Cavity-resonator-integrated guided-mode resonance band-stop reflector

Shogo Ura; Masahiro Nakata; Kenichi Yanagida; Junichi Inoue; Kenji Kintaka

doi:10.1364/OE.24.015120

1. Introduction

A guided-mode resonance (GMR) filter consists of a sub-wavelength grating in a thin film waveguide on a transparent substrate and shows a band-stop filtering for an incident wave from air [1–6]. A periodic structure in refractive index of the grating spreads in a waveguide plane in contrast to a multilayer dielectric mirror of which periodic structure is vertically formed on the plane. Since longer coupling length can give narrower bandwidth, the configuration of a GMR filter is much better for bandwidth narrowing than a multilayer dielectric mirror regarding fabrication process. A GMR filter is polarization sensitive when a normal one-dimensional grating is used. Another feature is a possibility of monolithic integration of filters of different filtering wavelengths at arbitrary locations on a substrate. On the other hand, a GMR filter needs an aperture size and incident beam diameter of hundreds of microns for narrowband filtering and is not suitable for high-density packaging.

Cavity-resonator integration was proposed and demonstrated [7–13] for aperture miniaturization of GMR filters. A cavity-resonator-integrated GMR filter (CRIGF) consists of a grating coupler (GC) in a waveguide cavity formed by a pair of distributed Bragg reflectors (DBRs) on a transparent substrate. The cavity-resonator integration provides wider acceptable angle [14,15]. It was also demonstrated that reflection spectra drastically depended on the location of the GC in the waveguide cavity [16,17], suggesting a possibility of a functional reflection filter in combination with an electro-optic material. One of applications of a CRIGF will be a key component in a next-generation surface-mount packaging of VCSELs. A CRIGF on a high-reflection substrate would serve as both an external mirror of surface-mounted half-VCSEL to stabilize its lasing operation and an input coupler launching a guided wave [18]. Another structure of CRIGF on a high-reflection substrate can be a band-stop reflector [19,20]. It is expected that reflection spectra strongly depends on the distance of the grating layer and the substrate because of an interference effect. This time, we investigated the dependence to obtain a narrow-band-stop reflector for an application to dense wavelength-division multiplexing systems, and obtained a stop band with –3-dB bandwidth of 0.15 nm. In this paper, theoretical prediction and experimental demonstration are reported.

2. Basic configuration and operation principle

Schematic cross-sectional views of a basic structure and wave coupling of a band-stop reflector are illustrated in Fig. 1. The reflector consists of a CRIGF, a high reflection substrate, and an optical buffer layer between them. Grating periods of the GC and DBR are Λ and Λ/2, respectively. Length of the phase adjusting gap between the centers of the end teeth of the GC and the DBR is determined to be 3Λ/8 so that the coupling efficiency of the GC is maximized.

Fig. 1 Basic configuration (a) and wave coupling (b) of a band-stop reflector with a vertical cavity consisting of a CRIGF and a high-reflection substrate.

Download Full Size | PDF

A CRIGF shows high reflection in resonance but normal Fresnel reflection in non-resonance, resulting in a narrowband reflection spectrum. The structure shown in Fig. 1 can be regarded as a kind of vertical cavity consisting of the CRIGF and the reflection substrate. The quality factor of the vertical cavity will be high when a vertical resonance wavelength happens to be the one giving high reflectance of the CRIGF. Optical power readily loses when a cavity shows a high quality factor, resulting in low reflectance. The vertical-resonance wavelength depends on the distance between the CRIGF and the substrate. In other words, the reflection spectrum of the reflector depends seriously upon the distance, namely optical-buffer-layer thickness.

3. Theoretical prediction

The rigorous coupled wave analysis (RCWA) is a popular numerical-simulation method for analyzing coupling characteristics of grating structures of infinite size. However, it is too complicated to apply the RCWA to limited size grating. The finite-difference time-domain (FDTD) method is another well-known numerical method and powerful tool to simulate electro-magnetic wave propagations in arbitrary structures, but consumes lots of machine time and memory. On the other hand, an analytical calculation model is useful for a device design. Device characteristics are explicitly described with structural or device parameters, and then performance can be easily predicted though some approximations may be included. We developed a calculation model [7,18] based on coupled mode theories to express reflection and transmission coefficients of CRIGFs.

A differential equation of the amplitude of the resonant guided mode g is written as

\frac{d g}{d t} = (j ω_{0} - \frac{1}{τ}) g - j 〈 \hat{κ} * | | a_{I N} 〉

where ω₀ and τ denote a resonance angular frequency and the lifetime of the resonance, respectively. An incident wave of limited beam waist is regarded as a superposition of an appropriate set of radiation modes, and its amplitude components on the basis of radiation modes are represented by

| a_{I N} 〉

. The second term in the right-hand side expresses resonant mode excitation from the incident wave with coupling coefficients

| \hat{κ} 〉

between g and

| a_{I N} 〉

. Out-going waves

| a_{O U T} 〉

from the GC are expressed by

| a_{O U T} 〉 = | a_{I N} 〉 - j g | \hat{κ} 〉,

Steady state solution of Eq. (1) is obtained for an incident wave of angular frequency ω with relation of dg/dt = jω g to be

g = - j \frac{〈 \hat{κ} * | | a_{I N} 〉}{j (ω - ω_{0}) + 1 / τ} .

A differential equation for the guided-mode intensity can be written as

\frac{d {| g |}^{2}}{d t} = - 2 \frac{1}{τ} {| g |}^{2} \equiv - 2 (\frac{1}{τ_{R a d}} + \frac{1}{τ_{D B R}} + \frac{1}{τ_{L o s s}}) {| g |}^{2} .

Liftime components due to radiation τ_Rad, transmission through DBRs τ_DBR, and excess loss τ_Loss other than the radiation and transmission losses are given by

\frac{1}{τ_{R a d}} = \frac{c}{N} \frac{L_{G C}}{L_{C a v}} α, \frac{1}{τ_{D B R}} = - \frac{c}{N} \frac{\ln (r_{D B R})}{L_{C a v}}, \frac{1}{τ_{L o s s}} = \frac{c}{N} κ_{L o s s},

where c, N, L_GC, L_Cav, α, r_DBR, and κ_Loss denote the speed of light in vacuum, the effective refractive index of the guided mode, GC coupling length, the effective length of the waveguide cavity [21], the radiation decay factor of the GC, the reflection coefficient of the DBR, and propagation loss of the guided mode.

We consider here a planar waveguide model for simplicity. Guided modes propagate to z or –z directions. The structure is assumed to be uniform along y-axis, and x-axis is along the normal of the waveguide plane. Electric field distribution E_νa(x) of air-radiation mode and E_νs(x) of substrate-radiation mode having a wave number of β_ν along z-direction can be written as

E_{ν a} (x) = {\begin{matrix} E_{ν a}^{0} {(t_{a} t_{s} - r_{a} r_{s}) \exp (- j k_{a x} x) - r_{s} \exp (j k_{a x} x)} \exp (- j β_{ν} z); in air, \\ E_{ν a}^{0} t_{a} \exp (- j k_{s x} x) \exp (- j β_{ν} z); in substrate \end{matrix}

E_{ν s} (x) = {\begin{matrix} E_{ν s}^{0} t_{s} \exp (j k_{a x} x) \exp (- j β_{ν} z); in air \\ E_{ν s}^{0} {- r_{a} \exp (- j k_{s x} x) + (t_{a} t_{s} - r_{a} r_{s}) \exp (j k_{s x} x)} \exp (- j β_{ν} z); in substrate \end{matrix},

where r_a, t_a, r_s, and t_s denote coefficients of reflection, transmission for incidence from the air, reflection and transmission from the substrate, respectively. Wave numbers k_ax and k_sx satisfy the relations of

k_{a} = \sqrt{k_{a x}^{2} + β_{ν}^{2}}, k_{s} = \sqrt{k_{s x}^{2} + β_{ν}^{2}},

where k_a and k_s are wave numbers in the air and substrate, respectively.

E_{ν a}^{0}

and

E_{ν s}^{0}

are real numbers and determined by normalization conditions of

\frac{π}{2 ω μ_{0}} E_{ν a}^{0}^{2} {k_{a} ({| t_{a} t_{s} - r_{a} r_{s} |}^{2} + {| r_{s} |}^{2}) + k_{s} {| t_{a} |}^{2}} = 1,

\frac{π}{2 ω μ_{0}} E_{ν s}^{0}^{2} {k_{a} {| t_{s} |}^{2} + k_{s} ({| t_{a} t_{s} - r_{a} r_{s} |}^{2} + {| r_{a} |}^{2})} = 1 .

where μ₀ is the vacuum permeability. A plane wave from the air can be given by a linear combination of radiation modes as

E_{A ν} (x) = {c_{a} E_{ν a} (x) + c_{s} E_{ν s} (x)} E_{A ν}^{0},

c_{a} = \frac{r_{a}}{t_{a} t_{s} - r_{a} r_{s}} \frac{1}{E_{ν a}^{0}}, c_{s} = \frac{t_{a}}{t_{a} t_{s} - r_{a} r_{s}} \frac{1}{E_{ν s}^{0}}

An incident wave and the out-going wave are expressed by

\begin{matrix} | a_{I N} 〉 = \end{matrix} (\begin{array}{l} c_{a} | E_{I N ν}^{0} 〉 \\ c_{s} | E_{I N ν}^{0} 〉 \end{array}) = (\begin{array}{l} c_{a} \\ c_{s} \end{array}) | E_{I N ν}^{0} 〉,

\begin{matrix} | a_{O U T} 〉 = (\begin{array}{l} d_{a} | E_{O U T ν}^{0} 〉 \\ d_{s} | E_{O U T ν}^{0} 〉 \end{array}) \end{matrix} = (\begin{array}{l} d_{a} \\ d_{s} \end{array}) | E_{O U T ν}^{0} 〉 .

We consider here the impulse response for an incident wave defined by

| E_{I N ν}^{0} 〉 = E_{I N 0}^{0} (ω) δ (β_{ν} = 0),

where δ indicates Dirac’s delta. By substituting Eqs. (13) and (14) to Eqs. (2) and (3) and integrating with respect to ν, we obtain

(\begin{matrix} d_{a} \\ d_{s} \end{matrix}) = (\begin{matrix} c_{a} \\ c_{s} \end{matrix}) - \frac{{\hat{κ}}_{0 a} * c_{a} + {\hat{κ}}_{0 s} * c_{s}}{j (ω - ω_{0}) + 1 / τ} (\begin{matrix} {\hat{κ}}_{a} \\ {\hat{κ}}_{s} \end{matrix}),

(\begin{matrix} {\hat{κ}}_{a} \\ {\hat{κ}}_{s} \end{matrix}) = \sqrt{\frac{2 π c}{N} \frac{L_{G C}}{L_{C a v}}} (\begin{matrix} κ_{ν a} \\ κ_{ν s} \end{matrix}),

(\begin{matrix} {\hat{κ}}_{0 a} * \\ {\hat{κ}}_{0 s} * \end{matrix}) = \sin c (\frac{Δ_{ω} L_{G C}}{2}) \sqrt{\frac{2 π c}{N} \frac{L_{G C}}{L_{C a v}}} (\begin{matrix} κ_{ν a} * \\ κ_{ν s} * \end{matrix}),

where Δ_ω represents phase mismatching defined by

Δ_{ω} = \frac{N}{c} (ω - ω_{0}) .

The spatial coupling coefficient κ_νi between the resonant guided mode g and a radiation mode νi is given by

κ_{ν i} = \frac{ε_{0} ω}{4} \int E_{ν i}^{*} (x) Δ ε E_{g} (x) d x (i = a or s),

where E_g(x) and E_νi(x) denote normalized electric field distributions of the guided and radiation modes, respectively. Finally, the reflection coefficient of the CRIGF is calculated to be

r_{C R I G F} = r_{a} - \sin c (\frac{Δ_{ω} L_{G C}}{2}) \frac{2 π c}{N} \frac{L_{G C}}{L_{C a v}} \frac{{| κ_{ν a} |}^{2} r_{a} + κ_{ν s} * κ_{ν a} t_{a} E_{ν a}^{0} / E_{ν s}^{0}}{j (ω - ω_{0}) + 1 / τ} .

Reflection spectra of the reflector were calculated with a resonance wavelength λ_C ( = 2πc/ω₀) of 1540 nm with TE polarization. A schematic cross-sectional view of the structure used for calculation is depicted in Fig. 2, where n indicates refractive index. A single-mode waveguide consisting of a Si-N guiding core of 600-nm thickness with Si-N/air grating of 80-nm thickness and a SiO₂ optical buffer of thickness t_b are formed on a Ta₂O₅/SiO₂ multilayer mirror. The mode index N of TE₀ mode was calculated to be 1.8304. The period Λ was 841.4 nm for vertical coupling of the incident wave. The length L_GC, namely the aperture size of the CRIGF, was chosen to be 10 μm. The length of the DBR was 240 μm each. The coupling coefficient of the DBR was calculated to be 33 mm^–1. Reflectance of the DBR at λ_C was estimated to be 1 with the bandwidth of 15 nm. The length L_Cav was calculated to be 40 μm.

Fig. 2 Schematic cross-sectional view of designed CRIGF.

Download Full Size | PDF

Calculated examples of reflection spectra with different t_b are shown in Fig. 3. The loss κ_Loss was assumed to be 0.5 and 2.0 dB/mm. An extinction ratio lower than –20 dB can be expected for appropriate t_b range. A bandwidth of –10 dB was predicted to be 0.01 and 0.04 nm for κ_Loss of 0.5 and 2.0 dB/mm with optimized t_b, respectively. Lower κ_Loss gives narrower bandwidth. On the other hand, fabrication tolerance in t_b becomes smaller for lower κ_Loss. An error of 1% in t_b is acceptable for obtaining –10 dB extinction ratio with κ_Loss of 2.0 dB/mm, while the tolerance becomes almost a half for a case of κ_Loss = 0.5 dB/mm.

Fig. 3 Reflection spectra calculated with κ_Loss of 0.5 dB/mm (a), and 2.0 dB/mm (b) for various buffer layer thicknesses t_b.

Download Full Size | PDF

3. Experimental characterization

Devices were fabricated and characterized. A substrate coated with Ta₂O₅/SiO₂ multilayer mirror was prepared. A SiO₂ optical buffer layer was deposited by plasma-enhanced chemical vapor deposition. A Si-N layer of 680-nm thickness was formed by a reactive sputtering. An electron-beam (EB) resist was coated. Groove patterns of a GC and DBRs and surrounding area of a waveguide channel of 10-μm width were exposed by EB direct writing and developed. The resist patterns were transferred to the Si-N layer by reactive-ion etching to form 80-nm height grating teeth of 10-μm width. An SEM image of the fabricated pattern is shown in Fig. 4. Fill factors of gratings of the GC and DBR were 0.5. Both sides of the waveguide channel were etched out with the same depth as grating grooves.

Fig. 4 SEM image of a part of the fabricated reflector.

Download Full Size | PDF

A wave from a wavelength-tunable diode laser was focused to have a beam waist of 10-μm diameter on the GC. The reflected wave was guided to an optical spectrum analyzer. An example of measured reflection spectrum is shown in Fig. 5. The extinction ratio was lower than –20 dB. The bandwidth at –10 dB was narrower than 0.04 nm, while that at –3 dB was 0.15 nm. The obtained spectrum shows good agreement with theoretical prediction for t_b = 2205 nm in Fig. 3(b), suggesting κ_Loss was happened to be ~2 dB/mm. This value is considerably larger than expected for a usual waveguide with grating structures. We are now under investigation of causes of the loss. One possible cause would be mismatch in electric field distribution of the guided mode among GC, DBR, and phase-adjusting regions. The guided mode experiences 50 times transitions between GC and DBR regions during 1 mm propagation for L_Cav = 40 μm. Even if transition loss is as low as 0.05 dB/time, κ_Loss due to the transition is counted up to 2.5 dB/mm.

Fig. 5 Measured reflection spectrum of the fabricated device.

Download Full Size | PDF

4. Conclusions

A band-stop reflector consisting of a CRIGF on a high reflection substrate was designed to have a narrow bandwidth. It was quantitatively shown that reflection spectra depended on optical-buffer-layer thickness. Excess loss in the cavity should be suppressed to provide narrow bandwidth. On the other hand, lower excess loss results in smaller fabrication tolerance. The reflector of 10-μm aperture was fabricated and characterized. The extinction ratio in reflectance was measured to be lower than –20 dB. The bandwidth at –3 dB was 0.15 nm. Thus a narrowband-stop reflector was theoretically predicted and experimentally demonstrated.

Acknowledgments

The authors would like to express our thanks to Prof. H. Kikuta and Mr. D. Yamashita in Osaka Prefecture University for their support on the EB patterning process. This work was supported in part by JSPS KAKENHI Grant Number 26420331.

References and links

1. L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55(6), 377–380 (1985). [CrossRef]

2. G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15(7), 886–887 (1985). [CrossRef]

3. S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32(14), 2606–2613 (1993). [CrossRef] [PubMed]

4. Z. S. Liu, S. Tibuleac, D. Shin, P. P. Young, and R. Magnusson, “High-efficiency guided-mode resonance filter,” Opt. Lett. 23(19), 1556–1558 (1998). [CrossRef] [PubMed]

5. S. M. Norton, T. Erdogan, and G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14(3), 629–639 (1997). [CrossRef]

6. D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997). [CrossRef]

7. S. Ura, J. Inoue, K. Kintaka, and Y. Awatsuji, “Proposal of small-aperture guided-mode-resonance filter,” in Proceedings of the 13th International Conference on Transparent Optical Networks (2011), paper Th.A4.4. [CrossRef]

8. K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20(2), 1444–1449 (2012). [PubMed]

9. J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of guided-mode resonance filter by cavity-resonator integration,” Appl. Phys. Express 5(2), 022201 (2012). [CrossRef]

10. K. Hatanaka, T. Majima, K. Kintaka, J. Inoue, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter consisting of curved gratings,” Electron. Lett. 48(12), 717–718 (2012). [CrossRef]

11. K. Kintaka, T. Majima, K. Hatanaka, J. Inoue, and S. Ura, “Polarization-independent guided-mode resonance filter with cross-integrated waveguide resonators,” Opt. Lett. 37(15), 3264–3266 (2012). [CrossRef] [PubMed]

12. J. Inoue, T. Ogura, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter in channel waveguide,” IEICE Electron. Express 10(15), 20130444 (2013). [CrossRef]

13. K. Kintaka, K. Hatanaka, J. Inoue, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter with nonuniform grating coupler for efficient coupling with Gaussian beam,” Appl. Phys. Express 6(10), 102203 (2013). [CrossRef]

14. X. Buet, E. Daran, D. Belharet, F. Lozes-Dupuy, A. Monmayrant, and O. Gauthier-Lafaye, “High angular tolerance and reflectivity with narrow bandwidth cavity-resonator-integrated guided-mode resonance filter,” Opt. Express 20(8), 9322–9327 (2012). [CrossRef] [PubMed]

15. N. Rassem, A.-L. Fehrembach, and E. Popov, “Waveguide mode in the box with an extraordinary flat dispersion curve,” J. Opt. Soc. Am. A 32(3), 420–430 (2015). [CrossRef] [PubMed]

16. K. Kintaka, K. Asai, T. Kondo, J. Inoue, and S. Ura, “Reflectance change by grating-position shifting in cavity-resonator-integrated guided-mode resonance filter,” in Technical Digest of Integrated Photonics Research, Silicon and Nano Photonics 2015 (2015), paper IM2A.3.

17. K. Asai, K. Kintaka, J. Inoue, and S. Ura, “Coupled-mode analysis of grating-position-shifted cavity-resonator-integrated guided-mode resonance filter,” in Technical Digest of 20th Microoptics Conference (2015), paper H-4. [CrossRef]

18. S. Ura, J. Inoue, T. Ogura, K. Nishio, Y. Awatsuji, and K. Kintaka, “Reflection-phase variation of cavity-resonator-integrated guided-mode-resonance reflector for guided-mode-exciting surface laser mirror,” in Proceedings of the 63th Electronics Components and Technology Conference (2013), pp. 1874–1879. [CrossRef]

19. M. Nakata, T. Kondo, K. Kintaka, J. Inoue, and S. Ura, “Reflection optical notch filtering behavior of cavity-resonator-integrated guided-mode resonance mirror,” in Proceedings of International Conference on Electronics Packaging 2015 (2015), paper 652–655. [CrossRef]

20. T. Kondo, S. Ura, and R. Magnusson, “Design of guided-mode resonance mirrors for short laser cavities,” J. Opt. Soc. Am. A 32(8), 1454–1458 (2015). [CrossRef] [PubMed]

21. J. Inoue, T. Kondo, K. Kintaka, K. Nishio, and S. Ura, “Determination of cavity length of cavity-resonator-integrated guided-mode resonance filter,” Opt. Express 23(3), 3020–3026 (2015). [CrossRef] [PubMed]

Cavity-resonator-integrated guided-mode resonance band-stop reflector

Abstract

1. Introduction

2. Basic configuration and operation principle

3. Theoretical prediction

3. Experimental characterization

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (21)

Optics Express