Combined effect of Raman and parametric gain on single-pump parametric amplifiers.

A. S. Y. Hsieh; G. K. L. Wong; S. G. Murdoch; S. Coen; F. Vanholsbeeck; R. Leonhardt; J. D. Harvey

doi:10.1364/OE.15.008104

1. Introduction

Raman scattering and parametric amplification in optical fibers have been the subject of extensive studies [1]. Investigations of these effects have shown that separately both are promising candidates for all-optical amplifiers [2, 3]. The combined effect of Raman scattering and parametric amplification has received less attention. Early work by Shen and Bloembergen [4] provides a theoretical description of the interaction, and Refs. [5–7] extend this theoretical treatment to the domain of optical fibers. However experimental investigations of these effects have been limited [5, 8–10]. In optical fibers, Refs. [8, 9] have examined the effect of the parametric wavevector mismatch on the combined interaction, and more recently Ref. [10] has studied the effect of the frequency detuning between the signal and the pump. This recent work demonstrated that the combined Raman-parametric gain is strongly dependent on the value of the real part of the Raman susceptibility at each detuning [10]. In this paper we extend the work of Ref. [10] and discuss its implications for the design of single-pump optical parametric amplifiers. We develop analytic expressions for both the small-signal gain of these amplifiers and the power asymmetry between the amplified signal and the generated idler. Using standard telecommunications dispersion-shifted fiber (DSF) we are able to measure, for the first time, the combined action of Raman and parametric gain for signal detunings ranging from 7 to 22 THz. These results clearly demonstrate the strong influence of the Raman susceptibility on the phasematched parametric gain. They show that to obtain a flat broadband parametric gain from a single-pump parametric amplifier it is not sufficient simply to carefully design the fiber’s dispersion, it is also necessary to consider the effect of Raman scattering.

2. Theory.

The starting point for our theoretical analysis is the generalized nonlinear Schrödinger equation that includes the effects of Raman scattering and dispersion to fourth order [1]:

\frac{\partial A (z, t)}{\partial z} = - i \frac{β_{2}}{2} \frac{\partial^{2} A (z, t)}{{\partial t}^{2}} + \frac{β_{3}}{6} \frac{\partial^{3} A (z, t)}{{\partial t}^{3}} + i \frac{β_{4}}{24} \frac{\partial^{4} A (z, t)}{{\partial t}^{4}} + iγ [(1 - f) {∣ A (z, t) ∣}^{2} + f \int_{- \infty}^{t} χ_{R}^{(3)} (t - t') | A (z, t') |^{2} dt'] A (z, t)

where A(z,t) is the electric field envelope, γ is the nonlinear interaction coefficient, χ_R ⁽³⁾(t) is the temporal Raman response function, and f is the fractional strength of the Raman susceptibility to the Kerr nonlinearity (for fused silica f ≈ 0.18 [11]). β_n = dⁿ β/dωⁿ is the n^th order dispersion coefficient. Closely following the approach of Ref. [8] we consider the interaction of three collinearly polarized monochromatic waves – a pump and two sidebands detuned by a frequency ±Q. Substituting this ansatz into Eq. (1) gives rise to a set of coupled mode equations for the three wave amplitudes [12]:

\frac{1}{iγ} \frac{{d A}_{a}}{d z} = {∣ A_{a} ∣}^{2} A_{a} + (1 + q (- 2 Ω)) {∣ A_{s} ∣}^{2} A_{a} + (1 + q (- Ω)) {∣ A_{p} ∣}^{2} A_{a} + q (- Ω) A_{p}^{2} A_{s}^{*} \exp (- iΔ kz)

\frac{1}{iγ} \frac{{d A}_{p}}{d z} = {∣ A_{p} ∣}^{2} A_{p} + (1 + q (- Ω)) {∣ A_{s} ∣}^{2} A_{p} + (1 + q (Ω)) {∣ A_{a} ∣}^{2} A_{p} + q (Ω) + q (- Ω) A_{a} A_{s} A_{p}^{*} \exp (- iΔ kz)

\frac{1}{iγ} \frac{{d A}_{s}}{d z} = {∣ A_{s} ∣}^{2} A_{s} + (1 + q (2 Ω)) {∣ A_{a} ∣}^{2} A_{s} + (1 + q (Ω)) {∣ A_{p} ∣}^{2} A_{s} + q (Ω) A_{p}^{2} A_{a}^{*} \exp (- iΔ kz)

where Δk is the linear wavevector mismatch and can be expressed in terms of the even orders of dispersion expanded around the pump frequency:

Δ k (Ω) = β_{2} Ω^{2} + \frac{β_{4} Ω^{4}}{12}

The parameter q(Ω) is defined as q(Ω) ≡ 1- f + f $\tilde{χ}$ _R ⁽³⁾ (-Ω) [9], with $\tilde{χ}$ _R ⁽³⁾(Ω) the complex Raman susceptibility (the Fourier transform of χ_R ^(t) normalized such that $\tilde{χ}$ _R ⁽⁰⁾ = 1 [11]). Typically the imaginary part of $\tilde{χ}$ _R ⁽³⁾ is obtained from direct measurement of the Raman gain as a function of signal detuning, and the real part calculated from the imaginary part via the Kramers-Kronig relations. In Fig. 1 we plot the measured complex Raman susceptibility of a fused silica optical fiber presented in Ref. [11].

Fig. 1. Measured real and imaginary parts of the complex Raman susceptibility of fused silica (after Ref. [11]).

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Equations (2) - (4) describe the evolution of three arbitrary amplitude waves. Restricting our analysis to the case of a strong pump and two weak sidebands (the undepleted pump approximation) reduces these equations to the three coupled mode equations presented in Ref. [8]. A standard small signal analysis of these equations yields analytic expressions for the evolution of the Stokes (low frequency) and anti-Stokes (high frequency) sidebands [1]:

B_{a}^{*} (z) = \frac{1}{2 R} {[(R - iq + iK) B_{a}^{*} (0) - iq B_{s} (0)] \exp (γRPz) + [(R - iq + iK) B_{a}^{*} (0) - iq B_{s} (0)] \exp (- γRPz)}

B_{s}^{*} (z) = \frac{1}{2 R} {[iq B_{a}^{*} (0) + (R - iq + iK) B_{s} (0)] \exp (γRPz) + [- iq B_{a}^{*} (0) + (R - iq + iK) B_{s} (0] \exp (- γRPz)}

where B_j = A_j exp(-i(γP - Δk/2)z), P is the pump power, R = √K(2q - K) , and K = -Δk/(2γP) is the normalized mismatch parameter. Phasematched parametric amplification corresponds to a value of K = 1, and the parametric gain bandwidth is set by the range of detunings that satisfy the condition 0 < K(Ω) < 2. These two equations, first derived in Ref. [8], form the basis for our investigation of the combined effects of the Raman and Kerr nonlinearities on single-pump optical parametric amplifiers. Equations (6) and (7) give the small-signal intensity gain of a signal injected into a single-pump parametric amplifier length L as:

G = {∣ \cosh (γRPL) \pm \frac{i (K - q)}{R} \sinh (γRPL) ∣}^{2}

where the sinh term has a positive coefficient if the signal is injected on the anti-Stokes side of the pump and a negative coefficient if the signal is injected on the Stokes side. For a high-gain amplifier (γP Re(R)L ≫ 1) Eq. (8) reduces to:

G = \frac{1}{4} {∣ 1 \pm \frac{i (K - q)}{R} ∣}^{2} \exp (2 γP Re (R) L)

This yields the combined Raman-parametric gain coefficient of Ref. [9]:

g = γP Re (R) .

There are three physically interesting scenarios in which Eq. (8) reduces to give a simple analytic result. The first is the case of an amplifier operated far from its phasematched condition (∣K∣ ≫ 1). Here the small-signal intensity gain of the amplifier, given by Eq. (8), reduces to:

G = \exp (2 γP Im (q) L) = \exp (2 γPf Im ({\tilde{χ}}_{R}^{(3)} (- Ω)) L)

which is the expected result for the small-signal gain of a Raman amplifier with the gain governed by the imaginary part of $\tilde{χ}$ _R ⁽³⁾. The second case of interest is the gain at small values of the linear wavevector mismatch (K → 0). Under these conditions exponential gain coefficient falls to zero and the signal experiences only a weak quadratic amplification. This prediction was experimentally verified in Ref. [9]. The third case of interest is the combined Raman-parametric gain at the phasematched frequency (K = 1). Here Eq. (8) reduces to:

G = ∣ \cosh (γ \sqrt{2 q - 1} PL) \pm \frac{i (1 - q)}{\sqrt{2 q - 1}} \sinh (γ \sqrt{2 q - 1} PL) ∣^{2}

As f is small we may expand the square root term as √2q -1 ≈ q via the binomial expansion which gives the phasematched gain of the amplifier in the high-gain limit (γP Re(R)L ≫ 1) as:

G = \frac{1}{4} {∣ 1 \pm \frac{i (1 - q)}{q} ∣}^{2} \exp (2 γP Re (q) L)

This expression is the product of a constant term that depends both on the real and imaginary parts of $\tilde{χ}$ _R ⁽³⁾ and an exponential gain that depends only on the real part. In the high-gain limit the exponential term dominates and so the phasematched gain is a strong function of the real part of $\tilde{χ}$ _R ⁽³⁾ and only a weak function of the imaginary part. In the absence of Raman scattering (f = 0) Eq. (13) reduces to G = 0.25exp(2γPL) which is the standard expression for the small-signal gain of a single-pump parametric amplifier [2]. The coefficient of the exponential term gives the phasematched gain coefficient g_pm as:

g_{p m} = 2 γP (1 - f + f Re ({\tilde{χ}}_{R}^{(3)} (- Ω)))

Equation (14) shows explicitly the dependence of the phasematched parametric gain on the real part of the Raman susceptibility. As shown in Fig. 1, the real part of the Raman susceptibility of fused silica is a strong function of detuning, and so large variations in the phasematched parametric gain with signal detuning are to be expected. For example, at 15.5 THz the value of the real part of the Raman susceptibility is strongly negative and Eq. (14) predicts an ∼40% (2f) reduction in the phasematched parametric gain coefficient at this detuning compared the gain coefficient at small detunings. The presence of this strong dip in the parametric gain at 15.5 THz was experimentally verified in Ref [10]. In the experimental section which follows we are able to measure the phasematched gain of a single-pump parametric amplifier for signal detunings ranging from 7 to 22 THz. To our knowledge this is the first time such measurements have been reported. We also note that Eq. (13) suggests a method for the direct measurement of the real part of the Raman susceptibility. This measurement can not be made from a single gain measurement due to the presence of the constant term at the front of Eq. (13) that depends both on the real and imaginary parts of $\tilde{χ}$ _R ⁽³⁾) . However if the phasematched parametric gain is measured at two different pump powers P ₁ and P ₂, then Eq. (13) gives the ratio of these two parametric gains as exp(2γ(P ₁-P ₂)Re(q)L). If γ and f are known the real part of the Raman susceptibility is directly accessible from this ratio.

In Fig. 2 we plot the normalized Raman-parametric gain coefficient g_n = g/(2γ P) = Re(R) as a function of the normalized mismatch K and the detuning Ω. Figure 2 clearly shows all the features of the above analysis. When the amplifier is strongly phase-mismatched (∣K∣ ≫ 1) the gain is given by the imaginary part of $\tilde{χ}$ _R ⁽³⁾ with a strong peak around 13.4 THz, when K = 0 the combined Raman-parametric gain falls to zero, and at the phasematched frequency (K = 1) the Raman-parametric gain coefficient mimics the frequency dependence of the real part of $\tilde{χ}$ _R ⁽³⁾ with the strong dip at 15.5 THz clearly evident. The implications of these results for the design of single-pump parametric amplifiers are clear. The strong dispersion of the real part of the complex Raman susceptibility around 15 THz means that irrespective of the dispersive properties of the fiber, large variations in the phasematched parametric gain are unavoidable at this detuning. For smaller detunings (below 10 THz) the dispersion of the real part of $\tilde{χ}$ _R ⁽³⁾ is much weaker and so these effects are less pronounced. Most of the current generation of single-pump optical parametric amplifiers operate in this regime, with bandwidths in the order of 2-5 THz [13,14]. These results show that for the design of ultra-broadband (larger than 10 THz bandwidth) parametric amplifiers the effect of the complex Raman susceptibility on the amplifier’s gain must be included.

Fig. 2. Normalized Raman-parametric gain coefficient g_n as a function of normalized wavevector mismatch K and the signal detuning Ω. This calculation uses the complex Raman susceptibility curve of Fig. 1. Inset is the graph of g_n versus Ω for a phasematched process (K = 1).

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In addition to the amplifier gain, Eqs. (6) and (7) also provide an analytic expression for the power asymmetry between the amplified signal and the generated idler. For a signal injected on the anti-Stokes side of the pump the ratio of these two powers is given by:

\frac{P_{a}}{P_{s}} = {∣ \frac{\frac{K - q - (iR)}{\tanh (γRPL)}}{q} ∣}^{2}

For a signal injected on the Stokes side of the pump the ratio is given by:

\frac{P_{a}}{P_{s}} = {∣ \frac{q}{\frac{K - q + (iR)}{\tanh (γRPL)}} ∣}^{2}

For high-gain amplifiers (γ PRe(R)L ≫ 1) the tanh terms of Eqs. (15) and (16) tend to unity and both these expression give the same fixed ratio between the Stokes and anti-Stokes powers [7,8]:

\frac{P_{a}}{P_{s}} = {∣ \frac{K - q - iR}{q} ∣}^{2}

References [7] and [8] have previously plotted this high-gain limit power asymmetry but only at the peak of the Raman gain 13.4 THz. In Fig. 3 we plot this ratio as a function of the normalized mismatch parameter K and the detuning Ω for the entire range of detunings over which the Raman susceptibility has a non-zero value. Figure 3 shows that far from the phasematch condition (∣k∣ ≫ 1) most of the power is to be found in the Stokes sideband as might be expected for a purely Raman amplifier. Reference [15] has experimentally investigated the power asymmetry in this strongly phase-mismatched regime. For a phasematched amplifier (K = 1) the ratio between the two sidebands depends primarily on the value of the imaginary part of $\tilde{χ}$ _R ⁽³⁾ (the Raman gain). Where the Raman gain is low (at detunings close to zero, and above 25 THz) the ratio is close to unity over the entire parametric gain bandwidth (0 < K < 2). Where the Raman gain is maximum (at a detuning of 13.4 THz) the power asymmetry is at its largest.

Fig. 3. Sideband power asymmetry as a function of normalized wavevector mismatch K and the signal detuning Ω. This calculation uses the complex Raman susceptibility curve of Fig. 1. Inset is the graph of the power asymmetry versus Ω for a phasematched process (K = 1).

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3. Experiment

We choose to investigate the combined effects of Raman and parametric gain on single-pump parametric amplifiers using commercial telecommunications dispersion-shifted fiber. The advantages of commercial DSF are several fold: Firstly these fibers have an extremely high uniformity so the parametric gain is not reduced by dispersion fluctuations along the length of the amplifier [16]; secondly for the amplifier lengths we wish to consider (tens of meters) these fibers are essentially lossless; and thirdly a zero dispersion wavelength (ZDW) around 1550 nm allows us to use telecommunications components and an Erbium doped fiber amplifier (EDFA) to construct our pump source. In order to obtain sufficient pump power for high-gain parametric amplification we have constructed the pulsed pump source shown in Fig. 4. A C-band CW tunable external cavity laser (ECL1) is followed by an intensity modulator. This modulator is driven by a short pulse (1.5 ns), low duty cycle (1/667) electrical pulse generator. The resulting optical pulse train is then amplified by a high-gain, high-power EDFA (Keopsys KPS-BT2-C-30-SP). This results in a high peak power (160 W) train of rectangular 1.5 ns pulses tunable between 1520 and 1560 nm. The pump is filtered by a free-space band-pass filter to remove the unwanted ASE component and then passed though the 99% port of a 99/1 coupler into the 40 m length of DSF (zero dispersion wavelength of 1556.5 nm) which forms our parametric amplifier. When required an optical signal (ECL2) can be injected into the amplifier via the 1% port of the coupler.

Fig. 4. Schematic diagram of the experimental setup: PC - fiber polarization controller, ISO -optical isolator, ECL - tunable external cavity laser.

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Phase matched parametric amplification around the zero dispersion wavelength of a DSF has previously been investigated in Refs. [17–21]. These works show that when the pump wavelength is at or above the ZDW the resulting region of parametric gain is reasonably broadband (∼ 5 THz for a 100 W pump) with a small detuning (∼ a few THz). As the pump wavelength is decreased (tuned into the normal dispersion regime) the region of parametric gain narrows and moves to larger detunings. According to Eq. (14) the available parametric gain at the phasematched frequency depends strongly on the real part of the Raman susceptibility and as such should be a strong function of detuning. We can qualitatively demonstrate this by measuring the spectra of the spontaneous (unseeded) parametric sidebands of the amplifier as a function of detuning. In Fig. 5 we plot the spectrum of the spontaneously generated sidebands for 8 pump wavelengths between 1558.0 and 1548.4 nm. The pump wavelengths correspond to a set of phasematched sideband frequency shifts ranging from 7 to 21 THz. For each spectrum the peak power of the pump was set to 82 W. The spectrum of each spontaneous gain curve is sufficiently narrow that the Raman susceptibility does not vary significantly over its bandwidth. This means the shape of each curve is not significantly distorted and the influence of the Raman susceptibility, specifically the real part, can be seen on the envelope of the set of spontaneous gain spectra. Figure 5 clearly shows the measured parametric gain of the sidebands changes significantly as the detuning is varied, with a large decrease in phasematched parametric gain beyond 10 THz detuning followed by a slow recovery after 15.5 THz. We can rule out the possibility that this effect is due to an increased temporal walk-off between the pump and the sidebands at large detunings by considering the experimental parameters. Even for the largest frequency detuning (21 THz) the walk-off between the pump and the sidebands is only ∼ 80 ps in 40 m of DSF. This is negligible when compared to pump pulse width of 1.5 ns. In addition to this we observe no measurable depolarization between the pump and the sidebands at any detuning. This allows us to discount the possibility of any change in the parametric gain with detuning due to a frequency dependent depolarization between the three waves. We attribute the absence of any measurable depolarization between the three highly detuned waves to the short length of fiber used and the very high quality (low PMD coefficient) of commercial DSF fiber.

Fig. 5. Spectrum of spontaneous parametric sidebands for eight pump wavelengths: 1558.0, 1556.0, 1554.5, 1553.7, 1552.5, 1551.9, 1550.6 and 1548.4 nm. The input pump peak power was 82 W. The smallest frequency shift sidebands correspond to the highest pump wavelength.

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Quantitative analysis of the gain of spontaneous parametric sidebands is difficult and so we next consider the measurement of seeded parametric gain. First we wish to measure the seeded parametric gain as a function of detuning for a fixed pump wavelength. This allows us to accurately determine the nonlinear and dispersive parameters of the fiber. For this experiment the pump wavelength was set to 1555.7 nm which gives a phasematched parametric gain detuning of 10.5 THz at a pump peak power of 82 W. A small collinearly polarized CW seed is then tuned across the anti-Stokes gain band and the on-off parametric gain is measured at each point. In order to avoid saturating the amplifier the power of the seed was kept 80 dB below that of the pump. The measured seeded parametric gains are plotted as circles in Fig. 6. The solid line is the prediction of Eq. (8) with the fitted dispersion parameters at the pump wavelength set to β ₂ = 0.095 ps²/km and β ⁴ = -5.5 × 10^-4 ps⁴/km, and the nonlinear parameter γ = 2.53 W^-1km^-1. The parameter f is set to its accepted value (f = 0.18 [11]). The agreement between the experimentally measured points and the predicted gain is excellent.

We next measure the seeded phasematched parametric gain of the amplifier as a function of detuning. For a pump peak power of 82 W and a range of pump wavelengths between 1558.0 and 1548.1 nm a small CW seed signal is injected at the phasematched detuning of the anti-Stokes gain band. This phasematched detuning can be easily determined as it corresponds to the peak of the spontaneous sideband. For each phasematched detuning the on-off parametric gain is measured. The experimentally measured phasematched gains are plotted as circles in Fig. 7 for a range of detunings from 7 to 22 THz. The solid line shows the prediction of Eq. (8). We note that this theoretical curve contains no free parameters, with the nonlinear interaction coefficient y set by the previous experiment and the complex Raman susceptibility taken from the data shown in Fig. 1 [11]. Figure 7 shows a good agreement between the predicted and measured results. The gain of the amplifier is clearly seen to closely map out the variation of the real part of the Raman susceptibility of fused silica, with the gain constant out to 10 THz, then falling sharply out to 15.5 THz before recovering at larger detunings as the real part of $\tilde{χ}$ _R ⁽³⁾ returns to zero.

Fig. 6. Measured parametric gain for an anti-Stokes seed as a function of seed detuning (circles). The solid line is the prediction of Eq. (8) with γ = 2.53 W^-1km^-1 and f = 0.18. The pump wavelength is 1555.7 nm, and the input pump peak power is 82 W.

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Fig. 7. Measured phasematched parametric gain for an anti-Stokes seed as a function of detuning (circles). The solid line is the prediction of Eq. (8) with γ = 2.53 W^-1km^-1 and f = 0.18. The input pump peak power is 82 W.

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4. Conclusion

In conclusion we have presented a theory for the combined effect of Raman and parametric gain on single-pump parametric amplifiers. The key prediction of this theory is that the phasematched parametric gain of such an amplifier depends strongly on the real part of the complex Raman susceptibility. Using a single-pump parametric amplifier constructed from 40 m of dispersion shifted fiber we have been able to verify this prediction and measure the available phasematched parametric gain for signal detunings ranging from 7 to 22 THz. The agreement between these measurements and the predicted phasematched gains is excellent and does not require the use of any free parameters. These results have important implications for the design of broadband optical parametric amplifiers. We have shown that in fused silica fibers, at detunings above 10 THz, the Raman susceptibility has a large effect of the available gain of these devices and cannot be neglected. In this paper we have considered only the simplest class of parametric amplifier – the single-pump parametric amplifier. Many of the current designs for broadband flat-gain parametric amplifiers use two pumps [22]. Initial calculations show that these two pump amplifiers are also subject to these effects. This topic is to be the subject of future work.

References and links

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16. J. S. Y. Chen, S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Effect of dispersion fluctuations on widely tunable optical parametric amplification in photonic crystal fibers,” Opt. Express 14, 9491–9501 (2006). [CrossRef] [PubMed]

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19. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Wideband tuning of the gain spectra of one-pump fiber optical parametric amplifiers,” IEEE J. Sel. Top. Quantum. Electron. 10, 1133–1141 (2004). [CrossRef]

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Combined effect of Raman and parametric gain on single-pump parametric amplifiers.

Abstract

1. Introduction

2. Theory.

3. Experiment

4. Conclusion

References and links

Cited By

Figures (7)

Equations (17)

Optics Express