## 1.

## Introduction

Broadband light sources with flat spectra and high-power outputs are highly desirable in many applications where the light intensity of different wavelengths are of concerned.^{1} With the advances in emitting materials, the use of light-emitting diodes (LEDs) with a proper fiber tapping-out structure provides a solution to achieve a high-power light output. However, the output spectra of these LEDs have a Gaussian shape. Therefore, equalization of the Gaussian spectrum of a LED becomes an important issue. Although many spectrum equalization techniques have been developed over the last years, most of them are for gain equalization of erbium-doped fiber amplifiers (EDFAs),^{2, 3} and few are reported on design of a spectrum equalizer for a light source with Gaussian-like output spectrum. The difficulty lies in the fact that spectrum equalization for a light source requires not only a flat output spectrum but also a higher power output.

Recently, fiber lattice filters were proposed for equalizing Gaussian spectra and their performance has systematically analyzed.^{4, 5} Although arbitrary flatness can be demonstrated theoretically with a lattice filter, a certain amount of input power is lost through the second output port of the lattice filter. This limits the power efficiency of the resultant light source. To overcome this problem, a fiber equalization filter with a feedback lattice structure is demonstrated with output power improvement of
$3\phantom{\rule{0.3em}{0ex}}\mathrm{dBm}$
per wavelength and 0.5-dB ripple over
$30\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
of bandwidth.^{6} As the filter requires two symmetric balanced Mach-Zehnder interferometers (MZIs) in the forward and feedback lattices, respectively, accurate control is required in the fabrication of the equal phase differential lengths and the symmetrical fiber coupling ratios. This creates difficulty when implementing such a spectrum equalization filter. In this paper, we demonstrate a simple fiber equalization filter that is easy to fabricate while retaining the same merit equalization performance as the equalization filter with a feedback lattice structure.

## 2.

## Design of Equalization Filter with a Reflection Structure

The proposed spectrum equalization filter is formed by an all-fiber
$n$
’th order MZI that is terminated by a loop, as shown in Fig. 1. The filter accepts the input light with a Gaussian-like spectrum at input port 1 through an isolator, and it outputs the equalized spectrum at port 2. The objective of the equalization filter design is to choose the
$n+1$
fiber coupling ratios in the MZI such that the passband of the equalized spectrum is as flat as possible and its central power is as high as possible. To this end, we use the same design strategy as in previous results^{5, 6} where the optimization of the central power incorporates the maximally flat conditions of an analytic function. We carry out the equalization filter design in the optical frequency domain characterized by
$w=\mathrm{\Delta}L\pi {n}_{\mathrm{ref}}\u2215\lambda $
where
$\lambda $
is the wavelength,
${n}_{\mathrm{ref}}$
is the refractive index of the optical fiber, and
$\mathrm{\Delta}L$
is the differential length of the MZI. In the optical domain, the input Gaussian-like spectrum is expressed by
$F\left(w\right)$
and the power transmission of the equalizer from port 1 to port 2 is denoted as
$P\left(w\right)$
. Thus, after applying the equalization filter to the spectrum
$F\left(w\right)$
, the equalized spectrum is expressed by
$P\left(w\right)F\left(w\right)$
.

The design of a desired equalization filter is done in two steps. First, the Gaussian-like spectrum is approximated by an analytic Gaussian function $F\left(w\right)=\beta \bullet \mathrm{exp}[-(w-{w}_{0}{)}^{2}\u22152{\delta}^{2}]$ , where ${w}_{0}$ corresponds to the central wavelength, $\beta $ the amplitude, and $\delta $ the deviation of the Gaussian function. The approximation can be done in a least-squares sense when the measurement of the original Gaussian-like spectrum is available. Second, the fiber coupling ratios of the MZI are obtained by optimizing the central power of $P\left(w\right)F\left(w\right)$ at $w={w}_{0}$ with constraints that specify the output spectrum to be as flat as possible in the neighborhood of ${w}_{0}$ . The constraints are obtained by applying the maximally flat conditions of the analytic function $P\left(w\right)F\left(w\right)$ at $w={w}_{0}$ . It is easy to verify that with the proposed filter structure, all the odd-order derivatives of the equalized spectrum $P\left(w\right)F\left(w\right)$ vanish at ${w}_{0}$ . Therefore, the maximal flatness of $P\left(w\right)F\left(w\right)$ is satisfied sufficiently by setting to zeros the highest possible number of even-order derivatives of $P\left(w\right)F\left(w\right)$ at $w={w}_{0}$ . Then, the equalization filter design is to obtain the appropriate fiber coupling ratios by solving a constrained optimization problem characterized by

## 1

$${\phantom{\mid}{\left[P\left(w\right)F\left(w\right)\right]}^{\left(l\right)}\mid}_{w={w}_{0}}=0\phantom{\rule{1em}{0ex}}\mathrm{for}\phantom{\rule{0.3em}{0ex}}l=2,4,6,\dots ,2n$$^{5}they are omitted here.

## 3.

## Performance of the Proposed Equalization Filter

The proposed spectrum equalizer is demonstrated by comparing its performance with that of the spectrum equalization filter using a feedback lattice structure.^{6} First, we examine the central output power and the bandwidth of a given flatness, which are two important specifications for spectrum equalization. Taking advantage of the maximally flat principle, the resultant equalized spectrum can be approximated around the central wavelength by a polynomial
$G\left(w\right)={c}_{0}+{c}_{1}{w}^{2n+2}$
where
${c}_{0}$
is the central power
$P\left({w}_{0}\right)F\left({w}_{0}\right)$
and
${c}_{1}$
is the
$2n+2$
’th derivative of
$P\left(w\right)F\left(w\right)$
at
$w={w}_{0}$
. Then, the bandwidth of the equalized spectrum can be evaluated by the
$\u03f5$
-dB bandwidth where
$\u03f5$
is a small value that can be specified by the users. It is defined as the distance between the two points where the equalized spectrum intersects the line with a power of
$\u03f5$
-dB level less than the maximal output power. Using the polynomial approximation, the
$\u03f5$
-dB bandwidth is obtained as
${w}_{b}=2\left[{c}_{0}\right(1-{10}^{-\u03f5\u221510})\u2215{c}_{1}{]}^{1\u2215(2n+2)}$
in the optical frequency domain and
$B{W}_{\u03f5}=2({\lambda}_{b}-{\lambda}_{c})=2[{\lambda}_{c}\mathrm{\Delta}L\pi {n}_{ref}\u2215(\mathrm{\Delta}L\pi {n}_{ref}-2{w}_{b}{\lambda}_{c})-{\lambda}_{c}]$
in the wavelength domain where
${\lambda}_{c}$
is the central wavelength of the Gaussian spectrum to be equalized and it is used to convert the spectrum from the wavelength domain to the optical frequency domain.

To compare the performance of the proposed equalization filter and the equalization filter with a feedback lattice structure, we consider the equalization of a Gaussian-like spectrum in Fig. 2 to have a fourth order flatness that requires $n=1$ . The spectrum is that of a superluminescent LED (SLED). It is also used in subsequent experiments. Using a least-square method, the spectrum can be approximated by choosing $\delta =0.638$ and ${w}_{0}=1533.8\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . Thus, we take $\mathrm{\Delta}L=18.5\phantom{\rule{0.3em}{0ex}}\mathrm{\mu}\mathrm{m}$ so that the central wavelength of the equalization filter sits at $1533.8\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ . For $n=1$ , the equalization filter is a first-order MZI consisting of two couplers. Since all odd-order derivatives of $P\left(w\right)F\left(w\right)$ resulting from the proposed equalizer vanish at $w={w}_{0}$ , it suffices to force $\left[P\right(w\left)F\right(w){]}^{\left(2\right)}=0$ at $w={w}_{0}$ to achieve the fourth order flatness. Applying the proposed design, the desired fiber coupling ratios of the MZI are ${\kappa}_{1}=0.74$ and ${\kappa}_{3}=0.74$ , respectively. The central output power and the 0.5-dB bandwidth of the equalized spectrum are calculated and listed in Table 1 where the corresponding results obtained by using the equalization filter with a feedback lattice structure are also given for comparison. The data show that the two equalization filters can achieve the same equalization results. However, it is noted that the previous equalization filter with a feedback lattice structure has to use two second-order MZIs consisting of six symmetric fiber couplers to achieve the same order of flatness. Therefore, the current proposed equalization filter has improved the efficiency of using fiber filter to equalize a Gaussian-like spectrum.

## Table 1

Central output power and the 0.5-dB bandwidth.

Normalized central power c0 | Higher derivative c1 | Flatness order 2n+2 | 0.5-dB bandwidth in wavelength | |
---|---|---|---|---|

Proposed equalizer | 0.69 | $-121.62$ | 4 | $36\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ |

Equalizer using feedback | 0.71 | $-179.35$ | 4 | $32\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ |

Next we examine the experimental performance of using the proposed equalizer to equalize the SLED spectrum as shown in Fig. 2. The equalization filter designed above is fabricated using the same technique as in previous literature.^{5, 6} After the MZI is fabricated, one of its ends is fused to form the loop. Connecting the filter to the output of the SLED with the Gaussian-like spectrum, the equalized spectrum is measured on OSA and it is shown in Fig. 2. For comparison, an equalization filter with a feedback lattice structure is also fabricated. Its equalization spectrum is measured and recorded in Fig. 2. It is shown that the proposed equalization filter can equalize the SLED spectrum with 0.5-dB ripples over a 30-nm bandwidth. This result is similar to that of the equalizer using a feedback lattice structure. But the proposed equalization filter is clearly simpler and it has an easy-to-control fabrication procedure, while the equalizer with a feedback lattice structure requires accurate control of the fabrication of six fiber couplers to achieve the symmetric feedback. The experimental results demonstrate the efficiency of the proposed equalizer.