## Far-field scattering microscopy applied to analysis of slow light, power enhancement, and delay times in uniform Bragg waveguide gratings

Optics Express, Vol. 15, Issue 4, pp. 1851-1870 (2007)

http://dx.doi.org/10.1364/OE.15.001851

Acrobat PDF (1246 KB)

### Abstract

A novel method is presented for determining the group index, intensity enhancement and delay times for waveguide gratings, based on (Rayleigh) scattering observations. This far-field scattering microscopy (FScM) method is compared with the phase shift method and a method that uses the transmission spectrum to quantify the slow wave properties. We find a minimum group velocity of 0.04c and a maximum intensity enhancement of ∼14.5 for a 1000-period grating and a maximum group delay of ∼80 ps for a 2000-period grating. Furthermore, we show that the FScM method can be used for both displaying the intensity distribution of the Bloch resonances and for investigating out of plane losses. Finally, an application is discussed for the slow-wave grating as index sensor able to detect a minimum cladding index change of 10^{-8}, assuming a transmission detection limit of 10^{-4}.

© 2007 Optical Society of America

## 1. Introduction

1. J. F. Lepage, R. Massudi, G. Anctil, S. Gilbert, M. Piche, and N. McCarthy, “Apodizing holographic gratings for the modal control of semiconductor lasers,” Appl. Opt. **36**,4993–4998 (1997). [CrossRef] [PubMed]

2. W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Tops. Quantum Electron. **11**,11–16 (2005). [CrossRef]

3. P. Madasamy, G. N. Conti, P. Poyhonen, Y. Hu, M. M. Morrell, D. F. Geraghty, S. Honkanen, and N. Peyghambarian, “Waveguide distributed Bragg reflector laser arrays in erbium doped glass made by dry Ag film ion exchange,” Opt. Eng. **41**,1084–1086 (2002). [CrossRef]

4. H. C. Wu, Z. M. Sheng, and J. Zhang, “Chirped pulse compression in nonuniform plasma Bragg gratings,” Appl. Phys. Lett. **87**,201502/1–3 (2005). [CrossRef]

5. D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic-bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B **18**,1326–1333 (2001). [CrossRef]

6. D. Faccio, F. Bragheri, and M. Cherchi, “Optical Bloch-mode-induced quasi phase matching of quadratic interactions in one-dimensional photonic crystals,” J. Opt. Soc. Am. B **21**,296–301 (2004). [CrossRef]

7. J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quantum Electron. **34**,455–470 (2002). [CrossRef]

*c*. A slow light wave has a longer interaction time with the material it is traveling through. This effect can be exploited for example for sensors [2

2. W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Tops. Quantum Electron. **11**,11–16 (2005). [CrossRef]

9. M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. **3**,211–219 (2004). [CrossRef] [PubMed]

10. V. I. Kopp, B. Fan, H. K. M. Vithana, and A. Z. Genack, “Low-threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals,” Opt. Lett. **23**,1707–1709 (1998). [CrossRef]

11. A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media **16**,293–382 (2006). [CrossRef]

13. W. C. L. Hopman, R. Dekker, D. Yudistira, W. F. A. Engbers, H. J. W. M. Hoekstra, and R. M. De Ridder, “Fabrication and characterization of high-quality uniform and apodized Si_{3}N_{4} waveguide gratings using laser interference lithography,” IEEE Photon. Technol. Lett. **18**,1855–1857 (2006). [CrossRef]

6. D. Faccio, F. Bragheri, and M. Cherchi, “Optical Bloch-mode-induced quasi phase matching of quadratic interactions in one-dimensional photonic crystals,” J. Opt. Soc. Am. B **21**,296–301 (2004). [CrossRef]

14. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**,4107–4121 (1996). [CrossRef]

7. J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quantum Electron. **34**,455–470 (2002). [CrossRef]

15. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys Rev Lett **94** (2005). [CrossRef] [PubMed]

16. C. E. Finlayson, F. Cattaneo, N. M. B. Perney, J. J. Baumberg, M. C. Netti, M. E. Zoorob, M. D. B. Charlton, and G. J. Parker, “Slow light and chromatic temporal dispersion in photonic crystal waveguides using femtosecond time of flight,” Phys. Rev. E **73**,016619/1–10 (2006). [CrossRef]

20. Y. A. Vlasov, M. O′Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature **438**,65–69 (2005). [CrossRef] [PubMed]

21. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. **87**,253902/1–4 (2001). [CrossRef] [PubMed]

22. X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor D′Yerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes,” Appl. Phys. Lett. **79**,2312–2314 (2001). [CrossRef]

13. W. C. L. Hopman, R. Dekker, D. Yudistira, W. F. A. Engbers, H. J. W. M. Hoekstra, and R. M. De Ridder, “Fabrication and characterization of high-quality uniform and apodized Si_{3}N_{4} waveguide gratings using laser interference lithography,” IEEE Photon. Technol. Lett. **18**,1855–1857 (2006). [CrossRef]

*Q*and also using the far-field scattering microscopy method. The modeling of the WGGs is presented in subsection 4.2 for verification of experimental results presented in subsection 4.3. In this subsection we will present our main experimental results, demonstrating

*Q*factors up to 10

^{5}, and showing how the “scatter-intensity” approach can be applied to slow-wave structures for estimating the corresponding group velocity. The results are verified with those obtained using the well-known RF phase shift method [17

17. R. S. Jacobsen, A. V. Lavrinenko, L. H. Frandsen, C. Peucheret, B. Zsigri, G. Moulin, J. Fage-Pedersen, and P. I. Borel, “Direct experimental and numerical determination of extremely high group indices in photonic crystal waveguides,” Opt. Express. **13**,7861–7871 (2005). [CrossRef] [PubMed]

23. K. Daikoku and A. Sugimura, “Direct measurement of wavelength dispersion in optical fibres-difference method,” Electron. Lett. **14**,149–151 (1978). [CrossRef]

26. W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in 1-D photonic crystal slabs,” Opt. Quantum Electron. **34**,195–203 (2002). [CrossRef]

27. R. Ferrini, R. Houdre, H. Benisty, M. Qiu, and J. Moosburger, “Radiation losses in planar photonic crystals: two-dimensional representation of hole depth and shape by an imaginary dielectric constant,” J. Opt. Soc. Am. B **20**,469–478 (2003). [CrossRef]

^{-8}). Finally we will present the main conclusions of this research in section 7.

## 2. Design and fabrication

_{3}N

_{4}guiding layer with refractive index

*n*= 1.981 and initial thickness

*d*= 275 nm, which was LPCVD-deposited on top of a 9 micrometer thick SiO

_{g}_{2}(

*n*= 1.445) buffer layer.

*t*= 5 nm was chosen for lateral confinement. The advantage of the resulting small lateral effective index contrast is that a 2D model provides sufficient accuracy for designing the grating and simulating its optical properties [7

7. J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quantum Electron. **34**,455–470 (2002). [CrossRef]

*w*= 7 μm could be chosen for the ridge waveguide to be still single moded. In order to have a stop band in the 1.5 μm wavelength region, we chose the period of the WGGs to be Λ = 0.47 μm. The gratings were defined using laser interference lithography. After reactive ion etching of the grating mask the etch depth was estimated to be

*d*= 50 nm (i.e. ∼19 % of the core-thickness), using both profilometry and fitting of the measurement data using a mode expansion technique in 2D. The gratings were characterized with both an air cladding (

_{e}*n*= 1) and a polymer cladding (

*n*≅ 1.5) on top. In this paper we consider three different lengths of the WGGs, determined by the number of periods: 500, 1000 and 2000 periods. Further details of the fabrication process of similar WGGs can be found in [13

13. W. C. L. Hopman, R. Dekker, D. Yudistira, W. F. A. Engbers, H. J. W. M. Hoekstra, and R. M. De Ridder, “Fabrication and characterization of high-quality uniform and apodized Si_{3}N_{4} waveguide gratings using laser interference lithography,” IEEE Photon. Technol. Lett. **18**,1855–1857 (2006). [CrossRef]

## 3. Measurement setup

28. S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express. **11**,2927–2939 (2003). [CrossRef] [PubMed]

_{3}N_{4} waveguide gratings using laser interference lithography,” IEEE Photon. Technol. Lett. **18**,1855–1857 (2006). [CrossRef]

29. M. Loncar, D. Nedeljkovic, T. P. Pearsall, J. Vuckovic, A. Scherer, S. Kuchinsky, and D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. **80**,1689–1691 (2002). [CrossRef]

30. D. J. W. Klunder, F. S. Tan, T. Van der Veen, H. F. Bulthuis, G. Sengo, B. Docter, H. J. W. M. Hoekstra, and A. Driessen, “Experimental and numerical study of SiON microresonators with air and polymer cladding,” J. Lightwave Technol. **21**,1099–1110 (2003). [CrossRef]

_{s}), we obtain “scatter images” for each selected wavelength. The wavelength dependent scattering intensities of the selected area (region of interest: ROI) could be obtained by image processing of the scatter data. A tunable laser (HP 8186c) in combination with a polarization controller was used to couple light into the TE

_{00}waveguide mode. The light source provided a wavelength range of 1470 nm to 1600 nm with a resolution of 1 pm. The transmission spectra of the waveguide gratings were normalized with respect to that of a reference waveguide without a grating, in order to cancel out any other wavelength dependencies in the optical path.

31. D. B. Hunter, M. E. Parker, and J. L. Dexter, “Demonstration of a continuously variable true-time delay beamformer using a multichannel chirped fiber grating,” IEEE Trans. Microwave Theory Tech. **54**,861–867 (2006). [CrossRef]

## 4. Resonances in waveguide gratings

### 4.1 Theory

*Q*factor, the scattered power, the free spectral range (FSR), and the group delay. In order to compare the methods experimentally, the first three parameters will be written as three separate expressions fro the delay times.

*ω*as the angular frequency and

*k*defined as the wave vector (2π/λ) of the optical wave. The group index is defined as

*N*=

_{g}*c*/

*v*. A resonance can be described by its quality factor

_{g}*Q*defined as,

*U*is the time averaged energy stored in a resonator at the resonant angular frequency

*ω*

_{0}, and (

*T*

_{0}

*dU*/

*dt*) is the energy loss in one period

*T*

_{0}(=2π/

*ω*

_{0}) of this frequency. Assuming an exponential decay of the resonator with decay time

*τ*=

_{c}*U*/(

*dU*/

*dt*), it follows

*Q*is by measuring the spectral transmission response of a resonator. The

*Q*factor can than be obtained using the expression:

*λ*

_{-3dB}as the full width at half maximum of the Lorentzian shaped spectral response of the resonator and with λ

_{0}as the resonance wavelength.

*Q*to the group velocity and to measurable scattered light, we will evaluate Eq. (2) using a power balance as illustrated by Fig. 3. To find a relation between the intensity at resonance and

*Q*and

*v*we first write the energy stored in the resonator as the product of the energy density

_{g}*W*with the modal volume of the resonator

_{r}*V*.

*A*and the cavity length

_{r}*L*. In a stationary state, the outgoing energy loss per period at the right hand side (

*S*) is exactly equal to the energy flux fed to the resonator [36

_{out}36. H. G. Winful, “The meaning of group delay in barrier tunnelling: A re-examination of superluminal group velocities,” New J. Phys. **8**,101/1–16 (2006). [CrossRef]

*S*). Therefore the time derivative of

_{sc}*U*can be written as,

*S*is the net energy flux density (power density) that is transferred to the resonator. We assume the scattering loss from the cavity to be relatively small, so that the scatter flux

_{i}*S*is small compared to the net intensity flux inside the resonator

_{sc}*S*. Then

_{r}*S*will be approximately constant and equal to

_{r}*S*, and its relation to

_{i}*v*is given by [11

_{g}11. A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media **16**,293–382 (2006). [CrossRef]

*Q*as a function of

*v*at the resonance wavelength, by using Eqs.(2), (5), (6) and (7),

_{g}*t*of the resonator is equal to its cavity decay time

_{d}*τ*, which was also derived in [36

_{c}36. H. G. Winful, “The meaning of group delay in barrier tunnelling: A re-examination of superluminal group velocities,” New J. Phys. **8**,101/1–16 (2006). [CrossRef]

*t*derived from the measured

_{q}*Q*[Eq. (4)] at the resonance wavelength as:

*S*, a lower group velocity implies a higher energy density

_{i}*W*. The ratio

_{r}*η*between the average energy in the cavity at two different wavelengths

*λ*and

*λ*, hence different group indices

_{ref}*N*and

_{g}*N*, can than be written as:

_{g,ref}*N*at a specific wavelength

_{g}*λ*, if both

*N*is known, and

_{g,ref}*η*is known at wavelength

*λ*. With the FScM approach we present a method for estimating the reference group index

*N*in order to measure

_{g,ref}*N*at wavelengths near the band edge. The scattered power

_{g}*I*(

_{sc}*λ*) is proportional to the probability that a photon is scattered, which is proportional to the number of photons in the resonator, hence to the total energy. This is true if the considered wavelength range is small (to limit the influence of the λ

^{-4}dependency of Rayleigh scattering), the number of scatterers is high and the distribution is random, which is the case in our WGGs as can be observed further in this paper, in Fig. 9(b). We assume that in the wavelength region of interest, the scatter efficiency is not strongly wavelength dependent. By measuring a reference scatter intensity

*I*(

_{sc,ref}*λ*) at a wavelength where a value for

_{ref}*N*is known (a method will be explained in section 4.2), an estimate for

_{g}*N*(

_{g}*λ*) can be found:

*t*can be calculated from the measured scatter ratio

_{sc}*η*by substituting (13) into (10):

*Q*from the scatter ratio (or visa versa) at the resonance wavelength

*λ*

_{0}, by substituting

*t*from (14) for

_{sc}*t*in (11):

_{q}*Q*, and calculated from measured scatter enhancement

*η*:

*t*≈

_{d}*t*≈

_{q}*t*.

_{sc}21. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. **87**,253902/1–4 (2001). [CrossRef] [PubMed]

22. X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor D′Yerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes,” Appl. Phys. Lett. **79**,2312–2314 (2001). [CrossRef]

*t*can be found.

_{FSR}*ω*—

*k*) dispersion curve for a small wavelength range close to the photonic band edge. For finite structures however, this curve never becomes completely flat, i.e. the group velocity will not approach to zero. The minimum

*v*can be calculated using the complex optical transfer function

_{g}*t*(

*ω*) of the WGG [14

14. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**,4107–4121 (1996). [CrossRef]

*ω*/d

*φ*, with

*φ*defined as the total phase

*kL*, with

*L*the device length which equals the number of periods

*N*times the period

*Λ*. Using

*φ*= tan

^{-1}(

*y/x*), the group velocity can be calculated from the (computed) real and imaginary parts of the transfer function, as follows [14

14. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**,4107–4121 (1996). [CrossRef]

### 4.2 Method & simulations

37. Olympios, “OlympIOs Integrated Optics Software,” C2V, http://www.c2v.nl/software/.

*T*(

*ω*) for TE polarization. The group index was derived as a function of the wavelength using Eq. (19), and the transmission of the WGG (

*T*) is derived from the absolute value of the complex transfer function, see Fig. 4. We chose a WGG having a total number of periods of 350 instead of the fabricated 500 or more, for computing time/resources convenience. The group index shows two major peaks near the stopband edges of almost equal height/value. A maximum value for

_{g}*N*= 7.3 at the smaller wavelength band edge is derived for this grating. Moving away from the band edge, the group index rapidly decreases to a value of ∼1.86 (at 1560 nm) approximately equal to the

_{g}*N*of the unperturbed ridge waveguide. This effect is used to define the reference group index (

_{g}*N*). At a wavelength around for example 1560 or 1510 nm, we can determine the scatter intensity, which we then relate to the reference scattering intensity

_{g,ref}*I*. However this approach is not used in this paper; we take the average of the oscillation present around these reference wavelengths. A value for the group index of the WGG can then be calculated using Eq. (13

_{sc,ref}_{3}N_{4} waveguide gratings using laser interference lithography,” IEEE Photon. Technol. Lett. **18**,1855–1857 (2006). [CrossRef]

*T*(

*λ*) and

*N*(

_{g}*λ*) curves [7

**34**,455–470 (2002). [CrossRef]

*L*the grating may act as an apodized grating. We found in our measurements (section 4.3) that for a maximum number of periods of 2000 (

*L*∼ 1 mm), the WGG still showed the clearly separated fringes as seen in the simulations.

*N*of a WGG, we can also find the longitudinal intra-cavity grating resonance patterns using the BEP method by modeling the cross-section of a WGG, see Fig. 5. These resonances are also verified experimentally in section 4.3 At the maximum of each fringe we can find a sinusoidal field distribution along the length of the grating, with nodes at its beginning and end; a simple mathematical derivation for this phenomenon can for example be found in Ref. [11

_{g}11. A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media **16**,293–382 (2006). [CrossRef]

_{3}N

^{4}slab. In the feeding waveguide (left) we see a small ripple indicating a small residual reflection. At the first minimum in the transmission curve, denoted by ‘Resonance I

^{+}’ [Fig. 5(b)], a strong reflection is evidenced by the large amplitude of the standing wave in the input waveguide. At the start of the grating, we now find a maximum and inside the grating we find both a minimum and a maximum. On the output of the grating (right) we observe a lowered value of the transmitted field, which is consistent with the calculated transmission shown in Fig. 4. Increasing the wavelength further, away from the stop band, we find the higher order resonances. Figures 5(c) and 5(d) show the higher order resonances II and III. The maximum H-field amplitude decreases with the resonance order, which is expected from Fig. 4, which shows equivalently a decrease in group index. This decrease in modal-amplitude can be explained by the decrease in reflectivity from the grating for wavelengths further away from the Bragg condition.

*Q*and the grating length has been found using coupled mode equations for a waveguide grating suitable for second harmonic generation [5

5. D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic-bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B **18**,1326–1333 (2001). [CrossRef]

*N*and

_{g}*L*. We have also verified this relation by rigorous BEP modeling [Fig. 7(b)]. For gratings shorter than 200-periods, we have estimated the

*Q*from the left half side of the resonance I, taking Δ

*λ*

_{3dB}as 2 times the value found left from the resonance wavelength at the -3dB point.

*N*is expected for the same device length. In Fig. 8 we have plotted the transmission as a function of the refractive index of the cladding and the wavelength for a 200-period WGG. The figure clearly shows how the stop band shifts with cladding index. As expected the stopband shifts to larger wavelengths when the refractive index of the cladding is increased.

_{g}### 4.3 Experimental

*Q*s of resonances I and −I (Fig. 4) are ∼8500 and 6000, respectively.

*I*) integrated over a vertical line in the dotted box shown in B as a function of the distance (horizontal in the box). Although the camera images cannot resolve the sub-wavelength details of the field distribution in the grating area, they clearly show the envelope of the intensity distribution of the Fabry-Perot-like resonances of the Bloch modes in the WGG. By integrating

_{sc}*I*over the area enclosed by the dotted box, it is possible to derive the

_{sc}*I*spectrum shown in Fig. 9(b).

_{sc}*η*) we obtain is about 7.4 taking the scattered intensity at 1514 nm as a reference. Using Eq. (13) and taking the reference index

*N*equal to the group index of the undisturbed ridge waveguide (

_{g,ref}*N*≈ 1.9) a value of

_{g,ref}*N*≈ 7.4×

_{g}*N*= 14 can be calculated. This corresponds to a group velocity of ∼0.07

_{g,ref}*c*. The estimated value of

*N*using Eq. (9) with a

_{g}*Q*of 8500 is about 8.8. The deviation between both numbers could be explained from the definition of

*Q*given in Eq. (4). The shape of the fringes near the stop band is not strictly Lorentzian, because the reflectance changes rapidly with wavelength. Therefore,

*Q*derived from the spectrum may be different from the actual

*Q*value defined by Eq. (2) at the resonance wavelength.

*Q*s are more accurately determined using Eq. (9) for the longer gratings. Also other factors may contribute to small differences, for example the presence of slab-light and/or noise by the input and output spot of the WGG (see Fig. 9(c) image D), to which the FScM-method is sensitive. For longer WGGs the influence of both contributions is lower, which results in a better signal-to-noise ratio and a better agreement between both methods as we will show. The group delay can now also be calculated using Eq. (11) and Eq. (14), resulting in values

*t*≈ 7 ps and

_{q}*t*≈ 11 ps.

_{s}*N*can be expected when the length of the selected WGG is increased [Fig. 7(b)] to 1000 periods. The measured transmission curve of this grating is shown in Fig. 10(a). The fringes in the airband are separately observable, whereas the fringes in the dielectric band show overlap near the edge, which is caused by fabrication errors inducing apodization, see Ref. [13

_{g}_{3}N_{4} waveguide gratings using laser interference lithography,” IEEE Photon. Technol. Lett. **18**,1855–1857 (2006). [CrossRef]

*Q*of 46000. Figure 10(b) shows the corresponding spectrum of the scattered intensity, from which a large

*N*≈ 24 is deduced indeed at this high

_{g}*Q*resonance, using Eq. (9). The noise level (due to slab light) in this particular grating was higher than in the other 2 gratings; therefore the background noise was subtracted from the scattered intensity collected from the grating area by selecting an equal sized area as reference next to the ridge waveguide at approximately 20 micrometer below the waveguide. From this scatter intensity enhancement we estimate

*η*to be about 12. Using Eq. (13) we then find

*N*≈ 27, corresponding to a group velocity of 0.04

_{g}*c*. For the corresponding group delays at the resonance wavelength (

*λ*= 1525.52 nm) calculated using the measured

*Q*and

*η*, we find

*t*≈ 38 ps and

_{q}*t*≈ 42 ps, respectively. Both methods agree quite well on the group index and consequently on the group delay at resonance −I (see Fig. 4).

_{sc}*t*≈ 50 ps is found, see Fig. 11. The difference between

_{d}*t*,

_{q}*t*and

_{sc}*t*can be explained from the inaccuracy in each method.

_{d}*X*,

*Y*and

*Z*,

*Q*-values of approximately 50 000, 70 000 and 110 000 respectively were found. With increasing

*Q*the transmission drops due to the longer lifetime of the resonant photons, which causes a larger fraction of the light to scatter out of the WGG. The spectrum of the scattered intensity is shown in Fig. 13(a). We find a maximum

*N*≈ 25 using Eq. (13), which results in a maximum group delay of

_{g}*t*≈ 78 ps [Eq. (14)]. For the maximum group delay calculated using the maximum

_{sc}*Q*of 110 000, a maximum

*t*≈ 88 ps is found. The estimated group delays using both indirect methods match within 13%. The group delay calculated using the phase shift method is shown in Fig. 13(b). For resonance −I a maximum delay

_{q}*t*≈ 75 ps is found. Although a larger

_{d}*N*should be expected from a longer grating, the maximum

_{g}*N*was found for the 1000-period grating, rather than for one with 2000 periods. Inaccuracies in the fabrication process limit the maximum achievable spatial coherence in the grating, which puts a limit on the maximum achievable

_{g}*Q*.

*Q*-values and plotted them as a function of

*η*, see Fig. 14(a). The graph shows a linear relationship which agrees with Eq. (15).

*Q*and

*η*show a fairly good agreement with the directly measured group delay. From these results we can conclude that both indirect methods can be used to determine the group delay taking into account the accuracy of the methods.

*N*can be determined easily over a broad spectrum, by cancelling the noise contribution by slab light in the scatter experiments. On the other hand, the FScM method provides

_{g}*N*also for wavelengths out of resonance. Figure 14(b) clearly shows that the FSR method [Eq. (17)] shows the strongest deviations from the values measured using the phase shift method, especially close to the band edge (at a

_{g}*t*of 75 ps we find a

_{d}*t*of ∼40 ps). A possible explanation is the invalidity of the assumption of linear dispersion that is made to derive this equation (linear Taylor expansion), whereas in reality the dispersion curve is approximately quadratic near the stop band edge (

_{FSR}*ω*versus

*β*).

## 5. Out of plane scattering

### 5.1 Simulations

39. M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express. **13**,7145–7159 (2005). [CrossRef] [PubMed]

40. D. Yudistira, H. Hoekstra, M. Hammer, and D. Marpaung, “Slow light excitation in tapered 1D photonic crystals: Theory,” Opt. Quantum Electron. **38**,161–176 (2006). [CrossRef]

*T*), reflection (

*R*) and loss curves. The out of plane loss is calculated for each wavelength using the relation OPS = 1 −

*T*−

*R*. The loss-curve shows interesting behavior. It has a global minimum near the dielectric band edge, where the reflection reaches its maximum; for shorter wavelengths (in the bandgap) the loss increases and reaches a global maximum close to the air band edge. The loss maxima coincide with the transmission maxima in the dielectric band, whereas in the airband they coincide with the reflection maxima, a possible explanation will be given in the section below.

### 5.2 Experimental

## 6. Sensor application

*Q*gratings presented in this paper. Among many other applications, the relatively small, simple and easy to fabricate WGGs are suited for integration into sensor arrays. Their relatively small dimensions compared to linear phase based sensors, such as Mach-Zehnder Interferometers [41

41. P. V. Lambeck, “Integrated optical sensors for the chemical domain,” Meas. Sci. Technol. **17**,R93–R116 (2006). [CrossRef]

2. W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Tops. Quantum Electron. **11**,11–16 (2005). [CrossRef]

*T*

_{0}= 10

^{-4}(determined by the quality peripheral equipment and the number of samples taken) and the following approximation:

^{-12}1 can be estimated for ∂

*λ*/∂

*T*using the transmission curve in Fig. 12 (the 2000-period WGG, resonance

*Z*). For ∂

*n*/∂

*λ*a value of 7.8×10

^{6}m

^{-1}(0.5 / 64 nm) is derived from the stopband shift with and polymer cladding, see also Fig. 8. Inserting these numbers in Eq. (20), we find a minimum detectable cladding index change Δ

*n*∼ 1×10

_{cladding}^{-8}. Such a grating sensor performs well compared to the state of the art MZI sensors (Δ

*n*∼ 10

_{cladding}^{-8}) [41

41. P. V. Lambeck, “Integrated optical sensors for the chemical domain,” Meas. Sci. Technol. **17**,R93–R116 (2006). [CrossRef]

## 7. Conclusion

*c*. The 1000-period WGG showed a

*v*as low as 0.04

_{g}*c*. For the 2000-period grating we measured a maximum

*Q*of 110.000 and a maximum delay of 75 ps. The minimum

*v*for this grating was estimated to be approximately equal to the value found for the 1000-period WGG: 0.04

_{g}*c*. The reason for this are the errors in fabrication.

*Q*” WGG as compact sensor, and calculated the minimum detectable change in cladding index of 1×10

^{-8}

## Acknowledgments

## References and links

1. | J. F. Lepage, R. Massudi, G. Anctil, S. Gilbert, M. Piche, and N. McCarthy, “Apodizing holographic gratings for the modal control of semiconductor lasers,” Appl. Opt. |

2. | W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Tops. Quantum Electron. |

3. | P. Madasamy, G. N. Conti, P. Poyhonen, Y. Hu, M. M. Morrell, D. F. Geraghty, S. Honkanen, and N. Peyghambarian, “Waveguide distributed Bragg reflector laser arrays in erbium doped glass made by dry Ag film ion exchange,” Opt. Eng. |

4. | H. C. Wu, Z. M. Sheng, and J. Zhang, “Chirped pulse compression in nonuniform plasma Bragg gratings,” Appl. Phys. Lett. |

5. | D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic-bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B |

6. | D. Faccio, F. Bragheri, and M. Cherchi, “Optical Bloch-mode-induced quasi phase matching of quadratic interactions in one-dimensional photonic crystals,” J. Opt. Soc. Am. B |

7. | J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quantum Electron. |

8. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

9. | M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. |

10. | V. I. Kopp, B. Fan, H. K. M. Vithana, and A. Z. Genack, “Low-threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals,” Opt. Lett. |

11. | A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media |

12. | H. J. W. M. Hoekstra, W. C. L. Hopman, J. Kautz, R. Dekker, and R. M. de Ridder, “A simple coupled mode model for near band-edge phenomena in grated waveguides,” accepted for publication in Opt. Quantum Electron. (2006). |

13. | W. C. L. Hopman, R. Dekker, D. Yudistira, W. F. A. Engbers, H. J. W. M. Hoekstra, and R. M. De Ridder, “Fabrication and characterization of high-quality uniform and apodized Si |

14. | J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E |

15. | H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys Rev Lett |

16. | C. E. Finlayson, F. Cattaneo, N. M. B. Perney, J. J. Baumberg, M. C. Netti, M. E. Zoorob, M. D. B. Charlton, and G. J. Parker, “Slow light and chromatic temporal dispersion in photonic crystal waveguides using femtosecond time of flight,” Phys. Rev. E |

17. | R. S. Jacobsen, A. V. Lavrinenko, L. H. Frandsen, C. Peucheret, B. Zsigri, G. Moulin, J. Fage-Pedersen, and P. I. Borel, “Direct experimental and numerical determination of extremely high group indices in photonic crystal waveguides,” Opt. Express. |

18. | M. C. Netti, C. E. Finlayson, J. J. Baumberg, M. D. B. Charlton, M. E. Zoorob, J. S. Wilkinson, and G. J. Parker, “Separation of photonic crystal waveguides modes using femtosecond time-of-flight,” Appl. Phys. Lett. |

19. | Y. A. Vlasov, S. Petit, G. Klein, B. Honerlage, and C. Hirlimann, “Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal,” Phys. Rev. E |

20. | Y. A. Vlasov, M. O′Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature |

21. | M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. |

22. | X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor D′Yerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes,” Appl. Phys. Lett. |

23. | K. Daikoku and A. Sugimura, “Direct measurement of wavelength dispersion in optical fibres-difference method,” Electron. Lett. |

24. | K. Hosomi, T. Fukamachi, T. Katsuyama, and Y. Arakawa, “Group delay of a coupled-defect waveguide in a photonic crystal,” Opt. Rev. |

25. | S. Ryu, Y. Horiuchi, and K. Mochizuki, “Novel chromatic dispersion measurement method over continuous Gigahertz tuning range,” J. Lightwave Technol. |

26. | W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in 1-D photonic crystal slabs,” Opt. Quantum Electron. |

27. | R. Ferrini, R. Houdre, H. Benisty, M. Qiu, and J. Moosburger, “Radiation losses in planar photonic crystals: two-dimensional representation of hole depth and shape by an imaginary dielectric constant,” J. Opt. Soc. Am. B |

28. | S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express. |

29. | M. Loncar, D. Nedeljkovic, T. P. Pearsall, J. Vuckovic, A. Scherer, S. Kuchinsky, and D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. |

30. | D. J. W. Klunder, F. S. Tan, T. Van der Veen, H. F. Bulthuis, G. Sengo, B. Docter, H. J. W. M. Hoekstra, and A. Driessen, “Experimental and numerical study of SiON microresonators with air and polymer cladding,” J. Lightwave Technol. |

31. | D. B. Hunter, M. E. Parker, and J. L. Dexter, “Demonstration of a continuously variable true-time delay beamformer using a multichannel chirped fiber grating,” IEEE Trans. Microwave Theory Tech. |

32. | J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase-locked electron-beam lithography,” J. Vac. Sci. Technol., B |

33. | J. F. Lepage and N. McCarthy, “Analysis of the diffractional properties of dual-period apodizing gratings: theoretical and experimental results,” Appl. Opt. |

34. | D. Wiesmann, C. David, R. Germann, D. Emi, and G. L. Bona, “Apodized surface-corrugated gratings with varying duty cycles,” IEEE Photon. Technol. Lett. |

35. | L. Xuhui, C. Xiangfei, Y. Yuzhe, and X. Shizhong, “A novel apodization technique of variable duty cycle for sampled grating,” Opt. Commun. |

36. | H. G. Winful, “The meaning of group delay in barrier tunnelling: A re-examination of superluminal group velocities,” New J. Phys. |

37. | Olympios, “OlympIOs Integrated Optics Software,” C2V, http://www.c2v.nl/software/. |

38. | C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B |

39. | M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express. |

40. | D. Yudistira, H. Hoekstra, M. Hammer, and D. Marpaung, “Slow light excitation in tapered 1D photonic crystals: Theory,” Opt. Quantum Electron. |

41. | P. V. Lambeck, “Integrated optical sensors for the chemical domain,” Meas. Sci. Technol. |

**OCIS Codes**

(120.5820) Instrumentation, measurement, and metrology : Scattering measurements

(120.7000) Instrumentation, measurement, and metrology : Transmission

(130.3120) Integrated optics : Integrated optics devices

(230.3990) Optical devices : Micro-optical devices

(230.5750) Optical devices : Resonators

(290.5820) Scattering : Scattering measurements

**ToC Category:**

Slow Light

**History**

Original Manuscript: December 20, 2006

Revised Manuscript: February 8, 2007

Manuscript Accepted: February 9, 2007

Published: February 19, 2007

**Virtual Issues**

Vol. 2, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

W. C. L. Hopman, H. J. W. M. Hoekstra, R. Dekker, L. Zhuang, and R. M. de Ridder, "Far-field scattering microscopy applied to analysis of slow light, power enhancement, and delay times in uniform Bragg waveguide gratings," Opt. Express **15**, 1851-1870 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-4-1851

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### References

- J. F. Lepage, R. Massudi, G. Anctil, S. Gilbert, M. Piche, and N. McCarthy, "Apodizing holographic gratings for the modal control of semiconductor lasers," Appl. Opt. 36, 4993-4998 (1997). [CrossRef] [PubMed]
- W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. Hoekstra, and R. M. de Ridder, "Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors," IEEE J. Sel. Tops. Quantum Electron. 11, 11-16 (2005). [CrossRef]
- P. Madasamy, G. N. Conti, P. Poyhonen, Y. Hu, M. M. Morrell, D. F. Geraghty, S. Honkanen, and N. Peyghambarian, "Waveguide distributed Bragg reflector laser arrays in erbium doped glass made by dry Ag film ion exchange," Opt. Eng. 41, 1084-1086 (2002). [CrossRef]
- H. C. Wu, Z. M. Sheng, and J. Zhang, "Chirped pulse compression in nonuniform plasma Bragg gratings," Appl. Phys. Lett. 87, 201502/1-3 (2005). [CrossRef]
- D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, "Photonic-bandgap structures in planar nonlinear waveguides: application to second-harmonic generation," J. Opt. Soc. Am. B 18, 1326-1333 (2001). [CrossRef]
- D. Faccio, F. Bragheri, and M. Cherchi, "Optical Bloch-mode-induced quasi phase matching of quadratic interactions in one-dimensional photonic crystals," J. Opt. Soc. Am. B 21, 296-301 (2004). [CrossRef]
- J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, "Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task," Opt. Quantum Electron. 34, 455-470 (2002). [CrossRef]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: Molding the flow of light (Princeton University Press, 1995).
- M. Soljacic, and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nat. Mater. 3, 211-219 (2004). [CrossRef] [PubMed]
- V. I. Kopp, B. Fan, H. K. M. Vithana, and A. Z. Genack, "Low-threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals," Opt. Lett. 23, 1707-1709 (1998). [CrossRef]
- A. Figotin, and I. Vitebskiy, "Slow light in photonic crystals," Waves Random Complex Media 16, 293-382 (2006). [CrossRef]
- H. J. W. M. Hoekstra, W. C. L. Hopman, J. Kautz, R. Dekker, and R. M. de Ridder, "A simple coupled mode model for near band-edge phenomena in grated waveguides," accepted for publication in Opt. Quantum Electron. (2006).
- W. C. L. Hopman, R. Dekker, D. Yudistira, W. F. A. Engbers, H. J. W. M. Hoekstra, and R. M. De Ridder, "Fabrication and characterization of high-quality uniform and apodized Si3N4 waveguide gratings using laser interference lithography," IEEE Photon. Technol. Lett. 18, 1855-1857 (2006). [CrossRef]
- J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996). [CrossRef]
- H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett 94 073903 (2005). [CrossRef] [PubMed]
- C. E. Finlayson, F. Cattaneo, N. M. B. Perney, J. J. Baumberg, M. C. Netti, M. E. Zoorob, M. D. B. Charlton, and G. J. Parker, "Slow light and chromatic temporal dispersion in photonic crystal waveguides using femtosecond time of flight," Phys. Rev. E 73, 016619/1-10 (2006). [CrossRef]
- R. S. Jacobsen, A. V. Lavrinenko, L. H. Frandsen, C. Peucheret, B. Zsigri, G. Moulin, J. Fage-Pedersen, and P. I. Borel, "Direct experimental and numerical determination of extremely high group indices in photonic crystal waveguides," Opt. Express. 13, 7861-7871 (2005). [CrossRef] [PubMed]
- M. C. Netti, C. E. Finlayson, J. J. Baumberg, M. D. B. Charlton, M. E. Zoorob, J. S. Wilkinson, and G. J. Parker, "Separation of photonic crystal waveguides modes using femtosecond time-of-flight," Appl. Phys. Lett. 81, 3927-3929 (2002). [CrossRef]
- Y. A. Vlasov, S. Petit, G. Klein, B. Honerlage, and C. Hirlimann, "Femtosecond measurements of the time of flight of photons in a three-dimensional photonic crystal," Phys. Rev. E 60, 1030-1035 (1999). [CrossRef]
- Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005). [CrossRef] [PubMed]
- M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902/1-4 (2001). [CrossRef] [PubMed]
- X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor D'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001). [CrossRef]
- K. Daikoku, and A. Sugimura, "Direct measurement of wavelength dispersion in optical fibres-difference method," Electron. Lett. 14, 149-151 (1978). [CrossRef]
- K. Hosomi, T. Fukamachi, T. Katsuyama, and Y. Arakawa, "Group delay of a coupled-defect waveguide in a photonic crystal," Opt. Rev. 11, 300-302 (2004). [CrossRef]
- S. Ryu, Y. Horiuchi, and K. Mochizuki, "Novel chromatic dispersion measurement method over continuous Gigahertz tuning range," J. Lightwave Technol. 7, 1177-1180 (1989). [CrossRef]
- W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, "Out-of-plane scattering in 1-D photonic crystal slabs," Opt. Quantum Electron. 34, 195-203 (2002). [CrossRef]
- R. Ferrini, R. Houdre, H. Benisty, M. Qiu, and J. Moosburger, "Radiation losses in planar photonic crystals: two-dimensional representation of hole depth and shape by an imaginary dielectric constant," J. Opt. Soc. Am. B 20, 469-478 (2003). [CrossRef]
- S. J. McNab, N. Moll, and Y. A. Vlasov, "Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides," Opt. Express. 11, 2927-2939 (2003). [CrossRef] [PubMed]
- M. Loncar, D. Nedeljkovic, T. P. Pearsall, J. Vuckovic, A. Scherer, S. Kuchinsky, and D. C. Allan, "Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides," Appl. Phys. Lett. 80, 1689-1691 (2002). [CrossRef]
- D. J. W. Klunder, F. S. Tan, T. Van der Veen, H. F. Bulthuis, G. Sengo, B. Docter, H. J. W. M. Hoekstra, and A. Driessen, "Experimental and numerical study of SiON microresonators with air and polymer cladding," J. Lightwave Technol. 21, 1099-1110 (2003). [CrossRef]
- D. B. Hunter, M. E. Parker, and J. L. Dexter, "Demonstration of a continuously variable true-time delay beamformer using a multichannel chirped fiber grating," IEEE Trans. Microwave Theory Tech. 54, 861-867 (2006). [CrossRef]
- J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, "Optical waveguides with apodized sidewall gratings via spatial-phase-locked electron-beam lithography," J. Vac. Sci. Technol. B 20, 2753-2757 (2002). [CrossRef]
- J. F. Lepage, and N. McCarthy, "Analysis of the diffractional properties of dual-period apodizing gratings: theoretical and experimental results," Appl. Opt. 43, 3504-3512 (2004). [CrossRef] [PubMed]
- D. Wiesmann, C. David, R. Germann, D. Emi, and G. L. Bona, "Apodized surface-corrugated gratings with varying duty cycles," IEEE Photon. Technol. Lett. 12, 639-641 (2000). [CrossRef]
- L. Xuhui, C. Xiangfei, Y. Yuzhe, and X. Shizhong, "A novel apodization technique of variable duty cycle for sampled grating," Opt. Commun. 225, 301-305 (2003). [CrossRef]
- H. G. Winful, "The meaning of group delay in barrier tunnelling: A re-examination of superluminal group velocities," New J. Phys. 8, 101/1-16 (2006). [CrossRef]
- Olympios, "OlympIOs Integrated Optics Software," C2V, http://www.c2v.nl/software/.
- C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, "Conversion efficiency for second-harmonic generation in photonic crystals," J. Opt. Soc. Am. B 18, 348-351 (2001). [CrossRef]
- M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, "Slow-light, band-edge waveguides for tunable time delays," Opt. Express. 13, 7145-7159 (2005). [CrossRef] [PubMed]
- D. Yudistira, H. Hoekstra, M. Hammer, and D. Marpaung, "Slow light excitation in tapered 1D photonic crystals: Theory," Opt. Quantum Electron. 38, 161-176 (2006). [CrossRef]
- P. V. Lambeck, "Integrated optical sensors for the chemical domain," Meas. Sci. Technol. 17, R93-R116 (2006). [CrossRef]

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