OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 4 — Feb. 21, 2005
  • pp: 1098–1106
« Show journal navigation

Design of discrete, nearly-uniform Bragg gratings in planar waveguides

George Ouyang and Amnon Yariv  »View Author Affiliations


Optics Express, Vol. 13, Issue 4, pp. 1098-1106 (2005)
http://dx.doi.org/10.1364/OPEX.13.001098


View Full Text Article

Acrobat PDF (152 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

In this paper we present an efficient method for designing discrete, nearly-uniform Bragg gratings in generic planar waveguides. Various schemes have already been proposed to design continuous Bragg gratings in optical fibers, but a general scheme for creating their discrete counterpart is still lacking. Taking a continuous Bragg grating as our starting point, we show that the same grating functionalities can also be realized in any planar waveguide by discretizing it into a series of air holes. The relationship between the two gratings is established in terms of grating strength and local grating period.

© 2005 Optical Society of America

1. Introduction

As a historic note, Laso et al. first proposed a simple discretization procedure in the microwave regime, in a similar effort to set up an equivalence relation between an FBG and a discrete air-hole grating in a microstrip waveguide [8

8. M. A. G. Laso et al., “Analysis and design of 1-D photonic bandgap microstrip structures using a fiber grating model,” Microwave Opt. Tech. Lett. 22, 223–226 (1999). [CrossRef]

]. This scheme, while simple to implement, works only for purely periodic structures and hence is limited in its applications. In the following we propose a discretization procedure that is much broader in scope and applies to both periodic and nearly-periodic gratings.

2. Background

An 1-D FBG is characterized by the evolution of its effective refractive index n(z) along its length:

n(z)=n0+Δn(z)cos(K0z+θ(z))
(1)

where n 0 is the average effective refractive index and K 0=(=2π/Λ), Δn(z), and θ(z) specify the grating parameters: Λ is the reference period, Δn(z) accounts for the local grating strength (apodization), and θ(z) determines its phase variation and local period.

Various schemes have been proposed to design continuous FBG’s that can accommodate almost any filter responses. But to the best knowledge of the authors a general scheme for designing discrete gratings is still lacking. In this paper we show that the same grating functionalities can also be realized in any planar waveguide by discretizing the continuous fiber grating into a series of air holes, and the connection between the two is established in terms of grating strength and local grating period.

3. Grating design

We draw our intuition from the discretization procedure outlined in a scheme called “layer peeling algorithm” [9

9. 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. 35, 1105–1115 (1999). [CrossRef]

], which, among others, provides a generic formalism for designing continuous, nearly-uniform gratings. Suppose we first design a continuous grating in the form of Eq. (1) (using the “layer peeling algorithm”, say) and then try to create its discrete counterpart by transforming it into a series of discrete air holes, distributed nearly uniformly along the center of a planar waveguide:

The refractive index profile should then be rewritten as:

n˜(x,y,z)=n˜0+Δn˜(x,y,z)
(3-a)

with

Δn˜(x,y,z)=δ(x)W(y)mamδ(zmΛ+bm)
(3-b)

where

ñ 0=the effective refractive index of the unperturbed waveguide,

am=(area of the mth air hole)×(1-nsubstrate),

bm=deviation of the mth air hole from z=mΛ,

W(y)=1 if

W(y)=1ift2<y<t2

(“t” is the thickness of the planar waveguide)

=0 otherwise,

i.e., W(y) is a windowing function that is nonzero only within the thickness of the waveguide. Also embedded in Eqs. (3-a) and (3-b) is the implicit assumption that the areas of the air holes, hence the am’s, are small, for otherwise we would not have been able to approximate the index perturbation by δ functions. Additionally the deviation of the air holes from their respective reference locations, i.e., the bm’s, are also taken to be small since the grating is assumed to be “nearly uniform”.

Fig. 1. Top view of a 3-D planar waveguide (the y-dimension is not shown). A series of air holes (with spacing ~Λ) drilled at the center of the waveguide serve as the perturbation. Note even though we have plotted here a conventional, index-confined rectangular waveguide, our analysis can also be extended to more exotic structures such as the multilayer Bragg waveguide or the 2-D photonic crystal slab waveguide.

Comparing the functional forms of n(z) in (1) and ñ(x,y,z) in Eq. (3) we arrive at the following intuition:

SizeofthemthairholeatzmΛΔn(mΛ)
(4-a)
Localperiodoftheairholes2πK0+θ'(z)
(4-b)

where θ′(z) is the first order derivative of θ(z).

To show that n(z) and ñ(x,y,z) give the same filter response we go back to the original wave equation. Assuming linearly polarized electric field, say E⃗=Ex, the perturbed wave equation can be written [10

10. A. Yariv, Optics Electronics (4th edition, Saunders College Publishing, 1991).

]:

2Exμε(r)2Ext2=μ2t2[Ppert(r,t)]
(5)

where

Ppert(r,t)Δ(n2)
(6)

is the electric polarization due to the perturbation. Since Ppert(r⃗,t) is the only term in the wave equation that provides coupling between the forward and backward propagating waves, n(z) and ñ(x,y,z) are equivalent as long as they give equivalent perturbative coupling term Ppert(r,t).

For the 1-D continuous grating, assuming Δn(z)≪n 0, Ppert(r⃗,t) is given by (1):

PpertΔ(n2)2n0Δn(z)cos(K0z+θ(z)).
(7)

For sufficiently slowly-varying Δn(z) and θ(z), Δ(n 2) consists of two distinct Fourier components centered respectively at ±K 0 (see Fig. 2). Without loss of generality we filter out the positive frequency component and apply the sampling theorem on its slowly-varying envelope (including the phase factor eiθ(z). The end result is an alternative but equivalent expression for the perturbative component at +K 0 :

Δ(n2)K0=n0π[mΔn(mΛ)eiθ(mΛ)sin[(zmΛ)Γ]zmΛ]eiK0z
(8)

where Λ is the sampling period and Γ is one half of the bandwidth of Δ(n2)K0 (see Fig. 2).

Fig. 2. Fourier spectrum of Δ(n 2). Valid when Δn(z) and θ(z) are slowly varying.

For the case of a discrete grating, we once again assume Δñ(x, y, z)≪ñ 0 and from Eqs. (3-a) and (3-b) we get:

PpertΔ(n˜2)2n˜0Δn˜(x,y,z)=2n˜0δ(x)W(y)mamδ(zmΛ+bm)
(9)

Ignoring δ(x) and W(y) for now and once again assuming am ’s and bm ’s are sufficiently slowly varying, we plot the Fourier spectrum of Δ(ñ 2) in Fig. 3. Note only the two components located at K 0 and -K 0 can provide coupling between the forward and backward propagating waves and hence are relevant in our analysis.

Fig. 3. Fourier spectrum of Δ(ñ 2). Valid when am ’s and bm ’s are slowly varying. Note only the two resonant components at K 0 and -K 0 can provide coupling between the forward and backward propagating waves.

Once again we filter out the component at +K 0 with a pass-band bandwidth of 2Γ :

Δ(n˜2)K0=2n˜0π[mameibmK0sin[(zmΛ+bm)Γ]zmΛ+bm]eiK0z.
(10)

Equating Δ(n2)K0 with Δ(n˜2)K0 and assuming bm ’s are small, we obtain the following relations:

am=n02n˜0Δn(mΛ)
(11-a)
bm=θ(mΛ)K0.
(11-b)

Since am is proportional to the area of the mth air hole, Eq. (11-a) confirms our first intuition in Eq. (4-a).

What about the second intuition in Eq. (4-b)? First of all we need to define what “local period” means. In Fig. 4 we plot a few air holes in the vicinity of the mth air hole.

Fig. 4. Local period at the mth air hole. The local period at a given air hole is defined to be the average of the distances to the two adjacent air holes, i.e., (Λ12)/2.

If we define the local period at the mth air hole as the average of Λ1 and Λ2, and recall K 0=2π/Λ, we get:

Localperiodatthemthairhole=Λ1+Λ22
=2Λbm+1+bm12
=Λ[1θ[(m+1)Λ]θ[(m1)Λ]4π]
Λ[12Λθ'(mΛ)4π]
=2πK0[1θ'K0]
2πK0+θ'
(12)

where we have substituted in (11-b) for bm+1 and b m-1 and made the slowly-varying assumption on θ(z). (12) indeed confirms our second intuition in Eq. (4-b).

One last thing that needs to be addressed is the δ(x) and W(y) functions that we have deliberately left out of Δ(ñ 2) in our derivation so far. We show in the following that the inclusion of the two functions requires the introduction of an extra scaling factor γ into Δñ(x, y, z), i.e.,

n˜(x,y,z)=n˜0+Δn˜(x,y,z)=n˜0+γδ(x)W(y)mamδ(zmΛ+bm)
(13)

where γ is presumably highly dependent on the geometry of the waveguide.

We compute γ by matching coupled mode equations. In the Appendix we derive the coupled mode equations for wave propagation in FBG’s (i.e., the continuous grating), and so long as the goal is to generate the same filter response via a discrete grating in a planar waveguide, we need to reproduce the same set of coupled mode equations for the new waveguide as well. Following through the derivation in the Appendix, we realize that we need to make the following substitution for the discrete grating:

Δ(n2)continuous=2n0Δn(z)cos(K0z+θ(z))

Δ(n˜2)discrete=γδ(x)W(y)[2n0Δn(z)cos(K0z+θ(z))]
(14)

where we have applied equality between Eqs. (7), (8) and (10). Then for the planar waveguide, equation (A-11) becomes:

eiωt2[bBEBK0βeiK0z2+bFEFK0βeiK0z2iK0bBEBeiK0z2+iK0bFEFeiK0z2]
=μ2t2{γδ(x)W(y)n0ε0Δn(z)cos(K0z+θ(z))[(bBEBeiK0z2eiωt+bFEFeiK0z2eiωt)]}
(15)

Since we are considering a TE wave (i.e., E⃗=Ex)), the orthonormality relation is given by [10

10. A. Yariv, Optics Electronics (4th edition, Saunders College Publishing, 1991).

]:

EmEn*dxdy=2ωμkmδmn
(16)

where the integral is taken over the entire transverse plane and km is the propagation constant of the mth mode.

As in the Appendix we replace the 2/∂t2 operator in Eq. (15) by -ω 2, multiply both sides by E*B(x,y)=E*F(x,y), integrate over the entire transverse plane, and then apply the orthonormality relation (16). Collecting resonant terms in the resultant equation such as the ones that vary as eiK0z2 and comparing them with Eq. (A-12), which is the target mode equation we are hoping to reproduce, we realize the following equality needs to be satisfied:

2ωμkm=γt+tEm(0,y)2dy
(17)

where t is the thickness of the planar waveguide and Em=EB=EF. Or in other words,

γ=2ωμkmt+tEm(0,y)2dy2ω0μK02t+tEm(0,y)2dy=2μcn˜0t+tEm(0,y)2dy
(18)

4. Conclusion

We presented in this paper an efficient scheme for designing discrete and nearly-uniform Bragg gratings in planar waveguides, which in principle are capable of performing any functionalities that can be achieved by continuous FBG’s. One limitation of our design, which we did not explore in this paper, is the potentially significant scattering losses that could be associated with the air holes. More theoretical and experimental work is needed on this subject.

Appendix

In this appendix we derive the coupled mode equations for wave propagation in FBG’s. The coupled mode equations provide a mathematical description for the coupling between two counter-propagating waves in the fiber: bB(z,β) (backward propagating) and bF(z,β) (forward propagating) [9

9. 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. 35, 1105–1115 (1999). [CrossRef]

]

dbB(z,β)dz+iβbB(z,β)=q(z)bF(z,β)
(A-1)
dbF(z,β)dziβbF(z,β)=q*(z)bB(z,β)
(A-2)

where β is the detuning parameter

β=2kn0K02
(A-3)

q(z)=iK02n0Δn(z)2ejθ(z)
(A-4)

where Δn(z) and θ(z) have been defined in (1).

To simplify the derivation we first assume the electric field is linearly polarized, say E⃗=Ex, and then expand the total field as:

Ex=12eiωt[bB(z,β)EB(x,y)eiK0z2+bF(z,β)EF(x,y)eiK0z2]
(A-5)

Plugging the above expression into the perturbed wave equation

2Exμε(r)2Ext2=μ2t2[Ppert(r,t)]
(A-6)

we get

eiωt2[bBeiK0z2(K024EB+2EBx2+2EBy2+ω2μεEB)+
bFeiK0z2(K024EF+2EFx2+2EFy2+ω2μεEF)
iK0bBEBeiK0z2+iK0bFEFeiK0z2]
=μ2t2Ppert(r,t)
(A-7)

where prime denotes derivative with respect to z. Note in the derivation above we have also made the following slowly-varying assumptions on wave amplitudes bB and bF :

bBK02bB;bFK02bF.
(A-8)

So all terms containing bB″ and bF have been omitted from (A-7).

From the unperturbed wave equation we know

(kn0)2EB+2EBx2+2EBy2+ω2μεEB=0
(A-9)

We then expand (kn 0)2 as

(kn0)2=(K02+β)2K024+K0β
(A-10)

since β≪K 0. Plugging (A-9) and (A-10) into (A-7), we find the first term in the square brackets becomes

bBEBK0βeiK0z2.

Similarly the second term in the square brackets becomes

bFEFK0βeiK0z2.

Putting them together, we get

eiωt2[bBEBK0βeiK0z2+bFEFK0βeiK0z2iK0bBEBeiK0z2+iK0bFEFeiK0z2]
=μ2t2Ppert
=μ2t2[Δ(n2)ε0Ey]
=μ2t2{2n0Δn(z)cos(K0z+θ(z))ε0×12[(bBEBeiK0z2eiωt+bFEFeiK0z2eiωt)]}
(A-11)

For a lossless fiber we have EB(x,y)=EF(x,y). So we can cross them out on both sides of (A-11). Also if we replace the ∂2/∂t 2 operator by -ω 2 and collect the resonant terms, say the ones that vary as

eiK0z2

, we get the following equation:

bB+iβbB=in0k0(ω2με0n02)Δn(z)eiθ(z)bF.
(A-12)

But we also know

ω2με0n02=n02ω2c2=(kn0)2=(K02+β)2K024.
(A-13)

Plugging into (A-12), we have

bB+iβbB=iK04n0Δn(z)eiθ(z)bF=q(z)bF
(A-14)

where q(z) has been defined in (A-4). Thus (A-14) gives back the original mode equation (A-1). In a similar fashion, by collecting resonant terms with

eiK0z2

dependence we could also derive (A-2) from (A-11).

Acknowledgments

References and links

1.

J. Zhang, P. Shum, and S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. 15, 1558–1560 (2003). [CrossRef]

2.

J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003). [CrossRef]

3.

H. P. Li, Y. L. Sheng, and Y. Li, et al. “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. 21, 2074–2083 (2003). [CrossRef]

4.

L. G. Sheu, K. P. Chuang, and Y.C. Lai, “Fiber bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photonics Technol. Lett. 15, 939–941 (2003). [CrossRef]

5.

M. Ibsen and R. Feced, “Fiber Bragg gratings for pure dispersion-slope compensation,” Opt. Lett. 28, 980–982 (2003). [CrossRef] [PubMed]

6.

A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. 27, 936–938 (2002). [CrossRef]

7.

W. Kuang and J.D. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. 29, 860–862 (2004). [CrossRef] [PubMed]

8.

M. A. G. Laso et al., “Analysis and design of 1-D photonic bandgap microstrip structures using a fiber grating model,” Microwave Opt. Tech. Lett. 22, 223–226 (1999). [CrossRef]

9.

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. 35, 1105–1115 (1999). [CrossRef]

10.

A. Yariv, Optics Electronics (4th edition, Saunders College Publishing, 1991).

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(060.2330) Fiber optics and optical communications : Fiber optics communications
(230.1480) Optical devices : Bragg reflectors

ToC Category:
Research Papers

History
Original Manuscript: November 19, 2004
Revised Manuscript: November 18, 2004
Published: February 21, 2005

Citation
George Ouyang and Amnon Yariv, "Design of discrete, nearly-uniform Bragg gratings in planar waveguides," Opt. Express 13, 1098-1106 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-4-1098


Sort:  Journal  |  Reset  

References

  1. J. Zhang, P. Shum, S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. 15, 1558-1560 (2003). [CrossRef]
  2. J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238-1245 (2003). [CrossRef]
  3. H. P. Li, Y. L. Sheng, Y. Li, et al. “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. 21, 2074–2083 (2003). [CrossRef]
  4. L. G. Sheu, K. P. Chuang, and Y.C. Lai, “Fiber bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photonics Technol. Lett. 15, 939-941 (2003). [CrossRef]
  5. M. Ibsen and R. Feced, “Fiber Bragg gratings for pure dispersion-slope compensation,” Opt. Lett. 28, 980-982 (2003). [CrossRef] [PubMed]
  6. A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. 27, 936-938 (2002). [CrossRef]
  7. W. Kuang and J.D. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. 29, 860-862 (2004). [CrossRef] [PubMed]
  8. M. A. G. Laso et al., “Analysis and design of 1-D photonic bandgap microstrip structures using a fiber grating model,” Microwave Opt. Tech. Lett. 22, 223-226 (1999). [CrossRef]
  9. 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. 35, 1105-1115 (1999). [CrossRef]
  10. A. Yariv, Optics Electronics (4th edition, Saunders College Publishing, 1991).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 
Fig. 4.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited