Abstract

A cavity-resonator-integrated guided-mode resonance filter (CRIGF) consists of a grating coupler inside a pair of distributed Bragg reflectors. A combination of a CRIGF with a high-reflection substrate can provide a new type of a band-stop reflector with a small aperture for a vertically incident wave from air. A narrow stopband was theoretically predicted and experimentally demonstrated. It was quantitatively shown that reflection spectra depended on optical-buffer-layer thickness. The reflector of 10-μm aperture was fabricated and characterized. The extinction ratio in reflectance was measured to be lower than –20 dB at a resonance wavelength. The bandwidth at –3 dB was 0.15 nm.

© 2016 Optical Society of America

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.

 figure: Fig. 1

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.

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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

dgdt=(jω01τ)gjκ^*||aIN,
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 |aIN. The second term in the right-hand side expresses resonant mode excitation from the incident wave with coupling coefficients |κ^ between g and |aIN. Out-going waves |aOUT from the GC are expressed by
|aOUT=|aINjg|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=jκ^*||aINj(ωω0)+1/τ.
A differential equation for the guided-mode intensity can be written as
d|g|2dt=21τ|g|22(1τRad+1τDBR+1τLoss)|g|2.
Liftime components due to radiation τRad, transmission through DBRs τDBR, and excess loss τLoss other than the radiation and transmission losses are given by
1τRad=cNLGCLCavα,1τDBR=cNln(rDBR)LCav,1τLoss=cNκLoss,
where c, N, LGC, LCav, α, rDBR, and κLoss 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 Eνa(x) of air-radiation mode and Eνs(x) of substrate-radiation mode having a wave number of βν along z-direction can be written as

Eνa(x)={Eνa0{(tatsrars)exp(jkaxx)rsexp(jkaxx)}exp(jβνz);inair,Eνa0taexp(jksxx)exp(jβνz);insubstrate
Eνs(x)={Eνs0tsexp(jkaxx)exp(jβνz);inairEνs0{raexp(jksxx)+(tatsrars)exp(jksxx)}exp(jβνz);insubstrate,
where ra, ta, rs, and ts denote coefficients of reflection, transmission for incidence from the air, reflection and transmission from the substrate, respectively. Wave numbers kax and ksx satisfy the relations of
ka=kax2+βν2,ks=ksx2+βν2,
where ka and ks are wave numbers in the air and substrate, respectively. Eνa0 and Eνs0 are real numbers and determined by normalization conditions of
π2ωμ0Eνa02{ka(|tatsrars|2+|rs|2)+ks|ta|2}=1,
π2ωμ0Eνs02{ka|ts|2+ks(|tatsrars|2+|ra|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
EAν(x)={caEνa(x)+csEνs(x)}EAν0,
ca=ratatsrars1Eνa0,cs=tatatsrars1Eνs0,
An incident wave and the out-going wave are expressed by
|aIN=(ca|EINν0cs|EINν0)=(cacs)|EINν0,
|aOUT=(da|EOUTν0ds|EOUTν0)=(dads)|EOUTν0.
We consider here the impulse response for an incident wave defined by
|EINν0=EIN00(ω)δ(βν=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
(dads)=(cacs)κ^0a*ca+κ^0s*csj(ωω0)+1/τ(κ^aκ^s),
(κ^aκ^s)=2πcNLGCLCav(κνaκνs),
(κ^0a*κ^0s*)=sinc(ΔωLGC2)2πcNLGCLCav(κνa*κνs*),
where Δω represents phase mismatching defined by
Δω=Nc(ωω0).
The spatial coupling coefficient κνi between the resonant guided mode g and a radiation mode νi is given by
κνi=ε0ω4Eνi*(x)ΔεEg(x)dx(i=aors),
where Eg(x) and Eνi(x) denote normalized electric field distributions of the guided and radiation modes, respectively. Finally, the reflection coefficient of the CRIGF is calculated to be

rCRIGF=rasinc(ΔωLGC2)2πcNLGCLCav|κνa|2ra+κνs*κνataEνa0/Eνs0j(ωω0)+1/τ.

Reflection spectra of the reflector were calculated with a resonance wavelength λC ( = 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 tb 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 LGC, 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 λC was estimated to be 1 with the bandwidth of 15 nm. The length LCav was calculated to be 40 μm.

 figure: Fig. 2

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

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

 figure: Fig. 3

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

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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.

 figure: Fig. 4

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

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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 tb = 2205 nm in Fig. 3(b), suggesting κLoss 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 LCav = 40 μm. Even if transition loss is as low as 0.05 dB/time, κLoss due to the transition is counted up to 2.5 dB/mm.

 figure: Fig. 5

Fig. 5 Measured reflection spectrum of the fabricated device.

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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]  

References

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  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]

2015 (3)

2013 (2)

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]

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]

2012 (5)

1998 (1)

1997 (2)

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]

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

1993 (1)

1985 (2)

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

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]

Awatsuji, Y.

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]

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]

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]

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]

Belharet, D.

Buet, X.

Daran, E.

Erdogan, T.

Fehrembach, A.-L.

Friesem, A. A.

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

Gauthier-Lafaye, O.

Golubenko, G. A.

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]

Hatanaka, K.

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]

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]

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]

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]

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]

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]

Inoue, J.

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]

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]

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]

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]

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]

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]

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]

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]

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]

Kintaka, K.

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]

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]

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]

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]

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]

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]

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]

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]

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]

Kondo, T.

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]

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]

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]

Liu, Z. S.

Lozes-Dupuy, F.

Magnusson, R.

Majima, T.

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]

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]

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]

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]

Mashev, L.

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

Monmayrant, A.

Morris, G. M.

Nakata, M.

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]

Nishii, J.

Nishio, K.

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]

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]

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]

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]

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]

Norton, S. M.

Ogura, T.

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]

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]

Popov, E.

Rassem, N.

Rosenblatt, D.

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

Sharon, A.

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

Shin, D.

Svakhin, A. S.

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]

Sychugov, V. A.

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]

Tibuleac, S.

Tishchenko, A. V.

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]

Ura, S.

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Wang, S. S.

Young, P. P.

Appl. Opt. (1)

Appl. Phys. Express (2)

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]

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]

Electron. Lett. (1)

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]

IEEE J. Quantum Electron. (1)

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

IEICE Electron. Express (1)

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]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

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

Opt. Express (3)

Opt. Lett. (2)

Sov. J. Quantum Electron. (1)

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]

Other (5)

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]

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.

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]

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]

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]

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Figures (5)

Fig. 1
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.
Fig. 2
Fig. 2 Schematic cross-sectional view of designed CRIGF.
Fig. 3
Fig. 3 Reflection spectra calculated with κLoss of 0.5 dB/mm (a), and 2.0 dB/mm (b) for various buffer layer thicknesses tb.
Fig. 4
Fig. 4 SEM image of a part of the fabricated reflector.
Fig. 5
Fig. 5 Measured reflection spectrum of the fabricated device.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

dg dt =( j ω 0 1 τ )gj κ ^ * || a IN
| a OUT =| a IN jg| κ ^ ,
g=j κ ^ * || a IN j( ω ω 0 )+1/τ .
d | g | 2 dt =2 1 τ | g | 2 2( 1 τ Rad + 1 τ DBR + 1 τ Loss ) | g | 2 .
1 τ Rad = c N L GC L Cav α , 1 τ DBR = c N ln( r DBR ) L Cav , 1 τ Loss = c N κ Loss ,
E νa ( x )={ E νa 0 { ( t a t s r a r s )exp( j k ax x ) r s exp( j k ax x ) }exp( j β ν z ) ; in air , E νa 0 t a exp( j k sx x )exp( j β ν z ) ; in substrate
E νs ( x )={ E νs 0 t s exp( j k ax x )exp( j β ν z ) ; in air E νs 0 { r a exp( j k sx x )+( t a t s r a r s )exp( j k sx x ) }exp( j β ν z ) ; in substrate ,
k a = k ax 2 + β ν 2 , k s = k sx 2 + β ν 2 ,
π 2ω μ 0 E νa 0 2 { k a ( | t a t s r a r s | 2 + | r s | 2 )+ k s | t a | 2 }=1 ,
π 2ω μ 0 E νs 0 2 { k a | t s | 2 + k s ( | t a t s r a r s | 2 + | r a | 2 ) }=1 .
E Aν (x)={ c a E νa (x)+ c s E νs (x) } E Aν 0 ,
c a = r a t a t s r a r s 1 E νa 0 , c s = t a t a t s r a r s 1 E νs 0
| a IN = ( c a | E INν 0 c s | E INν 0 )=( c a c s )| E INν 0 ,
| a OUT =( d a | E OUTν 0 d s | E OUTν 0 ) =( d a d s )| E OUTν 0 .
| E INν 0 = E IN0 0 (ω)δ( β ν =0),
( d a d s )=( c a c s ) κ ^ 0a * c a + κ ^ 0s * c s j( ω ω 0 )+1/τ ( κ ^ a κ ^ s ) ,
( κ ^ a κ ^ s )= 2πc N L GC L Cav ( κ νa κ νs ) ,
( κ ^ 0a * κ ^ 0s * )=sinc( Δ ω L GC 2 ) 2πc N L GC L Cav ( κ νa * κ νs * ) ,
Δ ω = N c ( ω ω 0 ).
κ νi = ε 0 ω 4 E νi * (x)Δε E g (x)dx ( i=a or s ) ,
r CRIGF = r a sinc( Δ ω L GC 2 ) 2πc N L GC L Cav | κ νa | 2 r a + κ νs * κ νa t a E νa 0 / E νs 0 j( ω ω 0 )+1/τ .

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