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The workshop "Stratification biomarkers in personalised medicine" that was held at the European Commission in Brussels on 10th and 11th June 2010 brought together experts from different fields and perspectives in order to stimulate an open discussion about the evolution of this rapidly expanding area, in which biomarkers are used to adapt therapies to the characteristics of patient populations. The purpose of the workshop was to clarify the scope for stratification biomarkers and to identify barriers in the process from biomarker discovery to their use in clinical trials and in medical practice. It aimed at building a...

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  1. CHAPTER 26 OPTICAL INTEGRATED CIRCUITS Hiroshi Nishihara, Masamitsu Haruna, and Toshiaki Suhara 26.1 FEATURES OF OPTICAL INTEGRATED CIRCUITS An optical integrated circuit (OIC) is a thin-film-type optical circuit designed to perform a function by integrating a laser diode light source, functional components such as switches / modulators, interconnecting waveguides, and photodiode detectors, all on a single substrate. Through integration, a more compact, stable, and functional optical system can be produced. The key components are slab [two-dimensional (2-D)] or channel [three dimensional (3-D)] waveguides. Therefore, the important point is how to design and fabricate good waveguides using the right materials and processes. Some theories and technologies have been investi- gated by many researchers, and published in several technical books.1–4 The features of OICs are4 1. Single-mode structure: waveguide widths are on the order of micrometers and are such that a single-mode optical wave propagates. 2. Stable alignment by integration: the device can withstand vibration and temperature change; that is the greatest advantage of OICs. 3. Easy control of the guided wave. 4. Low operating voltage and short interaction length. 5. Faster operation due to shorter electrodes and less capacitance. 6. Larger optical power density. 7. Compactness and light weight. 26.2 WAVEGUIDE THEORY, DESIGN, AND FABRICATION 26.2.1 2-D Waveguides The basic structure of a 2-D (or slab) waveguide is shown in Fig. 26.1 with the index profiles along the depth, where the indices of the cladding layer, guiding layer, and substrate are nc, nƒ , and ns, respectively. In the case that nƒ , ns nc, the light is confined in the guiding layer by the total internal reflections at two interfaces and propagates along a zigzag path, as shown in Fig. 26.1a. Such a confined lightwave is called a guided mode whose propagation constant along the z direction exists in the range of k0ns k0nƒ , where k0 2 / . 26.1
  2. 26.2 CHAPTER TWENTY-SIX FIGURE 26.1 2-D optical waveguides. (a) The basic optical-waveguide struc- ture; (b) the step-index type; and (c) the graded-index type. (From Ref 4.) Usually, the guided mode is characterized by the effective index N, where k0 N and ns N nƒ . N must have discrete values in this range because only zigzag rays with certain incident angles can propagate as the guided modes along the guiding layer. The dispersion characteristics of the guided modes in the 2-D waveguide with a step- index distribution are straightforward, being derived from Maxwell’s equations (see Fig. 26.1b). The 2-D wave analysis indicates that pure TE and TM modes can propagate in the waveguide. The TE mode consists of field components, Ey , Hx , and Hz, while the TM mode has Ex , Hy , and Ez. A unified treatment of the TE modes is made possible by introducing the normalized frequency V and the normalized guide index bE , defined as n2 n2 V k0T ƒ s N2 2 ns bE (26.1) 2 2 nƒ ns The asymmetric measure of the waveguide is also defined as (n2 n2) s c aE (26.2) (n2 n2) ƒ s when ns nc , aE 0. This implies symmetric waveguides. However, the 2-D waveguides are generally asymmetric (ns nc). By using the above definitions, the dispersion equation of the TEm modes can be expressed in the normalized form 1 bE 1 bE 1 1 V 1 bE (m 1) tan tan (26.3) bE bE aE The normalized dispersion curve is shown in Fig. 26.2, where m 0, 1, 2, . . . , which is the mode number corresponding to the number of nodes of the electric field distribution Ey (x). When the waveguide parameters, such as the material indices and the guide thick- ness, are given, the effective index N of the TE mode is obtained graphically. The wave- guide parameters are usually defined on the basis of cutoff of the guided mode, in which N ns (bE 0). From Eq. (26.3), the value of Vm at the cutoff is given by 1 Vm V0 m V0 tan aE (26.4) V0 is the cutoff value of the fundamental mode. If V ranges over Vm V Vm 1, the number of TE modes supported in the waveguide is m 1. In symmetric waveguides (ns nc , aE 0), the fundamental mode is not cut off. On the other hand, the dispersion equation of the TM mode is rather complex. In an actual waveguide, however, the index difference between
  3. 26.3 OPTICAL INTEGRATED CIRCUITS FIGURE 26.2 Dispersion curves of step-index 2- D waveguides. the guiding layer and the substrate is small enough that the condition (nƒ ns) ns is satisfied. Under this condition, all dispersion curves in Fig. 26.2 are made applicable to the TM modes simply by replacing the asymmetric measure aE with aM , defined as 4 nƒ n2 n2 s c aM (26.5) 2 2 nc nƒ ns Low-loss optical waveguides are usually fabricated by metal diffusion and ion-exchange techniques that provide a graded-index profile along the depth, as shown in Fig. 26.1c. Two analytical methods, the ray approximation5 and Wentzel-Kramers-Brillouin (WKB) methods, are often used to obtain the mode dispersion of such graded-index slab waveguides. The index distribution is generally given by x n(x) ns nƒ n nƒ ns (26.6) d where nƒ is the maximum index of the waveguide and d is the diffusion depth. The distri- bution function ƒ (x / d ) is assumed to be a function that decreases monotonically with x, and ƒ (x / d ) takes on values between 0 and 1. Using the normalized diffusion depth, defined as n2 n2 Vd k0d (26.7) ƒ s he dispersion equation is expressed in the normalized form 3 t 2Vd ƒ( ) bd 2m (26.8) 2 0 where x / d, t x t / d, and b ƒ ( t ), b has already been defined as Eq. (26.1). xt denotes the turning point, and is regarded as the effective waveguide depth. Equation (26.8) is also usable as long as the condition (nƒ N) (N nc) is satisfied. The mode dispersion is
  4. 26.4 CHAPTER TWENTY-SIX FIGURE 26.3 Disperison curves of graded-index 2- D waveguides with a Gaussian index profile. calculated from Eq. (26.8) if the index distribution ƒ ( ) is specified. Titanium-diffused LiNbO3 waveguides, for example, have the Gaussian index distribution, that is ƒ ( ) exp( 2). The Vd b diagram for the Gaussian index distribution is shown in Fig. 26.3 where ns is n0 or ne for the ordinary or extraordinary wave used as the guided mode. In addition, the values Vd for the guided-mode cutoff are found by putting b 0 and xt → , resulting in 3 Vdm 2 m (26.9) 4 26.2.2 3-D Waveguides Optical waveguide devices having functions of light modulation / switching require 3-D (or channel) waveguides in which the light is transversely confined in the y direction in addition to confinement along the depth. In 3-D waveguides, a guided mode is effectively controlled without light spreading due to diffraction on the guide surface. The 3-D waveguides are divided into four different types, as shown in Fig. 26.4. Among them, the buried type of 3- D waveguides, including Ti-diffused LiNbO3 and ion-exchanged waveguides, are more suit- able for optical waveguide devices. The reasons why this type of waveguide has advantages are that the propagation loss is usually lower than 1 dB / cm even for visible light and that planar electrodes are easily placed on the guide surface to achieve light modulation / switch- ing. On the contrary, ridge waveguides are formed by removing undesired higher-index film with dry etching and lift-off of deposited film. These waveguides tend to suffer a significant scattering loss due to waveguide wall roughness. This shortcoming, however, is overcome by deposition of rather thick lower-index material as a cladding layer on the waveguides. In the 3-D waveguides consisting of dielectric materials, pure TE and TM modes are not supported, and two families of hybrid modes exist. The hybrid modes are classified according to whether the main electric field component lies in the x or y direction (see Fig. 26.5). The mode having the main electric field Ex is called the E x mode. This mode resembles the TM pq mode in a slab waveguide; hence the E x mode is sometimes called the TM-like mode. The pq subscripts p and q denote the number of nodes of the electric field Ex in the x and y directions, respectively. Similarly, the E y mode (that is the TE-like mode) has the main electric field pq E y. To obtain the mode dispersion of 3-D waveguides, two approximate analyses are often used: (1) Marcatili’s method and (2), the effective index method.2 Both are available if the 6 guided mode is far from the cutoff and the aspect ratio W / T is larger than unity. In the
  5. 26.5 OPTICAL INTEGRATED CIRCUITS FIGURE 26.4 Basic structures of #-D optical waveguides. analytical model for the effective index method, as shown in Fig. 26.5, a buried 3-D wave- guide is divided into two 2-D waveguides, I and II. Consider here the E x mode having main pq field components Ex and Hy in a 3-D waveguide with step-index distribution. In a 2-D waveguide I, the dispersion equation (26.3) yields the effective index NI of the TM mode. In the symmetric 2-D waveguide II, the guided mode of interest is regarded as the TE mode FIGURE 26.5 Analytical model for the effective index method.
  6. 26.6 CHAPTER TWENTY-SIX which sees the effective index NI as the index of the guiding layer because it is mainly polarized along the x direction. The dispersion equation of the TE mode in the symmetric 2-D waveguide is easily derived by putting aE 0 in Eq. (26.3), resulting in 1 bII 1 VII 1 bII (q 1) 2 tan (26.10) bII where N2 n2 s 2 2 VII k0W NI ns and bII (26.11) N2 n2 I s The effective index method discussed here is also adopted even for graded-index 3-D wave- guides if the dispersion equation (26.8) is used. This method has thus an advantage over Marcatili’s method in that the mode dispersion is easily obtained by a short calculation. If the field distributions are required as well as the mode dispersion, however, Marcatili’s method must be chosen. Design of Single-Mode 3-D Waveguides. Usually, most waveguide devices consist of single-mode 3-D waveguides to attain highly efficient control of the guided mode. It, there- fore, is important to provide design consideration of single-mode waveguides. Once all waveguide parameters are specified, it is found on the basis of the effective index method described above that, in a buried, step-index 3-D waveguide with air cladding, single- mode propagation is restricted in the hatched area of Fig. 26.6a, showing the relation of the aspect ratio W / T and the normalized frequency VI in the 2-D waveguide I. If the 3-D wave- guide has Gaussian index profiles in both x and y directions, the diagram for single-mode propagation range is shifted as shown in Fig. 26.6b. In practical use, waveguide devices require single- mode waveguides in which light is as strongly confined as possible to min- imize scattering loss due to bending and branching. To meet this requirement, the aspect ratio and the normalized frequency should be close to the upper boundary of the shaded areas of Fig. 26.6a and b. FIGURE 26.6 Single-mode propagation range (indicated by the hatched area) in (a) step- index 3-D waveguides and (b) graded-index 3-D waveguides with Gaussian index profiles in both x and y directions.
  7. 26.7 OPTICAL INTEGRATED CIRCUITS 26.2.3 Waveguide Materials and Fabrication Higher-index guiding layers are formed on substrates by deposition, thermal indiffusion, ion exchange, epitaxial growth, and so on. The relatively popular materials and the relevant fabrication techniques for 2-D waveguides are summarized in Table 26.1. Furthermore, microfabrication techniques, including photolithography, dry or chemical etching, and lift- off techniques, are required for fabrication of 3-D waveguides, as shown in Fig. 26.4. Our attention is focused on two representative waveguide materials, LiNbO3 and glass, and their fabrication process will be described as well as the waveguide characteristics. LiNbO3 Waveguides. Low-loss 3-D waveguides can be formed near the surface of LiNbO3 by the lift-off of Ti stripes, followed by thermal indiffusion, as illustrated in Fig. 26.7. In most cases, Z-cut LiNbO3 is used as the substrate, and Ti is indiffused into the Z surface to prevent domain inversion. Both the thickness and width of Ti stripes depend on the ˚ wavelength of interest Ti stripes for instance, are 4 m wide and 400 A thick at the 0.8- m wavelength; in this case, the thermal indiffusion is performed in flowing oxygen gas or synthetic air at 1025 C for nearly 6 h. The use of moistened flowing gas is effective to suppress outdiffusion of Li2O, which leads to weak light confinement in Ti-diffused channel waveguides. Surface roughness of the diffused waveguides should be remarkably less if the LiNbO3 is loosely closed within a platinum foil or crucible. The resulting Ti-diffused waveguides provide single-mode propa- gation for both TE- and TM-like modes with propagation loss of 0.5 dB / cm or less. Single-mode waveguides for the use of the 1.3- or 1.5- m wavelength are also fabricated ˚ under the conditions that Ti stripes are 6 to 8 m wide and more than 700 A thick, and the diffusion time is above 8 h. In such Ti-diffused waveguides, the input power level should be limited to a few tens of microwatts by the optical damage threshold of the waveguide TABLE 26.1 Optical Waveguide Materials and Fabrication Techniques Waveguide materials LiNbO3, Nb2O5, Fabrication techniques Polymer Glass Chalcogenide LiTaO3 ZnO Ta2O5 Si3N4 YIG Deposition: Spin-coating Vacuum evaporation RF or dc sputtering CVD Polymerization Thermal diffusion Ion exchange Ion implantation Epitaxial growth: LPE VPE Often-used fabrication techniques CVD chemical vapor deposition LPE liquid-phase epitaxy VPE vapor-phase epitaxy
  8. 26.8 CHAPTER TWENTY-SIX itself, especially for visible light and the 0.8 m wavelength. To avoid this problem, MgO- doped LiNbO3 is used as the substrate, resulting in a hundredfold increase in the damage threshold. Another way is to use Z-propagating LiNbO3, where both TE and TM modes are ordinary waves that are much less influenced by optical damage. Besides Ti indiffusion, the other important fabrication technique of LiNbO3 waveguides is proton exchange, which pro- vides an extremely high index increment ( ne 0.13) only for the extraordinary wave; on the contrary, the index change ( no) for the ordinary wave is nearly 0.04. It is noted, however, that the electro-optic and acousto-optic effects of LiNbO3 itself are drastically re- duced by the proton exchange, and therefore the proton-exchanged waveguides have a lower susceptibility to optical damage by one- tenth or less compared to Ti-diffused waveguides. The proton exchange is usually performed by immersing the LiNbO3 in molten benzoic acid (C6H5COOH) or pyrophosphoric acid (H4P2O7). The waveguide depth is determined by the exchange time and temperature. The proton-exchanged waveguides exhibit significant scat- tering loss due to a large amount of H ions localized very close to the crystal surface. Therefore, an annealing is necessary after the proton exchange to obtain low-loss waveguides. The electro-optic effect is also recovered by the annealing. Typical fabrication conditions for proton-exchanged / annealed single-mode waveguides are as follows: a shallow high-index layer is formed on a X-cut LiNbO3 surface by exchanging in pure benzoic acid at 200 C for 10 min through a 3.5- m window of a Ta mask, followed by annealing the LiNbO3 at 350 C for 2 h. The resulting waveguide exhibits the propagation loss of 0.15 dB / cm at the 0.8- m wavelength.7 Glass Waveguides. The most popular glass waveguide fabrication technique is ion exchange in which, for instance, soda-lime glass is immersed in molten salt (AgNO3, KNO3, or TlNO3) to exchange Na ions with univalent ions such as Ag , K , or Tl . The index change n is greatly dependent on the electronic polarizability of metal ions; typically, n FIGURE 26.7 Fabrication procedure for Ti-diffused FIGURE 26.8 Fabrication process of high-silica NiNbO3 waveguides. single-mode waveguides using flame hydrolysis deposition (FHD).
  9. 26.9 OPTICAL INTEGRATED CIRCUITS 10 2 for Ag ions, and n 10 3 for K ions. 0.1 for Tl ions, n 2 to 8 8 to 20 Three-dimensional waveguides are easily fabricated by waveguide patterning of a suitable metal mask deposited on the glass substrate before the ion exchange. The Tl and Ag ion exchanges provide multimode waveguides because the index change is quite large. On the other hand, the K ion exchange is suitable for fabricating single-mode waveguides. A microscope slide, for example, is immersed in molten KNO3 at 370 C to be selectively exchanged through aluminum-film windows. The K ion exchange takes nearly 1 h to form 4- m-wide single-mode waveguides. The resulting waveguide has a propagation loss of less than 1 dB / cm, even for visible light. The ion exchange is sometimes performed under application of an electric field E; in this case, the exchanged ion density becomes nearly constant within the depth E , t where t is the exchange time and is the ion mobility, which depends on temperature. The electric-field-assisted ion exchange thus provides a rigid step- index waveguide. Sputtering is another popular technique for depositing waveguide films on a glass substrate such as Corning 7059 and Pyrex glass. Silicon is also used as a substrate instead of glass. In this case, thermal oxidation of Si is necessary before deposition of a waveguide film to form a SiO2 buffer layer nearly 2 m thick. Recently, a research group at NTT developed a promising fabrication technique for low-loss 3-D silica waveguides using the SiO2 / Si substrate. Their procedure is shown in Fig. 26.8. The propagation loss is as low as 0.1 dB / cm at 1.3 m.8 26.3 GRATING COMPONENTS FOR OPTICAL INTEGRATED CIRCUITS Periodic structures or gratings in waveguide are one of the most important elements for OICs, since they can perform various passive functions and provide effective means of guided-wave control.9,10 26.3.1 Coupling of Optical Waves by Gratings Classification of Gratings. Figure 26.9 illustrates examples of passive grating components for OIC. They include input / output couplers, interwaveguide couplers, deflectors, guided- beam splitters, reflectors, mode converters, wavelength filters and dividers, and guided- wavefront converters such as waveguide lenses and focusing grating couplers. Periodic mod- ulation of he refractive index can be induced through acousto-optic (AO) and electro-optic (EO) effects. They can be considered a controllable grating, and have many applications to functional devices. Optical coupling by a grating is classified as either guided-mode to guided-mode coupling or guided-mode to radiation-mode coupling, the former subdivided into collinear coupling and coplanar coupling. Gratings are also classified by structure into index-modulation and relief types, as shown in Fig. 26.10. Phase Matching Condition. Various grating structures can be described by the change in distribution of relative dielectric permittivity, , caused by attaching a grating to a wave- guide. Since the grating is periodic, can be written by Fourier expansion as (x, y, z) (x) exp ( jqK r) (26.12) q q using a grating vector K ( K K 2/, period). When an optical wave with propagation vector is incident in the grating region, space harmonics of propagation vectors qK are produced. The harmonics can propagate as a guided mode, if a coupling con- dition
  10. 26.10 CHAPTER TWENTY-SIX FIGURE 26.9 Passive grating components for optical integrated circuits. FIGURE 26.10 Various cross sections of gratings. qK q 1, 2, . . . (26.13) b a is satisfied between two waves, a and b, with propagation vectors a, b. In many cases, is nonzero only in the vicinity of the waveguide ( y-z) plane, and Eq. (26.13) need not be satisfied for the x component. Each part of Eq. (26.13) is called a phase matching condition, while the three-dimensional relation is called the Bragg condition. The relation can be de- picted in a wave vector diagram, which is used to determine the waves involved in the coupling. 26.3.2 Collinear Coupling Two guided modes propagating along the z axis couple with each other in a grating of vector K parallel to the z axis, as shown in Fig. 26.11, if a and b satisfy approximately the phase matching condition b qK. The interaction is described by coupled mode equations a for the amplitude A(z) and B (z) of modes a and b:
  11. 26.11 OPTICAL INTEGRATED CIRCUITS FIGURE 26.11 Collinear coupling of guided modes by a grating. d A(z) j * B(z) exp ( j2 z) ( 0) (26.14a) a dz d B (z) j A(z) exp ( j2 z) ( 0) (26.14b) b dz where is the coupling coefficient and the parameter 2 denotes the deviation from the exact phase matching. Codirectional Coupling. For coupling between two different modes propagating in the same direction ( a 0, b 0), Eq. (26.14), with boundary conditions A(0) 1, B (0) 0 gives a solution which indicates periodic transfer of the guided mode power. The efficiency for a grating of length L is given by 2 sin2 { 2 2 L B(L) (26.15) 2 2 A(0) 1 / 0), the efficiency is given by the sin2 When the phase matching is exactly satisfied ( function. Complete power transfer takes place when L equals the coupling length Lc /2 . Contradirectional Coupling. For the coupling of modes propagating in the opposite direc- tions ( a 0, b 0), Eq. (26.14) with A(0) 1, B(L) 0 gives a solution which shows a monotonous power transfer. The efficiency is given by 2 1 2 2 B(0) 1 / 1 (26.16) A(0) sinh2 ( 2 2 L) Complete power transfer takes place for L → , provided that . When 0, the efficiency is given by the tanh2 function; most of the power is transferred ( 0.84) when L Lc / 2 . A grating reflector, called a distributed Bragg reflector (DBR), exhibits a sharp wavelength selectivity. Coupling Coefficient. Coupling coefficient can be evaluated by integrating the multiple of index modulation profile and the profiles of modes a and b. The mathematical ex- pressions of depend on the polarizations of the coupling modes. For an index-modulation grating, can be written as a multiple of b n / . b is the value for coupling in a bulk medium, and a factor describing the effect of confinement in waveguide. From mode orthogonality, for uniform index modulation and well-guided modes, can be written as b ab, which implies that coupling with mode conversion hardly takes place and substantial coupling is limited to contradirectional coupling (reflection) of the same mode. For relief
  12. 26.12 CHAPTER TWENTY-SIX FIGURE 26.12 Brillouin diagrams for guided-wave coupling by a grating. The coupling occurs at a frequency indicated by *. gratings with groove depth much smaller than guiding layer thickness, a simple analytical expression of is given by approximating the mode profiles by the values at the guide surface. Coupling with mode conversion (TEm ↔ TEn, TMm ↔ TMn, m n) may take place. Brillouin Diagram. The dispersion of a waveguide grating can be illustrated by the Bril- louin diagram, i.e., an / c( k)- diagram, as shown in Fig. 26.12. Curve a shows the dispersion of a waveguide without grating. The curves for the q thorder space harmonics are obtained by shifting the curves in a by qK along the axis. Coupling occurs in the vicinity of the intersection with the original curve where phase matching holds. Curves b and c show the diagrams for co- and contradirectional couplings, respectively. Coupling occurs only at or in the vicinity of the wavelength corresponding to / c k 2 / indicated by *, and, therefore, gratings can be used as wavelength filters and dividers. 26.3.3 Coplanar Coupling In a planar waveguide (in the y-z plane), coupling is made to take place between guided waves propagating in different directions by using a grating (with length L in the z direction) of appropriate orientation. The q th-order Bragg condition for two waves of vector a and b can be written as Eq. (26.13). For the coupling, Eq. (26.13) must be satisfied exactly for the y component, but the z component need not be satisfied exactly; the allowance depends on K and L. Since the coupling exhibits different behavior for different values of K and L, a parameter Q defined by Q K 2L / is used for classification. Raman-Nath Diffraction. When Q 1, many diffraction orders appear, since the relatively small value of L allows coupling without exact matching for the z component. The solution of the coupled-mode equation can be written by using Bessel functions, and the diffraction efficiency for the qth order is given by q J 2(2 L). The fundamental efficiency 1 takes q the maximum value 0.339 at 2 L 1.84. The incident-angle dependence of the efficiency is small, and accordingly, gratings barely exhibit angular and wavelength selectivities. Bragg Diffraction. When Q 1, the coupling takes place only between waves at the Bragg condition because of the relatively large length L. As shown in Fig. 26.13, a diffracted wave of a specific order appears only when the incident angle satisfies the Bragg condition. The wave vector diagram to determine the diffraction angle is shown in Fig. 26.14, where the wave vectors of the incident and diffracted waves are denoted by and . The phase mismatch 2 can be correlated with the deviation of the incident angle from the Bragg angle. When the incident angle is fixed at the Bragg angle, changing the wavelength results in a
  13. 26.13 OPTICAL INTEGRATED CIRCUITS FIGURE 26.13 Bragg diffraction of a guided wave. deviation from the Bragg condition. The phase mismatch 2 also can be correlated to such a wavelength change. The coupled-mode equations are written as d CR R (z) j *S(z) exp ( j 2 z) (26.17a) dz d CS S (z) j R(z) exp ( j 2 z) (26.17b) dz where cR cos i , CS cos d , and coupling coefficient. Transmission Grating. Equation (26.17) is solved with the boundary conditions R(0) 1, S (0) 0, and cR 0, cS 0. The diffraction efficiency can be written as sin2( 2 2 1/2 ) L/ cR cS , L (26.18) 2 2 (1 / ) Under Bragg condition 0, the efficiency takes a maximum value of 100 percent at FIGURE 26.14 Propagation vector diagram for Bragg diffraction.
  14. 26.14 CHAPTER TWENTY-SIX / 2. Efficiency decreases with deviation from the Bragg condition. Since, for / 2, /0 0.5 at 1.25, the angular and wavelength selectivity can be evaluated by com- bining L 1.25 with relations between 2 and angular / wavelength deviations. Reflection Grating. Equation (26.14) is solved with R(0) 1, S (L) 0 and cR 0, cS 0. The diffraction efficiency can be written as 1 2 2 (1 / ) 1 L/ cR cS L (26.19) sinh2 ( 2 2 1/2 ) Under Bragg conditions, the efficiency increases monotonously with . The efficiency is 84.1 percent at / 2 and larger than 99.3 percent for / . The angular and wavelength selectivities depend on or 0, since the value giving / 0 0.5 depends on . For / 2, for example, / 0 0.5 at L 2.5. Coupling Coefficient. Coupling coefficient ( d , i) for coplanar coupling can be written as TE-TE cos di , TM-TM, and TM-TE sin di , for TE-TE, TM-TM, and TE-TM coupling, respectively, where is the coupling coefficient for collinear coupling and di d i denotes the diffraction angle. The coefficient for TE-TE depends on di , whereas that for TM-TM has very little dependence. When di / 2, the former coupling does not occur, since the electric vectors are perpendicular to each other. A grating of di / 2 serves as a TE-TM mode divider. It should also be noted that TE-TM mode conversion, which does not take place in collinear coupling, may occur when di 0, although TE-TE is considerably smaller than TE-TE or TM-TM. 26.3.4 Guide-Mode to Radiation-Mode Coupling Output Coupling. Figure 26.15 illustrates the coupling between a guided mode and radi- ation modes. Coupling takes place between waves satisfying phase matching for z compo- nents. When a guided wave of propagation constant 0 is incident, the qth harmonics radiate into air and / or substrate at angles determined by FIGURE 26.15 Guided-mode–radiation-mode coupling in a grating coupler.
  15. 26.15 OPTICAL INTEGRATED CIRCUITS (c) (s) nc k sin nsk sin Nk qK (26.20) q q q The number of radiation beams is determined by the number of real values of (s) and qc) ( q satisfying Eq. (26.20). An order results in radiation into either the substrate alone or both air and substrate. Figure 26.15a shows multibeam coupling where more than three beams are yielded and Fig. 26. 15b shows two-beam coupling where only a single beam for the fundamental order (q 1) is yielded in both air and substrate. Another possibility is one- beam coupling where a beam radiates only into the substrate. The amplitude of the guided and radiation wave decays as g(z) exp ( r z) due to the power leakage by radiation. Since the guided-wave attenuation corresponds to the power transferred to radiation modes, the output coupling efficiency for a grating of length L can be written as i P q {1 exp ( 2 r L)} (26.21) out for the qth-order (i) radiation, where i(equal to c or s) distinguishes air and substrate. Here i r denotes the radiation decay factor and P q is the fractional power to q-i radiation. Input Coupling. A guided wave can be excited through reverse input coupling of an ex- ternal beam incident on a grating. When the incident angle coincides with one of the angles satisfying Eq. (26.20), one of the produced space harmonics synchronizes with a guided mode and the guided mode is excited. Figure 26.16 correlates output and input couplings. A reciprocity theorem analysis shows that the input coupling efficiency can be written as [ gh dz]2 i P q I (g, h) I(g, h) (26.22) in g 2 dz h2 dz where h(z) is the input beam profile. The overlap integral I (g, h) takes the maximum value 1 when the beam profiles are similar [h (z) g(z)]. Practically, high efficiency can be 1, (2) making P iq achieved by (1) making a grating of r L 1 for one beam q, i, and (3) feeding an input beam satisfying h(z) g(z). For an input beam with Gaussian profile, the maximum value of I (g, h) is 0.801. Radiation Decay Factor. The radiation decay factor r can be calculated by various meth- ods, e.g., a coupled-mode analysis, a rigorous numerical analysis to calculate the complex propagation constant of normal modes by space harmonics expansion based on Floquet’s theorem, and approximate perturbation analyses based on a Green’s function approach or a transmission-line approach. Figure 26.17 illustrates typical dependence of the decay factor r of couplers of the relief type on the grating groove depth h. For small h, r increases FIGURE 26.16 Input and output coupling by a grating; (a) Output coupling, (b) Input coupling.
  16. 26.16 CHAPTER TWENTY-SIX FIGURE 26.17 Dependence of radiation decay factor on the grat- ing groove depth for grating coupler of the relief type. monotonously with h and is approximately proportional to h2. For larger h, the coupling saturates because of the limited penetration of the guided-mode evanescent tail into the grating layer. In the saturation region, interference of the reflection at upper and lower interfaces of the grating gives rise to a weak periodic fluctuation. High-Efficiency Grating Couplers. One-beam coupling is desirable to achieve high effi- ciency. Such coupling can be realized by using backward coupling by a grating of short period. Two-beam couplers, as shown in Fig. 26.15b, are more widely used, but they have the drawback that the power is halved for air and substrate. The drawback can be eliminated by inserting a reflection layer on the substrate side. Methods for confining the power into single q and i include use of the Bragg effect in a thick index-modulation grating, and use of the blazing effect in a relief grating having an asymmetrical triangular cross section. 26.3.5 Fabrication of Gratings Grating Patterning Two-Beam Interference. The most effective optical method for obtaining fine periodic patterns is holographic interference lithography, which utilizes the interference fringe re- sulting from interference of two coherent optical waves. The fringe is recorded in a photo- resist layer. Grating patterns of the desired period are obtained by appropriate choice of recording wavelength and incidence angles. The recording optics are arranged on a vibra- tionfree optical bench. An Ar laser or He-Cd laser is used for the light source. The laser output is divided into two beams by a beam splitter, and the beam angles are adjusted for the required grating period. To maximize fringe visibility, the two beams must have equal intensities and path lengths. A spatial filter (pinhole) is used in the beam expander lens to remove spatial noise and obtain a uniform pattern. Inclined periodic structures, e.g., Bragg- effect index-modulation gratings, can be formed in a thick recording layer. Recording of inclined fringes in a thin resist and development result in a sawtooth cross section useful for blazed grating fabrication. The minimum feasible grating period is half the recording wave- length. For shorter periods, a prism or liquid immersion is used. Fabrication of chirped / curved gratings is also possible by using an appropriate combination of spherical and cylin-
  17. 26.17 OPTICAL INTEGRATED CIRCUITS drical lenses, but the flexibility is limited. The advantages of the two-beam interference are fabrication of small period gratings with simple apparatus, good period uniformity, and easy fabrication of large-area gratings. Electron-Beam Writing. Since many gratings for OIC require very small periods, but have rather small areas, computer-controlled electron-beam (EB) writing can be used, it is more convenient and effective to use a system with a specialized scanning controller, in which digital control and analog signal processing are incorporated to enable writing of very smooth straight and curved lines. The EB writing area with submicrometer resolution is 3 mm2. Grating patterns can be written by (1) a painting-out method which typically 3 writes a half period by many scanning lines, (2) a line-drawing method which writes one period by a scanning line, and (3) a gradient-dose method which involves continuous changes in EB dose. Methods 1 and 2 are suitable for gratings with large and small periods, respec- tively. Method 3 allows fabrication of blazed gratings (gradient-thickness cross sections after development) and gradient-index gratings. Resolution of EB writing is limited by EB di- ameter and EB scattering in the resist. If the substrate is an insulator, a very thin (about 100- ˚ A) conductive (Au, Al) layer must be deposited to avoid the charging-up problem. The EB writing technique has features almost complementary to those of the interference technique. The advantages are extremely high resolution, large flexibility in fabrication of modulated gratings, and easy parameter change by computer control. Grating Processing. If a resist is used as a grating material, the patterning is the final process to obtain a relief grating. If gratings are fabricated in a waveguide material whose refractive index can be changed by light or EB irradiation, index-modulation gratings are obtained by the patterning. Usually the resist pattern is transferred to waveguide, cladding, or hard-mask layers. Gratings of the relief type are fabricated by etching the waveguide surface of a cladding layer, using the resist pattern as a mask. Although gratings can be produced by chemical etching, better results are obtained with dry etching, e.g., sputter etching, plasma etching, reactive ion etching, and (reactive) ion-beam etching. Another method to obtain relief gratings is deposition and lift-off patterning of a thin cladding layer on the waveguide. Techniques to obtain index-modulation gratings include ion (proton) exchange using a hard mask and indiffusion of a patterned metal layer. 26.4 PASSIVE WAVEGUIDE DEVICES OIC elements which exhibit static characteristics, i.e., those without optical-wave control by an external signal, are called passive devices. Although direct modification into two- dimensional versions from classic bulk components can be used in OIC, there are many cases where such implementation is difficult or results in poor performance. Implementation of waveguide components may require different structures and working principles, but novel functions and improved performance can possibly be obtained by effective use of wave- guides. 26.4.1 Optical Path-Bending Components Implementation of OIC by integration of several components often requires changing optical- path direction or translating paths. Elements for Planar Waveguide. A prism can be implemented by loading a thin film on a triangular region of a waveguide. The wavefront is refracted according to Snell’s law. The deflection angle, however, cannot be large, since the available mode-index difference is small. Large changes of path are realized with geodesic components, in which the ray travels along
  18. 26.18 CHAPTER TWENTY-SIX the geodesic on a concave part produced by deformation of the waveguide plane. Another element is a waveguide end-face mirror, prepared by polishing at a right angle with respect to the guide plane. The path can be bent by the total internal reflection (TIR) at the end face. For deflection larger than the critical angle, the end face should be coated with a reflective (metal) film. Similar TIR can be accomplished by a tapered termination of a guid- ing layer. Mirrors and beam splitters can be obtained by making a ridge in the waveguide, which produces a quasi-abrupt change of the mode index. Reflection- and transmission-type Bragg grating components can also be used for path bending. Bent Waveguides. Path bending for connecting channel-guide components can be accom- plished simply by bending the channel. Although the simplest method is to use corner-bent waveguides, the guided wave suffers a large scattering loss. The loss can be reduced by using a carefully designed multisection corner-bent waveguide. Another method often adopted for connecting two parallel channels with an offset is to use smoothly curved (S- shaped) waveguides. 26.4.2 Power Dividers Power dividers are an important component to divide an optical signal into many branches in optical-fiber subscriber networks. FIGURE 26.18 Basic structures of single-mode branching wave- guides.
  19. 26.19 OPTICAL INTEGRATED CIRCUITS FIGURE 26.19 Waveguide directional couplers. Single-Mode Power Dividers. Figure 26.18a and b shows two-branch waveguides. The waveguide should have a small branching angle and a tapered part for maintaining the fundamental-mode propagation of the incident wave. Although a multibranch waveguide (Fig. 26.18c) can perform multidividing, control of the branching ratio is easier in a tandem two-branch structure (Fig. 26.18d ). Directional couplers, shown in Fig. 26.19, are used as dividers of low insertion loss. The couplers are wavelength-sensitive because their operation is based on phase matching. Greater bandwidth and larger fabrication-error tolerances can be obtained in couplers modified to have variable spacing. Multimode Power Dividers. Multimode branching waveguides suffer from the problem that the dividing ratio is influenced by guided-mode excitation conditions, and, therefore, a mode- mixing region is required to stabilize the ratio. Figure 26.20 illustrates N N power dividers (star couplers) using ion-exchanged glass waveguides.11,12 26.4.3 Polarizers and Mode Splitters Since optical waves transmitted by a fiber are usually elliptically polarized, and many wave- guide devices are polarization-dependent, polarizers / mode splitters are required for filtering single-polarization / mode waves to avoid performance degradation. Polarizers. A metal-clad waveguide is used as a polarizer, since TE and TM modes are transmitted and absorbed, respectively, in the cladded part. The most suitable cladding ma- terial is Al, which has a large value for the imaginary part of the dielectric constant. A typical extinction ratio for a 5-mm-long cladding is 30 dB. Polarizers of smaller insertion loss are obtained by using an anisotropic crystal (calcite, etc.) for cladding as shown in Fig. 26.21.13 In such structures, one of the TE and TM waves is transmitted and the other leaks into the crystal. A high extinction ratio is feasible, although high-grade crystal polishing and complete contact with the guide surface is required.
  20. 26.20 CHAPTER TWENTY-SIX FIGURE 26.20 Multimode star couplers using ion- exchanged glass waveguides.11,12 FIGURE 26.21 Waveguide polarizers using anisotropic crys- tals.13
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