which a portion of signal is transferred from the probe into the pump beam. The transferred signal is detected by a near-infrared photodetector placed in the path of the pump beam. The photodetector is connected to a lock-in amplifier with the reference frequency provided by the chopper. The output voltage of the lock-in amplifier is termed IGA signal. This signal is collected as a function of the time delay between the pump and the probe beams using a IEEE488 bus and a Labview program.

 figure: Fig. 1

Fig. 1 IGA experimental setup consisting of time-delayed two-beam coupling in CdMnTe:V photorefractive crystal.

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The IGA technique has been used previously to accurately measure the nonlinear index of refraction (n2) of several silica fiber samples. These measurements were conducted with a 1064 nm Nd: YAG pulsed laser of about 50 psec pulsewidth at 100 MHz repetition rate [5–9]. In this work, we used a 1342 nm pulsed laser. The gain material is a Neodymium-doped Yttrium Vanadate crystal (Nd:YVO4) pumped with a 808 nm cw diode-laser bar. The laser cavity contains six mirrors and a semiconductor saturable absorber mirror (SESAM) placed at one end of the folded cavity to generate a train of 10 psec laser pulses at a 76 MHz repetition rate.

3.1 Nonlinear refractive index (n2) of single mode fiber with the IGA technique

The use of IGA techniques requires a well-characterized laser in order to accurately determine the n2 of optical fibers. Through reliance upon the specifications provided by the laser manufacturer, combined with the laser pulsewidth determination using the background free second harmonic generation autocorrelation (SHGA) technique and the laser spectrum obtained with a 0.1 nm resolution Optical Spectrum Analyzer (OSA), we define the temporal profile of the pulsed laser electric field to be a Gaussian field:

E(t)=E0exp(2ln2(t/τp)2i(ω0t+ϕ))
where E0 is the electric field amplitude, τp the pulsewidth, ω0 the angular frequency, and is ϕ the phase. With this assumption, we tested Corning fiber SMF-28, a well known single mode silica fiber in the telecommunications industry. Its mode field diameter (MFD) is 9.50 μm, measured at 1330 nm. The cladding diameter is 250 μm and the length of the fiber sample is 10 meters. The laser pulsewidth τp, measured by SHGA technique was 9.64 psec.

Figure 2 shows the theoretical IGA signal fitted to experimental IGA signal in the case of 500 mW average laser power, coupled into 10 meters of fiber.

 figure: Fig. 2

Fig. 2 Fitted IGA experimental data to early version of the theoretical IGA model.

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The above fitting figure shows a clear deviation between the theory and the experimental IGA signal. We fitted five more sets of experimental IGA signals to the theoretical IGA signal. The resulting self-phase modulation strengths (SPMS) are plotted as function of the power in Fig. 3.

 figure: Fig. 3

Fig. 3 Linear fitting of SPMS versus average power for 10 meter long SMF- 28 fiber.

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The slope of the linear regression is used in Eq. (10) to determine the value of n2 which is (6.22±0.54)×1017cm2/W. This value is about four times smaller than the published value of the nonlinear refractive index of silica fiber2.53×1016cm2/W [5]. The misfit of data as shown in Fig. 2, and the erroneous n2 values obtained from the analysis are the results of the inaccurate characterization of the laser pulse electric field.

3.2 Determination of the 1342 nm Nd:YVO4 laser electric field

We conducted a thorough investigation of the laser electric field spectrum with a high resolution optical spectrum analyzer (ANDO AQ6317). Figure 4 shows the laser spectral response graph in the frequency space. The two peaks centered at 1403.80×1012rad/sec and at 1403.60×1012rad/sec corresponding to 1342.08 nm and 1342.25 nm respectively. The dominant peak is about 3 times stronger than the second peak. However, the two peaks remain very distinctive.

 figure: Fig. 4

Fig. 4 OSA laser frequency response and fit to a combination of a hyperbolic secant and a Gaussian pulse temporal profile.

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We derived the electric field of the laser pulses by fitting the spectral intensity trace to the theoretical expressions of the spectral intensities of two temporally overlapping pulses with their center wavelengths corresponding to the two lines.

To obtain a good representation of the laser pulse’s electric field, we performed several fittings in frequency space with Gaussian and hyperbolic secant pulses. The best fit is obtained with a combination of a strong hyperbolic secant pulse, and a small Gaussian pulse Fig. 4. The electric field of the laser pulse is obtained by taking the Fourier transform of the expression obtained from the fitting:

E(t)=E1(t)+E2(t)=E01sech(2ln(1+2)t/τp)exp(iω01t)+E02exp(2ln2×(t/τp)2iω02t)
where E01 and E02 are the amplitudes of the hyperbolic secant and the Gaussian fields respectively; ω01 and ω02 are their angular frequencies, and τp the pulse width.

We subsequently modified the original IGA model to include the new electric field given by Eq. (12), and the new model is utilized to determine the nonlinear refractive indices of the optical fibers, in the research. Figure 5 shows the theoretical expression of the modified IGA signal fitted to the experimental IGA signal, in the case of 500mW average laser power coupled into 10 meters of Corning SMF-28 fiber.

 figure: Fig. 5

Fig. 5 Experimental fit of IGA signal to theory for 500 mW average power in 10 meters of Corning single mode silica fiber SMF-28.

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We notice a great improvement in the fitting of the IGA data to the modified IGA model, by comparing the insets of Figs. 2 and 5. For the same input average power of 500 mW, there is an increase in the SPMS, from 5.38 to 20.59.

Six sets of experimental IGA signal data were collected at different average power levels and fitted with the modified theoretical IGA signal. The values of the SPMS obtained from the fitting are plotted as a function of the average powers. Figure 6 shows the linear regression of the plot and the value of the slope.

 figure: Fig. 6

Fig. 6 Experimental fit of IGA signal to theory for 500 mW average power in 10 meters of single mode silica fiber SMF-28.

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Utilizing the slope of the linear regression (SPMS/Pavg) in Eq. (10), the calculated value of n2 of the single mode silica fiber (SMF-28) is: (2.49±0.07)×1016cm2/W. This n2 value is within 2% range of 2.53×1016cm2/W , the expected value of the nonlinear refractive index of silica glass optical fiber [5].

Comparing Figs. 3 and 6, we see an increase in the slope from 10.14 to 40.76. This increase changes the value of n2 from(6.22±0.54)×1017cm2/Wto (2.49±0.07)×1016cm2/W.

The improved accuracy in the measurement of n2 using the modified IGA is directly attributable to a more accurate description of the electric field of the laser pulse.

It is worth noting the sensitivity of IGA measurements to the temporal profile of the laser pulses electric field. The knowledge of the pulse shape and its phase are necessary for accurate determination of n2 using IGA technique [4,5]. Whether the pulse shape is Gaussian or hyperbolic-secant the peak intensity (Peak/Aeff) in the core of the fiber has a different expression [11,12]. n2 being derived from that expression, it is therefore imperative to know the temporal shape of the laser pulse. Many approaches exist for defining the temporal profile of ultra-short pulses [13], among them, the frequency-resolved optical gating (FROG) which can provide both the laser field envelope and its phase [14].

The second fiber sample tested with the modified IGA technique was the single mode fiber CS-97-2623 manufactured by 3M. The n2 value obtained from the IGA measurement on 5 meters of CS-97-2623 is(2.36±0.13)×1016cm2/W. The n2 value is within 7% of the published value of2.53×1016cm2/W [5]. These accurate modified IGA measurements were performed on two distinct silica single-mode fibers from different manufacturers and serves as a calibration of this technique. As a result, we have confidence to move forward and test this technique on a multimode silica glass fiber, FUD-3539 manufactured by Nufern.

The spatial profiles of the laser beam from the output of a meter long single mode fiber SFM-28 and multimode fiber FUD-3539 are shown in Figs. 7 and 8 respectively.

 figure: Fig. 7

Fig. 7 Beam profile of single mode fiber SMF-28 9.5 µm core diameter, 1m long.

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 figure: Fig. 8

Fig. 8 Beam profile of multimode fiber FUD-3539 17.0 μm core diameter, 1m long.

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The core diameter of the multimode fiber is 17μm, the cladding diameter is 140 μm and the numerical aperture (NA) is 0.08. The number of modes propagating in a multimode fiber is determined by the V number defined by V#=k0dNA, where k0 is the wavenumber and d the fiber core diameter. The V# of FUD-3539 is 2.99 at λ = 1342 nm. This number is greater than 2.405, which classifies the fiber as multimode fiber. It carries the first two lower transverse modes (LP01 and LP11) at 1342 nm [15–17].

LP01F1=er2a2[11]andLP11F2=J1(Ura)[cos(φ)sin(φ)]
For two modes case, the GNLSE is reduced to two coupled mode equations:
A1/z+β1(1)A1/t+0.5jβ2(1)2A1/2t+0.5αA1=[(f11spms1|A1|2+2f12spms|A2|22)A1]
A2/z+β2(2)A1/t+0.5jβ2(2)2A2/2t+0.5αA2=[(f22spms2|A2|2+2f21spms|A1|12)A2]
where fii=02π0+|Fi|4rdrdφ/(02π0+|Fi|2rdrdφ)2, spmsi are the self-phase modulation strengths of the fields (i = 1,2).

The solution of Eqs. (14) and (15) combined with Eq. (13), provides the electric field of the pulses at the output of the fiber as expressed in Eq. (1). The determined electric field is used in Eq. (9) to find the theoretical expression of the detected IGA signal [4–10]. To determine the nonlinear refractive index n2, or the nonlinear coefficient n2/Aeff we find the SPMS by fitting the experimental IGA data to the theoretical expression of the IGA signal. The IGA data are collected at several average power levels. The slope of the plot of SPMS versus the average power gives the ratio (SPMS/Pavg) which is used in Eq. (10) to determine n2 or n2/Aeff if Aeff is unknown. For multimode fibers the effective area Aeffcan be estimated as the fiber core area [18]. Therefore the core area of the multimode fiber FUD-3539, which value is:226.98×108cm2 was used as theAeffin this experiment.

Figure 9 shows the fitting of the experimental IGA signal to theoretical IGA signal for an average power of 260 mW measured at the output of 10 meters of Nufern multimode mode silica fiber FUD-3539.

 figure: Fig. 9

Fig. 9 Experimental fit of IGA signal to theory for 250 mW average power in 10 meters of Nufern multimode silica fiber FUD-3539.

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Seven sets of experimental IGA signals collected as a function of the output average power are fitted to the theoretical IGA signal given in Eq. (9). The resulting SPMS are plotted as a function of the power in Fig. 10.

 figure: Fig. 10

Fig. 10 SPMS versus average power plot and the slope of the linear fitting for 10 meters of Nufern multimode mode silica fiber FUD-3539.

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The n2 value obtained by using the above slope in the Eq. (10) is (2.68±0.17)1016cm2/W which is within 6% of the accepted value of n2 of silica glass fibers.

4. Conclusion

We have measured the n2 of the multimode silica fiber FUD-3539. The value obtained is within 6% of the accepted value for silica glass fibers [5].

The good agreement indicates that the modified IGA measurement using a dual-line (1342.25 nm & 1342.08 nm) 10 psec duration Nd: YVO4 laser (76 MHz repetition rate) is reliable for measuring n2 of optical fibers. This is the first time IGA technique has been used to investigate the nonlinear refractive index of a multimode fiber by modifying the standard IGA model designed for single mode fibers. The technique based on time resolved of two-beam coupling in photorefractive crystals has again proven to be a reliable tool for investigating nonlinearities in short length (10 meters or less) single mode and moderate multimode fibers.

Multimode nonlinear Schrodinger equation (MM-NLSE) has been derived [2]; it describes the propagation of ultra-short laser pulse in multimode fibers, including all the nonlinear effects known in single mode fibers. It has also been shown that only few of the lowest order modes dominate the nonlinear phenomena in multimode fibers. The electric fields of these modes can be determined by fitting experimental spectral broadening data to its theoretical expressions [3]. Following the same approach for finding the electric fields of these dominant modes, the modified IGA technique can be used to determine the n2 of multimode silica fibers with core diameter as large as 100 μm; assuming the moderate power level approximation is used, meaning only SPM is the nonlinear effect present in the fiber. In the presence of the SPM and SRS, the numerical solutions of MM-NLSE may be used in Eq. (9) to determine the theoretical expression of the IGA signal [6]. Fitting the experimental IGA data to the theoretical expression should provide simultaneously the nonlinear refractive index (n2) and the Raman gain coefficient (gR) of the multimode fiber under characterization. In the future, we will explore the use of IGA technique to simultaneously measure n2 and gR in multimode fibers.

References and links

1. R. Rokitski and S. Fainman, “Propagation of ultrashort pulses in multimode fiber in space and time,” Opt. Express 11(13), 1497–1502 (2003). [CrossRef]   [PubMed]  

2. F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,” J. Opt. Soc. Am. B 25(10), 1645–1654 (2008). [CrossRef]  

3. Q. Z. Wang, D. Ji, L. Yang, P. P. Ho, and R. R. Alfano, “Self-phase modulation in multimode optical fibers produced by moderately high-powered picosecond pulses,” Opt. Lett. 14(11), 578–580 (1989). [CrossRef]   [PubMed]  

4. A. M. Levine, E. Ozizmir, R. Trebino, C. C. Hayden, A. M. Johnson, and K. L. Tokuda, “Induced-grating autocorrelation of ultrashort pulses in a slowly responding medium,” J. Opt. Soc. Am. B 11(9), 1609–1618 (1994). [CrossRef]  

5. 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]  

6. 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]  

7. 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]  

8. H. Garcia, A. M. Johnson, F. A. Oguama, and S. Trivedi, “Pump-induced nonlinear refractive-index change in erbium- and ytterbium-doped fibers: theory and experiment,” Opt. Lett. 30(11), 1261–1263 (2005). [CrossRef]   [PubMed]  

9. F. A. Oguama, A. Tchouassi, A. M. Johnson, and H. Garcia, “Numerical modeling of the induced grating autocorrelation for studying optical fiber nonlinearities in the picoseconds regime,” Appl. Phys. Lett. 86(9), 091101 (2005). [CrossRef]  

10. R. Kuis, A. Johnson, and S. Trivedi, “Measurement of the effective nonlinear and dispersion coefficients in optical fibers by the induced grating autocorrelation technique,” Opt. Express 19(3), 1755–1766 (2011). [CrossRef]   [PubMed]  

11. R. R. Alfano, The Supercontinuum Laser source (Springer-Verlag, 1989), New York.

12. W. J. Tomlinson, R. H. Stolen, and C. V. Shank, “Compression of optical pulses chirped by self-phase modulation in fibers,” J. Opt. Soc. Am. B 1(2), 139–149 (1984). [CrossRef]  

13. J. Peatross and A. Rundquist, “Temporal decorrelation of short laser pulses,” J. Opt. Soc. Am. B 15(1), 216–222 (1998). [CrossRef]  

14. R. Trebino and D. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10(5), 1101–1111 (1993). [CrossRef]  

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

16. A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University Press, 1998).

17. K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2000).

18. S. F. Feldman, P. R. Staver, and W. T. Lotshaw, “Observation of spectral broadening caused by self-phase modulation in highly multimode optical fiber,” Appl. Opt. 36(3), 617–621 (1997). [CrossRef]   [PubMed]  

References

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  1. R. Rokitski and S. Fainman, “Propagation of ultrashort pulses in multimode fiber in space and time,” Opt. Express 11(13), 1497–1502 (2003).
    [Crossref] [PubMed]
  2. F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,” J. Opt. Soc. Am. B 25(10), 1645–1654 (2008).
    [Crossref]
  3. Q. Z. Wang, D. Ji, L. Yang, P. P. Ho, and R. R. Alfano, “Self-phase modulation in multimode optical fibers produced by moderately high-powered picosecond pulses,” Opt. Lett. 14(11), 578–580 (1989).
    [Crossref] [PubMed]
  4. A. M. Levine, E. Ozizmir, R. Trebino, C. C. Hayden, A. M. Johnson, and K. L. Tokuda, “Induced-grating autocorrelation of ultrashort pulses in a slowly responding medium,” J. Opt. Soc. Am. B 11(9), 1609–1618 (1994).
    [Crossref]
  5. 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]
  6. 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]
  7. 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]
  8. H. Garcia, A. M. Johnson, F. A. Oguama, and S. Trivedi, “Pump-induced nonlinear refractive-index change in erbium- and ytterbium-doped fibers: theory and experiment,” Opt. Lett. 30(11), 1261–1263 (2005).
    [Crossref] [PubMed]
  9. F. A. Oguama, A. Tchouassi, A. M. Johnson, and H. Garcia, “Numerical modeling of the induced grating autocorrelation for studying optical fiber nonlinearities in the picoseconds regime,” Appl. Phys. Lett. 86(9), 091101 (2005).
    [Crossref]
  10. R. Kuis, A. Johnson, and S. Trivedi, “Measurement of the effective nonlinear and dispersion coefficients in optical fibers by the induced grating autocorrelation technique,” Opt. Express 19(3), 1755–1766 (2011).
    [Crossref] [PubMed]
  11. R. R. Alfano, The Supercontinuum Laser source (Springer-Verlag, 1989), New York.
  12. W. J. Tomlinson, R. H. Stolen, and C. V. Shank, “Compression of optical pulses chirped by self-phase modulation in fibers,” J. Opt. Soc. Am. B 1(2), 139–149 (1984).
    [Crossref]
  13. J. Peatross and A. Rundquist, “Temporal decorrelation of short laser pulses,” J. Opt. Soc. Am. B 15(1), 216–222 (1998).
    [Crossref]
  14. R. Trebino and D. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10(5), 1101–1111 (1993).
    [Crossref]
  15. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971).
    [Crossref] [PubMed]
  16. A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University Press, 1998).
  17. K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2000).
  18. S. F. Feldman, P. R. Staver, and W. T. Lotshaw, “Observation of spectral broadening caused by self-phase modulation in highly multimode optical fiber,” Appl. Opt. 36(3), 617–621 (1997).
    [Crossref] [PubMed]

2011 (1)

2008 (1)

2005 (4)

2003 (2)

1998 (1)

1997 (1)

1994 (1)

1993 (1)

1989 (1)

1984 (1)

1971 (1)

Alfano, R. R.

Fainman, S.

Feldman, S. F.

Garcia, H.

Gloge, D.

Hayden, C. C.

Ho, P. P.

Horak, P.

Ji, D.

Johnson, A.

Johnson, A. M.

Kane, D.

Kuis, R.

Levine, A. M.

Lotshaw, W. T.

Oguama, F. A.

Ozizmir, E.

Peatross, J.

Poletti, F.

Reed, W. A.

Rokitski, R.

Rundquist, A.

Shank, C. V.

Staver, P. R.

Stolen, R. H.

Tchouassi, A.

F. A. Oguama, A. Tchouassi, A. M. Johnson, and H. Garcia, “Numerical modeling of the induced grating autocorrelation for studying optical fiber nonlinearities in the picoseconds regime,” Appl. Phys. Lett. 86(9), 091101 (2005).
[Crossref]

Tokuda, K. L.

Tomlinson, W. J.

Trebino, R.

Trivedi, S.

Wang, Q. Z.

Yang, L.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

F. A. Oguama, A. Tchouassi, A. M. Johnson, and H. Garcia, “Numerical modeling of the induced grating autocorrelation for studying optical fiber nonlinearities in the picoseconds regime,” Appl. Phys. Lett. 86(9), 091101 (2005).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (6)

Opt. Express (2)

Opt. Lett. (3)

Other (3)

R. R. Alfano, The Supercontinuum Laser source (Springer-Verlag, 1989), New York.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University Press, 1998).

K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2000).

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Figures (10)

Fig. 1
Fig. 1 IGA experimental setup consisting of time-delayed two-beam coupling in CdMnTe:V photorefractive crystal.
Fig. 2
Fig. 2 Fitted IGA experimental data to early version of the theoretical IGA model.
Fig. 3
Fig. 3 Linear fitting of SPMS versus average power for 10 meter long SMF- 28 fiber.
Fig. 4
Fig. 4 OSA laser frequency response and fit to a combination of a hyperbolic secant and a Gaussian pulse temporal profile.
Fig. 5
Fig. 5 Experimental fit of IGA signal to theory for 500 mW average power in 10 meters of Corning single mode silica fiber SMF-28.
Fig. 6
Fig. 6 Experimental fit of IGA signal to theory for 500 mW average power in 10 meters of single mode silica fiber SMF-28.
Fig. 7
Fig. 7 Beam profile of single mode fiber SMF-28 9.5 µm core diameter, 1m long.
Fig. 8
Fig. 8 Beam profile of multimode fiber FUD-3539 17.0 μm core diameter, 1m long.
Fig. 9
Fig. 9 Experimental fit of IGA signal to theory for 250 mW average power in 10 meters of Nufern multimode silica fiber FUD-3539.
Fig. 10
Fig. 10 SPMS versus average power plot and the slope of the linear fitting for 10 meters of Nufern multimode mode silica fiber FUD-3539.

Equations (15)

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E( r,z,φ,t )= j F j ( r,φ ) A j ( z,t )exp( i β 0j )
r 2 2 F/ r 2 +rF/ r+( U 2 r 2 / a 2 l 2 )F=0for0<r<a ( core )
r 2 2 F/ r 2 +rF/ r+( W 2 r 2 / a 2 l 2 )F=0forr>a ( cladding )
U=a ( k 0 2 n 1 2 β 2 ) 1/2 and W=a ( β 2 k 0 2 n 2 2 ) 1/2
F( r,φ )= A J l ( U ) J l ( Ur a )[ cos( lφ ) sin( lφ ) ]0<r<a(core)
F( r,φ )= A K l ( W ) K l ( Wr a )[ cos( lφ ) sin( lφ ) ]r>a(cladding)
A p ( z,t ) / z =i( β 0 ( p ) β 0 ) A p ( z,t )( β 1 ( p ) β 1 ) A p ( z,t ) / t i n2 β n ( p ) n! ( i/ t ) n A p ( z,t ) +( i n 2 ω 0 /c )( 1+i τ 0 / t ) l,m,n { Q plmn ( ω 0 )[ 2( 1 f R ) A l ( z,t ) A m ( z,t ) A n ( z,t ) +3 f R A l ( z,t ) dτh( τ ) A m ( z,tτ ) A n ( z,tτ ) ] }
SPMS= j | A j | 2 (spms) j / j | A j | 2
W det | K( τ ) | 2 = | + E( r,z,φ,t ) E ( r,z,φ,t+τ )dt | 2
n 2 / A eff =( n τ p cλ / t R L ) [ 32 π 2 ln2 /π 10 7 ] 1 ( SPMS / P avg )
E( t )= E 0 exp( 2ln2 ( t/ τ p ) 2 i( ω 0 t+ϕ ) )
E( t )= E 1 ( t )+ E 2 ( t )= E 01 sech( 2ln( 1+ 2 )t/ τ p )exp( i ω 01 t )+ E 02 exp( 2ln2× ( t/ τ p ) 2 i ω 02 t )
L P 01 F 1 = e r 2 a 2 [ 1 1 ]andL P 11 F 2 = J 1 ( Ur a )[ cos( φ ) sin( φ ) ]
A 1 / z + β 1 ( 1 ) A 1 / t +0.5j β 2 ( 1 ) 2 A 1 / 2 t +0.5α A 1 =[ ( f 11 spm s 1 | A 1 | 2 +2 f 12 spms | A 2 | 2 2 ) A 1 ]
A 2 / z + β 2 ( 2 ) A 1 / t +0.5j β 2 ( 2 ) 2 A 2 / 2 t +0.5α A 2 =[ ( f 22 spm s 2 | A 2 | 2 +2 f 21 spms | A 1 | 1 2 ) A 2 ]

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