## 1D periodic structures for slow-wave induced non-linearity enhancement

Optics Express, Vol. 16, Issue 5, pp. 3146-3160 (2008)

http://dx.doi.org/10.1364/OE.16.003146

Acrobat PDF (625 KB)

### Abstract

A comparison between several 1D periodic structures designed to enhance non-linear effects for high-speed all-optical applications is presented. These structures allow for a small group velocity of the propagating waves, so the light-matter interaction is increased, making the non-linear process to be more efficient. In addition, the propagating wave is compressed, making the field intensity to be higher in the non-linear material. Thus, a significant reduction in both the structure length and the input power needed to induce a particular phase shift is achieved. The selected 1D periodic structures are compared by means of properties such as the modal effective volume, coupling efficiency, mode bandwidth, group velocity dispersion, and easiness of fabrication, in order to determine the optimum configuration in terms of non-linear enhancement.

© 2008 Optical Society of America

## 1. Introduction

*v*) can enhance the efficiency of non-linear processes in optical materials [1

_{g}1. N. A. R. Bhat and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E **64**, 056604 (2001). [CrossRef]

*v*results in higher electric field intensities for the same total optical power. The improvement in the refraction index change induced by a third-order non-linearity has been quantified as (

_{g}*v*/

_{g,WG}*v*)

_{g,SW}^{2}, where

*v*and

_{g,WG}*v*are the group velocities of a conventional dielectric optical waveguide (no slow-wave structure is introduced) and a slow-wave optical waveguide, respectively [2

_{g,SW}2. Marin Soljacic, Steven G. Johnson, Shanhui Fan, Mihai Ibanescu, Erich Ippen, and J.D. Joannopoulos, “Photonic crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B **19**, 2052–2059 (2002). [CrossRef]

3. 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 (2001). [CrossRef] [PubMed]

*v*values (theoretically zero just at the band edge). In this work, different types of one-dimensional (1D) periodic structures created by inserting a strong periodic modulation in a high-index material with a third-order non-linearity are compared. Three-dimensional periodic structures are not considered because our interest is centered in planar photonic structures, whose fabrication is quite simple; two-dimensional periodic structures are not considered either because periodicity is only required in one direction to achieve the slow-wave behavior. Moreover, 1D periodic structures are expected to be easier to design and fabricate than 2D structures.

_{g}*v*giving rise to the enhancement of non-linear effects. Since values of

_{g}*v*as low as desired can be achieved for any periodic structure just by getting closer to the 1

_{g}^{st}BZ edge, other parameters of the guided mode are obtained and analyzed in order to make a comparison between the structures and choose the most suitable for non-linear applications. The application of these 1D periodic structures in a functional device (a XOR all-optical logic gate) is also commented at the end of this text.

## 2. Benefits of low group velocity in non-linear applications

- for a given power flow, the energy is temporarily compressed and the field amplitude is increased as the group velocity decreases,
- longer time is necessary for an electromagnetic wave to travel through a given length if the group velocity is small.

*v*/

_{g,WG}*v*)

_{g,SW}^{2}when a slow-wave structure is used [2

2. Marin Soljacic, Steven G. Johnson, Shanhui Fan, Mihai Ibanescu, Erich Ippen, and J.D. Joannopoulos, “Photonic crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B **19**, 2052–2059 (2002). [CrossRef]

*n*when an optical signal propagates along the structure. If we consider an optical waveguide structure of length

*L*made of this material, the propagating signal will experiment an additional non-linear phase shift Δ

*ϕ*that can be well approximated by the following expression [2

2. Marin Soljacic, Steven G. Johnson, Shanhui Fan, Mihai Ibanescu, Erich Ippen, and J.D. Joannopoulos, “Photonic crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B **19**, 2052–2059 (2002). [CrossRef]

*σ*is the fraction of energy of the guided mode inside the non-linear material. Eq. (1) indicates that the non-linear phase shift depends on the quantity of the field that is confined in the non-linear material of the structure and is inversely proportional to the group velocity of the guided mode. In addition, since Δ

*n*is caused by a Kerr effect Δ

*n*=

*n*

_{2}

*I*, where

*n*

_{2}is the Kerr coefficient and

*I*is the optical intensity, which is proportional to |

*E*|

^{2}. Due to the energy temporal compression in the slow-wave structure, it can be predicted that |

*E*|

^{2}is inversely proportional to

*v*[2

_{g}**19**, 2052–2059 (2002). [CrossRef]

*v*. So, the phase shift induced by a slow-wave section is (

_{g}*v*/

_{g,WG}*v*)

_{g,SW}^{2}higher than that of a conventional waveguide:

- a reduction of the total length of the structure needed to achieve a particular phase shift. This is mandatory to achieve large scale integration of photonic devices at low cost,
- the total power required to achieve a particular phase shift can be significantly reduced due to the field enhancement in the slow-wave structures, which results in power saving.

## 3. Criteria for the comparison between structures

_{g}is selected and other parameters/properties of the guided modes are compared.

**normalized effective volume parameter,**(effective volume per period): This parameter characterise how the energy of the mode is concentrated in the non-linear region. The lower*V*_{eff,n}*V*value, the higher the concentration of the energy in the non-linear region. This parameter is an equivalent of the effective area [4] for structures with discrete translational symmetry (the effective area is only valid for structures with continuous translational symmetry)._{eff,n}*V*is numerically calculated from the field profile of the guided mode as:_{eff,n}The optical intensity in the waveguide can be expressed as*I*=*P*/*V*, where_{eff,n}*P*is the optical power, so the induced index change Δ*n*will be inversely proportional to*V*._{eff,n}- Fig. 1.Coupling efficiency between structures can be increased by adiabatically changing the parameters of the structure. The structure shown in the example is proposed in [5].
**coupling efficiency:**Since the induced non-linearity is proportional to the optical power in this region, the more efficiently the light is coupled into the slow-waveguide, the less power will be required at the input of the structure in order to induce the same index change. Group index*n*of the guided mode in the periodic structure is very high when working close to the band edge, what provokes a great mode mismatch with external waveguides and increases coupling losses. The coupling efficiency for analyzed structures is estimated by means of 2D-FDTD calculations using an effective index for the material of the waveguide to take into account the vertical confinement. Since mode mismatch between access and slow-wave structures is mainly due to in-plane geometry variations, results obtained from 2D calculations are equivalent to those for 3D structures. Coupling techniques such as tapering or adiabatic transitions are evaluated for some structures in order to increase coupling efficiency, as it can be seen in Fig. 1._{g} **bandwidth, BW**Time limited signals (i.e., pulses) are considered to be transmitted along the slow-wave structure, so the guided mode must have enough bandwidth to allocate the spectrum of these signals. The distance between the wavelength with the selected group velocity and the edge of the band is defined to be BW_{max}:_{max}/2. This is a limiting factor for the structures, since those with a small bandwidth will not be able to propagate signals at determined bit rates. For example, for bit rates of 40 Gbit/s, signal energy has a spectral bandwidth (at 95%) around 0.8 nm, so structures with a BW_{max}higher than this value are required.**dispersion parameter,**: Pulse shape is desired to remain unperturbed after crossing the slow-wave structure, with only a phase variation due to the non-linear index change. Unfortunately, dispersion in periodic structures is extremely high when we get close to the band edge of the guided mode [3*D*].3. 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 (2001). [CrossRef] [PubMed]*D*is obtained at the working wavelength for the periodic structures analyzed, looking for low values of this parameter.*D*is expressed here in ps/(mm·nm) since very high values are obtained and sizes are small (*D*is usually expressed in ps/(km·nm) in fibre-optics communications systems [6]).Dispersion compensation mechanisms consisting in cascading two stages of slow-wave structures with opposite values of GVD are considered. In a 1D periodic structure this can be done easily since the 1^{st}and the 2^{nd}guided bands have opposite GVDs near their respective band edges. Fig. 2 shows schematically a usual band diagram of a generic 1D periodic structure where this fact can be seen, and how two cascaded structures can be used to compensate the GVD accumulated along the first periodic structure, if required.**easiness of fabrication**: This is a more subjective property of the structures but equally important. Propagation losses in photonic structures are mainly due to imperfections in the fabrication process (e.g., surface roughness or scatterers in the structures). A reduction in the group velocity of the mode also “enhances” the interaction between the field and the imperfections of the structure, increasing propagation losses. Some works state that losses scale with (*c*/*v*) or (_{g}*c*/*v*)_{g}^{2}[7, 87. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B

**72**, 161318(R) (2005). [CrossRef]], what makes them to grow dramatically in the small group velocity region. But we think that this relation is not so simple and it will greatly depend on the concrete structure [9]. Hence, we have only centred our work in the enhancement of non-linear effects due to the reduction of8. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of the fabrication disorder and photon group velocity,” Phys.Rev. Lett.

**94**, 033903 (2005). [CrossRef] [PubMed]*v*. Therefore, a very accurate fabrication process is mandatory in order to use slow-wave structures in non-linear applications, without the problem of losses, independently of their relation with_{g}*v*. Hence, the simpler the designed structure is, the more accurate the fabrication could be done._{g}

*D*will be selected in order to perform dispersion compensation if needed. Then, band diagrams for each configuration are calculated using the plane wave expansion (PWE) method [10

10. S.G. Johnson and J.D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-8-3-173. [CrossRef] [PubMed]

*c*is selected (any other value could have been selected), and the distance to the band edge (used to obtain BW

_{max}) and the dispersion parameter

*D*are obtained for the wavelength with this

*v*. Then, the normalized effective volume

_{g}*V*of the guided mode is determined from the electric field distribution in the structure (obtained from PWE calculations). Finally, the effective index of the core material for the selected band is obtained in order to perform 2D FDTD simulations to study transmission performance and coupling efficiencies.

_{eff,n}## 4. 1D periodic structures analyzed and results

*n*=2.74) as non-linear material, with a SiO

_{2}(

*n*=1.45) lowercladding and surrounded by air. CdTe is chosen as non-linear material for the calculations since it presents high Kerr effect at wavelengths around 1550 nm (

*n*

_{2}=5.23·10

^{-13}cm

^{2}/W) and small losses due to two-photon absorption (TPA) effect (

*β*=18.3 mm/GW) [11

11. Satoshi Tatsuura, Takashi Matsubara, Hiroyuki Mitsu, Yasuhiro Sato, Izumi Iwasa, Minquan Tian, and Makoto Furuki, “Cadmium telluride bulk crystal as an ultrafast nonlinear optical switch,” Appl. Phys. Lett. **87**, 251110 (2005). [CrossRef]

*n*=3.34/

*n*

_{2}=2.93·10

^{-13}cm

^{2}/W/

*β*=103 mm/GW) [11

11. Satoshi Tatsuura, Takashi Matsubara, Hiroyuki Mitsu, Yasuhiro Sato, Izumi Iwasa, Minquan Tian, and Makoto Furuki, “Cadmium telluride bulk crystal as an ultrafast nonlinear optical switch,” Appl. Phys. Lett. **87**, 251110 (2005). [CrossRef]

*w*=550 nm (width) and

*h*=350 nm (height) in order to be single mode and to have low effective area. These parameters have been used for the design of the slow-wave structures. It is seen in Fig. 3 that analyzed structures vary from structures with isolated defects ((a) and (b)) to structures where a periodicity is introduced in the previously designed strip waveguide ((c), (d), (e) and (f)).

- height of the structures is set to
*h*=350 nm, the same height as for the single mode waveguide designed, - in order to avoid leaky propagation for the guided modes, only those located below the light line of the silica (the cladding with higher refraction index) are considered. This condition determines the maximum period
*a*allowable:*a*<(*k*·_{edge}*n*)·_{cladd}*λ*→_{work}*a*<535 nm, - the minimum size of elements and the minimum distance between them is set to 100 nm in order to avoid problems in a future fabrication process. For example, in the rods configurations the minimum radius of them is set to 50 nm, while their maximum radius will be determined by the minimum distance between elements (100 nm) and the period selected.

## 4.1 CdTe rods chain with n=1

*d*between adjacent rows, as shown in Fig. 3(b). The use of several rows provides the advantage of a larger lateral size, which in principle would be advantageous for a more efficient coupling from/to external waveguides. Previous works [12

12. S. Fan, J. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B **12**, 1267–272 (1995). [CrossRef]

13. D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, “Nanopillars photonic crystal waveguides,” Opt. Express **12**, 617–622 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-4-617. [CrossRef] [PubMed]

*h*is fixed, a sweep for the lattice constant

*a*and the pillar radius

*r*is made, obtaining the wavelengths of the modes at the edge of the BZ with 3D PWE calculations. Calculations are made for lattice constant values between 400 nm and 535 nm, and radii values between 50 nm and 210 nm, due to dimension restrictions previously commented. All the combinations of these parameters for this configuration have their frequencies above the light line of the silica, being leaky modes. A higher percentage of dielectric material (CdTe) would be needed in order to shift modes below the light line, but fabrication constraints (mainly the minimum distance between pillars) as well as the small height of the columns prevent the modes to be non-leaky. Hence, no designs for the 1D periodic structure with

*n*=1 column are selected because all the configurations give leaky guided modes.

## 4.2 CdTe rods chain with n=3

*d*, is set to be equal to the lattice constant of the structure. Again, the limitations in the maximum and minimum values of the parameters are considered in the calculations.

*a*,

*r*). The grey shaded area denotes the configurations where the distance between adjacent columns is smaller than 100 nm (dimension restrictions are not accomplished). Configurations giving guided modes with their band edges located at 1550 nm are depicted by a black solid line. Solid triangles indicate the sign of the group velocity and polarization of the calculated mode for each parameters combination. Triangles pointing upwards indicate positive group velocity, while triangles pointing downwards indicate negative group velocity. The polarization of the mode is indicated by the filling color of the triangles: TE – red and TM – blue. It can be seen that the polarization of the first mode changes when modifying structure parameters, because the relative position between modes changes (the first TE mode can be above the first TM mode for some configuration and below for others). These wavelength maps are useful to determine the appropriate dimensions of the structure to allocate the edge of each band at 1550 nm.

## 4.3 CdTe waveguide with air holes

12. S. Fan, J. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B **12**, 1267–272 (1995). [CrossRef]

*a, r*). The Grey shaded area indicates the combinations (

*a, r*) where the distance between holes is smaller than 100 nm. White area in the third and forth bands indicate combinations where the frequency of the mode is above the light line of the silica (leaky mode). It can be seen now that appropriate combinations of parameters can be obtained in order to achieve guided modes with slow-wave behaviour near 1550 nm and with both signs for the group velocity. Configurations to obtain proper TE and TM modes with both signs of group velocity are selected and their parameters are shown in Table 1 and Table 2. These tables show the parameters (

*a, r*) selected for each configuration, the order of the mode in the band diagram, the sign of the group velocity, the bandwidth allowable when working at the wavelength with

*v*=0.07

_{g}*c*, the dispersion parameter

*D*at this wavelength, the volume of the non-linear region in the basic cell

*V*, the normalized effective volume

_{NL,n}*V*, and the percentage of electromagnetic energy in the non-linear region, α.

_{eff,n}12. S. Fan, J. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B **12**, 1267–272 (1995). [CrossRef]

*D*| ~10 ps/(mm·nm)), so propagated signal will only be slightly distorted along propagation. Moreover, modes with opposite sign of dispersion and very similar absolute values are obtained, which eases dispersion compensation if needed. Furthermore, energy is highly confined in the non-linear region, that gives normalized effective volume parameters with very low values (

*V*~0.5-1.5 µm

_{eff,n}^{2}). However, these low values of

*V*are a little deceptive since they are greatly influenced by the low volume of the non-linear material of the structure. The insertion of air holes in the strip CdTe waveguide significantly reduces the region where non-linearities can be induced. It can be seen that effective volumes are one order of magnitude higher than non-linear volumes of the structures.

_{eff,n}*D*| ~35–50 ps/(mm·nm)) and effective volume (

*V*~2–4 µm

_{eff,n}^{2}) are achieved.

^{nd}band (

*a*=500 nm,

*r*=100.6 nm,

*n*=2.28) is considered. Figure 6(a) shows the obtained transmission spectra. The blue curve shows the response for the holes waveguide without any coupling section. It can be appreciated that a strong ripple appears due to both the modal and the group velocity mismatching between the waveguide mode and the mode in the holes structure, which results in a typical Fabry-Perot-like response. Adiabatic transitions have been implemented at both ends of the holes chain by linearly diminishing the radius of the holes in N steps (N=3, 10 and 20). This adiabatic coupling technique is schematically shown in Fig. 6(b) for the input side of the chain of holes. The objective is to improve the coupling by an adiabatic transition and reduce the Fabry-Perot ripple. The period is kept constant, so a reduction of the radius is equivalent to increasing the index of the structure and the second band is shifted up to higher wavelengths, which results in an improved coupling. It can be thought that an almost perfect coupling could be obtained with a sufficiently long tapering structure. Nevertheless, a linear reduction of the radius of the holes seems to be difficult to achieve in the fabrication process. In fact, holes with radius smaller than 100 nm might be drilled.

_{eff}## 4.4 CdTe waveguide with adjacent CdTe rods

*n*=1 adjacent row of rods) what makes the influence of the adjacent rods to be almost negligible over the total response. An example of band diagram for this kind of structure is depicted in Fig. 8(b), where it can be seen the very slight splitting of the mode at the edge of the BZ. These bands have an almost negligible curvature, so low group velocities would be only obtained for frequencies very close to the band edge, having very small bandwidths and very high dispersion. This behavior has been obtained for all possible configurations of this structure (with

*n*=1 and

*n*=2 rows of rods), so this scheme of 1D periodic structure must be discarded for non-linear enhancing purposes.

## 4.5 CdTe corrugated waveguide

*a*are introduced along the waveguide in order to create the periodicity [5

5. 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-7145. [CrossRef] [PubMed]

5. 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-7145. [CrossRef] [PubMed]

*h*of strip waveguide (fixed from design of the single mode waveguide), period

*a*, and width

*w*and length

_{i}*w*of the transversal elements.

_{e}*a - w*) were too small. This could be solved by using an appropriate hard mask or lift-off technique in order to achieve a higher resolution.

_{i}*a, w*) need to be determined in order to design the structure to work around 1550 nm. This fact hugely increases the number of calculations to be done in order to determine these parameters. Therefore, elements length

_{i}, w_{e}*w*is fixed to a value of 2 µm. This value is expected to be enough to give proper results, since it is four times greater than the width of the waveguide

_{e}*w*and it will be theoretically around ten times greater than the width of the transversal elements

*w*.

_{i}*a, w*) and obtaining the wavelength maps for the modes of the waveguide, a band diagram of a corrugated waveguide is calculated. Fig. 9(a) shows the band diagram for both polarizations of a corrugated waveguide with

_{i}*a*=363.2 nm and

*w*=150 nm. Symmetries in the in-plane direction are also depicted. It can be seen that modes with several polarizations and symmetries are obtained, overlapping each other. Nevertheless, if this structure is accessed from the designed single mode waveguide, TE-polarized modes (solid line) with an odd symmetry in the transversal direction (crosses) are excited. Therefore, only modes with these symmetries must be taken into account, as depicted in Fig. 9(b). Moreover, it can be seen that the first two modes have positive and negative slopes at the edge of the BZ, what makes them suitable for dispersion compensation purposes.

_{i}*a*=371.7 nm and

*a*=390.1 nm need to be taken. Parameters of these two modes are shown in Table 3. Fig. 10 shows the band diagrams for these two configurations. It can be seen that only one mode is excited when working at the edge of the band for these two configurations (uprising part of the second band is over the light line of the silica).

*D*| ~45 ps/(mm·nm)). Therefore, dispersion compensation would be easily performed by using two sections of approximately the same length and equal but opposite

*D*. But the main advantage of this configuration is the high concentration of the EM field in the non-linear region. Concentration values around 90 % are obtained for both modes, what gives extremely low values of normalized effective volume (around 0.34 µm

^{2}). These values are very close to the effective area of the CdTe strip waveguide, that has been calculated to be 0.27 µm

^{2}. Therefore, a huge enhancement of the non-linear effect will be produced, which is its main advantage. Contrary to what occurred for the air holes configuration, the volume of non-linear material is very high, so the low values of

*V*are due to the high concentration of the EM field (as desired, notice that they are of the same order of magnitude than the non-linear volume).

_{eff,n}*P*≈0.1 W would be needed to induce a π-phase-shift in a corrugated waveguide of length

*L*=500 µm for any of the two selected modes. By using the same equation, we obtain that an input power of ~1.8 W would be needed to induce the same phase shift in the CdTe strip waveguide. Concerning dispersion, if pulses width of the 40 Gbps signal is 7.5 ps,

*D*values for this structure will cause a slight broadening of pulses to ~8.4 ps for a corrugated waveguide of this length (

*L*=500 µm) (classical broadening expression given in [6] has been used for the calculation).

*a*=371.7 nm and

*w*=100 nm, and length of N=50 transversal elements, when coupling from/to it with the CdTe strip waveguide. It can be seen that transmission efficiencies for the first band (wavelengths higher than 1550 nm) around 75 % are obtained when the corrugated waveguide is directly connected to the strip CdTe waveguide. The second band (wavelengths lower than 1500 nm) provides efficiencies around 65 %, but large Fabry-Perot resonances appear due to the reflections at the interfaces between both waveguides. Efficiencies are highly enhanced if a simple linear taper of 5 elements is introduced to connect both waveguides (see inset in Fig. 11). Efficiency values are now obtained near to 100 % for the first band and around 85 % for the second. It can be also observed that Fabry-Perot resonances are reduced due to smaller reflections at the interfaces of the waveguides. Contrary to what happens for other structures such as the chain of holes, a taper can be easily fabricated, since it is only based on reducing the length of the transversal elements and fabrication constraints are always accomplished. In [5

_{i}5. 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-7145. [CrossRef] [PubMed]

## 5. Application of non-linear enhancement for optical processing

**19**, 2052–2059 (2002). [CrossRef]

14. J.M. Martinez, J. Herrera, F. Ramos, and J. Marti, “All-optical correlation employing single logic XOR gate with feedback,” Electron. Lett. **42**, 1170–1171 (2006). [CrossRef]

15. J. E. Heebner and R. W. Boyd “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. **24**, 847–849 (1999). [CrossRef]

- non-linear enhancement is only achieved for the wavelength of resonance of the ring and the bandwidth provided is very narrow. This makes the operation of the structure very sensitive to any fabrication deviation,
- the complete MZI structure is wider (ring radius around 5 µm-10 µm for high index materials) than the one obtained using the 1D periodic structures proposed in this text (in the worst case, the corrugated waveguide, the width of the structure is increased until 2 µm). Hence, the capacity of integration is reduced.

## 6. Conclusions

## Acknowledgments

## References and links

1. | N. A. R. Bhat and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E |

2. | Marin Soljacic, Steven G. Johnson, Shanhui Fan, Mihai Ibanescu, Erich Ippen, and J.D. Joannopoulos, “Photonic crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B |

3. | 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. |

4. | G. P. Agrawal, Nonlinear Fiber Optics - Third Edition, Academic Press (2001). |

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

6. | G. P. Agrawal, Fiber-Optic Communication Systems, Wiley-Interscience, Ed. (1997). |

7. | E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B |

8. | S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of the fabrication disorder and photon group velocity,” Phys.Rev. Lett. |

9. | J. García, A. Martínez, and J. Martí, “Influence of Group Velocity on Roughness Losses for 1D Periodic Structures,” in |

10. | S.G. Johnson and J.D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

11. | Satoshi Tatsuura, Takashi Matsubara, Hiroyuki Mitsu, Yasuhiro Sato, Izumi Iwasa, Minquan Tian, and Makoto Furuki, “Cadmium telluride bulk crystal as an ultrafast nonlinear optical switch,” Appl. Phys. Lett. |

12. | S. Fan, J. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B |

13. | D. N. Chigrin, A. V. Lavrinenko, and C. M. Sotomayor Torres, “Nanopillars photonic crystal waveguides,” Opt. Express |

14. | J.M. Martinez, J. Herrera, F. Ramos, and J. Marti, “All-optical correlation employing single logic XOR gate with feedback,” Electron. Lett. |

15. | J. E. Heebner and R. W. Boyd “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.4310) Integrated optics : Nonlinear

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: November 6, 2007

Revised Manuscript: January 7, 2008

Manuscript Accepted: January 7, 2008

Published: February 21, 2008

**Citation**

Jaime Garcia, Pablo Sanchis, Alejandro Martinez, and Javier Marti, "1D periodic structures for slow-wave induced non-linearity enhancement," Opt. Express **16**, 3146-3160 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3146

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### References

- N. A. R. Bhat and J. E. Sipe, "Optical pulse propagation in nonlinear photonic crystals," Phys. Rev. E 64, 056604 (2001). [CrossRef]
- M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen and J. D. Joannopoulos, "Photonic crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002). [CrossRef]
- 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 (2001). [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear Fiber Optics Third Ed., (Academic Press, 2001).
- 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]
- G. P. Agrawal, Fiber-Optic Communication Systems, (Wiley-Interscience, 1997).
- E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318(R) (2005). [CrossRef]
- S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic optical scattering loss in photonic crystal waveguides: role of the fabrication disorder and photon group velocity," Phys.Rev. Lett. 94, 033903 (2005). [CrossRef] [PubMed]
- J. García, A. Martínez, and J. Martí, "Influence of Group Velocity on roughness losses for 1D Periodic Structures," in Slow and Fast Light (Optical Society of America, 2007), paper JTuA4.
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8, 173 (2001). [CrossRef] [PubMed]
- S. Tatsuura, T. Matsubara, H. Mitsu, Y. Sato, I. Iwasa, M. Tian, and M. Furuki, "Cadmium telluride bulk crystal as an ultrafast nonlinear optical switch," Appl. Phys. Lett. 87, 251110 (2005). [CrossRef]
- S. Fan, J. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, "Guided and defect modes in periodic dielectric waveguides," J. Opt. Soc. Am. B 12, 1267-272 (1995). [CrossRef]
- D. N. Chigrin, A. V. Lavrinenko, C. M. Sotomayor Torres, "Nanopillars photonic crystal waveguides," Opt. Express 12, 617-622 (2004). [CrossRef] [PubMed]
- J. M. Martinez, J. Herrera, F. Ramos, and J. Marti, "All-optical correlation employing single logic XOR gate with feedback," Electron. Lett. 42, 1170-1171 (2006). [CrossRef]
- J. E. Heebner and R. W. Boyd, "Enhanced all-optical switching by use of a nonlinear fiber ring resonator," Opt. Lett. 24, 847-849 (1999). [CrossRef]

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