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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 4444–4453
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Step-index optical waveguide produced by multi-step ion implantation in LiNbO3

G. B. Montanari, P. De Nicola, S. Sugliani, A. Menin, A. Parini, A. Nubile, G. Bellanca, M. Chiarini, M. Bianconi, and G. G. Bentini  »View Author Affiliations


Optics Express, Vol. 20, Issue 4, pp. 4444-4453 (2012)
http://dx.doi.org/10.1364/OE.20.004444


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Abstract

The refractive indexes, material attenuation and damage fractions of a multi-step ion implanted Lithium Niobate (LiNbO3) waveguide were analyzed as functions of the annealing temperatures. An almost flat damage depth profile was designed to reduce the uncertainties related to the indexes profile shape, thus providing a better test-case for the characterizations. The measurements performed on the fabricated optical waveguides confirmed the predicted step-index profiles showing that the light is confined inside the damaged layer. The low measured attenuation (less than 0.8 dB/cm @ 632.8 nm) makes the obtained waveguide attractive for device fabrication.

© 2012 OSA

1. Introduction

The growing interest in Integrated Optics for sensing, telecommunications and even complex opto-electronics is driving researchers to find solutions to the new challenges issued by the need of more and more fast, compact and performing devices.

Lithium Niobate is a well-known and extensively studied substrate for integrated optics mainly because of its electro-optic and piezo-electric properties [1

1. K. K. Wong, Properties of Lithium Niobate, EMIS Datareviews No. 28, INSPEC (The Institution of Electrical Engineers, London, 2002).

]. The technology most commonly used to fabricate waveguides on this material is still the Titanium diffusion into the crystal through thermal process.

Recently, thanks to both systematic experiments [4

4. H. Hu, F. Lu, F. Chen, B. R. Shi, K. M. Wang, and D. Y. Shen, “Extraordinary refractive-index increase in lithium niobate caused by low-dose ion implantation,” Appl. Opt. 40(22), 3759–3761 (2001). [CrossRef] [PubMed]

] and literature data analysis, some groups proposed mathematical [5

5. Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in Lithium Niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75(19), 195101 (2007). [CrossRef]

] or semi-empirical models [6

6. S. Sugliani, M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Menin, A. Malacarne, and L. Potì, “Refractive index tailoring in congruent Lithium Niobate by ion implantation,” Nucl. Instrum. Meth. B 268(19), 2911–2914 (2010). [CrossRef]

] to predict the LiNbO3 refractive index profiles induced by a given ion implantation recipe. Nevertheless the relationship between the damage caused by ion implantation in LiNbO3 and all its optical properties is still an open question and none of these models is conclusive.

This work gives a further contribution toward a better comprehension of this topic. In order to simplify the analysis of the experimental data, a step-like damage profile, which reasonably corresponds to a step-like refractive index distribution, was produced and then characterized.

2. Experimental

The optical properties of both x- and z-cut congruent Lithium Niobate 1 mm thick samples were modified by performing a multi-step carbon ion implantation process using the 1.7 MV Tandetron accelerator of CNR-IMM Laboratory in Bologna. Carbon was chosen because it is the heaviest ion generating mainly nuclear damage for the experimental conditions under examination [7

7. M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, A. Menin, G. B. Montanari, A. Nubile, and S. Sugliani, “Simulation of damage induced by ion implantation in Lithium Niobate,” Nucl. Instrum. Methods Phys. Rev. B 268(22), 3452–3457 (2010). [CrossRef]

].

Ten different energies and fluences were used in order to create a step-like nuclear damage Depth Profile (DP) in the samples. In Table 1

Table 1. Multi-Step Ion Implantation Recipe

table-icon
View This Table
the multiple implantation recipe is described. To avoid channeling effects each sample was rotated 7° around the y axis and then 22° around the normal axis. During the process the samples were clamped onto a massive copper holder kept just below room temperature by icy water.

Annealing processes at 100°C, 150°C, 200°C, 280°C and 415°C for 30 minutes were performed on different samples for each wafer cut in order to reduce the optical absorption induced by the implantation process.

Rutherford Back-scattering Spectrometry-channeling (RBS-c) analysis with 2.2 MeV hydrogen ions was performed to quantify the damage in the Nb sub-lattice on either the As-Implanted (AI) or Annealed (ANN) samples along x, y and z axes. The x and y axes measurements were performed on the x-cut pieces while z axis was observed on the z-cut ones.

The refractive index profiles of the twelve samples were investigated using the dark m-lines technique [8

8. R. Ulrich and R. Torge, “Measurement of thin film parameters with a prism coupler,” Appl. Opt. 12(12), 2901–2908 (1973). [CrossRef] [PubMed]

] at λ = 632.8 nm while the absorption spectra were estimated using an optical spectrometer in the visible and NIR regions.

The annealed samples at 280°C showed the best characteristics in terms of refractive index profile and material attenuation, so their waveguide losses and near field intensity distributions were also measured.

3. Results and discussion

3.1 Waveguide design

In Fig. 1
Fig. 1 Simulated Energy density released in nuclear collisions for the designed multi-step ion implantation.
the calculated DP of the energy density released in nuclear collisions (Ed) [9

9. M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Nubile, and S. Sugliani, “Defect engineering and micromachining of Lithium Niobate by ion implantation,” Nucl. Instrum. Methods Phys. Res. B 267(17), 2839–2845 (2009). [CrossRef]

] for the multiple implantation is shown. This profile is calculated using SRIM Binary Collision Monte Carlo program [10

10. J. F. Ziegler, J. P. Biersack, and U. Littmark, The stopping and ranges of ions in solids (Pergamon, 1985), http://www.srim.org.

] (SRIM version 2008.04) by adding up the Ed DP for each implantation step.

The Ed DP is very close to a step-like profile extending down to 2.75 μm from the surface, except for the shallow well in the first 0.5 μm. A completely flat profile was not achieved because the required fluences are not available at lower energies. The released energy density plateau value is Ed,p=(1.33±0.03)×1023eV/cm3.

Then the defective fraction (n*)DP for the multiple implantation along the three crystal axes was calculated [9

9. M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Nubile, and S. Sugliani, “Defect engineering and micromachining of Lithium Niobate by ion implantation,” Nucl. Instrum. Methods Phys. Res. B 267(17), 2839–2845 (2009). [CrossRef]

]. The plateau values are nx,p*=0.075±0.001, ny,p*=0.0379±0.0004and nz,p*=0.0343±0.0005for x, y and z axis respectively.

This multi-step implantation recipe has been designed just in order to have a step-like Ed DP with plateau value as near as possible to 1.35×1023eV/cm3, which corresponds to the maximum value of extraordinary refractive index profile according to the predictive curves reported in [6

6. S. Sugliani, M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Menin, A. Malacarne, and L. Potì, “Refractive index tailoring in congruent Lithium Niobate by ion implantation,” Nucl. Instrum. Meth. B 268(19), 2911–2914 (2010). [CrossRef]

].

3.2 RBS-c measurements

Figure 2
Fig. 2 Defective fractions measured by RBS-c as functions of annealing temperature.
shows the defective fractions along the three crystal axes provided by the RBS-c measurements as functions of the annealing temperature. Even with strong dechanneling, the surface damage value can be determinated in a reliable way.

The AI experimental nx,*, ny* and nz* refer to the damage in a surface layer 0.2 μm thick and are compatible with the lattice defects model conventionally assumed to describe the ion implantation effects in LN [11

11. S. Jetschke, H. Karge, and K. Hehl, “Anisotropic effects in N+-implanted LiNbO3,” Phys. Status Solidi A 77(1), 207–214 (1983). [CrossRef]

].

According to this model the generated interstitial atoms occupy vacant octahedral sites. These sites are better exposed if the RBS-c ion beam is directed along the x axis, while they are partially hidden for a beam along the y axis and almost totally undetectable for a beam along z axis (see Fig. 4
Fig. 4 Z-cut m-lines measurements for different annealing temperatures.
in [12

12. G. Götz and H. Karge, “Ion implantation into LiNbO3,” Nucl. Instrum. Methods 209-210, 1079–1088 (1983). [CrossRef]

]).

The defects are reduced at every annealing temperature but, within this experimental approach, it is not possible to determinate how they evolve.

3.3 M-lines spectroscopy and RCM simulation

Figure 3
Fig. 3 Comparison of m-lines measurements for the two samples cuts.
shows for both cuts the measured mode effective indexes neff (expressed as the percentage variations compared to the substrate values) as a function of the squared normalized mode number, i.e. [(m + 1)λ/neff]2.

Here only the AI and ANN 280°C samples are reported for sake of simplicity. As usual only the extraordinary index supports guided modes. Since the main observed difference between the two cuts concerns the ordinary index, the two slab waveguides can be considered equal from a practical point of view.

Such small variations probably have two causes connected with the crystal cut. First the induced and released stresses depend on the relative orientation between the implanted surface and the c-axis. Second the ions-matter interaction is affected by the crystal structure encountered by the impinging ions which is slightly different in the two cases. These effects are still under investigation.

Figure 4 shows the normalized neff for z-cut samples at different annealing temperatures. The annealing process acts in opposite directions on no and ne recovering the crystal birefringence [6

6. S. Sugliani, M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Menin, A. Malacarne, and L. Potì, “Refractive index tailoring in congruent Lithium Niobate by ion implantation,” Nucl. Instrum. Meth. B 268(19), 2911–2914 (2010). [CrossRef]

].

Figure 5
Fig. 5 RCM simulated extraordinary refractive index profiles for different annealing temperatures. The dashed curve represents the AI predicted profile according to [6].
and Fig. 6
Fig. 6 RCM simulated ordinary refractive index profiles for different annealing temperatures. The dashed curve represents the AI predicted profile according to [6].
report the RCM [13

13. P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” Opt. Acta (Lond.) 33(2), 127–143 (1986). [CrossRef]

] simulated profiles for ne and no respectively. The assumed step-like shape well describes the fabricated waveguides.

The discrepancy between predicted and simulated ordinary index profiles is acceptable considering that the predictive curves [6

6. S. Sugliani, M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Menin, A. Malacarne, and L. Potì, “Refractive index tailoring in congruent Lithium Niobate by ion implantation,” Nucl. Instrum. Meth. B 268(19), 2911–2914 (2010). [CrossRef]

] are based on empirical data. More important, the predicted and simulated extraordinary index profiles are in very good agreement.

In Fig. 7
Fig. 7 Percentage variations of ne and no plateau values as functions of annealing temperature.
the plateau values (np) of ne and no are reported as functions of the annealing temperature. The ordinary index percentage variation is almost twice the extraordinary one for each temperature value.

3.4 Material absorption measurements

However it’s reasonable to think that κe follows the same trend of κο, both in spectral and annealing temperature dependences. Through reflectance R (Fig. 8
Fig. 8 Measured ordinary reflectance spectra in the visible region for every z-cut sample. The reflectance spectrum of the virgin crystal is also reported for comparison. Residual peaks are artifacts due to the spectrometer Deuterium lamp.
) and transmittance T (Fig. 9
Fig. 9 Measured ordinary transmittance spectra in the visible region for every z-cut sample. The transmittance spectrum of the virgin crystal is also reported for comparison. Residual peaks are artifacts due to the spectrometer Deuterium lamp.
) measurements, together with the estimation of implanted layer thickness t, one can determinate κο.

The thickness was calculated by fitting the oscillations measured in the NIR region (see Fig. 10
Fig. 10 NIR measured and simulated ordinary transmittance spectra for the as implanted z-cut sample.
) with the software Optical [14

14. E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt. 44(35), 7532–7539 (2005). [CrossRef] [PubMed]

], using as refractive index for the implanted layer the one obtained by the RCM simulation at 632.8 nm scaled with Sellmeier dispersion equation reported in [15

15. Crystal Technology, Inc., Lithium Niobate Optical Crystals, Data sheet. http://www.crystaltechnology.com/docs/LNopt.pdf

]. Its value, t = 2.75 ± 0.01 μm, is in good agreement with the one predicted by SRIM (see Fig. 1).

Figure 11
Fig. 11 Material absorption spectra in the visible region for every z-cut sample. The absorption spectrum of the virgin crystal is also reported for comparison.
shows the absorption coefficient as a function of the wavelength at different annealing temperature, while Fig. 12
Fig. 12 Material absorptions at fixed wavelengths as functions of annealing temperature. The error analysis is also reported.
shows α as a function of the annealing temperature at fixed wavelength.

For annealing temperature under 280 °C the material absorption is too high for guiding while above 280 °C the correct estimation of the absorption coefficient is not achievable due to the spectrometer sensitivity limit (see Fig. 11 and Fig. 12).

Two slabs (ANN 280°C and ANN 415°C) are potentially suitable for integrated optics. Considering that the second one has a very low index contrast (see Fig. 7), only the first one has been further characterized.

3.5 Waveguide attenuation measurements

The first propagating mode of the ANN 280°C z-cut sample was coupled using a prism technique at the corresponding angle. The waveguide losses were measured acquiring with a CMOS camera the propagating streak along the y axis (see Fig. 13
Fig. 13 Fundamental mode propagation loss @ 632.8 nm for the ANN 280°C sample.
). The streak intensity was integrated along its width and fitted with an exponential decay along its length (about 3 cm). The waveguide loss was estimated to be less than 0.8 dB/cm.

3.6 Near Field measurements

The sidewalls of the z-cut slab waveguide annealed at 280°C were optically polished. Then the near field images were collected for both the first and the second mode using the butt-coupling technique at λ = 660 nm. Their intensity profiles were compared with the ones obtained by means of a homemade numerical simulator (BPM) [16

16. 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(1), 54–59 (2002). [PubMed]

].

The profiles adopted were rescaled from 632.8 to 660 nm using the two Sellmeier dispersion equations for LiNbO3. Figure 14
Fig. 14 Near Field intensity of the first propagating mode @ 660 nm for the ANN 280°C sample.
and Fig. 15
Fig. 15 Near Field intensity of the second propagating mode @ 660 nm for the ANN 280°C sample.
report the comparison between measures and simulations for the first and the second mode respectively.

Since the second mode neff is very close to the extraordinary substrate value (see Fig. 4) it has a very low confinement and it’s difficult to couple without exciting the substrate modes. The latter effect can explain the difference between the simulation and the measure. However, for what concerns the fundamental mode, results are in quite good agreement.

4. Conclusions

A step index ion implanted optical waveguide was fabricated and characterized in the visible region showing good performances.

The refractive indexes of the implanted region resulted in good agreement with the values predicted by the semi-empirical model [6

6. S. Sugliani, M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Menin, A. Malacarne, and L. Potì, “Refractive index tailoring in congruent Lithium Niobate by ion implantation,” Nucl. Instrum. Meth. B 268(19), 2911–2914 (2010). [CrossRef]

].

The annealing effects were investigated. Damage reduction was observed even at low temperatures.

The birefringence constantly recovers as the annealing temperature increases and the no percentage decrease is almost twice the ne percentage increase at every temperature. Material attenuations were reduced through proper thermal treatment making the waveguide useful for device fabrication.

The Near Field characterization confirmed the assumed step-index profiles showing that the light is confined inside the damaged layer.

Defect engineering through ion implantation and proper thermal process allows to find a trade-off between index contrast and waveguide attenuation depending on the specific application.

Further studies have to be accomplished to measure the electro-optic coefficients of the damaged layer.

Acknowledgments

A special thank goes to Caterina Summonte for the assistance in the spectroscopic measurements. This work was partially supported by Programma Operativo FESR 2007-2013 della Regione Emilia-Romagna – Attività I.1.1.

References and links

1.

K. K. Wong, Properties of Lithium Niobate, EMIS Datareviews No. 28, INSPEC (The Institution of Electrical Engineers, London, 2002).

2.

P. D. Townsend, “Ion implantation and integrated optics,” J. Phys. E Sci. Instrum. 10(3), 197–203 (1977). [CrossRef]

3.

F. Chen, “Photonic guiding structures in lithium niobate crystals produced by energetic ion beams,” J. Appl. Phys. 106(8), 081101 (2009). [CrossRef]

4.

H. Hu, F. Lu, F. Chen, B. R. Shi, K. M. Wang, and D. Y. Shen, “Extraordinary refractive-index increase in lithium niobate caused by low-dose ion implantation,” Appl. Opt. 40(22), 3759–3761 (2001). [CrossRef] [PubMed]

5.

Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in Lithium Niobate waveguides fabricated by ion implantation,” Phys. Rev. B 75(19), 195101 (2007). [CrossRef]

6.

S. Sugliani, M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Menin, A. Malacarne, and L. Potì, “Refractive index tailoring in congruent Lithium Niobate by ion implantation,” Nucl. Instrum. Meth. B 268(19), 2911–2914 (2010). [CrossRef]

7.

M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, A. Menin, G. B. Montanari, A. Nubile, and S. Sugliani, “Simulation of damage induced by ion implantation in Lithium Niobate,” Nucl. Instrum. Methods Phys. Rev. B 268(22), 3452–3457 (2010). [CrossRef]

8.

R. Ulrich and R. Torge, “Measurement of thin film parameters with a prism coupler,” Appl. Opt. 12(12), 2901–2908 (1973). [CrossRef] [PubMed]

9.

M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Nubile, and S. Sugliani, “Defect engineering and micromachining of Lithium Niobate by ion implantation,” Nucl. Instrum. Methods Phys. Res. B 267(17), 2839–2845 (2009). [CrossRef]

10.

J. F. Ziegler, J. P. Biersack, and U. Littmark, The stopping and ranges of ions in solids (Pergamon, 1985), http://www.srim.org.

11.

S. Jetschke, H. Karge, and K. Hehl, “Anisotropic effects in N+-implanted LiNbO3,” Phys. Status Solidi A 77(1), 207–214 (1983). [CrossRef]

12.

G. Götz and H. Karge, “Ion implantation into LiNbO3,” Nucl. Instrum. Methods 209-210, 1079–1088 (1983). [CrossRef]

13.

P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” Opt. Acta (Lond.) 33(2), 127–143 (1986). [CrossRef]

14.

E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt. 44(35), 7532–7539 (2005). [CrossRef] [PubMed]

15.

Crystal Technology, Inc., Lithium Niobate Optical Crystals, Data sheet. http://www.crystaltechnology.com/docs/LNopt.pdf

16.

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(1), 54–59 (2002). [PubMed]

OCIS Codes
(160.3730) Materials : Lithium niobate
(230.7390) Optical devices : Waveguides, planar

ToC Category:
Integrated Optics

History
Original Manuscript: September 22, 2011
Revised Manuscript: November 9, 2011
Manuscript Accepted: November 9, 2011
Published: February 8, 2012

Citation
G. B. Montanari, P. De Nicola, S. Sugliani, A. Menin, A. Parini, A. Nubile, G. Bellanca, M. Chiarini, M. Bianconi, and G. G. Bentini, "Step-index optical waveguide produced by multi-step ion implantation in LiNbO3," Opt. Express 20, 4444-4453 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4444


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References

  1. K. K. Wong, Properties of Lithium Niobate, EMIS Datareviews No. 28, INSPEC (The Institution of Electrical Engineers, London, 2002).
  2. P. D. Townsend, “Ion implantation and integrated optics,” J. Phys. E Sci. Instrum.10(3), 197–203 (1977). [CrossRef]
  3. F. Chen, “Photonic guiding structures in lithium niobate crystals produced by energetic ion beams,” J. Appl. Phys.106(8), 081101 (2009). [CrossRef]
  4. H. Hu, F. Lu, F. Chen, B. R. Shi, K. M. Wang, and D. Y. Shen, “Extraordinary refractive-index increase in lithium niobate caused by low-dose ion implantation,” Appl. Opt.40(22), 3759–3761 (2001). [CrossRef] [PubMed]
  5. Y. Jiang, K. M. Wang, X. L. Wang, F. Chen, C. L. Jia, L. Wang, Y. Jiao, and F. Lu, “Model of refractive-index changes in Lithium Niobate waveguides fabricated by ion implantation,” Phys. Rev. B75(19), 195101 (2007). [CrossRef]
  6. S. Sugliani, M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Menin, A. Malacarne, and L. Potì, “Refractive index tailoring in congruent Lithium Niobate by ion implantation,” Nucl. Instrum. Meth. B268(19), 2911–2914 (2010). [CrossRef]
  7. M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, A. Menin, G. B. Montanari, A. Nubile, and S. Sugliani, “Simulation of damage induced by ion implantation in Lithium Niobate,” Nucl. Instrum. Methods Phys. Rev. B268(22), 3452–3457 (2010). [CrossRef]
  8. R. Ulrich and R. Torge, “Measurement of thin film parameters with a prism coupler,” Appl. Opt.12(12), 2901–2908 (1973). [CrossRef] [PubMed]
  9. M. Bianconi, G. G. Bentini, M. Chiarini, P. De Nicola, G. B. Montanari, A. Nubile, and S. Sugliani, “Defect engineering and micromachining of Lithium Niobate by ion implantation,” Nucl. Instrum. Methods Phys. Res. B267(17), 2839–2845 (2009). [CrossRef]
  10. J. F. Ziegler, J. P. Biersack, and U. Littmark, The stopping and ranges of ions in solids (Pergamon, 1985), http://www.srim.org .
  11. S. Jetschke, H. Karge, and K. Hehl, “Anisotropic effects in N+-implanted LiNbO3,” Phys. Status Solidi A77(1), 207–214 (1983). [CrossRef]
  12. G. Götz and H. Karge, “Ion implantation into LiNbO3,” Nucl. Instrum. Methods209-210, 1079–1088 (1983). [CrossRef]
  13. P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” Opt. Acta (Lond.)33(2), 127–143 (1986). [CrossRef]
  14. E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt.44(35), 7532–7539 (2005). [CrossRef] [PubMed]
  15. Crystal Technology, Inc., Lithium Niobate Optical Crystals, Data sheet. http://www.crystaltechnology.com/docs/LNopt.pdf
  16. F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of index-guiding photonic crystal fibers and couplers,” Opt. Express10(1), 54–59 (2002). [PubMed]

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