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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 24 — Nov. 27, 2006
  • pp: 11848–11854
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Analysis of photonic crystal fibers: Scalar solution and polarization correction

Víctor Hugo Aristizabal, Francisco Javier Vélez, and Pedro Torres  »View Author Affiliations


Optics Express, Vol. 14, Issue 24, pp. 11848-11854 (2006)
http://dx.doi.org/10.1364/OE.14.011848


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Abstract

A numerical approach based on the scalar finite element method is applied to analyse the modal properties of photonic crystal fibers having a solid core and a cladding region with either circular or non-circular microstructured holes. A correction which accounts for the polarization effects due to the large refractive index difference between silica materials and air holes is included in the analysis. Numerical results show that the proposed technique is an efficient and accurate alternative to vector ones.

© 2006 Optical Society of America

1. Introduction

Photonic crystal fibers (PCFs) consisting of a central defect region surrounded by multiple air holes that run along the length of the fiber are attracting much attention in recent years because of unique properties which are not realized in conventional optical fibers. They can be designed to obtain: i) wide range of single mode operation [1

1. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Aktin, “All silica single mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1999). [CrossRef]

, 2

2. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

]; ii) small mode area, allowing nonlinear effects [3

3. W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, “Soliton effects in photonic crystal fibers at 850 nm,” Electron. Lett. 36, 53–55 (2000). [CrossRef]

]; iii) large mode area for generation of high-power optical beams [4

4. K. Furusawa, A. N. Malinowski, J. H. V. Price, T. M. Monro, J. K. Sahu, J. Nilsson, and D. J. Richardson, “Cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding,” Opt. Express 9, 714–720 (2001). [CrossRef] [PubMed]

]; iv) great controllability in chromatic dispersion by varying the hole diameter and hole pitch [5–7

5. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000). [CrossRef]

].

In this work, first, the scalar approximation to PCFs is revised by using the finite element method (FEM) that reduces the scalar Helmholtz equation into eigenvalue problem which is solved to find the guided modes and their corresponding propagation constants. Next, we propose to improve the scalar approximation to extend its accuracy at both long wavelength and large air filling fraction conditions, by including in the analysis the polarization effects due to the refractive index discontinuity. Evidently this approach is more efficient than the vector ones since it takes advantage from the simplicity of the scalar solution. In order to show how good this new approximation is, the effective index of fundamental mode for different PCFs were computed and compared with full-vector results reported in literature.

2. Analysis method

2.1 Scalar solution

The electric field of a mode of an arbitrary waveguide is expressible in the form

E(x,y,z)={Et+Ezẑ}exp(iβz),
(1)

{t2+k02n2β2}Et=t{Et·tlnn2},
(2)

where t2 is the Laplacian operator in transverse plane, n=n(x, y) is the refractive-index profile, and k 0=2π/λ the wave number in the vacuum, λ being the wavelength.

The scalar approximation results when one neglects the so-called vector term on the right hand side of Eq. (2):

{t2+k02n2β˜2}E˜t=0.
(3)

Here, t and β˜ are the scalar field and its corresponding propagating constant, respectively. This approximation is valid when coupling between orthogonal field components become negligible.

In order to solve Eq. (3) efficiently for PCFs, we use a nodal based FEM with first order triangular element. By applying the variational finite element procedure, the scalar wave Eq. (3) yields the matrix equation:

[A]{e}=β˜2[B]{e},
(4)

where the eigenvector {e} contains the values of the electric field at the vertices of the triangular elements used for discretization.

By using scalar formulation to solve Eq. (3), it is neglected the fact the electric fields are discontinuous at the interfaces to air holes. With first-order triangular elements, the continuity of the electric field is fulfilled automatically. This fact is also found in recent applications of the scalar treatment to PCFs in the short-wavelength regimen [2

2. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

, 20–22

20. J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, “A ‘poor man’s approach’ to modelling microstructured optical fibres,” J. Opt. A: Pure Appl. Opt. 5, 534–538 (2003). [CrossRef]

], where the fraction of the electric field in the air hole is vanishing, while vectorial effects of Eq. (2) become negligible. An approach very different to this scalar treatment was recently proposed by Mortensen [23

23. N. A. Mortensen, “Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers,” Opt. Lett. 30, 1455–1457 (2005). [CrossRef] [PubMed]

], suggesting that one can approximate the problem by imposing the boundary condition that the electric field is zero at the interface to the air holes, achieving an excellent agreement with PWE results up to a normalized wavelength λ/Λ=0.4, where Λ is the hole pitch.

2.2. Polarization correction

In order to correct the scalar propagation constant, we applied the relationship between β˜ and β as described in Ref. [24

24. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).

]

δβ2=β2β˜2=A(t·E˜t)Et·tlnn2dAAE˜t·EtdA,
(5)

where A is the fiber cross section.

To determine δβ 2 exactly we would have to solve the vector wave Eq. (2). However, using simple perturbations methods [24

24. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).

], we have that the polarization correction to x- or y-polarized mode, to first order, reduces to

βi2β˜i2=A1n2xni2xiEixiEidAAEi2dA,
(6)

where i and xi are equal to x or y for x and y polarized mode, respectively. This term is always negative and therefore tends to shift down the propagation constant obtained from the scalar analysis.

3. Results and discussion

As a direct way of testing the accuracy of the proposed calculations, we compare the computed effective index neff =β/k 0 of the fundamental mode for different PCFs with vector results reported in literature. A single-polarization single mode (SPSM) fiber is also included in our analysis, as a case to predict sensitive vector quantities such as birefringence. All results were obtained by using large computational domains with Dirichlet boundary conditions.

3.1. Fiber with triangular lattice cladding

The PCF first considered consists of a triangular lattice of air holes. An example of its cross section is shown in Fig. 1(a), where d is the hole diameter. The PCF symmetry allows just one quarter of the structure to be considered for the numerical simulation. The scalar and the corrected scalar approaches were tested by comparing the computed curves of the effective index for the HE11x mode of this fiber with results reported in Ref. [11

11. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” J. Quantum Electron. 38, 927–933 (2002). [CrossRef]

], where a full-vector BPM was used. Figure 1(b) shows results where d/Λ is taken as a parameter, over a wide range of wavelength from λ=0.4 µm to λ=2.0 µm. As it was expected, the scalar solution is valid in the shorter wavelength region (λ<0.5 µm), and begins to break down with increasing air filling fraction. On the other hand, the results that include the polarization correction show an overall good agreement with vectorial ones, hence justifying the proposed calculation scheme. For large hole size d/Λ=0.7, where the polarization effects are important, the maximum relative error at longer wavelength of the studied range is 0.12%, which underlines the relevance of our approximation.

Fig. 1. (a). PCF with triangular lattice of air holes. (b). Effective index of the fundamental mode for this structure where Λ=2.3 µm and d/Λ is taken as a parameter. The dotted and dash dotted curves are the predictions of scalar and corrected scalar solution, respectively. Comparison with vector results reported in Ref. [11] (solid lines).

3.2. Annular-shaped holes fiber

Fig. 2. (a). Cross section of the 3 annular-shaped holes PCF. (b). Effective index of the HE11- like mode for this structure calculated with the scalar and the corrected scalar schemes discussed in the text. Comparison with vector results reported in Ref. [25].

3.3. Cobweb fiber

An extreme example of a PCF with large air filling fraction is the cobweb fiber. Figure 3(a) shows the cross section of the analyzed fiber. The core diameter is about 1 µm and the silica bridges, which separate the holes, have a thickness of 0.12 µm. This fiber with such geometrical characteristics was experimentally evaluated in Ref. [5

5. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000). [CrossRef]

] and theoretically analyzed through a vector FEM in Ref. [26

26. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite-element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002). [CrossRef]

]. Here, it is necessary to carry out a very accurate fiber cross section representation, so the FEM is a powerful tool able to cope with this kind of geometry. The structure is symmetric, thus, one half of its cross section was considered. The numerical effective-index curves are compared in Fig. 3(b). As this fiber has a comparable air filling fraction as the preceding case, it is expected that the general trend of the obtained results is the same, i.e., non appropriated solution of the scalar approach and that perturbative correction to the scalar solution can be an important tool for observing the properties of PCF even on extreme conditions of air filling fraction. In this case, the relative error of our approximation goes from 0.05% to 0.8% over the range of wavelength studied.

3.4. SPSM fiber

Fig. 3. (a). Cross section of the cobweb PCF. (b). Effective index of the fundamental mode for this structure calculated with the scalar and the corrected scalar schemes discussed in the text. Comparison with vector results reported in Ref. [26].
Fig. 4. (a). Cross section of the SPSM-PCF. (b). Modal dispersion for the slow-axis and fastaxis modes of this structure calculated with the scalar and the corrected scalar schemes discussed in the text. Comparison with vector results reported in Ref. [27].

4. Conclusion

Acknowledgments

Financial support by Colciencias (1118-05-13632) and DIME (National University of Colombia, Medellín Campus) is gratefully acknowledged.

References and links

1.

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Aktin, “All silica single mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1999). [CrossRef]

2.

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

3.

W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, “Soliton effects in photonic crystal fibers at 850 nm,” Electron. Lett. 36, 53–55 (2000). [CrossRef]

4.

K. Furusawa, A. N. Malinowski, J. H. V. Price, T. M. Monro, J. K. Sahu, J. Nilsson, and D. J. Richardson, “Cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding,” Opt. Express 9, 714–720 (2001). [CrossRef] [PubMed]

5.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000). [CrossRef]

6.

W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express 10, 609–613 (2002). [PubMed]

7.

K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fiber: application to ultra-flattened dispersion,” Opt. Express 11, 843–852 (2003). [CrossRef] [PubMed]

8.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. 24, 276–278 (1999). [CrossRef]

9.

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. 18, 50–56 (2000). [CrossRef]

10.

F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of indexguiding photonic crystal fibers and couplers,” Opt. Express 10, 54–59 (2002). [PubMed]

11.

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” J. Quantum Electron. 38, 927–933 (2002). [CrossRef]

12.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002). [CrossRef]

13.

S. Campbell, R. C. McPhedran, C. Martijn de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919–1928 (2004). [CrossRef]

14.

Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers, “Opt. Express 10, 853–864 (2002). [PubMed]

15.

C. P. Yu and H. C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. 36, 145–163 (2004). [CrossRef]

16.

M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett. 30, 327–330 (2001). [CrossRef]

17.

K. Saitoh and M. Koshiba, “Numerical modeling of photonic crystal fibers,” J. Lightwave Technol. 23, 3580–3580 (2005). [CrossRef]

18.

T. N. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: An efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999). [CrossRef]

19.

C. E. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Express 7, 113–122 (2000). [CrossRef] [PubMed]

20.

J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, “A ‘poor man’s approach’ to modelling microstructured optical fibres,” J. Opt. A: Pure Appl. Opt. 5, 534–538 (2003). [CrossRef]

21.

V. H. Aristizabal, F. J. Vélez, and P. Torres, “Modelling of photonic crystal fibers with the scalar finite element method,” in 5th Iberoamerican Meeting on Optics and 8th Latin American Meeting on Optics, Laser and their Applications, A. Marcano and J. L. Paz, eds., Proc. SPIE5622, 849–854 (2004). [CrossRef]

22.

T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express 12, 69–74 (2004). [CrossRef] [PubMed]

23.

N. A. Mortensen, “Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers,” Opt. Lett. 30, 1455–1457 (2005). [CrossRef] [PubMed]

24.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).

25.

H. P. Uranus and H. J. W. M. Hoekstr, “Modelling of microstructured waveguides using a finite-elementbased vectorial mode solver with transparent boundary conditions,” Opt. Express 12, 2795–2809 (2004). [CrossRef] [PubMed]

26.

A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite-element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002). [CrossRef]

27.

K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal fibers,” IEEE Photon. Technol. Lett. 15, 1384–1386 (2003). [CrossRef]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2400) Fiber optics and optical communications : Fiber properties
(060.2430) Fiber optics and optical communications : Fibers, single-mode

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: September 18, 2006
Revised Manuscript: November 8, 2006
Manuscript Accepted: November 11, 2006
Published: November 27, 2006

Citation
Víctor Hugo Aristizabal, Francisco Javier Vélez, and Pedro Torres, "Analysis of photonic crystal fibers: Scalar solution and polarization correction," Opt. Express 14, 11848-11854 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-24-11848


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References

  1. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Aktin, "All silica single mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1999). [CrossRef]
  2. T. A. Birks, J. C. Knight, P. St. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  3. W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, "Soliton effects in photonic crystal fibers at 850 nm," Electron. Lett. 36, 53-55 (2000). [CrossRef]
  4. K. Furusawa, A. N. Malinowski, J. H. V. Price, T. M. Monro, J. K. Sahu, J. Nilsson, and D. J. Richardson, "Cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding," Opt. Express 9, 714-720 (2001). [CrossRef] [PubMed]
  5. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, "Anomalous dispersion in photonic crystal fiber," IEEE Photon. Technol. Lett. 12, 807-809 (2000). [CrossRef]
  6. W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002). [PubMed]
  7. K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, "Chromatic dispersion control in photonic crystal fiber: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003). [CrossRef] [PubMed]
  8. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, M. V. Andrés, "Full-vector analysis of a realistic photonic crystal fiber," Opt. Lett. 24, 276-278 (1999). [CrossRef]
  9. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Modeling large air fraction holey optical fibers," J. Lightwave Technol. 18, 50-56 (2000). [CrossRef]
  10. F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, "Full vectorial BPM modeling of index-guiding photonic crystal fibers and couplers," Opt. Express 10, 54-59 (2002). [PubMed]
  11. K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers," J. Quantum Electron. 38, 927-933 (2002). [CrossRef]
  12. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002). [CrossRef]
  13. S. Campbell, R. C. McPhedran, C. Martijn de Sterke, and L. C. Botten, "Differential multipole method for microstructured optical fibers," J. Opt. Soc. Am. B 21, 1919-1928 (2004). [CrossRef]
  14. Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers, "Opt. Express 10, 853-864 (2002). [PubMed]
  15. C. P. Yu and H. C. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004). [CrossRef]
  16. M. Qiu, "Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method," Microwave Opt. Technol. Lett. 30, 327-330 (2001). [CrossRef]
  17. K. Saitoh and M. Koshiba, "Numerical modeling of photonic crystal fibers," J. Lightwave Technol. 23, 3580-3580 (2005). [CrossRef]
  18. T. N. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Holey optical fibers: An efficient modal model," J. Lightwave Technol. 17, 1093-1102 (1999). [CrossRef]
  19. C. E. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, "Experimental and scalar beam propagation analysis of an air-silica microstructure fiber," Opt. Express 7, 113-122 (2000). [CrossRef] [PubMed]
  20. J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, "A ‘poor man’s approach’ to modelling micro-structured optical fibres," J. Opt. A: Pure Appl. Opt. 5, 534-538 (2003). [CrossRef]
  21. V. H. Aristizabal, F. J. Vélez, and P. Torres, "Modelling of photonic crystal fibers with the scalar finite element method," in 5th Iberoamerican Meeting on Optics and 8th Latin American Meeting on Optics, Laser and their Applications, A. Marcano and J. L. Paz, eds., Proc. SPIE 5622, 849-854 (2004). [CrossRef]
  22. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, "Scaling laws and vector effects in bandgap-guiding fibres," Opt. Express 12, 69-74 (2004). [CrossRef] [PubMed]
  23. N. A. Mortensen, "Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers," Opt. Lett. 30, 1455-1457 (2005). [CrossRef] [PubMed]
  24. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).
  25. H. P. Uranus and H. J. W. M. Hoekstr, "Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions," Opt. Express 12, 2795-2809 (2004). [CrossRef] [PubMed]
  26. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, "Holey fiber analysis through the finite-element method," IEEE Photon. Technol. Lett. 14, 1530-1532 (2002). [CrossRef]
  27. K. Saitoh and M. Koshiba, "Single-polarization single-mode photonic crystal fibers," IEEE Photon. Technol. Lett. 15, 1384-1386 (2003). [CrossRef]

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