## Design of a compact mode and polarization converter in three-dimensional photonic crystals |

Optics Express, Vol. 20, Issue 18, pp. 20356-20367 (2012)

http://dx.doi.org/10.1364/OE.20.020356

Acrobat PDF (2680 KB)

### Abstract

A mode and polarization converter is proposed and optimized for 3D photonic integrated circuits based on photonic crystals (PhCs). The device converts the index-guided TE mode of a W1 solid-core (SC) waveguide to the band-gap-guided TM mode of a W1 hollow-core (HC) waveguide in 3D PhCs, and vice versa. The conversion is achieved based on contra-directional mode coupling. For a 25μm-long device, simulations show that the power conversion efficiency is over 98% across a wavelength range of 16 nm centered at 1550 nm, whereas the reflection remains below –20dB. The polarization extinction ratio of the conversion is in principle infinitely high because both W1 waveguides operate in the single-mode regimes in this wavelength range.

© 2012 OSA

## 1. Introduction

*etc*[1

1. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. **65**(25), 3152–3155 (1990). [CrossRef] [PubMed]

8. K. Ishizaki and S. Noda, “Manipulation of photons at the surface of three-dimensional photonic crystals,” Nature **460**(7253), 367–370 (2009). [CrossRef] [PubMed]

9. L. Tang and T. Yoshie, “Monopole woodpile photonic crystal modes for light-matter interaction and optical trapping,” Opt. Express **17**(3), 1346–1351 (2009). [CrossRef] [PubMed]

10. P. Russell, “Photonic crystal fibers,” Science **299**(5605), 358–362 (2003). [CrossRef] [PubMed]

1. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. **65**(25), 3152–3155 (1990). [CrossRef] [PubMed]

8. K. Ishizaki and S. Noda, “Manipulation of photons at the surface of three-dimensional photonic crystals,” Nature **460**(7253), 367–370 (2009). [CrossRef] [PubMed]

5. S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. **77**(22), 3490–3492 (2000). [CrossRef]

6. M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, “A three-dimensional optical photonic crystal with designed point defects,” Nature **429**(6991), 538–542 (2004). [CrossRef] [PubMed]

*r*

_{H}

*=*0.293

*a*,

*h*

_{H}

*=*0.224

*a*,

*r*

_{R}

*=*0.115

*a*, and

*h*

_{R}

*=*0.353

*a*(

*h*

_{H}+

*h*

_{R}=

*a/*√3), where

*a*is the

*fcc*lattice constant, and the subscripts H and R denote hole and rod, respectively. With such settings, a complete band-gap of 21% is achieved in silicon PhCs [5

5. S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. **77**(22), 3490–3492 (2000). [CrossRef]

*r*

_{H}’ or

*r*

_{R}’) in individual layers of the 3D PhC have strong similarities in mode profiles and polarization to those in 2D PhCs or PhC slabs [11

11. M. L. Povinelli, S. G. Johnson, S. H. Fan, and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B **64**(7), 075313 (2001). [CrossRef]

12. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. **87**(25), 253902 (2001). [CrossRef] [PubMed]

13. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**(6961), 944–947 (2003). [CrossRef] [PubMed]

18. E. Lidorikis, M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Polarization-independent linear waveguides in 3D photonic crystals,” Phys. Rev. Lett. **91**(2), 023902 (2003). [CrossRef] [PubMed]

19. A. Talneau, P. Lalanne, M. Agio, and C. M. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Lett. **27**(17), 1522–1524 (2002). [CrossRef] [PubMed]

20. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B **64**(15), 155113 (2001). [CrossRef]

*Ω*, covers most of the overlap frequency range of the TE and TM modes, but it excludes the slow-light region of the TM mode.

*Ω*, shown by the shaded areas in yellow in Fig. 2(a) and 2(c), centers at

*ω*

_{0}= 0.55(2

*πc/a*) and has a bandwidth of Δ

*ω*= 0.015(2

*πc/a*), where

*c*is the speed of light in the vacuum; (ii) the conversion efficiency approaches unity throughout

*Ω*; (iii) the mechanism is based on contra-directional coupling, which is indicated by the opposite slopes of the two dispersion curves; (iv) the device has a small footprint. In the rest of this paper, we will first discuss the W1 waveguide mode properties, followed by the converter design principles. Then, FDTD simulations of the device and optimization strategies will be covered. Finally, the device performance will be evaluated.

## 2. Mode evolution of W1 waveguides

11. M. L. Povinelli, S. G. Johnson, S. H. Fan, and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B **64**(7), 075313 (2001). [CrossRef]

*i*=

*x*,

*y*,

*z*). For the TE mode of the W1 SC waveguide (

*r*

_{H}

*’*= 0), the energy ratio of the non-dominant

*E*is merely 1%, whereas the dominant

_{z}*E*has a ratio of ~81%, both of which are calculated at a fixed frequency of

_{x}*ω*

_{0}= 0.55(2

*πc/a*) (Fig. 3(a) ). The energy ratios of the magnetic field components

*RUh*(

_{i}*i = x, y, z*) are also defined and calculated for the same mode in Fig. 3(b), where

*H*dominates over other components with a ratio of ~76%. For the TM mode of the W1 HC waveguide (

_{z}*r*

_{R}’ = 0), the energy ratios of

*E*,

_{z}*E*and

_{x}*H*are calculated as ~62%, ~26% and ~10%, respectively, as shown in Fig. 3(c) and 3(d). Yet the polarization purity of the two modes still remain high, the non-trivial energy ratios of the minor components in the TE and TM modes can lead to a non-zero mode overlap, which is proportional to the inner product of the two modal fields [21]. The overlay can be even increased when the two waveguides are constructed in different layers of the crystal. Nevertheless, despite the minor mode overlap resulting from the breaking of symmetry in 3D PhCs, the very weak interaction between the fundamental modes of the W1 SC and HC waveguides is still prohibited in

_{z}*Ω*due to the significant phase mismatch. From Fig. 2(a) and 2(c), if the two dispersion curves are overlaid with each other, they are found to be quite far away, except one crossing point in the slow-light region of the TM mode.

*i.e.*the inter-layer power transfer possible, given that the phase-matching condition is satisfied. A general procedure for the TE to TM conversion could be: the SC waveguide mode is first converted to a TE mode of some intermediate W1 hole waveguide through a mode evolution; the power in the hole waveguide is then transferred to a W1 rod waveguide, given a sufficient mode interaction; finally, the obtained TM mode is converted to the desired HC waveguide mode through a second mode evolution. Thus, the key of the mode conversion process is to achieve efficient mode interaction between two intermediate W1 hole and rod waveguides, which requires both a considerable mode overlap and an accurate phase match.

*Ω*: at the anti-crossing point, the Bragg condition is satisfied [21]. It is known that increasing the defect hole radius

*r*

_{H}’ can raise the bands of a W1 hole waveguide towards the air band, which results from the reduced refractive index of the waveguide [22]. Similarly, increasing the defect rod radius

*r*

_{R}’ can lower the bands of a W1 rod waveguide towards the dielectric band. In Fig. 4 , at a fixed frequency of

*ω*

_{0}= 0.55(2

*πc/a*), the wavevectors increase with the defect sizes for both waveguides. Accordingly, the phase mismatch between the two modes is significantly reduced. We can then identify a range on both

*r*

_{H}’ and

*r*

_{R}’, within which the two waveguides have common wavevectors, as indicated by the shaded area in grey in Fig. 4.

*r*

_{H}’ and

*r*

_{R}’, which essentially promote the interaction between the TE and TM modes. For each point on the red curve in Fig. 4, the energy ratios are calculated correspondingly, as shown in Fig. 3(a). Based on both the electric and magnetic field calculations, it is found that the TE purity degrades with the increased

*r*

_{H}’. Interestingly, for the W1 rod waveguides, the trends are very different. As shown in Fig. 3(c) and 3(d), there’s no apparent degradation in polarization purity when

*r*

_{R}’ is increased. Nevertheless, the similarities between the TE and TM modes are still enhanced, according to the comparisons made between the energy ratios of corresponding components in Fig. 3.

*Ω*guarantees the phase-matching condition, whereas it is the increased similarities in mode profiles and polarization that essentially strengthen the mode interaction.

## 3. Mode converter design

*i.e.*,

*r*

_{H}’ and

*r*

_{R}’, are placed in the adjacent layers, as shown in the inset of Fig. 5(a) , power can be transferred from one waveguide to the other. In the language of the coupled-mode theory, such a waveguide pair forms a bi-layer compound waveguide, and the coupled modes constitute two super-modes [20

20. M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B **64**(15), 155113 (2001). [CrossRef]

23. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De la Rue, T. F. Krauss, U. Oesterle, and R. Houdre, “Mini-stopbands of a one-dimensional system: The channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B **63**(11), 113311 (2001). [CrossRef]

*ω*

_{gap}directly reflects the amplitude of the coupling strength between the two modes [24

24. S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, and R. Houdre, “Coupled-mode theory and propagation losses in photonic crystal waveguides,” Opt. Express **11**(13), 1490–1496 (2003). [CrossRef] [PubMed]

*n*

_{g,H}and

*n*

_{g,R}denote the mode group indices of the uncoupled W1 hole and rod waveguides, and

*c*is the speed of light in the vacuum. Light at any frequency within Δ

*ω*

_{gap}that propagates in one waveguide can couple to the other after a certain mode interaction distance. Thus, the inter-layer power transfer can be realized with the aid of the bi-layer compound waveguide coupler that has a large mode-gap.

*A*,

*B*and

*C*denote the TE input, TM output, and TE residual ports, respectively. The red and blue arrows, representing the TE and TM waves, illustrate how the light propagates, evolves, couples, and re-evolves in the device. Along the propagation direction of the TE wave from port

*A*to

*C*, the filled holes and the smaller holes at the bottom layer comprise, in sequence, the input W1 SC waveguide, the W1 hole mode-evolution waveguide and the bottom part of the bi-layer coupler. On the upper layer, along the propagating direction of the TM wave from port

*C*to

*B*, the smaller rods in light green and the line of missing rods constitute, in order, the top layer of the coupler, the W1 rod mode-evolution waveguide, and the output W1 HC waveguide. In a general mode conversion process, the TE wave, which is sent into the converter from port

*A*, first propagates in the SC waveguide. It then evolves to the TE mode of a W1 hole waveguide with

*r*

_{H}’

*= r*

_{H0}through a slow taper. Right behind the defect

*D*

_{1}in dark green, a uniform W1 rod waveguide with

*r*

_{R}’

*= r*

_{R0}emerges, which constitutes a bi-layer coupler with the W1 hole waveguide and harvests power from its neighbor through the contra-directional mode coupling. The obtained TM wave that propagates in –

*y*direction is then in turn redirected by two 60° bends, and transformed into the W1 HC waveguide mode through a second taper. Two 60° bends are employed here to separate the output channel from the input one.

*ω*

_{gap}is larger than

*Ω*. Second, a short coupler length

*L*requires a large coupling strength

*κ*

_{HR}for a fixed efficiency

*CE*[21],

*i.e.*,According to Eq. (2),

*κ*

_{HR}is determined by both the group indices and the coupler’s mode-gap size Δ

*ω*

_{gap}. By examining the dispersion curves of the fundamental modes of the W1 SC and HC waveguides in Fig. 2, we find that they remain quite straight throughout

*Ω*. It is further found that this conclusion holds for a wide range of

*r*

_{H}’ (or

*r*

_{R}’),

*i.e.*, dispersion curves of all these W1 hole (or rod) waveguides remain straight across

*Ω*, and their slopes change slightly with

*r*

_{H}’ (or

*r*

_{R}’). In this scenario, the group indices

*n*

_{g,H}and

*n*

_{g,R}can be both viewed as constants. As a result,

*κ*

_{HR}is solely determined by Δ

*ω*

_{gap}, which means the compact size and the wideband coupling can be achieved at the same time. As shown in Fig. 5, an optimized coupler design with

*r*

_{H0}= 0.56

*r*

_{H}and

*r*

_{R0}= 0.30

*r*

_{R}is found to have a large mode-gap of Δ

*ω*

_{gap}

*≈*0.015(2

*πc/a*), centered at

*ω*

_{0,gap}

*≈*0.551(2

*πc/a*), which satisfies our requirements. At the anti-crossing point

*k*= 0.28(2

_{y}*π/ā*), two group indices are computed as

*n*

_{g,H}= 6.17 and

*n*

_{g,R}= 6.06 [16

16. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**(3), 173–190 (2001). [CrossRef] [PubMed]

*κ*

_{HR}is calculated as 0.046(2

*π/a*). Thus, a coupling efficiency of

*CE*= tanh

^{2}(

*κ*

_{HR}

*·L*)

_{π}*=*tanh

^{2}(

*π*) = 99.3% is achievable within a distance of

*L*= 15.4

_{π}*ā*. At this

*k*point, the two propagation constants, |

*β*

_{H}| = 0.72(2

*π/ā*) and |

*β*

_{R}| = 0.28(2

*π/ā*), also satisfy the Bragg condition,

*i.e.*, |

*β*

_{H}| + |

*β*

_{R}| = 2

*π/ā*.

25. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **66**(6 Pt 2), 066608 (2002). [CrossRef] [PubMed]

*ā*is sufficient to achieve a loss below 1% for both types of waveguides.

*πc/a*) to 0.556(2

*πc/a*). The transmission is further improved to ~99% through tuning the radii of

*D*

_{1}and

*D*

_{2}(in dark green in Fig. 6) to be

*r*

_{D1}=

*r*

_{D2}= 1.6

*r*

_{R}. It remains close to unity when

*r*

_{D1}and

*r*

_{D2}vary around these values. As shown later, the tuning of

*r*

_{D1}and

*r*

_{D2}plays a crucial role in improving the converter’s performance in the presence of defects in the hole layer.

## 4. Converter FDTD simulation and optimization

26. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. **181**(3), 687–702 (2010). [CrossRef]

*ā*× 276

*ā*× 3√3

*a*(

*x*×

*y*×

*z*), where a sufficient number of periods of crystals are used in the transverse plane and the large size in

*y*guarantees the complete separations of the pulses of interest during the simulations. Perfect-matched layers (PMLs) are used in

*y*directions, whereas the periodic boundary condition (PBC) is adopted in the transverse

*x-z*plane.

*B*, TE/TM residuals at port

*C*, TE reflection to port

*A*, and their summation are plotted in Fig. 7 . The conservation of power guarantees the accuracy of the simulation, despite the challenge in simulating pulse propagation at frequencies close to the slow-light region of the TM mode. We observe a conversion efficiency of 80% above 0.547(2

*πc/a*), which is, however, limited by the severe reflection throughout the full frequency range. Meanwhile, the rapid increase of the residual signal below 0.547(2

*πc/a*) raises another concern.

*D*

_{1}, and to the waveguide impedance mismatch,

*i.e.*, the abrupt index change behind

*D*

_{1}. The problem of the scattering by

*D*

_{1}is solved through tuning

*D*

_{1}’s radius from 1.6

*r*

_{R}to 1.4

*r*

_{R}. To match the impedance, the entire W1 hole mode-evolution waveguide is shifted by 6

*ā*with respect to

*D*

_{1},

*i.e.*, 14 out of the 20 tapered holes are placed before

*D*

_{1}whereas the other 6 tapered holes replace the first 6 holes in the coupler.

*πc/a*) to 0.5532(2

*πc/a*) (Fig. 9(a) ). This corresponds to a bandwidth of ~16 nm centered at 1550 nm, if

*a*is chosen as 853 nm. In the same range, both the reflection and residual remain below −20dB, as plotted in Fig. 9(c). The bandwidth for a conversion efficiency of 95% is 20 nm. Given that the envelope of the TM wave in the coupler region is in a hyperbolic sinusoidal form,

*κ*

_{HR}is retrieved as 0.05(2

*π/a*), which is very close to the calculated value of 0.046(2

*π/a)*by Eq. (3). This value suggests that the efficiency of 98% is achieved within a coupling distance of shorter than 20

*ā*, which leads to a total converter length of less than 40

*ā*≈25μm. Moreover, the simulation for the TM to TE mode conversion is also performed, which clearly shows the reciprocal transmission of the device (see results in the linear scale in Fig. 9(b) and in the logarithm scale in Fig. 9(d), respectively).

27. M. R. Watts and H. A. Haus, “Integrated mode-evolution-based polarization rotators,” Opt. Lett. **30**(2), 138–140 (2005). [CrossRef] [PubMed]

32. H. Zhang, S. Das, Y. Huang, C. Li, S. Chen, H. Zhou, M. Yu, P. G. Lo, and J. T. L. Thong, “Efficient and broadband polarization rotator using horizontal slot waveguide for silicon photonics,” Appl. Phys. Lett. **101**(2), 021105 (2012). [CrossRef]

*Ω*. Moreover, the conversion here is between modes guided by different mechanisms,

*i.e.*, TE mode by total internal reflection and TM mode by photonic band-gap. Our device has a low insertion loss, yet is compact in size, both of which facilitate the integration of such devices with other functional modules. The converter bandwidth, which covers most of the targeted

*Ω*range, is also sufficient for a wide range of applications in 3D PhC integrated circuits.

*πc/a*). Further reduction on reflection might be possible if radius tuning is applied to individual rods and/or holes around

*D*

_{1}. Moreover, the problem of rapid growth of the residual wave below 0.5474(2

*πc/a*) still exists, which limits the conversion efficiency and the operation bandwidth considerably. This growth is partially due to the insufficient coupling in the low frequency range, where

*κ*

_{HR}drops quickly from the value at

*ω*

_{0,gap}. One solution is cascading another coupler with a lower center frequency, to cover a wider frequency range and to improve the overall value of

*κ*

_{HR}throughout

*Ω*.

## 5. Conclusion

## Acknowledgments

## References and links

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**OCIS Codes**

(250.5300) Optoelectronics : Photonic integrated circuits

(130.5296) Integrated optics : Photonic crystal waveguides

(130.5440) Integrated optics : Polarization-selective devices

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: June 25, 2012

Revised Manuscript: August 6, 2012

Manuscript Accepted: August 6, 2012

Published: August 21, 2012

**Citation**

Jian Wang and Minghao Qi, "Design of a compact mode and polarization converter in three-dimensional photonic crystals," Opt. Express **20**, 20356-20367 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20356

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### References

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