## Asymptotic analysis of silicon based Bragg fibers

Optics Express, Vol. 11, Issue 9, pp. 1039-1049 (2003)

http://dx.doi.org/10.1364/OE.11.001039

Acrobat PDF (204 KB)

### Abstract

We developed an asymptotic formalism that fully characterizes the propagation and loss properties of a Bragg fiber with finite cladding layers. The formalism is subsequently applied to miniature air-core Bragg fibers with Silicon-based cladding mirrors. The fiber performance is analyzed as a function of the Bragg cladding geometries, the core radius and the material absorption. The problems of fiber core deformation and other defects in Bragg fibers are also addressed using a finite-difference time-domain analysis and a Gaussian beam approximation, respectively.

© 2003 Optical Society of America

## 1. Introduction

*et al.*[1

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. **68**, 1196–1201 (1978). [CrossRef]

2. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. **17**, 2039–2041 (1999). [CrossRef]

3. S. G. Johnson et al., “Low loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express **9**, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

4. Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. **25**, 1756–1758 (2000). [CrossRef]

5. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. **20**, 428–440 (2002). [CrossRef]

6. J. Marcou, F. Brechet, and P. Roy, “Design of weakly guiding Bragg fibres for chromatic dispersion shifting towards short wavelengths,” J. Opt. A , **3**, S144–S153 (2001). [CrossRef]

7. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express, **10**, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

9. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science **285**, 1537–1539 (1999). [CrossRef] [PubMed]

2. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. **17**, 2039–2041 (1999). [CrossRef]

3. S. G. Johnson et al., “Low loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express **9**, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

_{3}N

_{4}, were recently fabricated by J. Fleming

*et al.*[8] using combinations of etching and CVD (chemical vapor deposition). The silicon based miniature Bragg fibers are developed for integrated optics applications such as thermo-optical switches and BioMEMS devices, which require properties quite different from other types of Bragg fibers intended for telecommunication applications. For example, Bragg fibers for integrated optics applications can tolerate propagation loss of the order of dB/cm, rather than <dB/km demanded by telecommunication fibers.

4. Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. **25**, 1756–1758 (2000). [CrossRef]

5. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. **20**, 428–440 (2002). [CrossRef]

3. S. G. Johnson et al., “Low loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express **9**, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

7. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express, **10**, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

_{3}N

_{4}cladding pairs are required to achieve propagation loss below 1dB/cm. From the asymptotic analysis results, we establish that the material absorption in the Si cladding layers has little influence on the propagation loss of the guided air-core modes. We obtain a simple formula that characterizes the exponential reduction of the modal propagation loss as the number of cladding pairs increases. We find that the quarter-wave stack cladding geometry, which is commonly used in the literature, does not necessarily lead to optimal guiding.

## 2. Asymptotic analysis

4. Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. **25**, 1756–1758 (2000). [CrossRef]

5. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. **20**, 428–440 (2002). [CrossRef]

_{r}+

*i*β

_{i}(see also Ref. [3

**9**, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

7. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express, **10**, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

_{r}determines the dispersion properties of the Bragg fiber modes, and the imaginary part β

_{i}gives the modal loss. We also allow the dielectric constants of the cladding media to be complex to account for the material absorption. The main advantages of the asymptotic formalism are its simplicity and physical transparency, especially when we need consider both TE and non-TE polarized modes. More specifically, in the asymptotic analysis, we approximate electromagnetic fields in the Bragg fiber cladding layers as exp(±

*ikr*)/√

*r*, which behave similarly to those in a planar Bragg stack. In Ref. [4

**25**, 1756–1758 (2000). [CrossRef]

*E*

_{z},

*H*

_{θ},

*H*

_{z}, and

*E*

_{θ}[1

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. **68**, 1196–1201 (1978). [CrossRef]

*m*, with the functional dependence of exp(

*i*ω

*t*-

*i*β

*z*)exp(

*im*θ). In the

*n*th dielectric layer, the radial dependence of the guided mode can be written as [1

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. **68**, 1196–1201 (1978). [CrossRef]

*M*

_{n}(

*r*) is a 4×4 matrix, and the field amplitude constants

*A*

_{n}

*, B*

_{n}

*, C*

_{n}, and

*D*

_{n}are constant within the

*n*th dielectric layer.

*M*

_{n}(

*r*) is defined as [1

**68**, 1196–1201 (1978). [CrossRef]

_{n}is the dielectric constant of the

*n*th layer, and

*k*

_{n}is

_{n}can be a complex number, with its imaginary part accounting for the material absorption in the

*n*th dielectric layer. If the

*n*th dielectric layer belongs to the “asymptotic solution” region, the 4×4 matrix

*M*

_{n}(

*r*) take the form of [4

**25**, 1756–1758 (2000). [CrossRef]

**20**, 428–440 (2002). [CrossRef]

*E*

_{z}and

*H*

_{θ}, are decoupled from the TE modes which consist of

*H*

_{z}and

*E*

_{θ}.

*A*

_{n+1},

*B*

_{n+1},

*C*

_{n+1}, and

*D*

_{n+1}) in the (

*n*+1)th layer can be derived from the corresponding quantities in the adjacent

*n*th layer by requiring the continuity of

*E*

_{z}

*, H*

_{θ},

*H*

_{z}, and

*H*

_{θ}across the interface between the two dielectric layers [5

**20**, 428–440 (2002). [CrossRef]

_{n}is the radius of the interface between the

*n*th dielectric layer and the (

*n*+1)th dielectric layer,

*M*

_{n}and

*M*

_{n+1}are defined according to Eq. (2) or Eq. (4), depending on whether the dielectric layer belongs to the inner “exact solution” region or to the outer “asymptotic solution” region. We can apply Eq. (5) iteratively and relate the field amplitude constants outside of the Bragg fiber (

*A*

_{out}

*, B*

_{out}

*, C*

_{out}, and

*D*

_{out}) to the corresponding quantities within the low index core (

*A*

_{core}

*, B*

_{core}

*, C*

_{core}, and

*D*

_{core}) via a 4×4 transfer matrix:

*B*

_{core}=

*D*

_{core}=0 and

*B*

_{out}=

*D*

_{out}=0, respectively. Substituting these two requirements into Eq. (6), we obtain:

**T**)=0, and the field distribution can be obtained from the eigenvector [

*A*

_{core}

*B*

_{core}] that corresponds to the zero eigenvalue of the 2×2 matrix

**T**.

## 3. Guiding in silicon-based miniature Bragg fibers

### 3.1 Dispersion and loss in a specific Bragg fiber geometry

*et al.*in Ref. [8]. The fiber consists of four pairs of Si/Si

_{3}N

_{4}mirror stacks, with Si being the innermost cladding layer. For the polysilicon layers, we use a refractive index

*n*

_{1}=3.5 and thickness L

_{1}=0.11µ

*m*. The refractive index of the Si

_{3}N

_{4}layer is

*n*

_{2}=2.0 and its thickness is L

_{2}=0.21µ

*m*. The air core radius is

*r*

_{co}=7.5µ

*m*. The bulk absorption in the polysilicon layer is estimated to be about 10dB/cm. In our analysis, we assume that the “exact solution” region consists of the five innermost dielectric layers, whereas the rest of the structure is described by the asymptotic solutions. With a large air core radius, the Bragg fiber supports multiple guided modes, which are labeled according to their “azimuthal quantum number”

*m*followed by the number of zeros of the electromagnetic field in the air core. We limit ourselves to the first two TE modes (TE01 and TE02), the fundamental TM mode (TM01), and the fundamental mixed polarization mode (HE11).

*E*

_{θ}field of the TE01 mode at the wavelength of 1.65µ

*m*. We notice that the

*E*

_{θ}component is zero at the interface between the air core and the Si layer, which has been explained by drawing an analogy to perfect metal [3

**9**, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

**10**, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

*E*

_{θ}=0 at the air/Si interface leads to maximum reduction of the

*E*

_{θ}field amplitude within the Si cladding layer [1

**68**, 1196–1201 (1978). [CrossRef]

*E*

_{θ}=0 at

*r*=

*r*

_{co}with the fact that TE modes take the form of

*x*

_{1i}is the

*i*th zero of the first order Bessel functions, i.e.

*J*

_{1}(

*x*

_{1i})=0. This relation has been given in Ref. [7

**10**, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

_{r}

*c*/ω) of the TE01, TE02, TM01, and HE11 modes and plot the results in Fig. 2(b). The effective indices of the TE01 and TE02 modes, predicted by Eq. (8), are also shown in Fig. 2(b) as solid lines. It is clear that Eq. (8) is in excellent agreement with the calculated dispersion of the TE modes.

**9**, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

*m*. The results are shown in Fig. 2(d). The dashed line is a linear fit of the asymptotic results.We find the linear coefficient is Γ=10

^{-4}, which clearly demonstrates that the cladding material absorption has minimal influence on the guidance of photons in the miniature Bragg fibers.

### 3.2 Loss dependence on Bragg cladding pair number and air core radius

*r*

_{co}=8µ

*m*and vary the number of Bragg cladding pairs from 3 to 8, with the rest of the fiber parameters held to the same value as those in Fig. 2. Using the asymptotic approach, we calculate the loss of the TE01, TE02, TM01, and HE11 mode at λ=1.55µ

*m*and show the results in Fig. 3(a). From the asymptotic results, we find that the modal loss dependence on the number of the cladding pairs N is given by:

**68**, 1196–1201 (1978). [CrossRef]

_{l}is the dielectric constant of the low index medium and ε

_{h}is that of the high index medium. For TM modes, the corresponding amplitude reduction factor is

*c*. Due to the large core area, this is an excellent approximation, as can also be seen from Fig. 2(b). Using ε

_{l}=4.0 and ε

_{h}=12.25 for the Si/Si

_{3}N

_{4}cladding, Eq. (10) gives Δ

_{TE}=0.27 and Δ

_{non-TE}=0.40, quite close to the values of Δ

_{TE}=0.29 and Δ

_{non-TE}=0.43 obtained from fitting Fig. 3(a) using Eq. (9). The difference might be due to the slight deviation of the Bragg cladding from a quarter-wave stack at λ=1.55µ

*m*.

_{3}N

_{4}cladding layers and investigate the variation of the modal loss as we change the air core radius. It was suggested in Ref. [3

**9**, 748–779 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748. [CrossRef] [PubMed]

*m*) is inversely proportional to

*m*, we find that the 1/

_{TE01}=3.61, α

_{TE02}=3.49, α

_{TM01}=2.20, and α

_{HE11}=2.64.

**10**, 1411–1417 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411. [CrossRef]

*x*

_{1i}λ/2π

*r*

_{co}according to Eq. (8). As shown in Fig. 4, the distance between two consecutive “bounces” should be 2

*r*

_{co}/θ, assuming a small θ. Denoting the amplitude reflection coefficient of the Bragg cladding is ℛ, we have exp(-4β

_{i}

*r*

_{co}/θ)=|ℛ|

^{2}. For |ℛ|

^{2}approaching unity, this relation can be rewritten in the dB unit as

*r*

_{co}and λ is µ

*m*. Comparing Eq. (12) with Eq. (11), we notice that the 1/

*r*

_{co}dependence of the TE modal loss comes from the reflection coefficient of the 1D Bragg stack. We apply Eq. (12) to estimate the TE modal loss and show the result in Fig. 3(b) as solid lines. It is clear that Eq. (12) provides an excellent approximation for most of the

*r*

_{co}values. The only exception is the TE02 modal loss at

*r*

_{co}=2µ

*m*, which might be due to a large θ and the reflection coefficient that is no longer close to unity.

### 3.3 Loss dependence on cladding layer thickness

_{Si}=0.98 for the Si layer and

_{3}N

_{4}layer at λ=1.5µ

*m*. At λ=1.7µ

*m*, the corresponding quantities are, respectively, Φ

_{Si}=0.87 and

*m*the Bragg fiber is about four times as lossy as at 1.7µ

*m*. This suggests that the quarter-wave stack condition may not lead to lowest propagation loss.

_{3}N

_{4}layer thickness from 0.21µ

*m*to 0.19µ

*m*, while keeping the rest of the fiber parameters the same as in Fig. 2(c). At λ=1.5µ

*m*, the new Si

_{3}N

_{4}layer thickness corresponds to

*m*and 1.7µ

*m*and plot the results in Fig. 5(b). For the purpose of comparison, we copy Fig. 2(c) as Fig. 5(a). From Fig. 5, it is clear that the reduction of the Si

_{3}N

_{4}layer thickness leads to both lower loss for the TE01 mode and larger loss penalty for the higher order modes. The solid lines are the loss estimate for the TE modes given by Eq. (12), which again are in excellent agreement with the asymptotic result.

## 4. Influence of the fiber deformation

*z*dependence of the guided mode as exp(-

*i*β

*z*), where β is the propagation constant. This allows us to take out the

*z*dimension in our FDTD calculations and consider in the simulation only the cross-section of the Bragg fibers. For more details on the FDTD algorithm, the readers can consult Ref. [10].

_{3}N

_{4}layer are 6 cells and 12 cells, respectively. By introducing a normalization factor, this FDTD structure represents a Bragg fiber with air core radius of 5.3µ

*m*, Si layer thickness 0.11µ

*m*, and Si

_{3}N

_{4}layer thickness 0.21µ

*m*. Using FDTD simulations, we calculate the dispersion and the

*B*

_{z}field for both a circularly symmetric Bragg fiber and a deformed Bragg fiber whose upper section is partially flattened. The dispersion of both fibers are plotted in Fig. 6(a), together with a theoretical dispersion curve calculated according to Eq. (8). The theoretical results are in excellent agreement with the FDTD results for the circularly symmetric Bragg fiber, which justify our choice of the simulation parameters. As shown in Fig. 6(a), the dispersion of the deformed fiber is almost the same as that of the cylindrically symmetric fiber, with a small “down-shift” due to the slightly smaller cross-section for the deformed fiber.

*B*

_{z}field distribution of the circular Bragg fibers and the deformed Bragg fibers at λ=1.55µ

*m*are given in Fig. 6(b) and Fig. 6(c). The

*B*

_{z}field in the “flattened” Bragg fiber is essentially that of a TE01 mode. Yet it also has components with non-zero “angular quantum number” m that tend to be much more lossier. In fact, there is clearly some radiation field outside of the deformed Bragg fiber in Fig. 6(c). As a result, a deformed Bragg fiber should have modal loss higher than what is predicted by the asymptotic theory for a cylindrically symmetric fiber, which as shown in Fig. 2(c) is between 0.5–2dB/cm.

*r*

_{1}and the gas inlet port has an air core radius of

*r*

_{2}. One method of calculating the throughput loss is to find the azimuthal component of the electric field at the beginning of the throughput port (denoted by

*E*

_{θ,t}in Fig. (7)). Since the electric field of the TE01 mode has only the azimuthal component

*E*

_{θ,TE01}, the throughput loss is simply given by 1-|∫

*E*

_{θ,t}

*E*

_{θ,TE01}

*rdrd*θ|

^{2}. The process of finding

*E*

_{θ,t}, however, can be very complicated. In the following discussion, we present a much simpler approach by drawing analogy with the free space diffraction of the Gaussian beam. Since the

*E*

_{θ,t}field results from the diffraction of the TE01 mode, we expect the following discussion should at least give us an order of magnitude estimate.

*w*

_{0}, the evolution of the beam spot size

*w*(

*z*) follows [11]:

*w*

_{0}. According to Eq. (8), the far field divergence angle θ of the TE01 mode is θ=

*x*

_{11}λ/(2π

*r*

_{1}), whereas the far field divergence angle of a Gaussian beam is given by θ=λ/π

*w*

_{0}[11]. Equalizing the two divergence angles, we find

*w*

_{0}/

*r*

_{1}=2/

*x*

_{11}≈0.52, which has the right order of magnitude since the minimal spot size

*w*

_{0}should be less than the Bragg fiber radius

*r*

_{1}. Substituting this result into Eq. (13), we find

*w*(

*z*)

^{2}is proportional to the cross-section area of the Gaussian beam, the amount of light that is coupled back into the throughput port can be approximated by [

*w*

_{0}/

*w*(

*z*)]

^{2}. Set

*z*=2

*r*

_{2}and use dB unit, we find:

*r*

_{1}=7.5µ

*m*,

*r*

_{2}=

*r*

_{1}/2, and λ=1.55µ

*m*, we find the throughput loss is 0.25dB. We notice that the the loss has a 1/

## 5. Conclusion

**25**, 1756–1758 (2000). [CrossRef]

**20**, 428–440 (2002). [CrossRef]

_{3}N

_{4}cladding pairs, which were fabricated using chemical vapor deposition (CVD) [8], and demonstrate that it is possible to achieve propagation loss below 1dB/cm with only 4 pairs of Si/Si

_{3}N

_{4}cladding layers. We find that the material absorption in the cladding layers has little impact on the fiber propagation loss. We also give a simple formula that describes the reduction of fiber propagation loss as we increase the number of cladding pairs. We use a finite-difference time-domain (FDTD) algorithm to investigate the impact of fiber deformation on the modal propagation characteristics. Finally, a simple Gaussian beam approximation is applied to evaluate the additional loss due to the existence of the gas inlet ports introduced in the CVD fabrication processes.

## References and links

1. | P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. |

2. | Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. |

3. | S. G. Johnson et al., “Low loss asymptotically single-mode propagation in large-core Omniguide fibers,” Opt. Express |

4. | Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. |

5. | Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. |

6. | J. Marcou, F. Brechet, and P. Roy, “Design of weakly guiding Bragg fibres for chromatic dispersion shifting towards short wavelengths,” J. Opt. A , |

7. | A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express, |

8. | J. G. Fleming, S. Y. Lin, and R. Hadley, in Proceeding of the solid-state sensor actuator and microsystems workshop, p.p. 173, (Hilton Head, S.C., 2002). |

9. | R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science |

10. | A. Taflove and S. C. Hagness, |

11. | See for example, A. Yariv, |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(220.0220) Optical design and fabrication : Optical design and fabrication

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 18, 2003

Revised Manuscript: April 25, 2003

Published: May 5, 2003

**Citation**

Yong Xu, Amnon Yariv, James Fleming, and Shawn-Yu Lin, "Asymptotic analysis of silicon based Bragg fibers," Opt. Express **11**, 1039-1049 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-9-1039

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### References

- P. Yeh, A. Yariv, and E. Marom, �??Theory of Bragg fiber,�?? J. Opt. Soc. Am. 68, 1196-1201 (1978). [CrossRef]
- Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, �??Guiding optical light in air using an all-dielectric structure,�?? J. Lightwave Technol. 17, 2039-2041 (1999). [CrossRef]
- S. G. Johnson et al., �??Low loss asymptotically single-mode propagation in large-core Omniguide fibers,�?? Opt. Express 9, 748-779 (2001). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748</a> [CrossRef] [PubMed]
- Y. Xu, R. K. Lee, and A. Yariv, �??Asymptotic analysis of Bragg fibers,�?? Opt. Lett. 25, 1756-1758 (2000). [CrossRef]
- Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, �??Asymptotic matrix theory of Bragg fibers,�?? J. Lightwave Technol. 20, 428-440 (2002). [CrossRef]
- J. Marcou, F. Brechet, and P. Roy, �??Design of weakly guiding Bragg fibres for chromatic dispersion shifting towards short wavelengths,�?? J. Opt. A, 3, S144-S153 (2001). [CrossRef]
- A. Argyros, �??Guided modes and loss in Bragg fibres,�?? Opt. Express, 10, 1411-1417, (2002). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1411</a> [CrossRef]
- J. G. Fleming, S. Y. Lin, and R. Hadley, in Proceeding of the solid-state sensor actuator and microsystems workshop, p.p. 173, (Hilton Head, S.C., 2002).
- R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St J. Russell, P. J. Roberts, and D. C. Allan, �??Single-mode photonic band gap guidance of light in air,�?? Science 285, 1537-1539 (1999). [CrossRef] [PubMed]
- A. Taflove, S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, (Artech house, Boston, 2000).
- See for example, A. Yariv, Optical electronics in modern communications, (Oxford university press, New York, 1997) Chapter 2.

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