## Abstract

The gain spectrum of a fiber optical parametric amplifier (OPA) can be controlled by imposing a temperature distribution along the fiber, which modulates the local fiber zero-dispersion wavelength *λ*
_{0}, and hence the parametric gain coefficient. We present simulations and experimental verification for various binary temperature distributions. The method should be applicable to fibers with realistic longitudinal variations of *λ*
_{0}.

© 2005 Optical Society of America

## 1. Introduction

The shape of the gain spectrum of a fiber optical parametric amplifier (OPA) is determined by the phase matching condition, and hence by the fiber dispersion properties, at every point along its length. The parametric gain spectra for fibers with constant dispersion properties along their lengths are well known [1,2]. Practical fibers, however, exhibit random longitudinal variations of the zero-dispersion wavelength *λ*
_{0}, which prevent their gain spectra from matching those of ideal fibers with uniform *λ*
_{0}.

Here we consider, for the first time to our knowledge, the possibility of using a controlled longitudinal distribution of temperature along a fiber OPA in order to modify its gain spectrum. While the method is in principle quite general, we illustrate it by concentrating on the relatively simple case of binary temperature distributions. We first intuitively describe the modifications expected from some simple distributions, and we then present simulations verifying these predictions, and providing spectral shapes expected for more complex cases. We then present experimental results for some selected cases which confirm our theoretical predictions. Finally, we discuss the possibility of extending this technique for shaping the gain spectrum of practical OPAs.

## 2. Theory and simulations

The gain spectrum of a one-pump fiber OPA is governed by the wavevector mismatch distribution Δ*β*(*z*)=*β*
_{2}(*z*)(Δ*ω*
_{s})^{2}+*β*
_{4}(*z*)(Δ*ω*
_{s})^{4}/12, where: *z* is the distance along the fiber; *β _{m}*(

*z*) is the

*m*th derivative of the local wavevector

*β*(

*ω*,

*z*) with respect to the angular frequency

*ω*, evaluated at the pump angular frequency

*ω*, and at

_{p}*z*; Δ

*ω*=

_{s}*ω*-

_{s}*ω*;

_{p}*ω*is the signal angular frequency. Generally fiber OPAs must be operated with a small Δ

_{s}*β*(

*z*), and therefore the pump wavelength

*λ*must be close to the zero-dispersion wavelength

_{p}*λ*

_{0}(

*z*) for all

*z*. For a given fiber we assume that

*β*

_{3}and

*β*

_{4}remain constant along the fiber, so that

$$\approx 2\pi c{\beta}_{3}\left(\frac{1}{{\lambda}_{p}}-\frac{1}{{\lambda}_{0}\left(z\right)}\right){\left(\Delta {\omega}_{s}\right)}^{2}+\frac{{\beta}_{4}}{12}{\left(\Delta {\omega}_{s}\right)}^{4}$$

This expression shows that Δ*β*(*z*) can in principle be given an arbitrary shape, by suitably controlling the shape of *λ*
_{0}(*z*). One way to do this is to have a corresponding temperature distribution along the fiber, because it is known that *λ*
_{0} varies approximately linearly with temperature: the rate of change is about 0.03 nm/°C in dispersion-shifted fiber (DSF) [3], and 0.06 nm/°C in highly-nonlinear dispersion-shifted fiber (HNL-DSF) [4]. (One may assume as a first approximation that *β*
_{3} and *β*
_{4} do not vary with temperature; small variations will not significantly alter the basic reasoning.) In principle, one could use this method to synthesize arbitrary dispersion profiles to control the gain spectra of fiber OPAs. In particular, this method could in principle be used to overcome the presence of random dispersion fluctuations in practical fibers.

Here we limit ourselves to relatively simple temperature profiles, which are physically realizable. In particular, binary temperature distributions can easily be obtained by having two regions in the laboratory maintained at two distinct temperatures, *T*
_{1} and *T*
_{2}, and by placing different regions of a fiber in these two regions. Such regions could for example be: (i) a temperature-controlled oven; (ii) the room at ambient temperature, i.e. about 25°C; (iii) a mixture of water and ice providing a temperature of 0°C. We chose to perform our experiments with (ii) and (iii) for reasons of convenience; the simulations correspond to these conditions.

In order to clearly see the influence of the temperature differences, we focus on a narrow feature of the OPA gain spectrum that we have recently investigated, and which can be readily shifted by modest temperature changes [5] (see Appendix). We assume for simplicity that the narrow gain spectra of individual fiber segments at *T*
_{1} and *T*
_{2} do not overlap.

Let us now consider some interesting particular cases of binary distributions, namely periodic functions with *N _{p}* periods along the fiber. Each period consists of a length

*L*

_{1}at temperature

*T*

_{1}, followed by a length

*L*

_{2}at

*T*

_{2}.

For *N _{p}*=1, the overall power gain spectrum is simply obtained as the sum (in decibels) of the individual power gain spectra of the two halves. This can be understood as follows:

If there is gain in the first section, then the second section only introduces phase shifts for the pump, signal and idler, which do not alter powers. Hence the overall gain is the same as if the second section was absent.

If there is no gain in the first section, then it only introduces a phase shift between pump and signal, and no idler is generated. This phase shift does not alter the power gain of the second section. Hence the overall gain is the same as if the first section was absent.

As a result, the power gain spectrum for the two segments combined should consist of two non-overlapping peaks, each corresponding simply to the power gain in each individual segment.

For *N _{p}*=2, we can view the overall OPA as a cascade of two OPAs with

*N*=1. The overall power gain spectrum, however, cannot be calculated as the sum of the spectra of the two halves, because these spectra are actually identical, and therefore can strongly interfere. In general pump, signal and idler emerge from the first half with finite amplitudes, and phases governed by the structure of the first half. These phases now affect the power gains in the second half. This situation is similar to that encountered in our previous work on fiber OPAs with periodic dispersion compensation, where each period consisted of two different fiber segments [6]. Therefore, we expect the overall gain spectrum to exhibit modulation due to these phase-dependent effects.

_{p}As *N _{p}* increases further, the gain spectrum continues to exhibit a complicated behavior, including rapid modulation. However, when

*N*becomes very large (in a sense to be defined shortly), we eventually enter a new regime where the spectrum is very simple again. This occurs when the spatial period of the modulation,

_{p}*L*

_{1}+

*L*

_{2}, becomes much shorter than the nonlinear length

*L*=1/(

_{NL}*γP*

_{0}), where

*γ*is the fiber nonlinearity coefficient, and

*P*

_{0}is the pump power. Then the waves essentially experience the local average of the dispersion, and the details of the rapid fluctuations are not important. In this situation, the OPA behaves as if the fiber had a uniform temperature, given by the average

*T*=(

_{av}*L*

_{1}

*T*

_{1}+

*L*

_{2}

*T*

_{2})/(

*L*

_{1}+

*L*

_{2}). Therefore, we expect the spectrum to consist of a single high lobe, shifted by an amount proportional to

*L*

_{1}/

*L*

_{2}. So by varying

*L*

_{1}/

*L*

_{2}it is possible to obtain the same spectra as one would have with a uniform-temperature OPA, but at different effective temperatures.

We have performed simulations with a commercially available package [7] to quantify these considerations. Sections II–IV in Ref. [7] describe in detail the equations used for modeling propagation in transmission fibers. The model includes dispersion, nonlinearity, and random birefringence, which are also required for studying propagation in fiber OPAs. In previous work, we have found that this package provides numerical results for OPAs that are generally in good agreement with experimental results.

The common parameters used here were the same as in our experiments, namely: Fiber: Corning DSF; *L*=200 m; *γ*=2.3 W^{-1} km^{-1}; *λ*
_{0}=1542.3 nm; *β*
_{3}=1.14×10^{-40} s^{3} m^{-1}; *β*
_{4}=-5×10^{-55} s^{4} m^{-1}; *λ*
_{p}=1540.43 nm; *P*
_{0}=10 W; *T*
_{1}=25°C; *T*
_{2}=0°C. Figure 1 shows gain spectra calculated for *N _{p}*=1, 2, 4, and 32. For

*N*=1, 2, 4, the appearance of the spectra is as expected from the previous qualitative description.

_{p}*N _{p}*=32 is the minimum value of

*N*for which the spectrum forms a single smooth peak; the shape of the spectrum remains the same as

_{p}*N*is further increased. This last result shows that all spatial dispersion variations due to temperature variations (and/or other phenomena) that have a period smaller than 1/(32

_{p}*γP*

_{0}), will affect OPA performance only by means of their local average, calculated over a length of the order of 1/(32

*γP*

_{0}).

## 3. Experiments

The experimental configuration is similar to the one we used in Ref. [5], and its schematic is shown in Fig. 2. The gain medium was a spool of DSF with the parameters given in the previous section. A tunable laser source, TLS1, served as the pump source, which was pulse-modulated (with pulse width of 4 ns and duty cycle of 1/256) to provide 12 W of peak power. Polarization controller PC1 aligned the pump’s state of polarization (SOP) with the transmission axis of a Mach-Zehnder intensity modulator (MZ-IM). The pump was then amplified by EDFA1 and filtered by a 0.35 nm bandwidth tunable bandpass filter (TBF). It was further amplified by EDFA2, with maximum average output power of 27 dBm. We were not able to suppress the EDFA ASE further by inserting a fiber Bragg grating (FBG) and circulator (CIR) as in Ref. [5], due to the limited power budget. DSF was followed by an isolator, which prevented any reflection from the variable optical attenuator (VOA). The output spectrum was observed by the optical spectrum analyzer (OSA). We measured OPA ASE as an indicator of OPA gain, as in Ref. [5].

Two different configurations were used for the DSF, as shown in Fig. 3. In Fig. 3(a), the 200-m long DSF is on a single spool with a horizontal axis. By lowering the spool into a tank of ice water, a fraction *y* of the diameter [corresponding to the fraction on *x*=cos^{-1}(1-2*y*)/*π* of the circumference] could be immersed in ice water.

Another configuration is shown in Fig. 3(b), where the original 200-m long fiber was re-spooled onto two separate 100-m spools. One spool was maintained at 0°C by filling with ice water the groove in which the fiber rested; the other spool was at ambient temperature.

## 4. Experimental results

Figure 4(a) shows the experimental ASE spectra obtained with the setup of Fig. 3(a), with different fractions *y* of the DSF fiber spool diameter submerged in the ice water. *λ*
_{p} was 1540.43 nm. The ASE peak is an indication of the OPA gain spectrum. As *y* increases from 0 to 2/3, the ASE spectrum shifts to the right, the peak wavelength increasing from 1462.98 nm to 1472.99 nm (i.e. Δ*λ*≈10 nm), consistent with what we have obtained in Ref. [5] with uniform temperature changes. Note that the ASE spectrum shift is due to the decrease of *λ*
_{0}, in good agreement with what we predicted in previous section.

Figure 4(b) shows simulated gain spectra, with the same parameters as in the experimental conditions, and *λ _{p}*=1540.43 nm. The circumference of the spooled fiber was

*C*=0.518 m, so that

*N*

_{p}=387. We see that there is excellent agreement with the experimental results: the locations of the peaks, as well as their widths at -20 dB match very well. In order to obtain a good match for the peak locations, we had to use the value

*dλ*

_{0}/

*dT*=0.034 nm/°C, which is close to typical values generally used for DSF [3].

Experimental results for the second configuration (with the single 200-m length of DSF re-spooled onto two 100-m spools) are shown in Fig. 5(a). It is clear that the single peak obtained with both spools at 25°C, shown as a dashed line, splits into two smaller peaks when one spool is completely submerged in ice water. The separation between the two peaks is about 10.2 nm, consistent with the previous result. We note that the amplitudes of the two Stokes peaks near 1620 nm are uneven; the same is true for the anti-Stokes peaks near 1470 nm. Also, the Stokes peaks are close to the maximum of the Raman gain spectrum due to the pump, which raises their levels compared to those of the anti-Stokes peaks.

Figure 5(b) shows simulated gain spectra, with the same parameters as in the experimental conditions, and *λ _{p}*=1540.43 nm. We see that the match with the experimental ASE spectrum is quite good. In particular the relative heights of the two adjacent Stokes peaks in the two-temperature case are also found by simulation to be uneven; the same is true for the anti-Stokes peaks.

We have also run the same simulation with the Raman gain turned off. In that case the two Stokes peaks became even, and so did the anti-Stokes peaks. This is in agreement with what is expected from our initial qualitative discussion, as well as from Fig. 1(a).

## 5. Discussion

The simple experiments reported here provide an illustration of the idea of fiber OPA spectrum shaping via temperature control, as well as a good match with theoretical predictions and numerical simulations. Let us now consider whether this method may also be useful in more practical situations. To do so, we will consider two examples: 1) the possibility of cancelling *λ*
_{0}(*z*) variations in a typical HNLF; 2) the possibility of flattening the broad gain spectrum in a one-pump fiber OPA.

HNLFs have a smaller core than DSF, and are more difficult to manufacture. Small variations in core size result in significant longitudinal variations of *λ*
_{0} along their length. In recent work, it has been shown that *λ*
_{0} can vary by as much as 4 nm over a 300-m length, and that this variation leads to significant modifications of the gain spectrum compared to what it would be with a uniform fiber [8]. In HNLF, *dλ*
_{0}/*dT* has been found to be of the order of 0.06/°C. Hence, to shift *λ*
_{0} by 4 nm, a temperature increase of about 67°C would be required. This is smaller than has been used for increasing SBS threshold [4], and should therefore be feasible without much difficulty in the laboratory. One could consider a number of heating and/or cooling mechanisms that could be used for obtaining a non-uniform longitudinal temperature distribution in the fiber. With such a system, one could for example heat the fiber everywhere (except at the maximum of *λ*
_{0}, *λ*
_{0},_{max}) by different amounts, so as to bring up *λ*
_{0} at every point to *λ*
_{0,max}. (The required temperature map would be directly related to the dispersion map obtained by the method of Ref. [8].) Ideally the result would be a fiber with the same *λ*
_{0} everywhere, which would then display a standard, well-understood gain spectrum when used as an OPA.

On the other hand, a uniform *λ*
_{0} actually may not be provide the most desirable gain spectrum shape, and one might thus want to obtain some other well-controlled *λ*
_{0} distribution. The relationship between *λ*
_{0}(*z*) and the spectrum shape is a complex one, and to date it has not received much attention. One exception is Ref. [9], in which it was shown that by using four fiber segments, each with a different *λ*
_{0}, it is possible to obtain a gain spectrum which is nearly rectangular, a desirable shape for such applications as wideband optical communication. If we assume that we start from a perfect fiber with uniform *λ*
_{0}, all we need to do is wind it onto four different spools, and bring them to different temperatures by heating or cooling; this is a simple extension of our setup in Fig. 3(b). A numerical example in Ref. [9] shows that with an HNLF, the four required values of *λ*
_{0} span a 5.5 nm range, which corresponds to a 92°C temperature range. This is larger than the range considered above, but is still smaller than the range already demonstrated for SBS suppression [4]. In a laboratory experiment, we could immerse one spool in ice water, another spool would be heated to 92°C, and the other two spools would be at intermediate temperatures.

Of course this experiment would not work well as described if the HNLF used had *λ*
_{0} variations as severe as in Ref. [8]. In that case, to obtain a final fiber with four uniform-*λ*
_{0} regions, one would need a distributed temperature control, to counteract the local *λ*
_{0} variations, and to impose the desired ones. In other words, the system would need to be the same as above. On the other hand, if progress in the manufacture of HNLF succeeds in greatly reducing *λ*
_{0} fluctuations, then the simple four-spool arrangement should be sufficient to obtain a rectangular OPA gain spectrum as proposed in Ref. [9].

We note that controlling *λ*
_{0}(*z*) by means of temperature has a considerable advantage over the alternative method of splicing together different fibers, namely the absence of splicing loss. There is essentially no limit to how many different regions one can have by temperature control, whereas splice loss limits the alternative approach to just a few segments, and modifies the spectrum compared to the ideal lossless case [6].

## 6. Conclusion

We have introduced a novel technique for controlling the shape of the gain spectrum of fiber OPAs, by imposing arbitrary temperature distributions along the fiber length. We have presented results of simulations and experiments illustrating the versatility of the method. It can in principle be extended to generate more complex temperature distributions, and is limited only by the practicality of the heating/cooling systems required to do so. This is a promising new technique, which has the potential for overcoming for the first time the random longitudinal dispersion variations existing in all practical fibers, and for tailoring OPA gain profiles.

## Appendix

The origin of the narrow gain region that we concentrate on in this paper can be understood as follows. One-pump fiber OPAs with uniform *λ*
_{0} along their length exhibit gain when -4*γP*
_{0}<Δ*β*<0. Hence we can find the location of the edge of the gain region(s) by solving the equation

which is of second-order in (Δ*ω*
_{s})^{2}. The simplest root is (Δ*ω*
_{sa})^{2}=0; it corresponds to the wide gain region containing the pump wavelength, 5 which has been studied extensively in recent years. The other root is

Eq. (A.2) shows that Δ*ω*
_{sb} can be increased arbitrarily by moving *λ*
_{p} away from *λ*
_{0}; hence we can obtain a second gain region, away from the pump. In Ref. [5] we also showed that the width of this gain region decreases as it moves away from the pump. Its location is quite sensitive to the difference between *λ _{p}* and

*λ*

_{0}, and therefore to the fiber temperature, as it modifies

*λ*

_{0}. For this reason, in this paper we focus on the effect of temperature on this narrow gain feature, rather than on the usual broader gain region containing

*λ*.

_{p}## Acknowledgments

This work was supported in part by National Science Foundation grant #ANI-0123441. The authors would like to thank Sprint Advanced Laboratories for the loan of equipment.

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