## Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching |

Optics Express, Vol. 18, Issue 22, pp. 23183-23195 (2010)

http://dx.doi.org/10.1364/OE.18.023183

Acrobat PDF (1099 KB)

### Abstract

This paper addresses the design of narrow band transmission co-directional couplers suitable for wavelength division multiplexing applications. The originality of the proposed asymmetric two-waveguide configuration stems from the use of Bragg gratings operated near band gap to simultaneously achieve high wavelength dispersion and selectivity as well as co-directional phase matching between guides which would be mismatched otherwise. Our theoretical analysis reveals the existence of a minimum Bragg grating coupling strength for co-directional phase matching. The threshold condition is analytically determined, and a coupled mode theory (CMT) four-wave model is successfully applied to describe the behavior of the investigated device. A numerical validation of CMT results is reported in the case of slab waveguides with Bragg grating assisted coupling. The proposed design is shown to be compatible with existing micro-nano-fabrication technology.

© 2010 OSA

## 1. Introduction

1. H. F. Taylor, “Frequency-selective coupling in parallel dielectric waveguides,” Opt. Commun. **8**(4), 421–425 (1973). [CrossRef]

2. C. Elachi and C. Yeh, “Frequency selective coupler for integrated optics systems,” Opt. Commun. **7**(3), 201–204 (1973). [CrossRef]

3. D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part I: Synchronous couplers,” J. Lightwave Technol. **5**(1), 113–118 (1987). [CrossRef]

5. W.-P. Huang, “Coupled mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A **11**(3), 963–983 (1994). [CrossRef]

6. R. C. Alferness and R. V. Schmidt, “Tunable optical waveguide directional coupler filter,” Appl. Phys. Lett. **33**(2), 161–163 (1978). [CrossRef]

12. A. Lupu, P. Win, H. Sik, P. Boulet, M. Carré, S. Slempkes, and A. Carenco, “Tunable filter with box like spectral response for 1.28/1.32 µm duplexer application,” Electron. Lett. **35**(2), 174–175 (1999). [CrossRef]

9. C. Bornholdt, F. Kappe, R. Müller, H.-P. Nolting, F. Reier, R. Stenzel, H. Venghaus, and C. M. Weinert, “Meander coupler, a novel wavelength division multiplexer/demultiplexer,” Appl. Phys. Lett. **57**(24), 2517–2519 (1990). [CrossRef]

12. A. Lupu, P. Win, H. Sik, P. Boulet, M. Carré, S. Slempkes, and A. Carenco, “Tunable filter with box like spectral response for 1.28/1.32 µm duplexer application,” Electron. Lett. **35**(2), 174–175 (1999). [CrossRef]

6. R. C. Alferness and R. V. Schmidt, “Tunable optical waveguide directional coupler filter,” Appl. Phys. Lett. **33**(2), 161–163 (1978). [CrossRef]

8. P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. **7**(7), 697–699 (1995). [CrossRef]

8. P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. **7**(7), 697–699 (1995). [CrossRef]

12. A. Lupu, P. Win, H. Sik, P. Boulet, M. Carré, S. Slempkes, and A. Carenco, “Tunable filter with box like spectral response for 1.28/1.32 µm duplexer application,” Electron. Lett. **35**(2), 174–175 (1999). [CrossRef]

15. B. E. Little, C. Wu, and W. P. Huang, “Synthesis of codirectional couplers with ultralow sidelobes and minimum bandwidth,” Opt. Lett. **20**(11), 1259–1261 (1995). [CrossRef] [PubMed]

3. D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part I: Synchronous couplers,” J. Lightwave Technol. **5**(1), 113–118 (1987). [CrossRef]

5. W.-P. Huang, “Coupled mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A **11**(3), 963–983 (1994). [CrossRef]

6. R. C. Alferness and R. V. Schmidt, “Tunable optical waveguide directional coupler filter,” Appl. Phys. Lett. **33**(2), 161–163 (1978). [CrossRef]

9. C. Bornholdt, F. Kappe, R. Müller, H.-P. Nolting, F. Reier, R. Stenzel, H. Venghaus, and C. M. Weinert, “Meander coupler, a novel wavelength division multiplexer/demultiplexer,” Appl. Phys. Lett. **57**(24), 2517–2519 (1990). [CrossRef]

15. B. E. Little, C. Wu, and W. P. Huang, “Synthesis of codirectional couplers with ultralow sidelobes and minimum bandwidth,” Opt. Lett. **20**(11), 1259–1261 (1995). [CrossRef] [PubMed]

_{0}with phase-matching condition is in the inverse ratio of the coupling length L and of the differential dispersion between the effectives indexes

*n*

_{i}of the two waveguide modes [14

14. K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. **9**(11), 1481–1492 (1991). [CrossRef]

16. C. Wu, C. Rolland, N. Puetz, R. Bruce, K. D. Chik, and J. M. Xu, “A vertically coupled InGaAsP/InP directional coupler filter of ultranarrow bandwidth,” IEEE Photon. Technol. Lett. **3**(6), 519–521 (1991). [CrossRef]

16. C. Wu, C. Rolland, N. Puetz, R. Bruce, K. D. Chik, and J. M. Xu, “A vertically coupled InGaAsP/InP directional coupler filter of ultranarrow bandwidth,” IEEE Photon. Technol. Lett. **3**(6), 519–521 (1991). [CrossRef]

19. B. R. West and A. S. Helmy, “Dispersion tailoring of the quarter-wave Bragg reflection waveguide,” Opt. Express **14**(9), 4073–4086 (2006). [CrossRef] [PubMed]

21. M. Dainese, M. Swillo, L. Wosinski, and L. Thylen, “Directional coupler wavelength selective filter based on dispersive Bragg reflection waveguide,” Opt. Commun. **260**(2), 514–521 (2006). [CrossRef]

22. A. Ankiewicz and G.-D. Peng, “Narrow bandpass filter using Bragg grating coupler in transmission mode,” Electron. Lett. **33**(25), 2151–2153 (1997). [CrossRef]

23. A. Ankiewicz, Z. H. Wang, and G.-D. Peng, “Analysis of narrow bandpass filter using coupler with Bragg grating in transmission,” Opt. Commun. **156**(1-3), 27–31 (1998). [CrossRef]

21. M. Dainese, M. Swillo, L. Wosinski, and L. Thylen, “Directional coupler wavelength selective filter based on dispersive Bragg reflection waveguide,” Opt. Commun. **260**(2), 514–521 (2006). [CrossRef]

*et al*[22

22. A. Ankiewicz and G.-D. Peng, “Narrow bandpass filter using Bragg grating coupler in transmission mode,” Electron. Lett. **33**(25), 2151–2153 (1997). [CrossRef]

23. A. Ankiewicz, Z. H. Wang, and G.-D. Peng, “Analysis of narrow bandpass filter using coupler with Bragg grating in transmission,” Opt. Commun. **156**(1-3), 27–31 (1998). [CrossRef]

24. P. Bienstman and R. Baets, “Optical Modelling of Photonic Crystals and VCSELs using Eigenmode Expansion and Perfectly Matched Layers,” Opt. Quantum Electron. **33**(4/5), 327–341 (2001). [CrossRef]

## 2. Bragg grating assisted co-directional coupling phase matching

26. P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. **19**(16), 2848–2855 (1980). [CrossRef] [PubMed]

37. A. Yesayan and R. Vallée, “Zero backreflection condition for a grating-frustrated coupler,” J. Opt. Soc. Am. B **20**(7), 1418–1426 (2003). [CrossRef]

30. L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. S. J. Russell, “Novel add–drop filters for wavelength division multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. **8**(12), 1656–1658 (1996). [CrossRef]

33. I. Baumann, J. Seifert, W. Nowak, and M. Sauer, “Compact all-fiber add-drop-multiplexer using fiber Bragg gratings,” IEEE Photon. Technol. Lett. **8**(10), 1331–1333 (1996). [CrossRef]

34. J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. **19**(3), 180–182 (1994). [CrossRef] [PubMed]

37. A. Yesayan and R. Vallée, “Zero backreflection condition for a grating-frustrated coupler,” J. Opt. Soc. Am. B **20**(7), 1418–1426 (2003). [CrossRef]

38. N. Imoto, “An analysis for contradirectional-coupler-type optical grating filters,” J. Lightwave Technol. **3**(4), 895–900 (1985). [CrossRef]

22. A. Ankiewicz and G.-D. Peng, “Narrow bandpass filter using Bragg grating coupler in transmission mode,” Electron. Lett. **33**(25), 2151–2153 (1997). [CrossRef]

23. A. Ankiewicz, Z. H. Wang, and G.-D. Peng, “Analysis of narrow bandpass filter using coupler with Bragg grating in transmission,” Opt. Commun. **156**(1-3), 27–31 (1998). [CrossRef]

39. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A **11**(4), 1307–1320 (1994). [CrossRef]

40. A. Arraf, L. Poladian, C. M. de Sterke, and T. G. Brown, “Effective-medium approach for counterpropagating waves in nonuniform Bragg gratings,” J. Opt. Soc. Am. A **14**(5), 1137–1143 (1997). [CrossRef]

*et al.*[41

41. A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. **13**(4), 233–253 (1977). [CrossRef]

*k*= 2π/λ, of the grating period Λ, of the BG coupling coefficient

*χ*, and of the waveguide effective index

*n*:

42. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B **22**(6), 1179–1190 (2005). [CrossRef]

*n*

_{1}and

*n*

_{2}are wavelength independent. Such a crude approximation does not alter the essential of the device physics while it greatly facilitates an insight into its optical properties. Moreover, since the coupling phenomenon occurs in a narrow spectral range, one can also neglect the wavelength dependence of the BG coupling coefficients

*χ*

_{1}and

*χ*

_{2}. All these parameters will be set constant in the rest of the paper.

*β*

*=*

_{Eq. (1)}*β*

*. From Eq. (2) it follows that:*

_{Eq. (2)}_{1}= Λ

_{2}or there is only one BG. It is worthwhile noticing that the case of two BGs with different periods and non-overlapping band gaps can be approximately treated as two distinct sub-cases with only one grating at a time (Λ

_{2}= ∞,

*χ*

_{2}= 0 and Λ

_{1}= ∞,

*χ*

_{1}= 0, respectively) provided that the dispersion variation induced by one grating is negligible in the spectral vicinity of the second grating band gap and vice-versa.

*χ*

_{1}and

*χ*

_{2}separately. Therefore, at this stage, the analysis of phase-matching conditions can be reduced to the case where there is only one BG. A further inspection of Eq. (4) shows that the solutions differ according as

*n*

_{1}<

*n*

_{2}or

*n*

_{1}>

*n*

_{2}. Each case has then to be separately analyzed.

*χ*

_{1}≠0 and

*χ*

_{2}= 0. The dispersion characteristics,

*ω = ω*(

*β*), obtained by means of Eq. (2), are displayed in Figs. 2(a) and 2(b) for several values of

_{eq}*χ*

_{1}and the two cases

*n*

_{1}>

*n*

_{2}and

*n*

_{1}<

*n*

_{2}, respectively. The difference between effective indexes (for instance,

*n*

_{1}= 3,

*n*

_{2}= 3.6 in Fig. 2b) and then the values of

*χ*

_{1}are voluntarily exaggerated to better illustrate the evolution of the dispersion characteristics. The black straight line corresponds to the dispersion characteristic of the uniform waveguide without grating (

*χ*

_{2}= 0). The family of hyperbolas represents the BG waveguide dispersion curves for different values of

*χ*

_{1}. The branches drawn in solid lines correspond to forward propagating waves. Those in dashed lines correspond to backward propagating waves shifted in abscissa by a grating vector Q = 2π/Λ. Colors from blue to light green in the graph correspond to increasing values of

*χ*

_{1}. The BG band gap frequency is centered at ω

_{Br}= πc/(Λ

*n*

_{1}). The width of the forbidden band is 2c

*χ*

_{1}/

*n*

_{1}.

*χ*

_{1}, the dispersion line of the uniform waveguide only intersects the dispersion curves of the BG waveguide in the region of backward propagating waves. This indicates a contra-directional phase-matching

*via*the Bragg exchange coupling mechanism.

**33**(25), 2151–2153 (1997). [CrossRef]

**156**(1-3), 27–31 (1998). [CrossRef]

26. P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. **19**(16), 2848–2855 (1980). [CrossRef] [PubMed]

38. N. Imoto, “An analysis for contradirectional-coupler-type optical grating filters,” J. Lightwave Technol. **3**(4), 895–900 (1985). [CrossRef]

*n*

_{1}<

*n*

_{2}, the situation is mostly similar. The corresponding dispersion characteristics are shown in Fig. 2(b). Co-directional phase-matching is achieved when the BG coupling coefficient

*χ*

_{1}is higher than a certain threshold value

*n*

_{1}>

*n*

_{2}) is that in the region of co-propagative waves, the dispersion characteristics of the two guides intersect at frequencies smaller than the band gap frequency. Another important difference is that the co-directional phase-matching condition is fulfilled only within a limited range of BG coupling coefficients:

*χ*

_{d}, for co-directional phase matching corresponds to a tangential contact between the dispersion characteristics of the two guides. Thus, for

*χ*

_{1}>

*χ*

_{d}the condition for co-directional phase-matching is no longer satisfied. It is worthwhile mentioning that contra-directional phase-matching is also excluded in this case.

*χ*

_{1}≠0 and

*χ*

_{2}≠0 with the assumption that:

*χ*

_{1}≥

*χ*

_{2}>0. The dispersion diagrams (not shown here) are obviously more complex than those of Figs. 2(a) and 2(b) since each of them is comprised of two series of dispersion curves instead of one. However, the general conclusions are unchanged. There is a minimum value of

*χ*

_{1}(

*χ*

_{1}=

*χ*

_{th}) for co-directional coupling whether

*n*

_{1}is larger or smaller than

*n*

_{2}. There is an upper limit

*χ*

_{1}=

*χ*

_{d}for co-directional coupling in the case where

*n*

_{1}<

*n*

_{2}. Expressions of

*χ*

_{th}and

*χ*

_{d}can be derived from Eqs. (2)–(4) using our simple model with constant index waveguides:

*n*

_{1}>

*n*

_{2}and

*n*

_{1}<

*n*

_{2}. Equation (5) also shows that

*χ*

_{2}≠0 and then the presence of a second BG increase the threshold value for co-directional phase matching. Such a situation with stronger BG index modulation is less favorable in practice. Co-directional phase-matching becomes even impossible when the two BGs are of equal strength in our asymmetric coupler geometry. Therefore, one can conclude that co-directional coupling is optimally achieved with the use of a BG in only one of the two ADC waveguides. The hypothesis

*χ*

_{2}= 0 will then be kept for the rest of the paper.

## 3. Coupled mode approach modeling

_{b}, and the refractive indices of waveguides 1 and 2 are N

_{1}and N

_{2}, respectively. Both waveguides are assumed to fulfill the single mode operation condition. The dashed line traced for the BG assisted waveguide indicates the original waveguide width without grating modulation. This width is used to determine the effective index and the corresponding propagation constant

43. R. R. A. Syms, “Improved coupled mode theory for codirectionally and contradirectionally coupled waveguide arrays,” J. Opt. Soc. Am. A **8**(7), 1062–1069 (1991). [CrossRef]

46. N. Izhaky and A. Hardy, “Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism,” J. Opt. Soc. Am. A **16**(6), 1303–1311 (1999). [CrossRef]

*z*in a series of parallel waveguide segments of lengths Λ

_{+}and Λ

_{-}(Λ = Λ

_{+}+ Λ

_{-}), delimited by the grating corrugations. Since all the coupling matrices are independent of

*z*within each section, the coupled-mode equations can be solved exactly. To define the elements of the coupling matrices, we use an approach based on the individual waveguides modes [29

29. R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. **19**(5), 273–287 (1987). [CrossRef]

45. A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. **34**(7), 1109–1116 (1998). [CrossRef]

46. N. Izhaky and A. Hardy, “Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism,” J. Opt. Soc. Am. A **16**(6), 1303–1311 (1999). [CrossRef]

**is the column vector**

*A***a**

*A*= [_{1}

^{+}, a

_{1}

^{-}, a

_{2}

^{+}, a

_{2}

^{-}

**]**with field amplitude elements and

^{T}**,**

*M*_{+}

*M*_{-}are 4 × 4 matrices with constant value coefficients:

*β*

_{1},

*β*

_{2}and the BG coupling coefficients

*χ*

_{1}and

*χ*

_{2}appearing in Eq. (8) have already been defined in Section 2. The newly appearing terms

*κ*and

*χ*

_{12}stand for the co-directional evanescent-coupling and contra-directional Bragg exchange evanescent-coupling, respectively. Using a standard diagonalization procedure, the system of linear differential Eqs. (7) and (8) can be reduced to the equations for the normal modes:where

**=**

*B***[**b

_{1}

^{+}, b

_{1}

^{-}, b

_{2}

^{+}, b

_{2}

^{-}

**]**is the new column vector and

^{T}**Γ**, the eigenvalue diagonal matrix:

_{±}*γ*

_{1}

^{+}and

*γ*

_{2}

^{±}are the eigenvalues of

**. In other words, they identify with the propagation constants of normal modes in the parallel coupler.**

*M*_{±}_{±}is the length of the ± section. The transfer matrix

**for one period Λ = Λ**

*T*_{1}_{+}+ Λ

_{-}is then:where

*O*_{±}is the eigenvectors matrix of

**in the ± section. The transfer matrix for the whole structure is then:**

*M*_{±}*R*-the thru-port direct reflection- and

^{||}*R*

^{×}-the drop-port exchange reflection-, the column vector associated to mode amplitudes at the device input writes:

*, T*

^{||}^{×}are the thru-port direct transmission and the drop-port exchange transmission, respectively. The system of linear Eq. (15) can be solved by elementary algebra. Final expressions for the transmission and reflection coefficients T

^{//}, T

^{x},

*R*

^{//},

*R*

^{x}are given in Appendix B.

*κ*and

*χ*

_{12}entering in the definition of matrix

**Eq. (8) need to be expressed versus the geometrical parameters of the coupler. A conventional expression corresponding to a complete crossover can actually be used for the co-directional evanescent coupling coefficient**

*M*_{±}*κ*:

*κ*is much smaller than the threshold coupling coefficient for co-directional coupling

*χ*

_{th}. On the other hand, the contra-directional Bragg exchange evanescent-coupling coefficient

*χ*

_{12}is approximated as follows:

*χ*

_{1}Λ<<1 in the following examples, Eq. (19) shows that

*χ*

_{12}<<

*κ*.

14. K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. **9**(11), 1481–1492 (1991). [CrossRef]

^{×}for several values of

*χ*

_{1}(with

*χ*

_{2}= 0) and for the two cases

*n*

_{1}>

*n*

_{2}and

*n*

_{1}<

*n*

_{2}, respectively.

*n*

_{1}>

*n*

_{2}), we take

*n*

_{1}= 3.255 and

*n*

_{2}= 3.25. The grating period is chosen to be Λ = 0.238µm to fix the Bragg wavelength at 1.55µm for a 50% grating duty ratio. The total grating length is 1000µm. The different values of

*χ*

_{1}are: 0.5, 3, 12, and 30

*χ*

_{th}, with

*χ*

_{th}= 0.032µm

^{−1}. The transmission spectra of Fig. 4(a) confirm that below threshold, the drop-port exchange transmission T

^{×}is low. Its amplitude is indeed of the same order as the amplitudes of secondary peaks. Just above threshold (

*χ*

_{1}= 3

*χ*

_{th}), a narrow transmission peak appears at a frequency slightly larger than the Bragg frequency, but the transmission does not reach its maximum. Despite the fact that the incident light is injected into the uniform waveguide, a fraction of it is reflected into this guide

*via*an indirect mechanism involving Bragg reflection combined with co-directional coupling. This will be illustrated later in Fig. 5 . As

*χ*

_{1}is increased far above threshold (

*χ*

_{1}= 12

*χ*

_{th},

*χ*

_{1}= 30

*χ*

_{th}) the transmission peak progressively broadens and shifts to higher frequencies. The peak amplitude is rapidly growing to its asymptotic limit while the phase matching frequency moves away from the Bragg frequency [see Fig. 2(a)].

*n*

_{1}<

*n*

_{2}), we keep

*n*

_{1}= 3.255 and set

*n*

_{2}= 3.26. The other parameters are unchanged. As seen in Fig. 4(b), below threshold, the transmission T

^{×}is low. Just above threshold, a narrow transmission peak appears at a frequency slightly smaller than the gap frequency. As

*χ*

_{1}is increased, the peak broadens and now shifts to the lower frequency side of the band gap. A still bigger difference with previous behavior of Fig. 4(a) occurs for very high values of

*χ*

_{1}(here

*χ*

_{1}= 30

*χ*

_{th}). Indeed, above the limit

*χ*

_{1}=

*χ*

_{d}, the phase matching condition is no more satisfied and the transmission T

^{×}is low again. One may admit, however, that

*χ*

_{d}is rather high and hardly attainable in practice. For practical applications, a good compromise between a high selectivity and a high transmission level will be to choose

*χ*

_{1}not too high, just a few times the threshold value.

## 4. CAMFR numerical modeling

_{1}= N

_{2}= 3.32 core index material, embedded in a background cladding material with refractive index N

_{b}= 3.17. As for the for CMT four-wave model, the Bragg wavelength is fixed at 1.55µm. The grating period is Λ = 0.238µm for a 50% grating duty ratio. The total grating length is 1000µm. For a rectangular grating profile, the relation between the BG coupling coefficient

*χ*

_{1}and the waveguide effective index modulation Δ

*n*

_{1}is:

*χ*

_{1}= 3.14

*χ*

_{th}. This imposes a grating profile with a minimal waveguide width of 0.41µm and a maximal waveguide width of 0.93µm. The width of the second waveguide is adjusted so as to explore each of the cases

*n*

_{1}>

*n*

_{2}and

*n*

_{1}<

*n*

_{2}. For

*n*

_{1}>

*n*

_{2}, the width of the second waveguide is 0.53µm. The mean separation distance between the waveguides is 1.7µm. The light is injected through the second waveguide without BG.

^{×}as in Fig. 4(a) for

*χ*

_{1}= 3

*χ*

_{th}. Co-directional coupling to the Bragg waveguide is achieved with around 80% transmitted power at maximum. The remaining fraction of power (20%) is essentially reflected into the input waveguide

*via*combined co-directional coupling and Bragg reflection mechanisms. The drop-port reflection

*R*

^{x}is low as well as the residual thru-port transmission T

^{//}. The selectivity obtained for a 1mm device length is approximately a factor of four higher than that in conventional approaches [16

16. C. Wu, C. Rolland, N. Puetz, R. Bruce, K. D. Chik, and J. M. Xu, “A vertically coupled InGaAsP/InP directional coupler filter of ultranarrow bandwidth,” IEEE Photon. Technol. Lett. **3**(6), 519–521 (1991). [CrossRef]

*n*

_{1}<

*n*

_{2}, the width of the second waveguide is 0.67µm while the BG parameters are let unchanged. The mean separation distance between the waveguides is 1.6µm. Figure 5(b) shows the transmission and reflection spectra. Results are similar to those of Fig. 5(a), except for the fact that the phase matching wavelength is now located on the long wavelength side of the band gap.

## 5. Summary and conclusions

**33**(25), 2151–2153 (1997). [CrossRef]

**156**(1-3), 27–31 (1998). [CrossRef]

*χ*. The only condition is that the grating coupling strength be sufficiently high: χL>>1. Because of their specificity, BGADCs then exhibit new properties that do not simply result from the individual properties of their constitutive elements.

## Appendix A. Threshold condition in Eq. (5) and upper limit of *χ*_{1} in Eq. (6)

*n*

_{1}>

*n*

_{2}. The reverse situation occurs for

*n*

_{1}<

*n*

_{2}. The wave-vector meeting the phase matching condition can then be directly determined from Eq. (2):

*n*

_{1}>

*n*

_{2}or

*n*

_{1}<

*n*

_{2}. Equation (5) in Section 2 is then obtained in a straightforward manner from Eq. (A1).

*χ*

_{1}=

*χ*

_{d}corresponds to a tangential contact between the dispersion curves of the two guides. In other words, there is only one solution for Eq. (4). Equation (6) in Section 2 is then readily obtained by elementary algebra.

## Appendix B. Transmission and reflection coefficients T^{//}, T^{x}, *R*^{//}, *R*^{x}

^{//}, T

^{x},

*R*

^{//},

*R*

^{x}are:

## Appendix C. Bragg exchange evanescent coupling coefficient in Eq. (19)

*χ*

_{12}is usually determined from the overlap integral between the forward propagating mode of one waveguide and the backward propagating mode of the other waveguide in the region where the index is modified by the Bragg grating. On the other hand, the co-directional coupling coefficient

*κ*is determined from the overlap between the mode propagating in one waveguide with the tail of the mode propagating in the adjacent waveguide. A fraction of this overlap, which corresponds to the to BG-perturbed-index region, is but the Bragg exchange evanescent coupling coefficient itself,

*χ*

_{12}. This fraction,

*χ*

_{12}/

*κ*, can then be approximated by the ratio of the grating index modulation to the waveguide effective index. Assuming that each waveguide includes a BG with period Λ, one can write:

## References and links

1. | H. F. Taylor, “Frequency-selective coupling in parallel dielectric waveguides,” Opt. Commun. |

2. | C. Elachi and C. Yeh, “Frequency selective coupler for integrated optics systems,” Opt. Commun. |

3. | D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part I: Synchronous couplers,” J. Lightwave Technol. |

4. | D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: Grating-assisted couplers,” J. Lightwave Technol. |

5. | W.-P. Huang, “Coupled mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A |

6. | R. C. Alferness and R. V. Schmidt, “Tunable optical waveguide directional coupler filter,” Appl. Phys. Lett. |

7. | R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, and C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” Appl. Phys. Lett. |

8. | P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. |

9. | C. Bornholdt, F. Kappe, R. Müller, H.-P. Nolting, F. Reier, R. Stenzel, H. Venghaus, and C. M. Weinert, “Meander coupler, a novel wavelength division multiplexer/demultiplexer,” Appl. Phys. Lett. |

10. | Y.-H. Jan, M. E. Heimbuch, L. A. Coldren, and S. P. DenBaars, “InP/InGaAsP grating-assisted codirectional coupler tunable receiver with a 30nm wavelength tuning range,” Electron. Lett. |

11. | A. Lupu, H. Sik, A. Mereuta, P. Boulet, M. Carré, S. Slempkes, A. Ougazzaden, and A. Carenco, “Three-waveguides two-grating codirectional coupler for 1.3-/1.3+/1.5µm demultiplexing in transceiver,” Electron. Lett. |

12. | A. Lupu, P. Win, H. Sik, P. Boulet, M. Carré, S. Slempkes, and A. Carenco, “Tunable filter with box like spectral response for 1.28/1.32 µm duplexer application,” Electron. Lett. |

13. | R. C. Alferness, “Optical directional couplers with weighted coupling,” Appl. Phys. Lett. |

14. | K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. |

15. | B. E. Little, C. Wu, and W. P. Huang, “Synthesis of codirectional couplers with ultralow sidelobes and minimum bandwidth,” Opt. Lett. |

16. | C. Wu, C. Rolland, N. Puetz, R. Bruce, K. D. Chik, and J. M. Xu, “A vertically coupled InGaAsP/InP directional coupler filter of ultranarrow bandwidth,” IEEE Photon. Technol. Lett. |

17. | B. Liu, A. Shakouri, P. Abraham, Y. J. Chiu, S. Zhang, and J. E. Bowers, “Fused InP–GaAs Vertical Coupler Filters,” IEEE Photon. Technol. Lett. |

18. | S.-K. Han, R. V. Ramaswamy, and R. F. Tavlykaev, “Highly asymmetrical vertical coupler wavelength filter in InGaAlAs/InP,” Electron. Lett. |

19. | B. R. West and A. S. Helmy, “Dispersion tailoring of the quarter-wave Bragg reflection waveguide,” Opt. Express |

20. | R. Das and K. Thyagarajan, “Strong dispersive features of dual Bragg cladding waveguides using higher-order mode-coupling mechanism,” J. Opt. A, Pure Appl. Opt. |

21. | M. Dainese, M. Swillo, L. Wosinski, and L. Thylen, “Directional coupler wavelength selective filter based on dispersive Bragg reflection waveguide,” Opt. Commun. |

22. | A. Ankiewicz and G.-D. Peng, “Narrow bandpass filter using Bragg grating coupler in transmission mode,” Electron. Lett. |

23. | A. Ankiewicz, Z. H. Wang, and G.-D. Peng, “Analysis of narrow bandpass filter using coupler with Bragg grating in transmission,” Opt. Commun. |

24. | P. Bienstman and R. Baets, “Optical Modelling of Photonic Crystals and VCSELs using Eigenmode Expansion and Perfectly Matched Layers,” Opt. Quantum Electron. |

25. | |

26. | P. Yeh and H. F. Taylor, “Contradirectional frequency-selective couplers for guided-wave optics,” Appl. Opt. |

27. | M. S. Whalen, M. D. Divino, and R. C. Alferness, “Demonstration of a narrowband Bragg-reflection filter in a single-mode fibre directional coupler,” Electron. Lett. |

28. | R. R. A. Syms, “Optical directional coupler with a grating overlay,” Appl. Opt. |

29. | R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. |

30. | L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. S. J. Russell, “Novel add–drop filters for wavelength division multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. |

31. | S. S. Orlov, A. Yariv, and S. Van Essen, “Coupled-mode analysis of fiber-optic add drop filters for dense wavelength-division multiplexing,” Opt. Lett. |

32. | T. Erdogan, “Optical add–drop multiplexer based on an asymmetric Bragg coupler,” Opt. Commun. |

33. | I. Baumann, J. Seifert, W. Nowak, and M. Sauer, “Compact all-fiber add-drop-multiplexer using fiber Bragg gratings,” IEEE Photon. Technol. Lett. |

34. | J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. |

35. | A.-C. Jacob-Poulin, R. Valle’e, S. LaRochelle, D. Faucher, and G. R. Atkins, “Channel-dropping filter based on a grating-frustrated two-core fiber,” J. Lightwave Technol. |

36. | A. Yesayan and R. Vallée, “Optimized grating-frustrated coupler,” Opt. Lett. |

37. | A. Yesayan and R. Vallée, “Zero backreflection condition for a grating-frustrated coupler,” J. Opt. Soc. Am. B |

38. | N. Imoto, “An analysis for contradirectional-coupler-type optical grating filters,” J. Lightwave Technol. |

39. | J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A |

40. | A. Arraf, L. Poladian, C. M. de Sterke, and T. G. Brown, “Effective-medium approach for counterpropagating waves in nonuniform Bragg gratings,” J. Opt. Soc. Am. A |

41. | A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. |

42. | B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B |

43. | R. R. A. Syms, “Improved coupled mode theory for codirectionally and contradirectionally coupled waveguide arrays,” J. Opt. Soc. Am. A |

44. | J. Hong and W. P. Huang, “Contra-directional coupling in grating-assisted guided-wave devices,” J. Lightwave Technol. |

45. | A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. |

46. | N. Izhaky and A. Hardy, “Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism,” J. Opt. Soc. Am. A |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: August 11, 2010

Revised Manuscript: September 30, 2010

Manuscript Accepted: October 10, 2010

Published: October 19, 2010

**Citation**

Kamal Muhieddine, Anatole Lupu, Eric Cassan, and Jean-Michel Lourtioz, "Proposal and analysis of narrow band transmission asymmetric directional couplers with Bragg grating induced phase matching," Opt. Express **18**, 23183-23195 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23183

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### References

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- C. Elachi and C. Yeh, “Frequency selective coupler for integrated optics systems,” Opt. Commun. 7(3), 201–204 (1973). [CrossRef]
- D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part I: Synchronous couplers,” J. Lightwave Technol. 5(1), 113–118 (1987). [CrossRef]
- D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: Grating-assisted couplers,” J. Lightwave Technol. 5(2), 268–273 (1987). [CrossRef]
- W.-P. Huang, “Coupled mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994). [CrossRef]
- R. C. Alferness and R. V. Schmidt, “Tunable optical waveguide directional coupler filter,” Appl. Phys. Lett. 33(2), 161–163 (1978). [CrossRef]
- R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, and C. A. Burrus, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” Appl. Phys. Lett. 60(26), 3209–3211 (1992). [CrossRef]
- P.-J. Rigole, S. Nilsson, L. Backbom, T. Klinga, J. Wallin, B. Stalnacke, E. Berglind, and B. Stoltz, “114-nm wavelength tuning range of a vertical grating assisted codirectional coupler laser with a super structure grating distributed Bragg reflector,” IEEE Photon. Technol. Lett. 7(7), 697–699 (1995). [CrossRef]
- C. Bornholdt, F. Kappe, R. Müller, H.-P. Nolting, F. Reier, R. Stenzel, H. Venghaus, and C. M. Weinert, “Meander coupler, a novel wavelength division multiplexer/demultiplexer,” Appl. Phys. Lett. 57(24), 2517–2519 (1990). [CrossRef]
- Y.-H. Jan, M. E. Heimbuch, L. A. Coldren, and S. P. DenBaars, “InP/InGaAsP grating-assisted codirectional coupler tunable receiver with a 30nm wavelength tuning range,” Electron. Lett. 32(18), 1697–1698 (1996). [CrossRef]
- A. Lupu, H. Sik, A. Mereuta, P. Boulet, M. Carré, S. Slempkes, A. Ougazzaden, and A. Carenco, “Three-waveguides two-grating codirectional coupler for 1.3-/1.3+/1.5µm demultiplexing in transceiver,” Electron. Lett. 36(24), 2030–2031 (2000). [CrossRef]
- A. Lupu, P. Win, H. Sik, P. Boulet, M. Carré, S. Slempkes, and A. Carenco, “Tunable filter with box like spectral response for 1.28/1.32 µm duplexer application,” Electron. Lett. 35(2), 174–175 (1999). [CrossRef]
- R. C. Alferness, “Optical directional couplers with weighted coupling,” Appl. Phys. Lett. 35, 109–126 (1978).
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- C. Wu, C. Rolland, N. Puetz, R. Bruce, K. D. Chik, and J. M. Xu, “A vertically coupled InGaAsP/InP directional coupler filter of ultranarrow bandwidth,” IEEE Photon. Technol. Lett. 3(6), 519–521 (1991). [CrossRef]
- B. Liu, A. Shakouri, P. Abraham, Y. J. Chiu, S. Zhang, and J. E. Bowers, “Fused InP–GaAs Vertical Coupler Filters,” IEEE Photon. Technol. Lett. 11, 93–95 (1995).
- S.-K. Han, R. V. Ramaswamy, and R. F. Tavlykaev, “Highly asymmetrical vertical coupler wavelength filter in InGaAlAs/InP,” Electron. Lett. 33, 31 (1999).
- B. R. West and A. S. Helmy, “Dispersion tailoring of the quarter-wave Bragg reflection waveguide,” Opt. Express 14(9), 4073–4086 (2006). [CrossRef] [PubMed]
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- M. Dainese, M. Swillo, L. Wosinski, and L. Thylen, “Directional coupler wavelength selective filter based on dispersive Bragg reflection waveguide,” Opt. Commun. 260(2), 514–521 (2006). [CrossRef]
- A. Ankiewicz and G.-D. Peng, “Narrow bandpass filter using Bragg grating coupler in transmission mode,” Electron. Lett. 33(25), 2151–2153 (1997). [CrossRef]
- A. Ankiewicz, Z. H. Wang, and G.-D. Peng, “Analysis of narrow bandpass filter using coupler with Bragg grating in transmission,” Opt. Commun. 156(1-3), 27–31 (1998). [CrossRef]
- P. Bienstman and R. Baets, “Optical Modelling of Photonic Crystals and VCSELs using Eigenmode Expansion and Perfectly Matched Layers,” Opt. Quantum Electron. 33(4/5), 327–341 (2001). [CrossRef]
- http://camfr.sourceforge.net .
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- M. S. Whalen, M. D. Divino, and R. C. Alferness, “Demonstration of a narrowband Bragg-reflection filter in a single-mode fibre directional coupler,” Electron. Lett. 22(12), 681–682 (1986). [CrossRef]
- R. R. A. Syms, “Optical directional coupler with a grating overlay,” Appl. Opt. 24(5), 717–726 (1985). [CrossRef] [PubMed]
- R. März and H. P. Nolting, “Spectral properties of asymmetrical optical directional couplers with periodic structures,” Opt. Quantum Electron. 19(5), 273–287 (1987). [CrossRef]
- L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. S. J. Russell, “Novel add–drop filters for wavelength division multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8(12), 1656–1658 (1996). [CrossRef]
- S. S. Orlov, A. Yariv, and S. Van Essen, “Coupled-mode analysis of fiber-optic add drop filters for dense wavelength-division multiplexing,” Opt. Lett. 22(10), 688–690 (1997). [CrossRef] [PubMed]
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- I. Baumann, J. Seifert, W. Nowak, and M. Sauer, “Compact all-fiber add-drop-multiplexer using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 8(10), 1331–1333 (1996). [CrossRef]
- J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19(3), 180–182 (1994). [CrossRef] [PubMed]
- A.-C. Jacob-Poulin, R. Valle’e, S. LaRochelle, D. Faucher, and G. R. Atkins, “Channel-dropping filter based on a grating-frustrated two-core fiber,” J. Lightwave Technol. 18(5), 715–720 (2000). [CrossRef]
- A. Yesayan and R. Vallée, “Optimized grating-frustrated coupler,” Opt. Lett. 26(17), 1329–1331 (2001). [CrossRef]
- A. Yesayan and R. Vallée, “Zero backreflection condition for a grating-frustrated coupler,” J. Opt. Soc. Am. B 20(7), 1418–1426 (2003). [CrossRef]
- N. Imoto, “An analysis for contradirectional-coupler-type optical grating filters,” J. Lightwave Technol. 3(4), 895–900 (1985). [CrossRef]
- J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11(4), 1307–1320 (1994). [CrossRef]
- A. Arraf, L. Poladian, C. M. de Sterke, and T. G. Brown, “Effective-medium approach for counterpropagating waves in nonuniform Bragg gratings,” J. Opt. Soc. Am. A 14(5), 1137–1143 (1997). [CrossRef]
- A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13(4), 233–253 (1977). [CrossRef]
- B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B 22(6), 1179–1190 (2005). [CrossRef]
- R. R. A. Syms, “Improved coupled mode theory for codirectionally and contradirectionally coupled waveguide arrays,” J. Opt. Soc. Am. A 8(7), 1062–1069 (1991). [CrossRef]
- J. Hong and W. P. Huang, “Contra-directional coupling in grating-assisted guided-wave devices,” J. Lightwave Technol. 10(7), 873–881 (1992). [CrossRef]
- A. Hardy, “A unified approach to coupled-mode phenomena,” IEEE J. Quantum Electron. 34(7), 1109–1116 (1998). [CrossRef]
- N. Izhaky and A. Hardy, “Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism,” J. Opt. Soc. Am. A 16(6), 1303–1311 (1999). [CrossRef]

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