In Chapter 3, detailed discussion of the polarization mode dispersion PMD and dispersion control in single-mode fibers are added together with the comprehensive treatment of the PCFs. Four-wave mixing FWM that has been added to Chapter 5 is an important nonlinear effect especially in wavelength division multiplexing WDM systems. High-index contrast PLCs such as Silicon-on-Insulator SOI waveguides are becoming increasingly important to construct optoelectronics integrated circuits.
In order to deal with high-index contrast waveguides, semi-vector analysis becomes prerequisite. Moreover, comprehensive treatment of the finite difference time domain FDTD method is introduced in Chapter 7. Readers will acquire comprehensive understanding of the operational principles in various kinds of flat spectral-response AWGs.
Origin of crosstalk and dispersion in AWGs are described thoroughly. I am indebted to a large number of people for the work on which this second edition of the book is based. First, I should like to thank the late Professor Takanori Okoshi of the University of Tokyo for his continuous encouragement and support.
I am also grateful to Prof. Finally, I wish to express my hearty thanks to my wife, Kuniko, and my sons, Hiroaki and Masaaki, for their warm support in completing the book. June Katsunari Okamoto Chapter 1 Wave Theory of Optical Waveguides The basic concepts and equations of electromagnetic wave theory required for the comprehension of lightwave propagation in optical waveguides are presented.
The light confinement and formation of modes in the waveguide are qualitatively explained, taking the case of a slab waveguide.
The refractive index of the core n1 is higher than that of the cladding n0. Therefore the light beam that is coupled to the end face of the waveguide is confined in the core by total internal reflection. Each mode is associated with light rays at a discrete angle of propagation, as given by electromagnetic wave analysis. Here we describe the formation of modes with the ray picture in the slab waveguide [1], as shown in Fig.
The phase fronts of the plane waves are perpendicular to the light rays. Before describing the formation of modes in detail, we must explain the phase shift of a light ray that suffers total reflection. The reflection coefficient of the totally reflected light, which is polarized perpendicular to the incident plane plane formed by the incident and reflected rays , as shown in Fig. Let us consider the phase difference between the two light rays belonging to the same plane wave in Fig.
Light ray PQ, which propagates from point P to Q, does not suffer the influence of reflection. On the other hand, light ray RS, Figure 1. Wave Theory of Optical Waveguides 4 propagating from point R to S, is reflected two times at the upper and lower core—cladding interfaces.
Substituting Eqs. The optical field distribution that satisfies the phase-matching condition of Eq. Figure 1. In the figure the solid line represents a positive phase front and a dotted line represents a negative phase front, respectively.
The electric field amplitude becomes the maximum minimum at the point where two positive negative phase fronts interfere. In contrast, the electric field amplitude becomes almost zero near the core—cladding interface, since positive and negative phase fronts cancel out each other. Formation of Guided Modes Figure 1. It is known from Fig. The waveguide operates in a single mode for wavelengths longer than c. When the electromagnetic fields e and h are sinusoidal functions of time, they are usually represented by complex amplitudes, i.
This expression is not strictly correct, so when we use this phasor expression we should keep in mind that what is meant by Eq. In most mathematical manipulations, such as addition, subtraction, differentiation and integration, the replacement of Eq.
However, we should be careful in the manipulations that involve the product of sinusoidal functions. In these cases we must use the real form of the function 1. The most common type of boundary condition occurs when there are discontinuities in the dielectric constant refractive index , as shown in Fig. Equations 1. There are also natural boundary conditions that require the electromagnetic fields to be zero at infinity.
Therefore, the left-hand side of Eq. In this equation, e and h denote instantaneous fields as functions of time t. Let us obtain the average power-flow density in an alternating field. Theory of Dielectric Optical Waveguides.
New York: Academic Press. Born, M. Principles of Optics. Oxford: Pergamon Press. Tamir, T. Integrated Optics. Berlin: Springer-Verlag. Marcuse, D. Light Transmission Optics. New York: Van Nostrand Rein-hold.
Stratton, J. Electromagnetic Theory. New York: McGraw-Hill. Chapter 2 Planar Optical Waveguides Planar optical waveguides are the key devices to construct integrated optical circuits and semiconductor lasers. Generally, rectangular waveguides consist of a square or rectangular core surrounded by a cladding with lower refractive index than that of the core.
Three-dimensional analysis is necessary to investigate the transmission characteristics of rectangular waveguides.
However, rigorous threedimensional analysis usually requires numerical calculations and does not always give a clear insight into the problem. Therefore, this chapter first describes twodimensional slab waveguides to acquire a fundamental understanding of optical waveguides. Then several analytical approximations are presented to analyze the three-dimensional rectangular waveguides.
Although these are approximate methods, the essential lightwave transmission mechanism in rectangular waveguides can be fully investigated. The rigorous treatment of three-dimensional rectangular waveguides by the finite element method will be presented in Chapter 6. Derivation of Basic Equations In this section, the wave analysis is described for the slab waveguide Fig.
Putting Slab Waveguides 15 these relations into Eqs. As shown in Eq. Since the electric field lies in the plane that is perpendicular to the z-axis, this electromagnetic field distribution is called transverse electric TE mode.
Since the magnetic field lies in the plane that is perpendicular to the z-axis, this electromagnetic field distribution is called transverse magnetic TM mode. Planar Optical Waveguides 16 2.
Here the derivation method to calculate the dispersion equation also called the eigenvalue equation and the electromagnetic field distributions is given. We consider the slab waveguide with uniform refractive-index profile in the core, as shown in Fig.
There is another boundary condition, that the magnetic field component Hz should be continuous at the boundaries. Hz is given by Eq. Neglecting the terms independent of x, the boundary Figure 2.
Using Eqs. Equations 2. As is known from Fig. Rewriting the dispersion Eq. Computation of Propagation Constant First the graphical method to obtain qualitatively obtain the propagation constant of the symmetrical slab waveguide is shown, and then the quantitative numerical method to calculate accurately the propagation constant is described.
The relationship between u and w for the symmetrical slab waveguide, which is shown in Eq. Transverse wavenumbers u and w should satisfy Eq. This relation is Planar Optical Waveguides 20 Figure 2.
The solutions of the dispersion equation are then given as the crossing points in Fig. The propagation constant or eigenvalue is then obtained by using Eqs.
In Fig. Generally, the cutoff v-value for the TE mode is given by Eq. However, in order to obtain accurate solution of the dispersion equation, we should rely on the numerical method. Here, we show the numerical treatment for the symmetrical slab waveguide so as to compare with the previous graphical method. We first rewrite the dispersion Eq. The solution of Eq. Here, the subroutine program of the most simple bisection method is shown in Fig.
The normalized propagation constant b is calculated for each normalized frequency v. Figure 2. Planar Optical Waveguides 22 Figure 2. The mode number is expressed by the subscript m, such as TEm or TMm mode. The parameter in Fig. Electric Field Distribution Once the eigenvalue of the waveguide is obtained, the electric field distribution given by Eq.
Constant A is determined when we specify the optical power P carried by the waveguide. Power P is expressed, by using Eq. For the TE mode we can rewrite Eq. The power Figure 2. Slab Waveguides 25 confinement factor in the core is important to calculate the threshold current density Jth of semiconductor lasers [4]. The confinement factor is calculated by using Eqs. The vertical lines in the figure express the single mode core width.
We first express the magnetic field Figure 2. Rectangular Waveguides 27 modes. It is known that the normalized propagation constant b for the TM mode is smaller than that for the TE mode with respect to the same v.
That means the TE mode is slightly better confined in the core than the TM mode. The power carried by the TM mode is obtained, from Eqs. Basic Equations In this section the analytical method, which was proposed by Marcatili [5], to deal with the three-dimensional optical waveguide, as shown in Fig. The important assumption of this method is that the electromagnetic field in the shaded area in Fig.
Then we do not impose the boundary conditions for the electromagnetic field in the shaded area. On the other y hand, the modes in Eqs. In the following section, the solution method x of the dispersion equation for the Epq mode is described in detail, and only the y results are shown for the Epq mode. Rectangular Waveguides 29 y 2. To the contrary the mode number m in Eq.
Therefore in the mode definition by Marcatili, integers p and q represent the number of local electric field peaks along the x- and y-axis directions. In other words, the hybrid modes in the rectangular waveguides are approximately analyzed by separating into two independent slab waveguides as shown in Fig. It is well understood x mode with when we compare the dispersion Eqs.
Equation 2. Kumar et al. Although the approximation is good, there still remains the small difference in the refractive-index expression of Eq. Rectangular Waveguides 33 from the actual value. In the present method, the correction is made by using the perturbation method as shown in the following. Dividing Eq. The line integral of Eq. Then 0 is the eigenvalue for the dispersion Eqs. The eigenvalue which is given by the first-order perturbation is therefore expressed from Eqs.
The normalized propagation constant is obtained from Eqs. Analysis by the point matching method [7] gives the most accurate value among four of the analyses and is used as the standard for the comparison of accuracy.
The effective index method will be described in the following section. For practicality, however, the effective index method is a very important method to analyze, for example, ridge waveguides which require a numerical method such as the finite element method. Rectangular Waveguides 37 Figure 2. After Ref. Effective Index Method The ridge waveguide, such as shown in Fig. In order to analyze ridge waveguides, we should use numerical methods, such as the finite element method and finite difference method.
The effective index method [8]—[9] is an analytical method applicable to complicated waveguides such as ridge waveguides and diffused waveguides in LiNbO3. In the following, the effective index x method of analysis is described, taking as example the Epq mode in the ridge waveguide.
Planar Optical Waveguides 38 Figure 2. Substituting Eq. We first solve Eq. The dispersion equations for the four-layer slab waveguide shown in Fig. The solution of the wave equation 2. The radiation field is different from the field in the waveguide.
Therefore it is important to know the profile of the radiation field for efficiently coupling the light between two waveguides or between a waveguide and an optical fiber. In this section, we describe the derivation of the radiation field pattern from the rectangular waveguide. Fresnel and Fraunhofer Regions We consider the coordinate system shown in Fig.
Therefore if z is larger than, for example, 1 mm, any term higher than the fourth term in the right-hand side of Eq. The radiation field where this condition is satisfied is called the farfield region or Fraunhofer region.
On the other hand, when z is not so large, we should take into account up to the fourth term in Eq. The radiation field in which this condition holds is called the near-field region or Fresnel region. However, we should note that even the Fresnel approximation is not satisfied in the region close to the waveguide endface.
The fourth term in the extreme right of Eq. When we apply the Fraunhofer approximation to r, Eq. Radiation Field from Waveguide 43 2. Radiation Pattern of Gaussian Beam It has been described that the radiation pattern from the waveguide is expressed by Eq. Therefore it is not easy to calculate the radiation pattern from the rectangular waveguide analytically.
Here we approximate the electric field distribution in the rectangular waveguide by a Gaussian profile to obtain the radiation pattern analytically. Since the integral in Eq. The electric field distribution of the waveguide is calculated by using the finite element method waveguide analysis, which will be described in Chapter 6, and is Gaussian fitted to obtain the spot sizes w1 and w2 along x- and y-axis directions.
The key structure of an MMI device is a waveguide designed to support a large number of modes. The width, thickness and length of the multimode region are W , 2d and L, respectively.
Single-mode waveguide with core width 2a and thickness 2d is connected to the multimode waveguide. Refractive indices of the core of single-mode and multimode waveguides are equal to n1 and the refractive index of the cladding is n0.
Three-dimensional waveguide structure can be reduced to a two-dimensional problem by using an effective index method. The effective index neff of the core is calculated by solving the eigen-mode equation along y-axis. Electric field in the multimode waveguide is calculated by using Eqs. Here waveguide parameters n1 , a and ns in Eqs. Then the electric field amplitude Am is obtained from Eq. Since there is a Goos-Hanshen effect, light field slightly penetrates into cladding region.
Therefore, it is known that the excess loss of each output port is less than 1 dB. BPM simulation is required in order to accurately determine the 3-dB coupler configuration.
Planar Optical Waveguides 52 Input Figure 2. It is known from the figure that coupling ratio of MMI 3-dB coupler is almost insensitive to wavelength as compared to codirectional coupler. This is a great advantage of MMI 3-dB coupler over a codirectional coupler. However, 1. First, insertion loss of MMI is not zero even in theoretical simulation. Theoretical insertion loss of the MMI 3-dB coupler is about 0.
Second, the insertion loss of MMI coupler rapidly increases as wavelength departs from the optimal wavelength. Complex electric field gi in the i-th input waveguide is g0 g15 a g0 g15 b Figure 2. In the BPM simulation shown in Fig. The device was fabricated using silica-based planar lightwave circuits. Phase shift was introduced to the waveguide by the thermo-optic effect. It is confirmed that the collective summation of complex electric fields is obtained by using MMI combiner.
New York: Van Nostrand Reinhold. Theory of Optical Waveguides. Unger, H. Planar Optical Waveguides and Fibers. Oxford: Clarendon Press.
Kressel, H. Marcatili, E. Dielectric rectangular waveguide and directional coupler for integrated optics. Bell Syst. Kumar, A. Thyagarajan and A. Analysis of rectangular-core dielectric waveguides—An accurate perturbation approach. Goell, J. A circular harmonic computer analysis of rectangular dielectric waveguides. Knox, R. Integrated circuits for the millimeter through optical frequency range. Bryngdahl, O. Image formation using self-imaging techniques.
Ulrich, R. Image formation by phase coincidences in optical waveguides. Niemeier, T. Veerman, F. Schalkwijk, E. Pennings, M. Smit and B. An optical passive 3-dB TMI-coupler with reduced fabrication tolerance sensitivity. IEEE J. Lightwave Tech. Bachmann, M. Besse and H. Heaton, J. General matrix theory of self-imaging in multimode interference MMI couplers.
IEEE Photon. Soldano, L. Optical multi-mode interference devices based on self-imaging: Principles and applications. Okamoto, K. Yamada and T. Chapter 3 Optical Fibers Silica-based optical fibers are the most important transmission medium for long-distance and large-capacity optical communication systems. The most distinguished feature of optical fiber is its low loss characteristics. The lowest transmission loss ever achieved is 0.
This means that the signal intensity of light becomes half of the original strength after propagating 20 km along the optical fiber. Together with such low loss properties, low dispersion is also required for signal transmission.
Signal distortion due to dispersion of the fiber is closely related to the guiding structure of optical fibers. In order to realize low-dispersion fibers, it is necessary to understand the transmission characteristics of fibers and to design and analyze the arbitrarily shaped guiding structures. In this chapter, first a rigorous analysis of step-index fiber is presented, to understand the basic properties of optical fibers. Then linearly polarized LP modes, which are quite important mode designations for the practical weakly guiding fibers, will be described.
The derivation of dispersion equations is explained in detail in order to understand the dispersion characteristics of fibers. Signal transmission bandwidths of graded-index multimode fibers and single-mode fibers are discussed and compared in connection with their dispersion values. Finally, the principle of polarization-maintaining properties in birefringent fibers and their polarization characteristics are described. In the following, electromagnetic fields, dispersion equations, and propagation characteristics of optical fibers are described in detail for step-index fibers as shown in Fig.
Then it is known that integer n should be zero for Eq. The solutions for Eq. Hybrid Modes In hybrid modes the axial electromagnetic field components Ez and Hz are not zero. Therefore solutions for Eq. Also it is known from Eq. Although Eq. In these cases Eq. The constant C in the electromagnetic field expressions 3. TE Modes The transmission power in the core and cladding are calculated from Eqs.
TM Modes Substituting Eqs. Hybrid Modes The analytical expressions of the power flow for the hybrid modes are rather complicated. Linearly Polarized LP Modes 71 3. This approximation allows us to simplify the analysis of optical fibers drastically and enables us to obtain quite clear results.
Therefore this approximation is called weakly guiding approximation. We will study LP modes in detail in the following sections. This equation is rewritten by using the recurrence relation of Bessel functions 3. The modes corresponding to the plus sign of Eq.
The mode designations of EH and HE have no particular rationale and are merely conventional. Historically, microwave engineers first referred to the lowest-order mode mode without cutoff in a dielectric rod waveguide as the HE11 mode [3, 8]. Later, designations of HE and EH modes were derived in accordance with this custom. It is worth stating, however, that in EH modes the axial magnetic field Hz is relatively strong, whereas in HE modes the axial electric filed Ez is relatively strong.
Therefore equations representing the electromagnetic fields 3. First, Eq. Then the recurrence relations for Bessel functions 3. Considering the similarity between the LP modes in Table 3. It is known from Eq. Of course, since they are not the strict modes, the eigenvalues are slightly different in the rigorous dispersion equations.
Therefore LP modes are approximate modes classified by the eigenvalues. Table 3. This means that the propagation constants of these mode groups are nearly degenerate. Figure 3.
Linearly Polarized LP Modes 75 respectively. It is known that the intensity profiles of transverse electric fields Ex or Ey belonging to the same LP mode have the same distribution.
Dispersion Characteristics of LP Modes The propagation constant of the step-index fiber is obtained by solving the dispersion equation in Table 3. The crossing points of the vertical curves and the semicircle give the set of eigenvalues u and w. The cutoff v-values are obtained by the following Figure 3.
They are calculated by numerically solving the dispersion equations Table 3. The horizontal axis of Fig. When we use the mode parameter m defined by Eq. Therefore it is indispensible to understand the propagation characteristics of the HE11 mode in order to grasp the signal transmission properties of single-mode fibers.
Here the dispersion characteristics and electromagnetic field distributions of the HE11 mode are investigated in detail. In order to rewrite Eqs. Equation 3. Fundamental HE 11 Mode 81 a. It is known that Ex is the dominant mode in Eq.
Therefore, the polarization mode in Eq. Actually, in the LP mode weakly guiding approximation , those minor components are neglected. Though the eigenvalue of the HE11 mode is given by the numerical solution of Eq. The error of the eigenvalue u given by Eq. The variation of group velocity with the signal frequency wavelength is called group velocity dispersion GVD.
It should be noted here that even though the optical power itself is not dissipated, the signal can be lost by the fiber dispersion as shown in Fig. In practical fibers, signal energy is further lost by the attenuation of the fiber. Therefore there are two factors limiting the transmission distance or repeater spacing of the signal in the fiber: dispersion and attenuation. Two cases are considered, depending on the dominance of the effects.
First is a case where attenuation is larger than dispersion. For example, for the case shown in Fig. In contrast is the second case, where dispersion is larger than attenuation. As shown in Fig.
The reduction in the attenuation of optical fibers owes much to improvements in fabrication technologies, and the ultralow attenuation of 0. In contrast, the 86 Optical Fibers Figure 3. The spectral intensity distribution of the pulse repetition sequence [Fig.
One is a RZ return-to-zero pulse waveform as shown in Fig. From the relation 3. It is known that the bandwidth occupation in the NRZ pulse is half that in the RZ pulse when transmitting the same bitrate B. When the signal delay time is different with respect to the different spectral components, which is caused by modulation or spectral broadening of the source itself, the signal waveform is distorted at the receiver end of the single-mode fiber.
In the multimode fiber, the group velocity itself may be different from mode to mode, and this causes the pulse broadening. There are four kinds of delay-time dispersion of fibers. Dispersion Characteristics of Step-index Fibers Figure 3. Multimode Dispersion Multimode dispersion is the delay-time dispersion caused by the difference of group velocity of the various modes [the first term in rightmost equation of Eq.
Therefore multimode dispersion is caused by the difference of inclination between many modes. The slight birefringence in the single-mode fiber is caused by the non concentricity of the core and the elliptical deformation of the core.
The next two dispersions exist even when the fiber is truly single-moded. They are caused by the last term in Eq. Dispersion Characteristics of Step-index Fibers 91 3. Material Dispersion Material dispersion is a delay-time dispersion caused by the fact that the refractive index of the glass material changes in accordance with the change of the signal frequency or wavelength.
Optical Fibers 92 3. Waveguide Dispersion Waveguide dispersion is a delay-time dispersion caused by the confinement of light in the waveguide structure. Therefore waveguide dispersion is an essential dispersion that inevitably exists in waveguides. Derivation of Delay-time Formula Let us calculate the first term of Eq. The refractive index of silica glass is well approximated by the Sellmeier polynomial [12] " 3!
For example, the Sellmeier coefficients of Ge O2 doped glass with 6. Group indices N1 and N0 are calculated from Eqs. Ge O2 -doped glass and a pure silica cladding. First, the unified dispersion equation 3. The differentiation of both terms in Eq. In such a condition, it is known from Eqs. However, there are several hundreds of modes [refer to Eq. Moreover, the optimum refractive-index profile of a multimode fiber, in which multimode dispersion becomes a minimum, is a quadratic-index profile rather than a step-index profile.
The WKB Wentzel—Kramers—Brillouin method is a suitable way to analyze the propagation characteristics and dispersion properties of graded-index fibers. Analysis of graded-index fibers using the WKB method will be described in Section 3. Chromatic Dispersion The sum of material dispersion and waveguide dispersion is called chromatic dispersion. Group delay-time dispersion is given, as shown in Eq.
Differentiation of Equation 3. In Eq. However, the third term in Eq. Therefore, here the third term is defined as the waveguide dispersion and the sum of the first and second terms is defined as the material dispersion, respectively.
Although the pulse broadening due to chromatic dispersion is given Dispersion Characteristics of Step-index Fibers Figure 3. Dotted lines indicate the refractive indices of core and cladding. As shown in Figs. The dependence of the waveguide dispersion on the normalized frequency v, which is shown in Fig. Dotted lines indicate the group indices of core and cladding. Dotted lines indicate the normalized dispersions of core and cladding.
Zero-dispersion Wavelength Chromatic dispersion in a single-mode fiber is the sum of material dispersion and waveguide dispersion, as shown in Eqs. As described in the previous section, the waveguide dispersion can be controlled by proper choice of the waveguide parameters, while the material dispersion is almost independent of these parameters.
Figures 3. Basic Equations and Mode Concepts in Graded-index Fibers First, the basic equations for wave propagation in graded-index fibers are derived. The detailed treatment is described in Ref. Waves satisfying condition 1 correspond to EH modes, and those satisfying condition 2 correspond to HE modes, which are identified by comparing the electromagnetic fields of these modes with those of step-index fibers. Optical Fibers Comparing Eqs. Therefore, it is usually much more convenient to solve the wave equation in the core and cladding separately an analytical solution is obtained in the cladding, since the refractive index is uniform and to match those solutions at the core—cladding boundary in accordance with the physical boundary conditions.
The WKB method is suitable for the analysis of multimode fibers of large v-value. On the other hand, the FEM is suitable for the analysis of single-mode graded-index fibers, since it requires a numerical calculation. Therefore, we can assume in Eq. Then Eq. Henceforth, we proceed as if Eq. From Eq. This relation is illustrated in Fig. The factor 4 in the right-hand side of Eq. Note also that the summation for m is replaced by Optical Fibers Figure 3. Exchanging the order of integration, we can rewrite Eq.
The usefulness of this profile lies in the fact that it approximates the actual profiles found in many fibers. Optical Fibers Substituting Eqs. Dispersion Characteristics of Graded-index Fibers The group delay time in optical fibers is given by Eq. In multimode fibers, the first term in the right-hand side equation has a dominant effect over multimode dispersion.
Wave Theory of Graded-index Fibers More precisely, the multimode dispersion is a function of the delay time t. Ge O2 -doped core fiber. Therefore, the transmission bandwidth of graded-index fiber refer to the Section 3. As described in Section 3. Combining Eqs. We will investigate the signal transmission capacity for several typical cases in multimode and single-mode fibers.
There are two possible situations, depending on the magnitude of spectral broadening due to the laser itself and the signal modulation. The signal frequency range of an NRZ pulse is given by Eq.
Two Orthogonally-polarized Modes in Nominally Single-mode Fibers In the axially symmetric single-mode fiber, there exist two orthogonally polarized modes, as shown in Section 3.
They are known as HEx11 and HEy11 modes in accordance with their polarization directions. If the fiber waveguide structure Birefringent Optical Fibers is truly axially symmetric, these orthogonally polarized modes have the same propagation constants and thus they are degenerate. In such fibers, the state of polarization SOP of the output light y randomly varies, since the mode coupling take place between HEx11 and HE11 modes, which is caused by fluctuations in core diameter along the z-direction, vibration and temperature variations.
Therefore, such fibers cannot be used for applications in optical fiber sensors and in coherent optical communications in which SOP and interference effects are utilized. Birefringent fibers have been proposed and fabricated to solve such polarization fluctuation problems [19].
The difference of the propagation constants y between HEx11 and HE11 modes are intentionally made large in birefringent fibers.
Birefringent fibers are also called polarization-maintaining fibers. The strength of the mode coupling is proportional to the magnitude of the spatial frequency component of the fluctuation.
The power spectrum of the core fluctuation is considered to be the same as that of Fig. It is known that the power spectrum of fluctuation is high for the low-spatial-frequency components and decreases rapidly for the high-spatial-frequency components.
Then, if we make a birefringent fiber having Figure 3. Birefringent fibers are classified into two categories: 1 geometrical birefringence type and 2 stress-induced birefringence type [21]. In the geometrical birefringence type, the birefringence is produced by the axially asymmetrical core or core vicinity structures. To the contrary, birefringence in the stress-induced birefringence type is generated by a nonsymmetric stress in the core.
In the geometrical birefringence fiber, however, there generally exists nonsymmetrical stress distributions if there is an axial nonsymmetry in the waveguide structure. Therefore, practically it is quite difficult to realize the purely geometrical birefringence type fibers. On the other hand, Figs. The dark regions in these figures are stress-applying parts in which doped silica glasses having large thermal expansion coefficients are inserted. When stressapplying parts are cooled after the fiber drawing, they apply a large stress or strain to the core since they shrink tightly.
Generally, a large tensile force is generated along the x-axis direction and a compressive force along the y-axis direction [21], respectively, since stress-applying parts are allocated in the xaxis direction. The refractive index of the glass under stress is changed by the photoelastic effect [28]. Therefore the refractive indices of the core for HEx11 and HEy11 modes become different due to the difference in the stress in the core region. The power densities of the core fluctuations for such highspatial-frequency components are confirmed to be quite small in Fig.
Therefore, mode coupling is well suppressed in birefringent fibers. For the strict analysis of such waveguides, we should rely on the numerical analysis using, for example, the finite element method [15]. Although FEM analysis gives rigorous results, it does not always give a clear insight since it requires numerical calculations. Therefore, here the analytical method based on the perturbation method will be described.
Here, the geometrical effect of nonsymmetrical core structure is Optical Fibers considered in K. The solutions of Eq. The second term represents the birefringence caused by the nonsymmetrical stress distribution and is called stress-induced birefringence.
On the other hand, stress-induced birefringence Bs is calculated by the finite element stress analysis. Substituting this into Eq. Elliptical-core Fibers We first define the geometry of an elliptical-core fiber, as shown in Fig. The core radii along the x- and y-axis directions are ax and ay , where ax is assumed to be larger than ay.
The refractive indices of core and cladding are denoted by n1 and n0. The rigorous analysis is called the vector analysis, since it rigorously considers the orientation of electric field vector. The collocation method [30] and vectorial finite element method are known as vectorial analyses.
The perturbation method analysis [33] is also applicable to calculate the geometrical birefringence, under the restriction that the ellipticity of the core is sufficiently small. Even for a relatively large ellipticity, the Eq. Normalized geometrical birefringences in Fig. The stress-induced birefringence Bs is obtained from Eqs. Bg and Bs in the elliptical-core fiber have the same sign and then contribute to the total birefringence additively.
It is seen from Figs. Optical Fibers Figure 3. Therefore, the elliptical-core fiber is not always the geometrical birefringence type of fiber. Polarization Mode Dispersion The polarization mode dispersion, which is given by the delay time difference between HEx11 and HEy11 modes, is expressed from Eqs. Behavior of pulse waveform in the two-mode optical fibers, in which two modes exchange their energy through mode coupling, was investigated in Ref. For the two limiting Optical Fibers Figure 3.
In the limit of negligible mode coupling the principal states of polarization become the polarization modes of the fiber, and the differential delay time between the principal states is determined simply by the difference in group velocity for the two polarization modes.
In the limit of extensive mode coupling, the principal states and differential delay time in a fiber are no longer correlated with local fiber properties since they depend on the collective effects of the random mode coupling over the entire propagation path.
From Figures 3. It is known that when we connect Dispersion-shifted Fibers Dispersion-shifted fibers based on step-index profile are already described in Section 3. Therefore it is susceptible to large bending loss because light confinement to the core becomes weak. Moreover, since step-index DSF has small core diameter, spot size of the light field becomes small.
Nonlinear optical effect in optical fibers depends on intensity of the light field as described in Chapter 5. Nonlinear optical effects normally cause signal waveform distortion through self-phase modulation and four-wave mixing effect, which will be described in Chapter 5. In order to enlarge the effective core area of DSFs, various kinds of refractiveindex configurations have been proposed.
Dual-shape core fiber has pedestal region and segmented core fiber has outer ring region so as to tightly confine the light field. The role of these regions is twofold: one is to reduce the bending loss and the other is to enlarge the effective core area. In large dispersion-slope fibers, chromatic dispersion quickly becomes large when signal wavelength goes apart from the zero-dispersion wavelength.
Then high bit-rate signals, which have broad frequency spectra, tend to suffer signal distortions. Large dispersion slope fibers also cause problem in wavelength division multiplexing WDM systems.
Signal channels that are separated from the zerodispersion wavelength suffer higher signal distortion due to large chromatic dispersions. Dispersion Flattened Fibers As discussed in the previous section, minimizing the dispersion slope is important for high bit-rate transmissions and WDM systems.
Optical fibers that have very small dispersion over a wide wavelength range are called dispersion flattened fibers DFFs. The first proposal of DFF was made by Okamoto [41, 42] using single-mode fibers with W-type refractive-index profiles [43].
In order to enlarge the effective core area while maintaining low dispersion over a wide spectral range, a quadruple-clad QC DFF was proposed [44]. It is confirmed that dispersion slopes in 1. In WDM systems, however, inter-channel nonlinear optical effects such as four-wave mixing refer to Section 5.
Crossphase modulation is always accompanied by self-phase modulation SPM and occurs because the effective refractive index of a wave depends not only on the intensity of that wave but also on the intensity of other co-propagating waves in WDM.
Broadly dispersion compensating fibers BDCFs [46, 47] were proposed to reduce the nonlinear signal impairment while maintaining the flatness of dispersion. Chromatic dispersion of BDCF is designed such that it almost completely cancels the dispersion of SMF when two fibers are connected with a certain ratio of lengths.
Then, nonlinear interactions between WDM channels are minimized. Though local dispersions are large, the total dispersions for all WDM channels are made almost zero as shown in Fig. Therefore, ultra-high bit-rate signals can be transmitted over long distances without suffering inter-channel nonlinear signal impairments. Yeh et al. In Bragg fibers, light cannot penetrate into cladding since light is reflected by the Bragg condition.
Then, light beam, which is coupled into Bragg fiber at the input end, propagates along the fiber. Since light is confined by the Bragg condition, refractive index of the core could be lower than that of the cladding or core could be air [49, 50]. The forbidden frequency ranges in periodic dielectric structures of cladding are called photonic bandgaps.
Bragg fibers use a one-dimensional transverse periodicity of concentric rings. There is another class of fibers that use a two-dimensional transverse periodicity [51, 52].
A solid-core PCF refracts light at steep angles of incidence on the core—cladding boundary. When the angle is shallow enough, light is trapped in the core and guided along the fiber. A hollow-core PCF with a proper cladding can guide light at angles of incidence where a photonic band gap operates. Therefore, hollow-core PCF requires strict control of the periodic cladding structures.
Photonic Crystal Fibers Figure 3. Light guiding principle of the solid-core PCF is basically similar to that of the traditional optical fibers. However, there is no definite boundary between solid-core region and air-hole cladding region. It was found that the azimuthally discontinuous core—cladding interface has a very important effect on the guidance of light field [53].
The FSM is the fundamental mode of the infinite photonic crystal cladding if the core is absent, so FSM is the maximum allowed in the cladding. Mode field distribution was calculated by using FEM refer to Chapter 6. The chromatic dispersion characteristics of PCF are distinctively different from that of the traditional optical fibers.
This is mainly due to the fact that the waveguide dispersion in PCF becomes very large. In Figure 3. Inset of Fig. Bulk silica glass has normal negative dispersion in this range. As the air-holes become bigger the core becomes increasingly isolated and begins to look like a strand of silica glass sitting in the air.
The waveguide dispersion of such a strand is very strong compared with the material dispersion. Origin of large waveguide dispersion in PCF with large air-hole diameter and the step-index air-clad silica fiber can be explained to be large refractive-index difference between silica core and air cladding [56].
One of the most interesting and useful applications of PCF is to the generation of broad-band supercontinuum light from short-pulse laser systems [57]. This is made possible by a combination of ultra-small cores and dispersion zeros that can be shifted to coincide with the pump laser wavelength.
Broad-band supercontinuum light is very important not only in telecommunications but also in applications such as optical coherence tomography [58] and frequency metrology [59]. In the derivation of Eqs. In order to further simplify Eqs. The first term in Eq. Since electromagnetic field Ep and Hp become zero when line integral is carried out on the peripheral sufficiently far from the core center, Eq.
In case of the circular symmetric fiber, HEx11 and HEy11 are degenerated and have the same propagation constants. Therefore, we can not prove the orthogonality relation from Eq. In such a case, however, we can derive the orthogonality relation by directly putting the electromagnetic fields for HEx11 and HEy11 modes [Eqs.
Kanamori, Y. Ishiguro, G. Tanaka, S. Tanaka, H. Takada, M. Watanabe, S. Suzuki, K. Yano, M. Hoshikawa, and H. Ultra low-loss pure silica core single-mode fiber and transmission experiment. Digest of Opt. Fiber Commun. Cylindrical dielectric waveguide modes. Optical Fibers. Handbook of Mathematical Functions. New York: Dover Publications. Theory of Bessel Functions. New York: Cambridge University Press. Asymptotic expression for eigenfunctions and eigenvalues of dielectric optical waveguides.
He joined Ibaraki Electrical Communication Laboratory, Nippon Telegraph and Telephone Corporation, Ibaraki, Japan, in , and was engaged in the research on transmission characteristics of multimode, dispersion-flattened single-mode, single-polarization PANDA fibers, and fiber-optic components.
As for the dispersion-flattened fibers DSF , he first proposed the idea and confirmed experimentally. From September to September , he joined Optical fiber Group, Southampton University, Southampton, England, where he was engaged in the research on birefringent Bow-tie optical fibers.
He published more than papers and authored or co-authored 8 books. A new chapter on arrayed-waveguide grating discusses the principles of operation, fundamental characteristics and analytical treatment of the grating demultiplexing properties. We are always looking for ways to improve customer experience on Elsevier.
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