Pulse distortion and the square of the degree of coherence in the presence of second- and third-order dispersions

Nori Shibata; Kimitaka Watanabe; Masaharu Ohashi; Kazuhiko Aikawa

doi:10.1364/OE.25.032640

1. Introduction

White-light interferometry is known to provide methods for measuring the dispersive nature of fiber. And these methods make it possible to measure the differential group delay (DGD) between guided modes and/or chromatic dispersion in short lengths of fiber [1–7]. Furthermore, a white-light source used in combination with a two-beam interferometer provides the same information about the dispersive nature of optical fibers as a femtosecond pulse from a laser source [8]. This means that white-light interferometry techniques have the potential to simulate ultra-short pulses propagating through a fiber. Based on an analogy between the square of the degree of coherence, | γ |², and dispersion-induced pulse broadening, it has been confirmed that a two-beam interferometer composed of nondispersive and dispersive arms makes it possible to obtain dispersion-induced pulse broadening for an unchirped Gaussian pulse [9]. In this work, the propagation constant was expanded in a Taylor series and terms up to the second order dispersion (SOD) were retained as in previous theoretical treatments [10–12]. The theoretical treatment must take the third-order dispersion (TOD) into consideration, when a fiber operates near the zero chromatic-dispersion wavelength. In an early experiment [7], interference patterns exhibited asymmetry due to SOD and TOD for a quadruple-clad dispersion-flattened fiber and a dispersion-shifted fiber (DSF).

In this article, we theoretically and experimentally investigate the relationship between the interference pattern with respect to | γ |² and the dispersion-induced pulse shape near the zero chromatic-dispersion wavelength. The theoretically derived | γ |²　is expressed taking SOD and TOD into consideration. Experimental results are also shown for a low-differential group delay few-mode fiber (FMF) [13] and a DSF at their zero chromatic-dispersion wavelengths in the vicinity of 1300 and 1550 nm, respectively.

2. Interferometer and theoretical background

Figure 1 shows the interferometer we used to measure | γ |² for the interference between waves traversing an air path and a fiber under test. The light from a low-coherence source (LCS) is intensity modulated by a mechanical chopper (MC). The modulated light illuminates the two-beam interferometer. Polarizers P₁ and P₂ are adjusted to give maximum interference visibility. The interference signal appears as a chopped signal at a pin-photodiode, and this signal component is detected with a lock-in amplifier. The output waveform from the lock-in amplifier gives the intensity distribution yielded by the interference for the fiber guided-mode and a wave propagating in the air. The intensity distributions are measured by moving mirror M₃ backwards.

Fig. 1 Interferometer model for measuring the dispersion-induced pulse shape by evaluating the | γ |².

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When the two beams have the same intensity I₀, the intensity I at the pin-photodiode is expressed as,

I = I_{0} {1 + Re (γ)} .

Here γ represents the complex degree of coherence given as [14]

γ (L, 2 d / c) = \frac{\int_{- \infty}^{\infty} S (ω) \exp [i {[β (ω) - k (ω)] L - ω (2 d / c)}] d ω}{\int_{- \infty}^{\infty} S (ω) d ω},

where i = √−1 is the imaginary unit, ω is the angular frequency, S(ω) is the light source spectrum, β(ω) is the propagation constant, k(ω) is the free space wavenumber given as k(ω) = ω/c (c is the velocity of light in free space), L is the fiber length, and 2d is the optical path difference between the nondispersive and dispersive paths in the two-beam interferometer. We derive the square of the degree of coherence | γ |² as a function of 2d/c for a single Gaussian spectrum function given as

S (ω) = exp [- {(ω - ω_{c})}^{2} / 2 Δ ω^{2}] / {(2 π)}^{1 / 2} Δ ω,

where ω_c is the center angular frequency and Δω is the spectral half-width which is related to the coherence time τ_c = 1/Δω. We expand β(ω) in a Taylor series about ω_c and retain terms up to the third order in ω −ω_c

\begin{array}{l} β (ω) = β_{0} (ω_{c}) + (ω - ω_{c}) β_{1} (ω_{c}) + {β_{2} (ω_{c}) / 2} (^{ω - ω_{c}) 2} + {β_{3} (ω_{c}) / 6} (^{ω - ω_{c}) 3} \\ = ω_{c} / v_{p} + (ω - ω_{c}) / v_{g} + {β_{2} (ω_{c}) / 2} (^{ω - ω_{c}) 2} + {β_{3} (ω_{c}) / 6} (^{ω - ω_{c}) 3}, \end{array}

where β_n(ω) = d ⁿβ/dω ⁿ, and v_p and v_g are the respective phase and group velocities of the guided mode. The propagation constant difference β(ω)−k(ω) in Eq. (2) is then expressed as

\begin{array}{l} β (ω) - k (ω) \\ = (1 / ν_{P} - 1 / c) ω_{c} + (1 / ν_{g} - 1 / c) (ω - ω_{c}) + {β_{2} (ω_{c}) / 2} (^{ω - ω_{c}) 2} + \\ {β_{3} (ω_{c}) / 6} (^{ω - ω_{c}) 3} . \end{array}

Substituting Eqs. (3) and (5) into Eq. (2) yields the following relationships

γ (L, T) = \frac{A}{π^{1 / 2}} \int_{- \infty}^{\infty} \exp (- \frac{i T Ω}{p} - Ω^{2} + \frac{i B Ω^{3}}{3}) d Ω

where p = {(τ_c²−iβ₂L)/2}^1/2, Ω = (ω−ω_c)p, T = 2d/c−(1/v_g−1/c)L, A = exp[i{(1/v_p−1/c)L−2d/c}], and B = β₃L/(2p³). Using the transformation Ω = B^−1/3u−i/B [15], we have

\begin{matrix} γ (L, T) = \frac{A}{π^{1 / 2} B^{1 / 3}} \exp (\frac{2 p - 3 B T}{3 p B^{2}}) \int_{- \infty}^{\infty} \exp [i {(\frac{p - B T}{p B^{4 / 3}}) u + \frac{u^{3}}{3}}] d u \\ = \frac{2 π^{1 / 2} A}{B^{1 / 3}} \exp (\frac{2 p - 3 B T}{3 p B^{2}}) Ai (\frac{p - B T}{p B^{4 / 3}}) \end{matrix}

where Ai(s) is an Airy function of the first kind. The form of Eq. (7) is equivalent to that of the normalized amplitude of a Gaussian pulse [15], so the complex degree of coherence γ corresponds to the Gaussian pulse amplitude. It is of some interest to consider the behavior of Ai(s). There is a need to know how a given Ai(s) behaves for large values of the argument. The asymptotic forms of Ai(s) are a convenient way to obtain the behavior. The asymptotic approximations are given as follows [16];

Ai(s) \sim {\begin{cases} \frac{1}{2 π^{1 / 2} s^{1 / 4}} \exp [- \frac{2}{3} s^{3 / 2}] (s \to \infty) \\ \frac{1}{π^{1 / 2} {(- s)}^{1 / 4}} \cos [\frac{2}{3} (^{- s) 3 / 2} - \frac{π}{4}] (s \to - \infty) \end{cases}

where s = (p−BT)/(pB^4/3). Here, s→∞ and −∞ correspond to T→ −∞ and ∞, respectively. The square of the degree of coherence | γ |² is then derived from Eq. (8),

{| γ (L, T) |}^{2} \sim {\begin{cases} \frac{1}{{| B |}^{2 / 3} {| s |}^{1 / 2}} \exp [- (^{\frac{τ_{c} T}{2 {| p |}^{2}}) 2} {1 + (\frac{τ_{c}^{4} - 3 β_{2}^{2} L^{2}}{48 {| p |}^{8}}) β_{3} L T}] (T \to - \infty) \\ \frac{2}{{| B |}^{2 / 3} {| s |}^{1 / 2}} \exp [- 2 τ_{c}^{2} (\frac{T}{β_{3} L} - \frac{τ_{c}^{4} - 3 β_{2}^{2} L^{2}}{3 β_{2}^{2} L^{2}})] [\cosh {\frac{4}{3} Im (^{- s) 3 / 2}} + \\ \cos {\frac{4}{3} Re (^{- s) 3 / 2} - \frac{π}{4}}] (T \to \infty) \end{cases}

The | γ |²-response expressed by Eq. (9) is asymmetric and exhibits an oscillatory structure near the edge of the | γ |²-curve due to the presence of TOD. For the special case β₂ = 0, we find from Eq. (9a) that

{| γ (L, T) |}^{2} \sim \frac{1}{{(1 - \frac{2 β_{3} L T}{τ_{c}^{4}})}^{1 / 2}} \exp [- {(\frac{T}{τ_{c}})}^{2} (1 + \frac{β_{3} L T}{3 τ_{c}^{4}})]

When β₃ = 0 or L = 0, Eq. (10) can be simplified to give

{| γ (L, T) |}^{2} \sim \exp [- {(\frac{T}{τ_{c}})}^{2}]

Equation (11) is valid for the interference of waves traversing the two air paths in a balanced Michelson interferometer [9], and | γ(0,T)| ² corresponds to an incident Gaussian pulse whose envelope is given by |E(t)|² = exp{−(t/t_in)²}, where t is time and t_in is the incident pulse width. The | γ |²-response measured with the two-beam interferometer shown in Fig. 1 gives the output pulse shape for the wave traversing a fiber of length L in the group velocity approximation [17].

3. Experimental results

We used super-luminescent diodes (SLDs) operating at center wavelengths of λ_c = 1297 and 1551 nm. The light spectra emitted from the SLDs and their Gaussian fitting curves are shown in Fig. 2. The emitted spectra match the Gaussian fitted curves well, and the FWHM spectral widths are 68.4 and 48.4 nm at λ_c = 1297 and 1551 nm, respectively. We prepared two test fibers. One was a depressed-cladding FMF whose effective cutoff wavelengths were 2299 and 1455 nm for the LP₁₁ and LP₂₁ modes, respectively, and the other was a DSF with dual-shape core profile. The modal interferometer method [13], [18] was employed to measure the chromatic dispersion D and dispersion-slope dD/dλ. Figure 3 shows the measured variations in D and dD/dλ with wavelength λ for the test fibers. As regards the zero chromatic-dispersion wavelengths λ₀ for the LP₀₁ and LP₁₁ modes guided in the FMF and for the LP₀₁ mode guided in the DSF, λ₀ = 1305.3, 1311.0 and 1553.7 nm are experimentally obtained, respectively. The variations in β₂ and β₃ can be experimentally estimated from the results shown in Fig. 3 through the relationships β₂ = −(2πc/ω²)D and β₃ = (4πc/ω³)D + (2πc/ω²)²(dD/dλ). Figure 4 shows the measured β₂ and β₃ values as a function of λ. Then, we can estimate the values of β₂ and β₃ for the fibers at λ_c = 1297 and 1551 nm.

Fig. 2 Spectrum distributions of SLDs.

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Fig. 3 Wavelength dependences of D and dD/dλ for test fibers.

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Fig. 4 Variation of β₂ and β₃ with wavelength for test fibers.

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Figure 5 shows the intensity distributions we obtained for the interference of waves traversing the two air paths in a Michelson interferometer and | γ(0,T)| ² as a function of time T. In Fig. 5, the intensity I₀ corresponds to that expressed in Eq. (1). The experimentally obtained and theoretically predicted results are in good agreement with the magnitudes and shapes of the | γ |²-curves. Figure 5 gives FWHM widths of τ _FWHM = 52 and 87 fs for SLDs operating at λ_c = 1297 and 1551 nm, respectively. Then coherence times τ_c = τ _FWHM/2(ln2)^1/2 of τ_c = 31 and 52 fs are obtained experimentally. The measured τ_c, β₂ and β₃ values are listed in Table 1.

Fig. 5 Intensity I and | γ |² as a function of time T for the interference of waves traversing the two air paths in a Michelson interferometer.

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Table 1. Values of τ_c, β₂ and β₃ used for calculations

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Figure 6 shows the time dependences of the intensity distributions and | γ |² for various fiber-lengths of the FMF and the DSF. As predicted from the theoretical treatment, the interference patterns are asymmetric, and oscillatory structures are clearly observed near the trailing edge of | γ |²-curves. In Fig. 6, dotted curves represent the | γ |²-responses, which were numerically calculated using the values listed in Table 1. In this numerical calculation, Eq. (7) expressed by Ai(s) is used to estimate | γ |². Figure 6 reveals that the theoretically obtained curves well reflect those obtained experimentally for the shapes of the | γ |²-curves. The FWHM widths of the | γ |²-response curves are τ_FWHM(L) = 65 and 75 fs for the LP₀₁ mode guided in FMFs with L = 1.185 and 1.519 m, respectively, and they are τ_FWHM(L) = 65 and 78 fs for the LP₁₁ mode. Transit time delay differences (199 and 250 fs) between the two modes give a low DGD value of 166.2 ± 1.7 ps/km at λ_c = 1297 nm. On the other hand, FWHM widths of the | γ |²-response curves are τ_FWHM(L) = 97 and 114 fs for the LP₀₁ mode guided in DSFs with L = 1.175 and 1.990 m, respectively, at λ_c = 1551 nm.

Fig. 6 Intensity distributions and | γ |² as a function of time T for FMFs with (A) L = 1.185 m and (B) L = 1.519 m, and for DSFs with (C) L = 1.175 m and (D) L = 1.990 m.

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Since the pulse evolution along the fiber depends on the relative magnitudes of β₂ and β₃, we here estimate the dispersion lengths associated with the SOD and TOD [15], [19,20]. In our experiments, the dispersion lengths can be defined as L_D₂ = τ_c²/| β₂| and L_D₃ = τ_c³/| β₃|. It is known that the TOD effects play a significant role only if L_D₃ ≤ L_D₂ [15]. Using the values of τ_c, β₂ and β₃ listed in Table 1, we obtain L_D₃ = 0.42, 0.42 and 1.15 m, and L_D₂ = 1.42, 0.83 and 11.51 m for the LP₀₁ and LP₁₁ modes of FMF and for the LP₀₁ mode of DSF, respectively. These L_D₂ and L_D₃ values satisfy the inequality L_D₃≤L_D₂, and then the TOD effects play a significant role as regards test fibers operated in the vicinity of zero chromatic-dispersion wavelengths.

Next, we estimate the broadening factor defined as σ/σ₀, where σ and σ₀( = τ_c/√2) are the root-mean-square (RMS) width of the pulse at fiber-length of L and that of the incident Gaussian pulse, respectively. The broadening factor σ/σ₀ of unchirped Gaussian pulse in the vicinity of λ₀ is related to L_D₂ and L_D₃ as follows [15], [21],

σ / σ_{0} = {1 + {(L / L_{D 2})}^{2} + {(L / 2 L_{D 3})}^{2}}^{1 / 2} .

Using Eq. (12), and the values of L_D₂ and L_D₃ mentioned above, we can obtain σ/σ₀ as a function of the normalized distance L/L_D3. Figure 7 shows variation of the broadening factor σ/σ₀ with the normalized distance L/L_D3 for test fibers. Broken, dotted, and solid lines represent the broadening factor σ/σ₀ calculated by Eq. (12) for the LP₀₁ and LP₁₁ modes guided in the FMF, and for the LP₀₁ mode guided in the DSF, respectively. As the other broadening factor, the ratio τ _FWHM(L)/τ _FWHM of FWHM widths of the | γ |²-response curves as a function of L/L_D3 is also shown in Fig. 7. Open and filled circles, and triangles represent the broadening factors τ _FWHM (L)/τ _FWHM, which are obtained from the results shown in Figs. 5 and 6, for the LP₀₁ and LP₁₁ modes guided in the FMF, and the LP₀₁ mode guided in the DSF, respectively. Values of τ _FWHM(L)/τ _FWHM at L/L_D3 = 2.82 and 3.62 for the FMF are approximately two times smaller than those of σ/σ₀. This discrepancy reveals that the FWHM is not a true measure of the width of pulses such as complicated pulse shapes having oscillatory structures shown in Figs. 6(A) and 6(B). On the contrary, values of τ _FWHM(L)/τ _FWHM and σ/σ₀ at L/L_D3 = 1.02 and 1.73 for the DSF are 1.11 and 1.13, and 1.31 and 1.33, respectively, and then they are in good agreement with each other. A possible explanation for the matching is that | γ |²-response curves shown in Figs. 6(C) and 6(D) can be approximated by Gaussian profile, even though small oscillations appear near the trailing edge of the | γ |²-response curves.

Fig. 7 Normalized distance dependence of the broadening factors for test fibers.

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4. Conclusions

We proposed the use of the two-beam interferometer composed of nondispersive and dispersive paths to simulate a dispersion-induced pulse distortion in the presence of SOD and TOD. And we investigated the analogy between the square of the degree of coherence and the dispersion-induced pulse shape in the presence of SOD and TOD. A theoretically derived equation describing | γ |² well reflects the experimentally obtained results for FMF and DSF in the vicinity of their zero chromatic-dispersion wavelengths. Furthermore, the validity of the theoretical treatment is shown using the asymptotic forms of an Airy function of the first kind. The salient feature of the present technique based on the measurement of the | γ |²-response is that it makes it possible to simulate the ultra-short pulse evolution along a fiber.

References and links

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SLD		FMF				DSF
SLD		LP₀₁mode		LP₁₁mode		LP₀₁mode
Wavelength λ_c(nm)	τ_c(fs)	β₂(ps²/m)	β₃(ps³/m)	β₂(ps²/m)	β₃(ps³/m)	β₂(ps²/m)	β₃(ps³/m)
1297	31	6.75 × 10⁻⁴	7.10 × 10⁻⁵	1.16 × 10⁻³	7.14 × 10⁻⁵	2.39 × 10⁻²	1.42 × 10⁻⁴
1551	52	-2.34 × 10⁻²	7.87 × 10⁻⁵	-2.58 × 10⁻²	1.38 × 10⁻⁴	2.35 × 10⁻⁴	1.22 × 10⁻⁴

Pulse distortion and the square of the degree of coherence in the presence of second- and third-order dispersions

Abstract

1. Introduction

2. Interferometer and theoretical background

3. Experimental results

4. Conclusions

References and links

Cited By

Figures (7)

Tables (1)

Equations (12)

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