## Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3D anisotropy

Optics Express, Vol. 15, Issue 24, pp. 15797-15811 (2007)

http://dx.doi.org/10.1364/OE.15.015797

Acrobat PDF (575 KB)

### Abstract

A full-vectorial finite element method based eigenvalue algorithm is developed to analyze the band structures of two-dimensional (2D) photonic crystals (PCs) with arbitray 3D anisotropy for in-planewave propagations, in which the simple transverse-electric (TE) or transverse-magnetic (TM) modes may not be clearly defined. By taking all the field components into consideration simultaneously without decoupling of the wave modes in 2D PCs into TE and TM modes, a full-vectorial matrix eigenvalue equation, with the square of the wavenumber as the eigenvalue, is derived. We examine the convergence behaviors of this algorithm and analyze 2D PCs with arbitrary anisotropy using this algorithm to demonstrate its correctness and usefulness by explaining the numerical results theoretically.

© 2007 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

6. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B **54**, 7837–7842 (1996). [CrossRef]

11. I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B **48**, 5004–5012 (1993). [CrossRef]

13. C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Phys. Rev. B **72**, 045133 (2005). [CrossRef]

14. S. M. Hsu, M. M. Chen, and H. C. Chang, “Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm,” Opt. Express15, 5416–5430 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-9-5416 [CrossRef] [PubMed]

14. S. M. Hsu, M. M. Chen, and H. C. Chang, “Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm,” Opt. Express15, 5416–5430 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-9-5416 [CrossRef] [PubMed]

15. G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and D. den Engelsen, “Symmetry properties of two-dimensional anisotropic photonic crystals,” J. Opt. Soc. Am. A **23**, 2002–2013 (2006). [CrossRef]

16. G. E. Antilla and N. G. Alexopoulos, “Scattering from complex three-dimensional geometries by a curvilinear hybrid finite-element-integral equation approach,” J. Opt. Soc. Am. A **11**, 1445–1457 (1994). [CrossRef]

17. L. Zhang and N. G. Alexopoulos, “Finite-element based techniques for the modeling of PBG materials,” Electromagnetics **19**, 225–239 (1999). [CrossRef]

18. D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. **47**, 2059–2074 (1999). [CrossRef]

17. L. Zhang and N. G. Alexopoulos, “Finite-element based techniques for the modeling of PBG materials,” Electromagnetics **19**, 225–239 (1999). [CrossRef]

## 2. Formulation

### 2.1. The governing equation

*t*) dependence of the form exp(

*jωt*) being implied, Maxwell’s curl equations can be expressed as

*ω*is the angular frequency,

*µ*

_{0}and

*ε*

_{0}are the permeability and permittivity of free space, and [

*µ*

*] and [*

_{r}*ε*

*] are, respectively, the relative permeability and permittivity tensors of the medium given by*

_{r}**E**or the magnetic field

**H**, and the tensors [

*p*] and [

*q*] are, respectively, given by

**E**and

**H**.

*a*is the lattice distance and

*r*is the radius of the parallel cylinders. The square region, enclosed by sides I, II, III, and VI, in Fig. 1(a) indicates the unit cell of the 2D PC with square lattice, while the hexagonal region, surrounded by sides I, II, …, VI, in Fig. 1(b) denotes the unit cell of the 2D PC with triangular lattice. For these 2D PCs which are uniform along the z direction and periodic in the x-y plane, the field distribution Φ of the wave modes in the PCs for the in-plane propagation can be expressed as

*and Φ*

_{t}*, both assumed to be function of*

_{z}*x*and

*y*, are the transverse and longitudinal field components of Φ, respectively. Substituting Eq. (10) into Eq. (5) and using

*and ∇*

_{t}*are, respectively, the transverse and longitudinal parts of the ∇ operator, Eq. (5) can be separated into its transverse component*

_{z}## 2.2. The FEM based matrix eigenvalue equation

20. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. **18**, 737–743 (2000). [CrossRef]

*ϕ*

_{t1}-

*ϕ*

_{t8}, based on linear tangential and quadratic normal (LT/QN) vector basis functions and a quadratic nodal element with six variables,

*ϕ*

_{z1}-

*ϕ*

_{z6}, are employed for the transverse and longitudinal field components, respectively. The vector-based shape function for the curvilinear edge element,

*x*̂{

*U*}+

*ŷ*{

*V*}, and the scalar-based shape function for the curvilinear nodal element, {

*N*}, are listed in Table 1, where {

*ϕ*

^{e}*} and {*

_{t}*ϕ*

^{e}*} are, respectively, the edge- and nodal-variable vectors for each element,*

_{z}*L*

*(*

_{i}*i*=1,2,3) are the simplex coordinates, |

*J*| is the Jacobian, the determinant of the Jacobian matrix, [

*J*], defined by

_{t}*L*

*|*

_{i}*and |*

_{j}*J*|

*are, respectively, the values of |∇*

_{j}

_{t}*L*

*| (*

_{i}*i*=1,2,3) and |

*J*| at the nodal point

*j*(

*j*=1,2, …, 6).

*f*(

*x*,

*y*) in the Cartesian coordinate system can be performed in the simplex coordinate system through

*T*denotes transpose. Applying Galerkin’s method, assembling all element matrices, and incorporating the periodic boundary conditions (PBCs) [14

14. S. M. Hsu, M. M. Chen, and H. C. Chang, “Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm,” Opt. Express15, 5416–5430 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-9-5416 [CrossRef] [PubMed]

*k*

*and*

_{x}*k*

*are, respectively, the wavenumbers in the*

_{y}*x*and

*y*directions, and the subscript denotes the side label of the unit cell, as defined in Fig. 1(a) and (b), Eqs. (12) and (13) can be transformed into the matrix form as

*ϕ*} and the matrices, [

*K*] and [

*M*], are given by

*extends over all different elements. Equation (33) is the desired matrix eigenvalue equation in the proposed full-vectorial algorithm for the analysis of 2D anisotropic PCs with arbitrary permittivity and permeability tensors.*

_{e}## 3. Numerical results

## 3.1. Isotropic PCs

*ε*=8.9 and radius

*r*=0.2

*a*in the air, and the second consists of triangle-arranged dielectric cylinders with relative permittivity

*ε*=11.4 and radius

*r*=0.2

*a*in the air. The first BZ for the 2D PC with square lattice is shown in Fig. 3(a), in which four sub-zones are marked. For isotropic PCs, considering all the possible directions of the wave vector

**k**in any single sub-zone, identical to the IBZ, is sufficient for the construction of complete band structures. Similarly, the six sub-zones in the first BZ for the 2D PC with triangular lattice, as shown in Fig. 3(b), are exactly the same as isotropic PCs. Consequently, the band structures of the two isotropic PCs considered here can be completely constructed from sub-zones Γ-X-M and Γ-M-K, respectively. The band structures of these two PCs obtained from the full-vectorial algorithm are, respectively, shown in Fig. 4(a) and (b), in which the band structures constructed from the scalar algorithm [14

**k**is fixed at the X point in the first BZ and various numbers of elements are used to search the eigen frequencies of several bands for both TE and TM modes. The convergence behaviors of the first and third bands for the TE mode are, respectively, shown in Fig. 5(a) and (b), and those for the TM mode are, respectively, shown in Fig. 5(c) and (d). In Fig. 5(a) and (b), the lines with blue triangles, blue squares, and red circles represent the results calculated from the

*E*-formulation full-vectorial algorithm, the

*H*-formulation full-vectorial algorithm, and the TE scalar algorithm, respectively. It can be seen that the eigen frequencies obtained from the

*E*- and

*H*-formulation full-vectorial algorithms will converge to the same value with enough number of elements used, although their convergence behaviors are obviously different from each other. On the other hand, the agreement of the convergence behaviors between the

*H*-formulation full-vectorial algorithm and the TE scalar algorithm provides a powerful support for the correctness of this full-vectorial algorithm. According to the definitions in [14

*H*

*,*

_{z}*E*

*, and*

_{x}*E*

*components, and the unknown field component to be solved in the scalar algorithm for the TE mode is*

_{y}*H*

*. Therefore, when the*

_{z}*H*-formulation is employed in the full-vectorial algorithm,

*H*

*will appear in the unknown field vector explicitly, and it is quite reasonable that the results from these two algorithms would be consistent with each other.*

_{z}*E*

*, it is no surprise that its convergence behaviors agree well with those of the*

_{z}*E*-formulation full-vectorial algorithm.

**k**is fixed at theMpoint in the first BZ, and the counterparts of Fig. 5(a), (b), (c), and (d) are displayed in Fig. 6(a), (b), (c), and (d), respectively. We can observe that the convergence behaviors of the

*E*- and

*H*-formulation full-vectorial algorithms, and the scalar algorithms for the TE and TM modes for triangular lattice are similar to those for square lattice. Hence, the discussions for the square lattice can be applied to the triangular lattice as well, and the correctness of the proposed full-vectorial algorithm for solving the band structures of isotropic 2D PCs is verified again.

## 3.2. Anisotropic PCs

*n*

_{o}=4.8 and

*n*

*=6.2, can be used to provide a chance of obtaining the absolute band gap [21*

_{e}21. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. **81**, 2574–2577 (1998). [CrossRef]

13. C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Phys. Rev. B **72**, 045133 (2005). [CrossRef]

*r*of the LC columns is 0.45

*a*and the components of the relative permittivity tensor [

*ε*

*] of the nematic LCs (5CB) [22] we use are given as*

_{r}*n*

*=1.5292 and*

_{o}*n*

*=1.7072 are, respectively, the ordinary and extraordinary refractive indices of the nematic LCs,*

_{e}*θ*

*is the angle between the crystal*

_{c}*c*-axis and the

*z*-axis, and

*ϕ*

_{c}represents the angle between the projection of the crystal

*c*-axis on the

*x*-

*y*plane and the

*x*-axis, as defined in Fig. 7(b).

*ϕ*

*=30°. Notice that for the special cases, i.e.*

_{c}*θ*

*=0° or 90°, the scalar algorithm in [14*

_{c}*θ*

*=0° and 90°, respectively, not only the results from the full-vectorial algorithm, represented by the blue solid lines, but also those from the scalar algorithm, represented by the red and green circular dots, are displayed. It can be seen that, even for anisotropic PCs, the results of these two algorithms still agree with each other very well provided that the wave modes can be decoupled into the TE and TM modes. The significance of the full-vectorial algorithm is conspicuously revealed from Fig. 8(b), (c), and (d). These band structures can not be obtained from the scalar algorithm because the wave modes in the PC are hybrid ones when*

_{c}*θ*

*is not exactly 0° or 90°. However, by taking advantage of the full-vectorial algorithm, we can construct these band structures smartly.*

_{c}*x*-

*y*plane of this PC, and the shadowed regions represent ignorable parts as a result of the symmetry conditions and the basic principle of periodic structures [14

*θ*

*’s when*

_{c}*ϕ*

*=30°. To systematically understand the influences of θc and ϕc on the absolute band gaps and clearly demonstrate how the anisotropy affects the wave modes in the PC structures, we analyze the band structures of more different*

_{c}*θ*

*’s and*

_{c}*ϕ*

*’s. Fig. 9 shows the normalized frequency range of the absolute band gap versus ϕc for five different*

_{c}*θ*

*’s from 0° to 90°. We can find out that the absolute band gap can be noticeably tuned by changing*

_{c}*θ*

*regardless of the value of*

_{c}*ϕ*

*, and the upper limit of the absolute band gap is much dependent on*

_{c}*θ*

*than the lower limit for this PC. On the other hand, it is seen that, for every fixed*

_{c}*θ*

*, the normalized frequency range of the absolute band gap almost keeps constant when*

_{c}*ϕ*

*increases from 0° to 90°.*

_{c}*θ*

*=0° or 90°, the above phenomenon is absolutely reasonable and can be directly understood by the aid of Fig. 8(a) and (e), from which we can observe that the normalized frequency range of the absolute band gap is determined by the TM modes which are independent of the*

_{c}*ωa*/2

*πc*|

_{max}and

*ωa*/2

*πc*|

_{min}are, respectively, the maximum and minimum values of the limits for

*ϕ*

*from 0° to 90°. The results after the transformation of Eq. (51) are shown in Fig. 10. It is seen that the absolute band gap limits indeed vary with*

_{c}*ϕ*

*for*

_{c}*θ*

*=30°, 45°, and 60°. We also observe that the upper limit is still much dependent on*

_{c}*ϕ*

*than the lower limit for this PC, just as their dependence on*

_{c}*θ*

*. Another interesting phenomenon is that the variations of the upper and lower limits are opposite, i.e. when the upper limit reaches the maximum value, the lower limit will reach the minimum value, and vice versa. Finally, it is noticed that the variation of the limits has a period of 60° over*

_{c}*ϕ*

*. This phenomenon can be reasonably accepted because the unit cell of a PC with triangular lattice is a hexagon and rotating the LC molecules around the*

_{c}*z*-axis by 60° dose not really change the PC structure due to the rotation symmetry. Based on these numerical results and theoretical discussions, the correctness and usefulness of the proposed full-vectorial algorithm have been clearly demonstrated.

## 4. Conclusion

*θ*

*and*

_{c}*ϕ*

*on the behaviors of the absolute band gap limits are discussed in detail. By considering the numerical results carefully and explaining the observed phenomena theoretically, we demonstrate the performance and significance of this full-vectorial algorithm, and see that a deeper insight about 2D anisotropic PCs can be obtained through this full-wave analysis.*

_{c}## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

4. | J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. |

5. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

6. | P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B |

7. | M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. |

8. | L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, “An efficient finite-element method for the analysis of photonic band-gap materials,” in 1999 IEEE MTT-S Dig. |

9. | C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express12, 1397–1408 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-7-1397 [CrossRef] [PubMed] |

10. | P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E |

11. | I. H. H. Zabel and D. Stroud, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B |

12. | Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. |

13. | C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Phys. Rev. B |

14. | S. M. Hsu, M. M. Chen, and H. C. Chang, “Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm,” Opt. Express15, 5416–5430 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-9-5416 [CrossRef] [PubMed] |

15. | G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and D. den Engelsen, “Symmetry properties of two-dimensional anisotropic photonic crystals,” J. Opt. Soc. Am. A |

16. | G. E. Antilla and N. G. Alexopoulos, “Scattering from complex three-dimensional geometries by a curvilinear hybrid finite-element-integral equation approach,” J. Opt. Soc. Am. A |

17. | L. Zhang and N. G. Alexopoulos, “Finite-element based techniques for the modeling of PBG materials,” Electromagnetics |

18. | D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. |

19. | J. Jin, |

20. | M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. |

21. | Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gap in 2D anisotropic photonic crystals,” Phys. Rev. Lett. |

22. | P. Yeh and C. Gu, |

**OCIS Codes**

(160.3710) Materials : Liquid crystals

(230.3990) Optical devices : Micro-optical devices

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: August 28, 2007

Revised Manuscript: November 12, 2007

Manuscript Accepted: November 13, 2007

Published: November 14, 2007

**Citation**

Sen-ming Hsu and Hung-chun Chang, "Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3D anisotropy," Opt. Express **15**, 15797-15811 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15797

Sort: Year | Journal | Reset

### References

- E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
- S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486- 2489 (1987). [CrossRef] [PubMed]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995).
- J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
- S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999). [CrossRef]
- P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996). [CrossRef]
- M. Qiu and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys. 87, 8268-8275 (2000). [CrossRef]
- L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, "An efficient finite-element method for the analysis of photonic band-gap materials," in 1999 IEEE MTT-S Dig. 4, 1703-1706 (1999).
- C. P. Yu and H. C. Chang, "Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals," Opt. Express 12, 1397-1408 (2004). [CrossRef] [PubMed]
- P. J. Chiang, C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007). [CrossRef]
- I. H. H. Zabel and D. Stroud, "Photonic band structures of optically anisotropic periodic arrays," Phys. Rev. B 48, 5004-5012 (1993). [CrossRef]
- Z. Y. Li, B. Y. Gu, and G. Z. Yang, "Large absolute band gap in 2D anisotropic photonic crystals," Phys. Rev. Lett. 81, 2574-2577 (1998). [CrossRef]
- C. Y. Liu and L.W. Chen, "Tunable band gap in a photonic crystal modulated by a nematic liquid crystal," Phys. Rev. B 72, 045133 (2005). [CrossRef]
- S. M. Hsu, M. M. Chen, and H. C. Chang, "Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm," Opt. Express 15, 5416-5430 (2007). [CrossRef] [PubMed]
- G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and D. den Engelsen, "Symmetry properties of two-dimensional anisotropic photonic crystals," J. Opt. Soc. Am. A 23, 2002-2013 (2006). [CrossRef]
- G. E. Antilla and N. G. Alexopoulos, "Scattering from complex three-dimensional geometries by a curvilinear hybrid finite-element-integral equation approach," J. Opt. Soc. Am. A 11, 1445-1457 (1994). [CrossRef]
- L. Zhang and N. G. Alexopoulos, "Finite-element based techniques for the modeling of PBG materials," Electromagnetics 19, 225-239 (1999). [CrossRef]
- D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, "High-impedance electromagnetic surfaces with a forbidden frequency band," IEEE Trans. Microwave Theory Tech. 47, 2059-2074 (1999). [CrossRef]
- J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, Inc., New York, 2002).
- M. Koshiba and Y. Tsuji, "Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems," J. Lightwave Technol. 18, 737-743 (2000). [CrossRef]
- Z. Y. Li, B. Y. Gu, and G. Z. Yang, "Large absolute band gap in 2D anisotropic photonic crystals," Phys. Rev. Lett. 81, 2574-2577 (1998). [CrossRef]
- P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons, Inc., New York, 1999).

## 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.

« Previous Article | Next Article »

OSA is a member of CrossRef.