4382
JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 12, DECEMBER 2005
Optimizing Fiber Bragg Gratings Using a Genetic Algorithm With Fabrication-Constraint Encoding Guillaume Tremblay, Jean-Numa Gillet, Yunlong Sheng, Martin Bernier, and Gilles Paul-Hus
Abstract—We propose an advanced genetic algorithm (GA) to design fiber Bragg gratings (FBGs) with given fabrication constraints. Our GA is enhanced by a new Fourier-series-based real-valued encoding to obtain more degrees of freedom and a rank-based fitness function. The new GA enables us to remove phase shifts in the gratings. The designed minimum-dispersion bandpass grating has a dispersion of ±28 ps/nm in the 0.33-nm flat-top passband. The grating is fabricated using a phase mask without phase shift.
Compared with the recent design of dispersion-free FBGs [15], our design with the fabrication constraints excludes any phase shifts, so that the designed FBG can be fabricated with a conventional high-quality holographic phase mask without requiring custom-made phase masks. As a tradeoff, the FBG’s dispersion increases slightly. The designed FBG has been fabricated to confirm its performances.
Index Terms—Fabrication constraints, fiber Bragg grating (FBG), genetic algorithm (GA), numerical optimization, optical fiber.
II. G ENETIC A LGORITHM (GA)
I. I NTRODUCTION
R
ECENTLY, the optimization of fiber Bragg gratings (FBGs) has received strong research interest. In [1]–[6], the FBGs are not simply synthesized by the inverse scattering procedures, such as the Gelfand–Marchenko methods and the layer-peeling algorithm (LPA) [7], [8], but are optimized with tradeoffs among a variety of constraints. Indeed, using powerful optimization algorithms such as the iterative LPA [2], [3], [5], simulated annealing, and genetic algorithm (GA), one can include requested fabrication constraints such that the designed FBGs not only achieve the required spectral performance but are also easy to fabricate. The GA optimizes a population of chromosomes by recursive processes of selection, reproduction, and mutation, based on the principles of natural selection [9]–[11]. The GA is a globalsearch algorithm and therefore, it is more likely to find the global optimum [9] than local-search approaches such as the iterative LPA [2] and simulated annealing with a fast cooling schedule [12]. The GA was already used in FBG design [13], [14]. In this paper, we introduce a new Fourier-seriesbased real-valued encoding technique for FBG coupling coefficients to include fabrication constraints in the GA. We enhance the GA with a rank-based selection and a new fitness function based on a cumulative normal distribution [10]. We show a design example of the bandpass filter with minimum dispersion.
Manuscript received February 10, 2005; revised March 31, 2005. G. Tremblay, Y. Sheng, M. Bernier, and G. Paul-Hus are with the Department of Physics, Physical Engineering and Optics, and the Center for Optics, Photonics and Laser, Université Laval, Québec, QC G1K 7P4, Canada (e-mail:
[email protected];
[email protected]). J.-N. Gillet was with the Department of Physics, Physical Engineering and Optics, and the Center for Optics, Photonics and Laser, Université Laval, Québec, QC G1K 7P4, Canada. He is now with the Department of Engineering Physics, École Polytechnique de Montréal, Montréal, QC H3C 3A7, Canada (e-mail:
[email protected]). Digital Object Identifier 10.1109/JLT.2005.858228
A. Real-Valued Encoding In most GA implementations, the solutions (chromosomes) in a population are coded with strings of binary numbers (genes). For designing an FBG, one can divide the grating length into a number of short sections and encode values of the local coupling coefficients in each section as a string of binary genes [13]. The main drawback of this encoding technique is that, in the GA, the crossover and mutation operations would change the coupling coefficients in the neighboring FBG sections independently and randomly, so that the resulting apodization profile could contain many inflections. Such a solution is not appropriate for an FBG. Our goal is to optimize an apodized coupling-coefficient profile q(z) of an FBG that is smooth along the fiber length, satisfies the requested spectral specifications, and is easy to fabricate. An additional fabrication constraint can be to remove all the phase shifts in the FBG so that its coupling-coefficient function is all positive valued. In this way, the designed FBG can be fabricated using a conventional holographic phase mask without specific phase shifts. For this purpose, we propose a new coding method based on the Fourier series. We describe the coupling coefficient q(z) of an FBG as the square of a Fourier series, which is positive valued
q(z) = AB
n−1 m=1
am cos
mπz L
2 (1)
where −L/2 ≤ z ≤ L/2, L being the length of the grating, A = qmax = π∆nmax /λB with the maximal index modulation ∆nmax , the Bragg wavelength λB , a normalization constant B, and we choose n = 13 as the number of real-valued genes in a chromosome. Thus, q(z) is the square of a summation of (n − 1) = 12 cosine modes. One can observe that by setting a1 = 1 and ai = 0 for 2 ≤ i ≤ (n − 1), we obtain the squared cosine profile. The parameters to be optimized are the n − 1 = 12 weights of the sinusoidal modes in the Fourier series, −1 ≤ am ≤ 1 with 1 ≤ m ≤ n − 1, which control the shape
0733-8724/$20.00 © 2005 IEEE
TREMBLAY et al.: GENETIC ALGORITHM WITH FABRICATION-CONSTRAINT ENCODING FOR OPTIMIZING FBGs
of the coupling coefficient q(z). The last gene a13 is used to calculate A. Note that the number of 13 parameters in the encoding of (1) is much higher than that in the previous GA for FBG optimization [14], where only three parameters were optimized: the grating length, period, and the weight ∆nmax of the given squared cosine apodization function. The high number of degrees of freedom is useful for our GA to contain the strong fabrication constraints such as removing all phase shifts in the FBG. Moreover, this representation of q(z) ensures that q(−L/2) = q(L/2) = 0 and that q (−L/2) = q (L/2) = 0. Indeed, when taking the first derivative of q(z), one obtains n−1 n−1 mπz pπz dq 2π =− am cos ap p sin dz L m=1 L L p=1
We define φ(p) as a cumulative Gaussian normal distribution [10] so that p √ −(y − µ)2 −1 exp dy φ(p) = 1 − ( 2πσ) σ2
(4)
−∞
where µ = N0 /2 and σ = N0 /6. The selection is performed based on a roulette game. In each game, a random number h, with 0 ≤ h ≤ 1, is generated with a uniform probability distribution. The chromosome with the rank p is selected for reproduction if CΦ (p − 1) < h ≤ CΦ (p)
(5)
(2) where CΦ (p) is the cumulative selection probability
which is always equal to 0 when m is an integer at z = ±L/2. At the initial generation k = 0, a population of N chromosomes is randomly generated. For the real-coded GA, a usual rule is to use a number N of chromosomes, which is six or seven times larger than the number n of genes in the chromosomes. As the chromosomes are made up of n = 13 genes, we set N = 102 to obtain enough genetic diversity. After each chromosome’s generation, the FBG spectrum is computed with the transfer-matrix method and a cost function is evaluated as a measure of errors in the actual spectrum with respect to the target spectrum.
CΦ (p) =
p
Φ(p ).
(6)
p =1
C. Reproduction We use a multiple crossover method to generate offspring chromosomes. The offspring c is reproduced by randomly selecting genes from parents a and b as follows: a = [a1 a2 b = [b1 b2
a3 b3
··· ···
an ] bn ]
c = [x1
x2
x3
...
xn ] (7)
B. Selection and Pairing For the next generations, k = 1, 2, 3, . . ., a number of chromosomes are selected for reproduction. The N0 = 100 best chromosomes are all taken as the first parents, which are then paired with N0 chromosomes selected as the second parents from the N0 best chromosomes. The two worst chromosomes in each generation are never implicated in the reproduction. This pairing technique increases genetic diversity since all the N0 best chromosomes will transmit genes to the next generations. In the classical GA, each pair generates two offspring chromosomes while in our GA, each pair is meant to generate only one offspring. The conventional GA often leads to a premature convergence because the selection probability is calculated according to its cost-function value, which can suffer from large numerical fluctuations. We used a selection based on the ranks p of the chromosomes instead of the cost-function values [10]. The ranks 1 < p < 100 of the chromosomes are given according to their ascending cost-function values. Then, the selection probability Φ(p) is defined as the normalized fitness function φ(p): Φ(p) =
4383
φ(p) N0
(3)
φ(p )
p =1
where the fitness function φ(p) is a positive number describing the reproductive success of a chromosome according to its rank.
where every “xi ” with 1 ≤ i ≤ n, in c has an equal probability of being an “ai ” or a “bi .” On the other hand, a linear crossover of the real-valued chromosomes is also used. The two crossover procedures are performed with an equal probability. In the GA with real-valued coding, the linear crossover usually generates three offspring chomosomes c1 , c2 , and c3 from two parents a and b [13] c1 = −0.5a + 1.5b
(8)
c2 = 0.5a + 0.5b
(9)
c3 = 1.5a − 0.5b.
(10)
Then, in the chromosomes’ space, the three offspring chromosomes are located at predefined intervals on a line segment passing through a and b. One keeps the best two of the three for the next generation and eliminates the worst one. Our linear crossover generates the new chromosome c from the parents a and b with c = (−0.5 + g)a + (1.5 − g)b
(11)
where g is a random number chosen in the interval 0 ≤ g ≤ 2. In our linear crossover, the offspring chromosome can be located randomly anywhere on the line segment passing through a and b. Moreover, as only one offspring chromosome is generated in every reproduction, the elimination process is skipped.
4384
JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 12, DECEMBER 2005
The reproduction probability was set to 0.9. In the case where reproduction does not occur, the offspring is cloned from its second parent. To introduce more genetic diversity, the mutation of a chromosome element m is also performed to generate mm as mm = 0.25(2h − 1) + 0.75m
(12)
where h is a random value and 0 ≤ h ≤ 1 such that the mutated value is kept close to the original value within an interval controlled by h . Only one randomly chosen chromosome element is mutated each time, with a mutation probability of 0.4. This value is a little higher than typical values of 0.15–0.3 for a real-valued GA [16] in order to increase genetic diversity. D. Elitism Elitism is necessary for the GA to converge [10]. The best genome in the population should not be altered or destroyed by reproduction processes. Elite chromosomes are transmitted to the next generation without crossovers or mutations. In our GA, a new generation consists of 100 new offspring individuals and two cloned elite individuals. Typically, elites are the individuals having the lowest total cost-function values. Usually, these individuals are similar to each other so that the use of too many elites often results in premature convergence. We select our two elites differently: the first one being selected according to the total cost value and the second one according to the partial cost value from the reflectivity spectrum. This technique improves genetic diversity in the population because there are two different elites distributing their genes into the population in each generation. The two elites are usually quite different in the early generations. Then, they tend to converge to the global optimum in the later generations.
R drops below a preset value of −0.5 dB to ensure a flat-top passband (R > −0.5 dB) at least 0.3 nm wide so that γ(λB ± 0.15) =
0, G,
for R(λ) ≥ −0.5 dB . otherwise
(14)
In the bands where 0.3 ≤ |λ − λB | ≤ 0.8 nm, the cost value is defined for the isolation error of the FBG so that γ(λ) =
|−40 − R(λ)|3 , for R(λ) ≥ −40 dB . 0, otherwise
(15)
We do not constrain the reflectivity spectrum in the bandgaps between the control points λ = λB ± 0.15 nm to λ = λB ± 0.3 nm, respectively, to provide some degrees of freedom for the GA to converge. The dispersion in the flat-top passband is to be minimized. The contribution of the dispersion to the cost function ΓD is the maximum dispersion within the R > −0.5 dB band. The total cost function Γ consists of the two contributions from the cost of the reflectivity error and that of the dispersion. One usually suggests Γ = βΓR + (1 − β)ΓD , where 0 < β < 1 is a weighting parameter [13]. This technique however relies on the determination of parameter β to properly weight the contributions of the two partial cost functions. However, in many GA applications including FBG optimization, the mean value and the variance of the statistical distribution of each partial cost contribution can vary greatly over generations. The weight of each contribution is thus affected by the degree of convergence of the population. This can make the choice of a suitable weight parameter β difficult. We thus define the total cost function according to the partial ranks as Γi = wpRi + pDi
(16)
III. D ESIGN As an example, we designed a bandpass FBG filter with minimum dispersion. The dispersion-free filter is extensively studied. In a preceding design with the LPA [15], the FBG contained eight π phase shifts. Our GA uses the all-positivevalued coding of the coupling coefficients. The designed FBG contains no π phase shifts and can be fabricated with a conventional holographic phase mask. As a price to pay, we expect a slightly deteriorated dispersion spectrum compared with that of the previous design [15]. The grating length was set to 1 cm and the target bandpass width was set to 0.4 nm. The FBG reflectivity and dispersion spectra are calculated using the transfer-matrix method with a subdivision of the grating into M = 200 uniform sections. The contribution of the reflectivity spectrum error ΓR to the cost function is a sum of three cost values, which are defined at the central wavelength λ = λB as 0, for R(λB ) ≥ 0.0436 dB (= 99%) γ(λB ) = (13) G, otherwise where R is the reflectivity and the number G → ∞. At the two control points λ = λB ± 0.15 nm, the cost value is set to G if
where pRi and pDi are the rank of the chromosome i according to its reflectivity and dispersion costs ΓRi and ΓDi , respectively, and w is a weighting constant. The total cost function defined in (16) would not be affected by the statistical distribution of the partial costs ΓR and ΓD . We choose a rational weight w = 1.001 such that the product w × p is never an integer in order to prevent equalities between different Γi ’s. The total cost Γi is then used to determine the rank p of the chromosome i, and the fitness φ(p) in (3) and cumulative probability CΦ (p) in (6) for selection. The GA recursive search process was stopped after only 325 generations. Fig. 1 shows the coupling coefficient q(z) of the designed FBG. Many runs of our GA were performed leading to the same results, which could be a sign that we achieved the global optimum. Simulations show that the grating has maximum reflectivity of 0.99, isolation of −33.3 dB, dispersion of ±28 ps/nm in the 0.33 nm passband, and bandwidth utilization ratio, which is the ratio of the bandwidths of the −0.5 dB passband over the −30 dB stopband, BWU = 0.59. Experimental results were obtained using a frequency-doubled argon laser of 500 mW and a holographic uniform phase mask at the central wavelength of 1556.5 nm to write the FBG in
TREMBLAY et al.: GENETIC ALGORITHM WITH FABRICATION-CONSTRAINT ENCODING FOR OPTIMIZING FBGs
4385
IV. C ONCLUSION We have successfully included the fabrication constraints in the genetic algorithm (GA) with the real-valued encoding for the optimization of a low-dispersion bandpass fiber Bragg grating (FBG) without phase shifts. The GA is enforced by a higher number of degrees of freedom and a rank-based fitness function. The designed FBG contains no phase shift and has a dispersion of ±28 ps/nm in the 0.33-nm flat-top passband. The designed FBG was fabricated with a conventional holographic phase mask. Experimental results confirm the designed FBG performances. Fig. 1.
Coupling coefficient q(z) of the designed FBG.
Fig. 2. Simulated (thin curve) and experimental (thick curve) reflection spectra of the designed FBG.
Fig. 3. Simulated (thin curve) and experimental (thick curve) group delay of the designed FBG.
an H2 loaded Ge-doped Corning SMF28 fiber. A piezoelectric actuator is used in the scanning system to control the index variation profile related to q(z). Numerical and experimental results are shown by the thin and thick curves, respectively, in Figs. 2 and 3, and present a very good agreement. The experimental curves were obtained using an Agilent chromatic-dispersion measurement system model HP86037C, with a scanning resolution of 4 pm and a frequency modulation of 250 MHz. It is apparent that experimental results are in good agreement with simulations, as shown in Figs. 2 and 3. The fabricated grating has a dispersion of ±33 ps/nm and an isolation of −22 dB in the 0.33-nm passband.
R EFERENCES [1] N. Plougmann and M. Kristensen, “Efficient iterative technique for designing Bragg gratings,” Opt. Lett., vol. 29, no. 1, pp. 23–25, Jan. 2004. [2] Y. OuYang and Y. Sheng, “Design fiber Bragg grating using iterative layer-peeling algorithm,” presented at the Optical Fiber Communication Conf. (OFC), 2004, Paper MF32. [3] H. Li, T. Kumagai, K. Ogusu, and Y. Sheng, “Advanced design of a multichannel fiber Bragg grating based on a layer-peeling method,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 21, no. 11, pp. 1929–1938, Nov. 2004. [4] K. Aksnes and J. Skaar, “Design of short fiber Bragg gratings by use of optimization,” Appl. Opt., vol. 43, no. 11, pp. 2226–2230, Apr. 2004. [5] K. Y. Kolossovski, R. A. Sammut, A. V. Buryak, and D. Y. Stepanov, “Three-step design optimization for multi-channel fibre Bragg gratings,” Opt. Express, vol. 11, no. 9, pp. 1029–1038, May 2003. [6] A. V. Buryak, “Iterative scheme for the ‘mixed’ scattering problem,” presented at the Bragg Gratings, Photosensitivity and Poling in Glass Waveguides (BGPP), Monterey, CA, 2003, MB3. [7] J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg grating by layer peeling,” IEEE J. Quantum Electron., vol. 37, no. 2, pp. 165–173, Feb. 2001. [8] R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron., vol. 35, no. 8, pp. 1105–1115, Aug. 1999. [9] W. S. Klug and M. R. Cummings, Concepts of Genetics, 6th ed. Upper Saddle River, NJ: Prentice-Hall, 2000. [10] J. N. Gillet and Y. Sheng, “Multiplexed computer-generated holograms with polygonal-aperture layouts optimized by genetic algorithm,” Appl. Opt., vol. 42, no. 20, pp. 4156–4165, Jul. 2003. [11] J. H. Holland, “Genetic algorithms,” Sci. Amer., vol. 267, no. 1, pp. 66–72, Jul. 1992. [12] S. Kirckpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, “Optimization by simulated annealing,” Science, vol. 220, no. 4598, pp. 671–680, May 1983. [13] J. Skaar and K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightw. Technol., vol. 16, no. 10, pp. 1928–1932, Oct. 1998. [14] G. Cormier, R. Boudreau, and S. Thériault, “Real-coded genetic algorithm for Bragg grating parameter synthesis,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 18, no. 12, pp. 1771–1776, Dec. 2001. [15] J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” J. Quantum Electron., vol. 39, no. 10, pp. 1238–1245, Oct. 2003. [16] A. H. Wright, “Genetic algorithm for real parameter optimization,” in Foundation of Genetic Algorithms, G. J. E. Rawlins, Ed. San Mateo, CA: Morgan Kaufmann, 1991.
Guillaume Tremblay received the B.Ing and M.Sc. degrees in physics from the Université Laval, Québec City, QC, Canada, in 2002 and 2005, respectively. He is currently pursuing the Ph.D. degree. His research interests involve fiber Bragg gratings (FBGs).
4386
Jean-Numa Gillet received the Master’s degree (second cycle) in physical engineering from Université Libre de Bruxelles, Brussels, Belgium, in August 1996 and the Ph.D. degree (cum laude) in physics, with optics as the subject, from Université Laval, Québec City, QC, Canada, in December 2001. From January 2002 to December 2003, he was a Postdoctoral Fellow (January 2002 to January 2003, Université Laval, and January 2003 to December 2003, École Polytechnique de Montréal, Montréal, QC, Canada). Currently, he holds the Faculty position of Research Fellow in the Department of Engineering Physics at the École Polytechnique de Montréal. He is a renowned researcher in varying fields of optics and photonics, such as diffractive optics, holography, laser processing, and fiber Bragg gratings (FBGs). He also performed other highly valuable research in other fields of physics, mathematics, and artificial intelligence, such as modeling, numerical physics, simulated annealing, genetic algorithms (GAs), semiconductor physics, microelectronics, nanotechnology, and X-ray photoelectron spectroscopy (XPS).
Yunlong Sheng received the B.S. degree from the University of Sciences and Technology of China in 1964. He received the M.Sc., Doctor, and Doctor ès Science Physique degrees from the Université de Franche-Comté, Besançon, France, in 1980, 1982, and 1986, respectively. Since 1985, he has been with the Centre d’Optique, Photonique et Laser, Université Laval, Québec City, QC, Canada, and is now a Full Professor. His research interests involve diffractive optics, fiber Bragg gratings (FBGs), nanooptics optical signal processing, pattern recognition, and image fusion. He is the author and coauthor of more than 180 technical papers, books, and book chapters and holds two patents on the FBG fabrication. Dr. Sheng is a Fellow of the Optical Society of America (OSA) and a Fellow of The International Society for Optical Engineering (SPIE). Dr. Sheng serves as a Guest Editor of Diffractive Optics in Optical Engineering, and a Cochair of a number of International Conferences.
JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 12, DECEMBER 2005
Martin Bernier received the B.Ing degree and Master’s degrees in physics from Laval University, Québec City, QC, Canada, in 2003 and 2004, respectively. Currently, he is pursuing the Ph.D. degree at the Center for Optics, Photonics and Laser at Laval University. His research interests involve fiber Bragg gratings, diffractive optics and femtosecond science. He also performed a technical position at National Optical Institute, Québec City, QC, Canada, from 1999 to 2001 and worked among other things on the development of the holographic phase mask technology.
Gilles Paul-Hus received the M.Sc. degree in physics form Laval University, Québec City, QC, Canada, in 1993. He was an Invited Scholar within the context of the German Academic Exchange Service from 1995 to 1997 at Dresden University of Technology, Dresden, Germany. From 1998 to 2002, he was with Jones, Duck, and Sinclair (JDS) Uniphase and Alcatel Optronics in the Ottawa/Gatineau area, where he was involved in the development of fiber Bragg grating (FBG) technology. Since 2002, he has been the Research Assistant in charge of the Nanophotonics Infrastructure at the Centre d’Optique, Photonique et Laser at Laval University. In addition, he has worked with graduate students on the development of passive optical component.
