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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 17 — Aug. 13, 2012
  • pp: 18545–18554
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Unidirectionally optical coupling from free space into silicon waveguide with wide flat-top angular efficiency

Kun Li, Guangyuan Li, Feng Xiao, Fan Lu, Zhonghua Wang, and Anshi Xu  »View Author Affiliations


Optics Express, Vol. 20, Issue 17, pp. 18545-18554 (2012)
http://dx.doi.org/10.1364/OE.20.018545


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Abstract

A grating coupling scheme from free-space light into silicon waveguide with a remarkable property of wide flat-top angular efficiency is proposed and theoretically investigated. The coupling structure is composed of two cascaded gratings with a proper distance between their peak angular efficiencies. A quantitative semi-analytical theory based on coupled-mode models is developed for performance prediction and validated with the fully vectorial aperiodic Fourier modal method (a-FMM). With the theory, wide flat-top angular response is achieved and the conditions are pointed out. Proof-of-principle demonstrations show that the −1 dB angular width, a figure of merit to evaluate the flat-top performance, is broadened to almost 3 to 4 times, and meanwhile the −3 dB angular width, i.e., angular-full-width-half-maximum (AFWHM), is widened to nearly more than twice, compared with the reference gratings composed of the same number of periodic defects. We believe this work will find applications in biological or chemical sensing and novel optical devices.

© 2012 OSA

1. Introduction

Before we address this challenge, let us retrospect the angular resolution of a diffraction limited system, i.e. the Rayleigh criterion [19

19. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

], as illustrated in Fig. 1. When two objects are separated by a small angle, the diffraction patterns overlap as shown in Fig. 1(a). It is able to resolve the two objects as long as the central peaks in the two diffraction patterns don’t overlap as illustrated in Fig. 1(c). The minimum resolvable detail is when one central peak falls below the first minimum of the other diffraction pattern as shown in Fig. 1(b). If the plots are for angular coupling efficiency, it is clear that the case of Fig. 1(b) with a proper separation between angular peaks will lead to wide flat-top response and wide AFWHM.

Fig. 1 The overlapping diffraction patterns: (a) light waves from two objects are unresolved; (b) Rayleigh criterion; (c) light waves from two objects are resolved.

Fig. 2 (a) Schematic of the proposed grating coupler. (b) Schematic of the global model, which treats each array of grating as a ‘black box’. The first (or the second) ‘black box’ is composed of N1 (or N2) periodic defects with period p1 (or p2). The defects in both ‘black boxes’ are of constant height hr and width wr. The structural distance between two ‘black boxes’ is d. (c,d) show the main elementary scattering processes for the ‘black box’ of Nm defects, i.e., the excitation coefficients βNm±, reflection coefficients rNm± and transmission coefficient tNm under illumination of plane wave (c) and waveguide modes (d). (e) schematic of the nested model of an isolated ‘black box’. (f,g) show the scattering coefficients of a single defect under illumination of plane wave (f) and waveguide mode (g). The vertical blue-dashed lines in (b,e) indicate the zero phase of the incident plane wave.

2. Quantitative semi-analytical theory

Figure 2(a) illustrates the schematic of proposed unidirectional coupling structure. The grating coupling structure is based on a silicon-on-insulator (SOI) wafer, consisting of a 240 nm silicon waveguide layer and a 900 nm buried oxide layer on a silicon substrate. First, following the deposition of a blanket SiO2 hard mask, the silicon overlay is locally defined by epitaxial silicon growth [2

2. G. Roelkens, D. Vermeulen, D. V. Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008). [CrossRef]

] to obtain 250nm silicon overlay thickness in the grating region. The taper and the photonic wire are then defined using e-beam lithography [21

21. D. Taillaert, F. V. Laere, M. Ayre, W. Bogaerts, D. V. Thourhout, P. Bienstman, and R. Baets, “Grating couplers for coupling between optical fibers and nanophotonic waveguides,” Jpn. J. Appl. Phys. 45, 6071–6077 (2006). [CrossRef]

] and etched 240 nm deep by inductively coupled plasma (ICP). After that, one may use the gray-scale electron-beam lithography [22

22. L. Dong, S. Iyer, S. Popov, and A. Friberg, “3D fabrication of waveguide and grating coupler in SU-8 by optimized gray scale electron beam lithography,” in Proceedings of ACP (2010).

] followed by ICP etching [23

23. A. Sure, T. Dillon, J. Murakowski, C. Lin, D. Pustai, and D. W. Prather, “Fabrication and characterization of three-dimensional silicon tapers,” Opt. Express 11, 3555–3561 (2003). [CrossRef] [PubMed]

] to produce triangular pattern of the grating. In practice, the triangular pattern are approximated by 20-step staircases. As the design and fabrication of the taper and photonic wire has a well-rounded study [8

8. F. V. Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. V. Thourhout, M. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. Light-wave Technol. 25, 151–156 (2007). [CrossRef]

, 9

9. L. Vivien, D. Pascal, S. Lardenois, D. Marris-Morini, E. Cassan, F. Grillot, S. Laval, J. M. Fédéli, and L. E. Melhaoui, “Light injection in SOI microwaveguides using high-efficiency grating couplers,” J. Lightwave Technol. 24, 3810–3815 (2006). [CrossRef]

, 24

24. S. Lardenois, D. Pascal, L. Vivien, E. Cassan, S. Laval, R. Orobtchouk, M. Heitzmann, N. Bouzaida, and L. Mollard, “Low-loss submicrometer silicon-on-insulator rib waveguides and corner mirrors,” Opt. Lett. 28, 1150–1152 (2003). [CrossRef] [PubMed]

], here we focus on the design of the grating, which can be fabricated using state-of-the-art techniques just as stated above, to achieve wide flat-top angular efficiency.

In the case of single-mode operation, a semi-analytical theory is developted in form of two nested models: a global model that treats the protuberance arrays as two ‘black boxes’ exciting, reflecting and transmitting waveguide modes (WMs), and focuses on the ‘box-to-box’ distance d, as illustrated in Figs. 2(b)–2(d); and a nested model on the excitation, reflectance and transmittance coefficients of WMs by the ‘black boxes’, as shown in Figs. 2(e)–2(g). The nested theoretical models bridge the scattering coefficients of the finite-size protuberances to those of a single one, resulting in a great reduction of the computational cost.

The global model is shown in Fig. 2(b), where B1 (or B2) and A1 (or A2) are the respective complex amplitudes of magnetic field Hy of the backward- and forward-going WMs away from the first (or the second) ‘black box’. The main elementary scattering processes involved in the proposed model are shown in Figs. 2(c) and 2(d): βNm+ and βNm are the excitation coefficients of forward- and backward-going WMs by the ‘black box’ with Nm defects under the plane wave illumination, respectively; rNm+ and rNm are the reflectance coefficients of forward- and backward-going WMs, respectively; tNm is the transmittance coefficient. The coupled-mode equations lead to:
B1=βN1+tN1uB2+rN1+uA0
(1a)
A1=βN1++rN1uB2+tN1uA0
(1b)
B2=wβN2+rN2+uA1
(1c)
A2=wβN2++tN2uA1
(1d)
where w = exp[ik0(p1N1p1 + wr + d)sinθ] is the phase shift introduced by the incident plane wave. This is because the zero phase is assumed to be at the leftmost side of the first ‘black box’ for the calculation of βN1± (x = z = 0); whereas it is at the leftmost side of the second ‘black box’ for the calculation of βN2± (x = p1N1p1 +wr +d, z = 0). d is the structural distance between the two ‘black boxes’. u = exp(ik0neffd) with neff being the complex effective refractive index of the waveguide mode. Note that tNm=tNm+=tNm according to the principle of optical reversibility. We emphasize the propagation losses of the WMs have been embodied via complex neff. To calculate the WMs excitation coefficients, βN1N2+=A2, βN1N2=B1, one sets A0 = 0; whereas to calculate the WMs reflectance and transmittance coefficients, rN1N2 = B1/(uA0) and tN1N2 = A2/uA0, one sets βN1±=0 and βN2±=0. Then the WMs excitation, reflection and transmission coefficients of the cascaded ‘black boxes’ are obtained after a series of algebraic operation:
βN1N2+=wβN2++tN2urN1uwβN2+βN1+1rN1rN2+u2,
(2a)
βN1N2=βN1+tN1uwβN2+rN2+uβN1+1rN1rN2+u2,
(2b)
rN1N2=rN1++rN2+tN12u21rN1rN2+u2,
(2c)
tN1N2=tN1tN2u1rN1rN2+u2,
(2d)
where βN1N2+ is dominated by wβN2+ in Eq. (2a) as |tN2| is usually small when |βN2+| is optimized. Similarly, βN1 dominates in βN1N2 in Eq. (2b).

Because the second ‘black box’ is designed to unidirectionally couple free-space light into the desired waveguide direction and suppress coupling in the opposite direction (i.e., |βN2||βN2+|), it is reasonable to ignore the term wβN2urN1utN2. Moreover, the multiple reflections between the two ‘black boxes’ rN1rN2+u2 are negligible because |rN1rN2+u2|1. Then Eq. (2a) is simplified into
βN1N2+tN2uβN1++wβN2+
(3)

The physical interpretations of Eq. (3) is intuitively meaningful as illustrated in Fig. 3. It means the interference between the forward-going WMs by the transmission of the excitation of the first ‘black box’ tN2uβN1+ (blue arrow) and by the excitation of the second ‘black box’ wβN2+ (green arrow). This recalls the above-mentioned concept to realize wide flat-top angular response as shown in Fig. 1(b). In Eq. (3), tN2 and u play key roles of phase modulation of βN1+. If βN1+ and βN2+ are first designed properly with comparable incoupling coefficients and suitable angular interval between peak incoupling coefficients, it is possible to obtain wide flat-top angular response for βN1N2+ provided the structural distance between the two ‘black boxes’ is given by
arg(tN2)+k0Re(neff)d2mπ
(4)
where the functions ‘arg’ and ‘Re’ refer to the argument and the real part, respectively, and m is an integer.

Fig. 3 Physical interpretations of Eq. (3).

Now let us consider the excitation, reflectance and transmittance coefficients of WMs by the ‘black box’, which may be consist of N aperiodic defects with arbitrary distances dj(j = 1,...,N − 1) as shown in Fig. 2(e). Following our pervious work [25

25. G. Li, F. Xiao, L. Cai, K. Alameh, and A. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects vis field decomposition,” New J. Phys. 13, 073045 (2011). [CrossRef]

, 26

26. G. Li, L. Cai, F. Xiao, Y. Pei, and A. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express 18, 10487–10499 (2010). [CrossRef] [PubMed]

], the corresponding coefficients of an isolated ‘black box’ of N defects are given by:
βN+=wN1(β1++t1rN1uN12β1)1t1uN1wN11,
(5a)
rN=rN1++r1+tN12uN121r1+rN1uN12,
(5b)
tN=t1tN1uN11r1+rN1uN12.
(5c)
Where uN−1 = exp(ik0neffdN−1), wN1=exp[ik0(Nwrwr+1N1dj)sinθ]. In some cases, the equations will be reduced into simplified forms. For example, for periodic symmetric defects, uj = u, wj = wj, and rj=rj=rj+ for j = 1,··· ,N − 1, |r1+rN1uN12|1, Eq. (5) is then reduced into
βN+=wN1(β1++t1rN1u2β1)1t1uw1N
(6a)
rN=rN1+r1tN12u2,
(6b)
tN=t1tN1u.
(6c)

We emphasize that the nested model is versatile for a general grating composed of periodic or aperiodic defects, where the defect may be of various geometries and refractive index profiles, as their influences have been embodied via β1±, r1± and t1.

For simplicity, hereafter we restrict ourselves to periodic defects, i.e., d1 = ... = dN−1 = pwr. βN+, rN and tN can be calculated recursively starting from β1±, r1± and t1, thus the geometry optimization for N periodic defects is reduced into that for a single one. The constructive interference condition of βN+ is obtained from Eq. (5a):
arg(t1)+k0Re(neff)pk0n0psinθ=2mπ
(7)
We refer to Eq. (7) as the generalized grating equation. Compared with the conventional grating equation, the generalized one is versatile for a general grating composed of various defects. Specially, for gratings composed of high-index dielectric protuberances, the additional term arg(t1) is quite important since it may be relatively large in this case.

3. Results and discussions

In this section, the grating coupling scheme to achieve wide flat-top response by properly cascading two gratings is validated with proof-of-principle demonstrations. The semi-analytical theory is quantitatively validated using exhaustive calculations with various grating parameters by comparing with simulations using the a-FMM. With the theory, wide flat-top angular efficiency is achieved and the conditions and versatility for various shaped defects are pointed out.

The incident plane wave is normalized such that its Poynting vector is unitary [20

20. H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008). [CrossRef]

]. Under this normalized condition, the incoupling cross section |βN+|2 means the power flow carried by the forward-going WMs, and the coupling efficiency ηN+=|βN+|2/D with D being the grating cross section represents the ratio of the forward-going WMs power flux to the incident power flux that launches into the gratings. To evaluate the performance of wide flat-top angular coupling efficiency, we adopt −1 dB angular width Φ and −3 dB angular width Θ as the figures of merit (FoMs), which are defined as the angular range between two specified angle cut-off points that are −1 dB and −3 dB below the peak angular efficiency, respectively. Specifically, Φ is used to evaluate flat-top angular response while Θ is AFWHM.

Figure 4 compares coupling efficiency ηN1+N2+ of the reference grating with N1 + N2 defects of period p1 (blue-solid line) or p2 (black-solid line) in (b,e,h,k) with the model predictions (green-dashed line with circles) and the a-FMM data (red-solid line) on the above-mentioned ηN1N2+ of the cascaded gratings. As shown in Fig. 4, by properly cascading two isolated gratings (a,d,g,j), flat-top angular response of ηN1N2+ is realized and the AFWHM is broadened in (c,f,i,l): Φ is broadened to almost 3 to 4 times and meanwhile Θ is widened to nearly more than twice compared with the reference ones (b,e,h,k). The angular coupling efficiency peaks up to values of 0.28, 0.36, 0.33, 0.31 average for (b,e,h,k) and 0.16, 0.18, 0.15, 0.14 for (c,f,i,l), respectively. More specifically, compared with the reference grating, Φ of the cascaded grating are broadened to almost 3 to 4 times at an acceptable cost of decreasing the peak coupling efficiency only by nearly half. It is also clearly shown that when the cascaded gratings are optimized for other different incidence angles ([d,e,f],[g,h,i]) or of a higher number of defects (j,k,l), the grating coupling scheme is still valid. Moreover, since the incoupling cross section |βN1N2+|2 is dominated by |βN2+|2, as we have stated previously, the second angular peak position (black-dashed line) of ηN1N2+ (c,f,i,l) remains almost consistent with that of ηN2+ (a,d,g,j) as expected, whereas for the first angular peak with regard to ηN1+, there is an offset introduced by tN2.

Fig. 4 Comparisons among coupling efficiency of the isolated grating ηN+ with N1 or N2 defects (top row), the reference grating ηN1+N2+ with N1 + N2 defects of period p1 (or p2) (middle row), and the cascaded gratings ηN1N2+ with N1 defects of period p1, N2 defects of period p2, and ‘box-to-box’ structural distance d (bottom row) by model predictions (green-dashed lines with circles) and a-FMM computational data (red-solid lines). The vertical black-dashed lines indicate the peak angular positions. The calculation of the four columns from left to right by a-FMM are performed for: hr = 250 nm, wr = 520 nm, (a,b,c) p1 = 567 nm, N1 = 8, p2 = 675 nm, N2 = 4, and d = 240 nm; (d,e,f) p1 = 617 nm, N1 = 7, p2 = 740 nm, N2 = 6, and d = 1150 nm; (g,h,i) p1 = 675 nm, N1 = 7, p2 = 811 nm, N2 = 6, and d = 1195 nm. (j,k,l) p1 = 567 nm, N1 = 14, p2 = 675 nm, N2 = 6, and d = 720 nm; Φ is broadened from 6° or 5° to 20° (b,c), from 7° or 5° to 15° (e,f), from 5° or 6° to 16° (h,i), and from 5° or 4° to 24° (k,l), respectively; Θ is widened from 12° or 10° to 27° (b,c), from 11° or 10° to 20° (e,f), from 9° or 10° to 20° (h,i), and from 10° or 7° to 29° (h,i), respectively.

Figure 5 illustrates the importance of the conditions for wide-flat top angular efficiency, i.e., the two ‘black boxes’ should be of proper structural distance d (a,b), comparable incoupling cross sections (c,d) and suitable angular interval (e,f). There will be no wide flat-top angular coupling efficiency if d deviates from the model prediction given by Eq. (4) (a,b), peak values of incoupling cross sections are not comparable (c,d), or the angular interval between the peaks of ηN1+ and ηN2+ is too small (e,f) or too large (not shown).

Fig. 5 Influential elements on wide-flat top angular efficiency: (a,b) structural distance, (c,d) peak incoupling cross sections, and (e,f) the interval between angular peaks. The vertical black-dashed lines indicate the peak angular positions. The calculation of the three columns from left to right by a-FMM are performed respectively for: hr = 250 nm, wr = 520 nm, (a,b) p1 = 567 nm, p2 = 675 nm, N1 = 8, N2 = 4, and d = 490 nm; (c,d) p1 = 617 nm, p2 = 740 nm, N1 = 6, N2 = 8, and d = 905 nm; (e,f) p1 = 567 nm, p2 = 617 nm, N1 = 9, N2 = 6, and d = 495 nm, respectively.

Fig. 6 The calculations are performed for deeply etched rectangular grating with groove depth hr = 100 nm and width wr = 150 nm. (a–d) depict the comparison of the model predictions (blue-dashed line with circles) and the a-FMM computational data (red line) for 10 periodic grooves on (a) |r10|2, (b) arg(r10), (c) |t10|2, and (d) arg(t10) as functions of the period p. (e,f) show the comparison between the reference rectangular grating ηN1+N2+ with N1 + N2 defects of period p1 or p2 (e), and the cascaded rectangular grating ηN1N2+ with N1 defects of period p1, N2 defects of period p2, and ‘box-to-box’ structural distance d (f). p1 = 809 nm, p2 = 1057 nm, N1 = 6, N2 = 4, and d = 630 nm.

4. Conclusions

In conclusion, we have proposed and investigated a grating coupling scheme from free-space light into silicon waveguide with a remarkable property of wide flat-top angular efficiency. A semi-analytical theory in form of two nested coupled-mode models has been developed for performance prediction and parameters optimization. Comparisons of the model predictions with a-FMM calculations have shown that all the salient feature is quantitatively captured by the model. The theoretical model is versatile for a general grating composed of periodic or aperiodic defects, where the defect may be of various geometries and refractive index profiles. With the theory, the conditions for wide flat-top angular response have been pointed out, i.e., the cascaded gratings should be of proper structural distance, comparable peak incoupling cross sections and suitable angular interval. Proof-of-principle demonstrations have shown that: compared with the reference gratings composed of the same number of periodic defects, the −1 dB angular width Φ, a figure of merit to evaluate the flat-top performance, is broadened to almost 3 to 4 times; and meanwhile the −3 dB angular width Θ, i.e., angular-full-width-half-maximum (AFWHM), is widened to nearly more than twice, at an acceptable cost of decreasing angular coupling efficiency only by half. We believe that this work may be of great interest for use in novel optical devices and biological or chemical sensing.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under grant 61107065, China Postdoctoral Science Foundation, and the State Key Laboratory of Advanced Optical Communication Systems and Networks, China.

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H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008). [CrossRef]

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D. Taillaert, F. V. Laere, M. Ayre, W. Bogaerts, D. V. Thourhout, P. Bienstman, and R. Baets, “Grating couplers for coupling between optical fibers and nanophotonic waveguides,” Jpn. J. Appl. Phys. 45, 6071–6077 (2006). [CrossRef]

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L. Dong, S. Iyer, S. Popov, and A. Friberg, “3D fabrication of waveguide and grating coupler in SU-8 by optimized gray scale electron beam lithography,” in Proceedings of ACP (2010).

23.

A. Sure, T. Dillon, J. Murakowski, C. Lin, D. Pustai, and D. W. Prather, “Fabrication and characterization of three-dimensional silicon tapers,” Opt. Express 11, 3555–3561 (2003). [CrossRef] [PubMed]

24.

S. Lardenois, D. Pascal, L. Vivien, E. Cassan, S. Laval, R. Orobtchouk, M. Heitzmann, N. Bouzaida, and L. Mollard, “Low-loss submicrometer silicon-on-insulator rib waveguides and corner mirrors,” Opt. Lett. 28, 1150–1152 (2003). [CrossRef] [PubMed]

25.

G. Li, F. Xiao, L. Cai, K. Alameh, and A. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects vis field decomposition,” New J. Phys. 13, 073045 (2011). [CrossRef]

26.

G. Li, L. Cai, F. Xiao, Y. Pei, and A. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express 18, 10487–10499 (2010). [CrossRef] [PubMed]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(130.3120) Integrated optics : Integrated optics devices

ToC Category:
Diffraction and Gratings

History
Original Manuscript: April 18, 2012
Revised Manuscript: July 20, 2012
Manuscript Accepted: July 22, 2012
Published: July 30, 2012

Citation
Kun Li, Guangyuan Li, Feng Xiao, Fan Lu, Zhonghua Wang, and Anshi Xu, "Unidirectionally optical coupling from free space into silicon waveguide with wide flat-top angular efficiency," Opt. Express 20, 18545-18554 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-17-18545


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