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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 19, NO. 6, MARCH 15, 2007 435

Grating Design of Oppositely Chirped FBGs forPulse Shaping

Miguel A. Preciado, Vctor Garca-Muoz, and Miguel A. Muriel, Senior Member, IEEE

AbstractIn this letter, we analyze and develop the requiredbasis for a precise grating design in a scheme based on twooppositely chirped fiber Bragg gratings, and apply it in severalexamples which are numerically simulated. We obtain the inter-esting result that the broader bandwidth of the reshaped pulse,the shorter gratings required.

Index TermsGratings, optical fiber dispersion, optical filters,optical pulse shaping.

I. INTRODUCTION

OPTICAL pulse shaping and manipulation are critical fea-tures for ultrafast optics, playing a central role in the area

of optical communication. In this letter, we focus our attention

in optical pulse shaping using a scheme based on two oppo-

sitely chirped fiber Bragg gratings (FBGs). As it can be seen

in Fig. 1, this scheme includes two chirped FBGs connected by

optical circulators. Note that we can use two circulators of three

ports or a single circulator of four ports. The first FBG, FBG , is

the spectral shaper, and provides the spectral response for pulse

shaping, and the second one, FBG , cancels the dispersion in-

troduced by the first grating. Obviously, the order of the FBGs

can be arbitrarily selected.

This scheme has been previously proposed, and experimen-tally demonstrated in [1] and [2]. In [1], phase-shifts are intro-

duced in the shaper FBG to generate spectral-phase-encoded bit.

In [2], a bandpass Gaussian FBG optical filter in which the band-

width can be continuously adjusted is presented. We have found

the necessity of exhaustively analyzing and developing the re-

quired basis to make a precise design. Three examples of design

are developed, with corresponding numerical simulations.

II. THEORY

Suppose a linearly chirped FGB with reflected spectral re-

sponse , where is the angular

frequency, is the reflectivity, and is the phase. The

refractive index can be written as

(1)

where represents the average refractive index of the

propagation mode, describes the maximum refractive

Manuscript received November 21, 2006; revised January 16, 2007. Thiswork was supported by the Spanish Ministerio de Educacion y Ciencia underProject Plan Nacional de I+D+I, TEC2004-04754-C03-02.

The authors are with ETSI Telecomunicacin, Universidad Politcnica deMadrid (UPM), 28040 Madrid, Spain (e-mail: miguel.preciado@tfo.upm.es).

Digital Object Identifier 10.1109/LPT.2007.892901

Fig. 1. Schematic diagram of the system.

index modulation, is the normalized apodization function,

is the fundamental period of the grating, describes the

additional phase variation (chirp), and is the spatial

coordinate over the grating, with the length of the grating.

In the following, we consider a constant average refractive

index , where is the effective

refractive index of the propagation mode. The additional phase

variation can be expressed as , whererepresents the chirp factor, and can be calculated from [3]

(2)

where is the first-order dispersion coef-

ficient. Moreover, the length of the grating can be obtained

from the following expression [3]:

(3)

where is the light vacuum speed, and is the grating band-

width. It is well known that when a chirped FBG introduces a

high enough dispersion, the apodization profile maps its spec-

tral response [4]. It can be deduced that the dispersion condition

of real-time Fourier transform [3] can be applied. Effectively, if

this condition is met, we have the same envelope in both the

spectral response and the impulse response. This condition can

be expressed as

(4)

where is the temporal length of the inverse Fourier trans-

form of the FBG spectral response without the dispersive term,

which is approximately equal to the temporal length of the

pulse reshaped. Furthermore, in the weak-grating limit (Born

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436 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 19, NO. 6, MARCH 15, 2007

approximation), the apodization profile maps the impulse re-

sponse envelope, so it also maps the spectral response. Notice

that this high dispersion condition is not required if the FBG

response is eigenfunction of the continuous Fourier transform

(e.g., Gaussian and Hermite-Gaussian functions), because then

the apodization profile maps its spectral and impulse response

for any dispersion value. From (3) and (4) we can deduce

(5)

where TBP is approximately equal to the time-

bandwidth product of the desired pulse. So from (5) we obtain

that, for a given desired pulse shape, the shorter temporal length

of the pulse, the broader its spectrum, and the shorter minimum

length of the grating. The bandwidth fixes the product chirp-

length of the grating and the shorter temporal length of the

pulse, the less dispersion required, the shorter length and the

greater chirp of the grating.

When this scheme is used for a CDMA approach similar to[1], we can use (5) with TBP , where TBP is

the time-bandwidth product of one spectral chip, and number

of chips, so . In the case

of designing a bandwidth tunable system as used in [2], we must

apply (5) with the minimum tunable bandwidth, . So,

we obtain . Note that this

condition was not required in [2] since, as it have been said, a

Gaussian response does not require a minimum dispersion con-

dition.

If condition (4) is met and the FBG operate in the weak-

grating limit (Born approximation), we can obtain the apodiza-

tion profile which corresponds to a desired reflectivity

[5], that can be written as

(6)

where is central angular frequency and is related to the

apodization function as

(7)

where the sign of is equal to the sign of . Inthe case of high

reflectivity an approximate function [6] must be applied over

the desired reflectivity . In particular, here a logarithmic-based function is proposed

(8)

III. EXAMPLES AND RESULTS

As instance we design three examples in which Gaussian

pulses from a short pulse source are reshaped in triangular ones.

We develop exhaustively the first example, and more briefly the

second and third ones. For all the examples, we assume a car-

rier frequency of 193 THz, and an effective refractiveindex for FBG .

In the fist example, we suppose that each Gaussian

input pulse has an full-width at half-maximum (FWHM)

of 0.7496 ps (spectral standard deviation of Thz),

and the total desired width for the reshaped triangular

pulse is 10 ps. Thus, the spectral function for the input

pulse and output pulse are proportional to

and , respectively,where rad/s and rad/s.

Notice that and , as well as all the spectral func-

tions in the following, are described as analytical signals (only

defined at ). We consider a band of interest of

2 THz centered at . The

spectral response of the system meets the following condition:

(9)

where and are the spectral response and thephase of the system, , , , are the

reflectivity and phase of both FBGs. Thus, we obtain

(10)

We suppose an ideal flat-top response for the FBG , so the

shape of the reflectivity is influenced by FBG solely, and we

find that

(11)

where is designed to get a maximum reflectivityvalue of 10% at . Using expression (4), we have the

dispersion parameter of FBG , s rad,

where have been used ps. We choose

s rad. Moreover, using (8) for FBG with

(11) at (where is imposed), we ob-

tain , , and

nm. Also, we make use of (2) to calculate the

values rad/m . From (3), we obtain

cm, where have been assumed.

Using (7), (8), and (11) we derive

(12)

Additionally, the FBG must be designed as a dispersion

compensator with s rad, and

a flat top response in the band of interest. Fig. 2(a) shows

the output pulse of the system in temporal domain obtained

from numerical simulation. Notice that it exhibits the desired

triangular shape.

As a second example, suppose we have the same signals asin the first example, but we intentionally choose a dispersion

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PRECIADO et al.: GRATING DESIGN OF OPPOSITELY CHIRPED FBGs FOR PULSE SHAPING 437

Fig. 2. Simulation results. Plots (a) and (b) show the temporal domain results:(a) input pulse (dashed line) and output pulse for first (solid line) and second(dotted line) examples;(b) input pulse (dashed line) and output pulse (solidline)for third example. Plots (c) and (d) show the FBG spectral response: (c) ideal(solid line) and simulated for first (undistingable from ideal) and second (dottedline) examples; (d) ideal (solid line) and simulated for third example (dottedline).

value s rad, which is not large enough tomeet (4). Imposing again a maximum reflectivity of 10%, we

obtain , ,

nm, rad/m . The apodiza-

tion profile is again described by (12), with cm.

Fig. 2(a) shows the output pulse of the system in temporal do-

main obtained by numerical simulation. As it can be seen, we

obtain neither the desired shape nor width in the output pulse.

Finally, as a third example, we assume we have scaled ver-

sions of the same signals (ten times shorter), so we have an input

Gaussian pulse with an FWHM of 74.96 fs, and a desired re-

shaped triangular pulse with a total width of 1 ps. Following

the former process, we obtain s rad,and we choose s rad. Imposing again

a maximum reflectivity 10%, we obtain

, , nm,

rad/m . The apodization profile is described by (12), with

cm. As predicted by (5), compared with the first ex-

ample, we have a ten times shorter length for a ten times broader

spectrum. Fig. 2(b) shows the output pulse of the system in tem-

poral domain obtained by simulation, where we get the desired

triangular shape.

Fig. 2(c) and (d) compares the simulated and ideal spectral

response of FBG . As can be seen, in the first and third exam-

ples, the ideal and simulated responses are very similar (undis-

tinguible for first example). However, in the second example,

FBG does not map properly the spatial profile on the spectral

response because of a bad choice of grating length.

In our simulations, we have supposed ideal cancellation ofdispersions of both FBGs. In practice, this requires a careful

monitoring of the chirp profile of each grating to avoid excessive

phase ripple. This has been achieved in [1], even with tunable

chirp in [2]. Considerations about dispersion and phase ripple

tolerance can be found in [2] and [7].

IV. CONCLUSION

We have developed a theoretical basis to design a scheme

based on two oppositely chirped FBGs. The numerically simu-

lated examples show how to use it, and validate this theoretical

work. We have deduced that the broader the handled spectrum,

the shorter the minimum length of the grating, which can be ob-served in the examples results. Note also that the length of the

grating is not fixed, so it is possible to set a length to obtain the

most realizable technological parameters.

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