Optical fibers

PC421 Winter 2013 Chapter 3. Optical Waveguides Slide 1 of 57

Chapter 3.Optical Waveguides


Most of the photonic devices that we will study in PC421 incorporate opticalwaveguides - structures that confine light in one or two dimensions while allowingfor propagation in the remaining dimension(s). This geometry permits light to besplit, re-routed, and recombined. In this chapter, we will explore the relation betweena waveguides geometry (size and refractive index) and the properties of the opticalfields that it guides.Most of the chapter will be concerned with slab waveguides, in which confinementexists in only one dimension. These are relatively easy to analyze - although this stillrequire numerical techniques (Matlab alert!) Towards the end of the chapter, we willencounter the more common channel waveguides, in which 2D confinementoccurs. Some approximate solutions will be presented.Much of the material in this chapter is not in Numais text; we draw it from a variety ofsources.


Outline

Table of Contents

1 Symmetric Slab WaveguidesRay AnalysisWave AnalysisGraphical Solution for the Characteristic EquationsNumerical Solution for the Characteristic EquationsMode CutoffLow- and High-Frequency LimitsNormalized ParametersNormalization ConstantMode OrthogonalityConfinement FactorConfinement Limits - Mode Field DiameterTM Modes

2 Asymmetric Slab WaveguidesTE ModesCutoff Condition

3 Bending Loss4 Waveguiding in a Lossy or Gain Medium5 Multilayer Waveguides


Outline

6 Channel WaveguidesMarcatili MethodEffective Index Method (EIM)Examples


Symmetric Slab Waveguides

Symmetric Slab WGs - Ray Analysis TN 3.1 3.2In your earlier optics courses, you learned about Snells law that relates the incidentand transmitted angles of rays at a boundary between media of different refractiveindices n1 and n2:

n1 sin1 = n2 sin2, or 2 = sin1(n1n2

sin1). (3.1)

Suppose that we are incident from medium 1, which has the higher refractive index:n1 > n2. For the inverse sine function above to have a real solution, we require thatthe incident angle be less than a critical angle

c = sin1(n2/n1). (3.2)For larger incident angles, we have a condition of total internal reflection (TIR), inwhich 100% of the the rays power reflects from the boundary.Now, consider the case of a high-index dielectric core of width h sandwichedbetween low-index dielectrics (for historical reasons, these are called the claddingand substrate; in general they can both be thought of as cladding layers) (Fig. 1).We now have two parallel boundaries. If the ray angle is greater than c at bothboundaries, then the ray will be trapped within the high-n material; it propagates byzigzagging between the two boundaries. This is the concept of a dielectricwaveguide, on which all photonic devices in PC421 are based. For the remainder ofthis chapter, we will explore the properties of this guided wave.



Film

Cladding Layer

Substrate

Fig. 1: Cross section of a 2D optical waveguide. TN Fig. 3.3.

(a) (c)(b)

Fig. 2: Propagation modes: (a) radiation mode; (b) substrate radiation mode; (c) guided mode. TN Fig. 3.5.



Symmetric Slab WGs - Ray Analysis cont TN 3.1 3.2We can identify three types of propagation modes, depending on whether or not TIRexists at one or more interfaces (it is assumed that nf > ns > nc , such that if TIRexists at the film-substrate interface, it also exists at the film-cover interface). Thesemodes are shown in Fig. 2. While a few exotic devices make use of substrateradiation modes, we will only be concerned with guided modes in PC421.Although there are a continuum of possible ray angles that produce TIR, there areactually only a discrete number of possible guided modes in any waveguide -possibly just a single one.Consider Figs. 3 and 4. A ray propagating at an angle f to the surface normal willhave a wavenumber k0nf in the core material. The component of this wavenumber inthe direction of the waveguide axis (z) is called the propagation constant :

= k0nf sinf . (3.3)

Now, lets keep track of the accumulated phase shifts as the ray completes one fullround trip between the two interfaces. These are equal to k0nf h cosf (propagationfrom substrate boundary to cover boundary), 2c (the phase shift upon TIR at thefilm-cover interface), k0nf h cosf (propagation from cover boundary to substrateboundary), and 2s (the phase shift upon TIR at the film-substrate interface).



Fig. 3: Coordinate system for a guided mode. TN Fig. 3.7.

Fig. 4: Definition of the propagation constant. TN Fig. 3.8.



Symmetric Slab WGs - Ray Analysis cont TN 3.1 3.2To obtain a lightwave propagating through the optical waveguide without decay, thetotal phase shift in the round trip must be a multiple of 2:

2k0nf h cosf 2c 2s = 2m (m = 0,1,2, ). (3.4)Now, it would seem that there would be an infinite combination of (f , s , c) valuesthat would solve this equation. However, the reflection angles s,c depend on f ,according to the Fresnel equations. Therefore, f uniquely specifies s,c , and thepreceding equation has a finite number of discrete solutions, indexed by the modenumber m. The phase shifts upon reflection can be found on p. 44 of the text.Importantly, they depend on the polarization of the ray. Therefore, so do theallowed values of f .Once the allowed ray angles (which are denoted as m) are found, Eq. (3.3) and Fig.4 can be used to find the allowed propagation constants m, transverse wavenumbers, and effective indices:

m = k0nf sinm, m = k0nf cosm, Nm =mk0

(3.5)

(for the latter, another common notation is neff). It is trivial to show thatnf > N > (nc , ns) in a TIR waveguide.



Symmetric Slab WGs - Wave AnalysisIn this section, we will analyze the waveguiding properties of a dielectric slabwaveguide using Maxwells equations. This structure provides the foundation for allother waveguides that utilize total internal reflection (TIR).Figure 5 shows a simplified view of a slab waveguide. It consists of a high-index coreof thickness h surrounded on both sides by identical semi-infinite lower-indexcladdings of index ncl. We will assume that the waveguide is infinite in extent in they- and z directions, with propagation along z. This implies that the dependence ofthe E and H fields on y is negligible (/y = 0).

Fig. 5: A symmetric slab waveguide



Symmetric Slab WGs - Wave Analysis contIn the last chapter (Eq. 2.20), we derived from Maxwells equation the waveequation,

2E = 00r 2Et2

= 2Et2

. (3.6)For time-harmonic fields, 2/t2 , so we can rewrite the wave equation as

(2 + 2)E = 0. (3.7)By placing the origin (x = 0) at the center of the waveguide, we will exploit thesymmetry of the structure to find even and odd mode solutions. The refractiveindices are ncl for |x | h/2 (the cladding), and nf for |x | < h/2 (the core).The guided modes can have two polarizations, TE (transverse electric) and TM(transverse magnetic). In the former, the E field is parallel to the core-claddingboundary, while in the latter, the H field is parallel to this boundary.A waveguide mode - also called a normal mode or eigenmode - is defined as awave solution to Maxwells equations with all boundary conditions satisfied, for whichthe transverse spatial profile of the fields and the polarization remain unchangedduring propagation. As we have seen, each mode is characterized by a propagationconstant m. It is also characterized by a particular spatial distribution of the EMfield; this latter information is not provided by the ray method of waveguide analysis.



Symmetric Slab WGs - Wave Analysis contFor TE polarization, the electric field has only a y component: E = Ey(x , y , z)y. Aswe have said, the mode propagates along z and has no y-dependence. Therefore, itcan be written E = Ey(x)e iz y (3.8)(the time dependence eit is suppressed here and for most of PC421). Weautomatically know the H field as well, using Maxwells equations for atime-harmonic field - specifically, E = iH, which gives us

H = 1i0

(iEy x +

Eyx

z

)e iz , (3.9)

where we assume that = 0 everywhere. Even and odd modes are sketched in thefigure below.

Fig. 6: Electric field profiles of the first three guided modesin a waveguide. This particular guide is asymmetric(nc , ns ), so the field profiles are asymmetric as well. TNFig. 3-12.



Symmetric Slab WGs - Wave Analysis contTE Even ModesWe can see from eq. (3.7) that the Laplacian (i.e. 2nd derivative) of E is proportionalto -E in each medium. Therefore, E must be oscillatory or exponentiallygrowing/decaying (depending on the sign of the proportionality constant). For awave to be guided, it must have a field solution that is oscillatory inside the coreand exponentially decaying in the cladding. For the even modes, the oscillatoryportion must be a cosine function. Using our slab geometry, we can write

Ey(x) ={

C0e(|x |h/2) |x | h/2,C1 cosx |x | h/2, (3.10)

By substituting this solution into eq (3.7), we see that the variables , , and mustsatisfy

2 + 2 = 200n2f =

2

c2n2f = k 20 n2f (3.11)

2 + 2 = 200ncl2 = 2

c2ncl

2 = k 20 ncl2; (3.12)

notice that the first of these equations agrees with the ray-analysis result, Eq. (3.5).C0 and C1 are constants. Their values are very important! The ratio of C0 to C1 canonly take a discrete set of values while still satisfying the boundary conditions on thefield components.



Symmetric Slab WGs - Wave Analysis contThe first boundary condition is continuity of tangential E field, which in the presentcase means that Ey is continuous at the core-cladding interface. This tells us that

C0 = C1 cos(h2

)(3.13)

(by symmetry, we only need to care about either - but not both - of the core-claddingboundaries). The second boundary condition is continuity of tangential H field,which in the present case means that Hz is continuous at the boundary. This tells usthat

i0C0 =

i0

C1 sin(h2

), (3.14)

which simplifies a bit to

0C0 =

0C1 sin

(h2

). (3.15)

We now have two equations in the two unknowns C0 and C1. These can be written[1 cos h20

0 sinh2

] [C0C1

]=

[00

]. (3.16)



Symmetric Slab WGs - Wave Analysis contThis matrix equation has a solution only if the determinant of the 2 X 2 matrix isequal to zero. Forcing this condition, we obtain the characteristic equation

= tan(h2

). (3.17)

Because the tangent function is periodic, there may exist multiple solutions (that is,pairs of [, ]) to this equation. Each represents a guided TE mode with evensymmetry. The first two of these even modes - m = 0 and 2 - are illustrated in Fig. 6.By the way, we now know the ratio of C0 to C1, but we dont know either oftheir absolute values. Knowing the ratio allows us to eliminate one of theseconstants. The remaining constant is proportional to the amplitude of the fieldthat exists in the waveguide, and is irrelevant to this discussion.



Symmetric Slab WGs - Wave Analysis contTE Odd ModesFor the odd modes, we can write

Ey(x) =

C0e(xh/2) x h/2,C1 sinx |x | h/2,C0e(x+h/2) x h/2.

(3.18)

Using the same procedure of matching the tangential field components (Ey and Hz)at x = h/2, we obtain the characteristic equation

= cot(h2

). (3.19)

This is very similar, but not identical, to the characteristic equation for the TE evenmodes.Before the advent of modern computing, the characteristic equations were solvedgraphically. We will now show how this is done. Then, we will introduce an algorithmfor numerical solution.



Graphical Solution for the Characteristic EquationsThe propagation constant of the guided modes can be found by the followingprocedure. First, multiply the characteristic equations by h/2:

h2 =

h2 tan

(h2

), TE even modes,

h2 cot(h2

), TE odd modes.

(3.20)

Now, subtract Eqn. (3.12) from Eqn. (3.11) to eliminate , and multiply by (h/2)2.This results in(

h2

)2+

(h2

)2= 200(n

2f ncl2)

(h2

)2=

(k0h2

)2(n2f ncl2), (3.21)

where k0 = /c. Now we will plot the curves of Eqns. (3.20) and (3.21) on the samegraph, with normalized coordinates X = h/2 and Y = h/2; that is, we plot

Y ={

X tan X , TE even modes,X cot X , TE odd modes and X

2 + Y2 = R2. (3.22)

Note that the latter equation appears as a circular arc of radius R = k0h2

n2f ncl2.Every point at which the lines intersect represents a guided TE mode, alternatingeven and odd and labeled, in turn, TE0, TE1, TE2, and so on, the index indicating thenumber of zeros in the field profile.



0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

h / 2

h /

2

Fig. 7: A graphical solution to determine and for TE modes. This particular waveguide supports five TE modes, shownby the intersection of the black line with the blue (even) and red (odd) lines.



Numerical Solution for the Characteristic EquationsFrom Eqns. (3.11) and (3.12), we can write

= k0

n2f N2, = k0

N2 ncl2. (3.23)Therefore, finding the discrete Xm or Ym from the graphical method leads easily tothe effective indices Nm and thus the propagation constants m.Also, as we shall soon see, the effective index is bounded below by ncl and above bynf . Therefore, the easiest numerical solution proceeds as follows:

1 Create a vector of trial N that extends from nf to ncl with the desired resolution(usually 104). Call this N.

2 Use the above equations to create corresponding vectors and . These willboth be functions of N.

3 Using Eqns. (3.17) or (3.19), create vectors equal to tan

(h2

)and + cot

(h2

)(3.24)

4 Use any appropriate root finder to locate the values of N for which either of theabove equations are equal to zero. These values of N are the effective indicesof the guided modes, Nm. Note that the functions in step 3 have poles. It maytake a while for you to figure out how to deal with these poles when solving forthe roots.



Mode CutoffA mode is said to be cut off when the wavelength is sufficiently long (i.e. issufficiently low) that the mode is no longer supported by the waveguide. From thegraph, it is clear that the cutoff condition for the TEm mode occurs at R = m/2.That is, k0h

2

n2f ncl2 = m

2 , m = 0,1,2, . (3.25)The implication is that the fundamental (TE0) mode has no cutoff frequency; itexists for any values of h, ,nf , and ncl, as long as the WG is symmetric.In most applications, we desire single-mode operation. That is, we require that ourwaveguide can support exactly one TE mode. In this case, we must have

k0h2

n2f ncl2 N2 > > ncl. (3.30)



Mode Cutoff contBelow cutoff, there exist radiation modes. They are not discrete as are the guidedmodes; rather, they form a continuum. With radiation modes, N < ncl. This meansthat is imaginary. Since the field in the cladding has the form exp(x), animaginary implies that the field is oscillatory in the cladding, rather thanexponentially decaying. This allows power to flow out of the guide duringpropagation. That is, the modes radiate. We wont study radiation modes in PC421,although we will briefly mention them in the next chapter when we discussmultimode interference devices.In your fiber optics course, you may have heard of cladding modes. These aremodes which are guided by TIR at the cladding/air or cladding/jacket interface (thatis, the cladding + core forms an effective waveguide core, with the environment as itscladding). We will not consider these modes any further, as we are dealing withintegrated waveguides, for which cladding modes dont exist unless our waveguide isspecifically designed to support these modes.Cladding modes in optical fiber are very useful for sensor applications, however. Asmall change in the refractive index of the environment around the fiber can lead to anoticeable change in the effective index of the cladding modes.



Low- and High-Frequency LimitsIn the low-frequency limit, all modes m > 0 can be cut off. At the cut-off point, wehave (k0h/2)

n2f ncl2 = m/2 and = 0, as can be seen from the graphical

solution. At this point, h/2 = m/2, and from eq. (3.11), =

00ncl = ncl/c. (3.31)

Therefore, the propagation constant of the mode approaches that which wouldbe encountered in the bulk cladding material. This is to be expected; at cutoff,the mode power is no longer localized at the core - nearly all of it propagates in thecladding.In the high-frequency limit, R . The graphical solution shows that, for the TEmmode,

(h2

) (m+1)2 and

(h2

). Therefore, from eq. (3.11), =

00nf = nf/c. (3.32)

Here, virtually all of the power is localized to the core, so the mode propagationconstant is identical to that of the bulk core material.These two extreme cases, and the transition between them, are illustrated on thefollowing slide.



Fig. 8: Dispersion curves of the guided TEm modes. They are confined to the region between the light lines of the claddingand core material. In the low-frequency limit, they approach the former and in the the high-frequency limit, they approach thelatter. The light lines should actually be slightly curved because the core and cladding material is dispersive. The region abovethe cores light line is forbidden, as such modes would not have an oscillatory profile in the core. The region below thecladdings light line represents the continuum of radiation modes. The cutoff frequencies of modes 1 and 2 are also shown.



Normalized ParametersEarlier, we saw that the propagation constant is related to the effective index as

N =

k0. (3.33)

Furthermore, we have just seen that nf > N > ncl; that is, the effective index isbounded by the cladding and core material indices. This is a property of all TIRwaveguides.Another widely-used parameter is the normalized propagation constant b. It takesa value between zero and one:

b = N2 ncl2

n2f ncl2. (3.34)

This parameter is plotted on the next slide, as a function of the normalizedfrequency V , where

V = k0h

n2f ncl2. (3.35)



Fig. 9: (a) Effective index of the TEm modes vs. frequency. The cut-off frequencies of the TE1 and TE2 modes areindicated. Normalized propagation parameter b vs. normalized frequency V for the TEm modes.



Normalization ConstantMany slides ago, we learned that the ratio of C0 to C1 must take certain values inorder to satisfy the boundary conditions on the fields at the core-cladding interface.The absolute value of these constants does not affect the mode condition. However,it is sometimes helpful to set the constants such that the modes are normalized.That is, so that their total guided power is unity.Because the mode propagates in the z-direction, we can write this power using thePoynting vector:

P = 12Re

(E H) zdx = 1. (3.36)As mentioned previously, for the TE modes we have Hx = Ey , and thus

P = 12

EyHxdx =

20

Ey 2 dx . (3.37)Carrying out the integral (which must be separated into its core and claddingportions), we find that for both even and odd modes,

C1 =

4(h + 2

) . (3.38)



Mode OrthogonalityThe mode solutions of a waveguide possess the property of orthogonality. Wewont prove it here, but it was probably covered in an earlier math class - for arelated example, review your PC321 notes where it was shown that the solutions tothe Schrdinger equation are also orthogonal. Mathematically, orthogonality of themodes is expressed as

12

"

(Em Hn) z dx dy = mn, (3.39)where m and n are the indices of any two modes and mn is the Kronecker deltafunction (equal to 1 if m = n and 0 otherwise).Physically, the result of mode orthogonality is that we can consider all guided modesto evolve along the propagation direction independently from each other. If wecouple light into only the nth mode of a waveguide, only the nth mode will emergefrom the other end. On the other hand, an arbitrary input can be decomposed into alinear combination of the orthogonal modes, propagated independently, and thenre-composed at the output. Since the modes have different m, they will de-phaseduring propagation, and the re-composed output field profile will no longer resemblethe input profile. In the next chapter, we will see that many devices operate on theprinciple of modal interference, or modal coupling (wherein a z-dependent geometryproduces a controlled coupling among modes).



Confinement FactorThe optical confinement factor is the fraction of the total power residing in the core.This is a very important concept in active photonic devices as it is generally only thecore material that provides gain, while the cladding is transparent. is calculated as:

=

12

coreRe(E H) z dx

12

totalRe(E H

) z dx=

core

E(x)2 dxtotal

E(x)2 dx . (3.40)Substituting the expressions for E and H from the beginning of this chapter, we findthat

=

1 +(

2h

)cos2(h/2)[1 + sin(h)h

]1

even TE modes

=

1 +(

2h

)sin2(h/2)[1 sin(h)h

]1

odd TE modes.

(3.41)

We see that the confinement factor is a function of the core width h, the materialindices, and the wavelength. It is implicitly a function of the mode number m, since and are m-dependent. For a mode that is very far from cutoff (such as thefundamental mode in a highly multimode waveguide), is very large and approaches 1. On the other hand, close to cutoff, rapidly becomes small and ishighly reduced; here, much of the guided mode extends into the cladding.



Confinement Limits - Mode Field DiameterFor many applications, we wish for the guided mode(s) to have as small a spatialextent as possible. There are many reasons for this, most of which are beyond thescope of this course. However, it should be obvious that if the guided fields arenarrow, we can have a denser packing of devices on our optical chips.Because the modes exhibit an exponential decay in the cladding, they technicallyhave an infinite extent. However, we can quantify the mode field diameter as beingthe core width plus the 1/e distance on either side. That is,

MFD = h + 2

or MFD = h + 11

+12

, (3.42)

for symmetric and asymmetric guides, respectively (the latter case will be discussedin a moment). These equations are valid for any mode (as long as you use theappropriate ), but are commonly applied only to the fundamental mode.When the core width h is several , the MFD scales with h. However, this is not thecase for core widths that are on the order of or smaller. Here, there is a rapidincrease in MFD, as shown on the next 2 slides. The reasons for this increase aretwofold. First, the portion of the field in the core is subject to the diffraction limit:MFD > /(2nf ). Second, as the core width decreases, the WG becomes weaker,increasing .



0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

h/

MFD

/

Fig. 10: Mode field diameter (blue) vs. core thickness (both normalized by wavelength). Here, nf = 1.55 and ncl = 1.54.The green line is the limit of MFD=h. The exact position of the minimum MFD is a function of the core-cladding indexdifference, but it generally lies around h/ = 1.



Fig. 11: As the core width (yellow shaded area) decreases, the spatial extent of the mode decreases as well, up to a limit.Further decrease results in a rapid expansion of the mode, with a loss of confinement within the core.



TM ModesRecall from the beginning of this chapter that for TE polarization, we had

E = Ey(x)e iz y and H =1

ii

(iEy x +

Eyx

z

). (3.43)

For TM polarization, the magnetic field has only a y component: H = Hy(x)e iz y. ByAmpres equation,

H = Et

, (3.44)we can write

E = 1i

(iHy x

Hyx

z

). (3.45)

The guided modes are found as before, by matching the tangential fields (in thiscase, Hy and Ez) at the core-cladding interface. The resulting characteristicequations are

h2 =

(nclnf

)2 (h2

)tan

(h2

)even TM modes (3.46)

h2 =

(nclnf

)2 (h2

)cot

(h2

)odd TM modes



TM Modes and BirefringenceThese equations look identical to those for the TE modes, other than the (ncl/nf )2factor. In fact, in the TE case, we had ratios of cl/f = 0/0, which cancelled out.The graphical and numerical methods for finding the TM effective indices areidentical to those used in the TE case, as long as the (ncl/nf )2 factor is included.Because ncl < nf by definition (otherwise we wouldnt have internal reflection), it canbe seen by comparing the characteristic equations that we will always have, for agiven mode number m, NTEm > NTMm . (3.47)We can therefore define the waveguides birefringence,

Bm = NTEm NTMm . (3.48)This indicates that if we input a signal with random polarization into the waveguide,the TE and TM components will drift out of phase as the signal propagates.Although a few devices exploit this fact, most photonic devices - particularly passiveones - are desired to operate in a polarization-insensitive fashion. For thesedevices, we need to design our waveguides with B as low as possible.The figure on the next slide shows the confinement factors of the first few modes forboth TE and TM polarizations; note that there is a slight difference between them,and that the difference reduces to zero at the low- and high-frequency limits. The TEmode always has slightly better confinement within the core.



Fig. 12: Optical confinement factors for the TEm modes (solid) and TMm modes (dashed).


Asymmetric Slab Waveguides

Asymmetric Slab Waveguides - TE ModesIt is not always the case that the dielectric constant in the upper and lower claddingis equal. For example, if we deposit a thin layer of high-index cover material onto alow-index substrate, then this forms a waveguide in which the upper cladding is air.From a technological standpoint, this is the simplest type of slab waveguide. It canbe analyzed using ray analysis, but we will only use wave analysis in PC421.Consider the waveguide shown in Fig. 13, which has different cladding indiceslabeled ns and nc . Due to the asymmetry, the field solutions will not be even orodd modes. It no longer simplifies the situation to place the origin in the middle ofthe core. Therefore, we place it at one boundary. The general solution for TE modesis of the form

Ey(x) =

C1e1x x 0,C2 cos(x) + C3 sin(x) 0 x h,C4e2(xh) x h.

(3.49)

Because Ey satisfies the wave equation in each of the three regions, we have

21 + 2 = 200n2s = (k0ns)2

2 + 2 = 200n2f = (k0nf )

2

22 + 2 = 200n2c = (k0nc)2 .(3.50)



Asymmetric Slab Waveguides - TE Modes contHere, we have four unknowns (C1,C2,C3, and C4). The four equations we use tosolve these modes are the continuity of tangential fields Ey and Hz at both of of thecore-cladding boundaries. This can be set up as a 4 4 determinant equation, but itcan also be solved step-by-step as shown here.At x = 0, continuity of Ey simply gives us C1 = C2, while continuity of Hz results inC3 = (1/)C1.

Fig. 13: An asymmetric dielectric slab waveguide.



Asymmetric Slab Waveguides - TE Modes contWe can re-write the general solution as

Ey(x) =

C1e1x x 0,C1

[cos(x) + 1 sin(x)

]0 x h,

C4e2(xh) x h,(3.51)

where we can identify C1 as the Ey amplitude at the x = 0 interface. Next, we applycontinuity of Ey at x = h, which requires that C4 = C1

[cos(h) + 1 sin(h)

],

allowing us to update the solution as

Ey(x) = C1

e1x x 0,[cos(x) + 1 sin(x)

]0 x h,[

cos(h) + 1 sin(h)]e2(xh) x h.

(3.52)

Finally, we impose continuity of Hz at x = h, which requires that

sin(h) + 1 cos(h) = 2[cos(h) +

1

sin(h)]. (3.53)

Dividing both sides by cos(h) and rearranging, we are left with the characteristicequation

tan(h) =1 + 2

(1 12

2

) . (3.54)



Asymmetric Slab Waveguides - TE Modes contThe preceding equation can be inverted, to remind us that there may be multiplesolutions - the TEm modes:

h = tan1(1

)+ tan1

(2

)+ m, m = 0,1,2, . (3.55)

If we require that the modes be normalized to unit power, then

C1 =

40(h + 11 +

12

) . (3.56)Without proof, we provide the characteristic equations for TM modes:

tan(h) =(

n2fn2s1 +

n2fn2c2

)

2 n4f

n2s n2c12

(3.57)

h = tan1(n2f 1

n2s

)+ tan1

(n2f 2

n2c

)+ m, m = 0,1,2, . (3.58)



Asymmetric Slab Waveguides - Cutoff ConditionFor the wave to be guided, nf must be greater than both ns and nc . Lets assume thatns > nc . When reducing until = k0ns , 1 will vanish before 2 does. Physically,we have reached a point where TIR only occurs at one of the core-claddingboundaries (the one with nc ; the mode is leaky at the other boundary). At this point,

h = tan1(2

)+ m. (3.59)

The tan1 term represents a shift (to the right) of the curves that we found in thegraphical solution (Fig. 7). As a result, for an asymmetric waveguide, there existsa cutoff frequency even for the TE0 mode.Dispersion curves for the guided modes of an asymmetric waveguide are shown onthe next slide. Note that the curves are confined between the light lines of the coreand the higher-index cladding.



Fig. 14: Dispersion curves for modes of an asymmetric dielectric waveguide. The light lines should actually be curvedbecause the core and cladding material is dispersive.


Bending Loss

Bending LossA straight waveguide made from lossless materials will in theory exhibit nopropagation loss - this is why the numerical mode solving technique producedreal-valued or N. Whenever a waveguide is curved, we have what is known asbending loss. There are two ways to describe this phenomenon qualitatively. Fig.15 uses the ray picture, in which a ray that bounces along the core-cladding interfacein a curved waveguide will eventually hit the interface at an angle that does not resultin TIR; therefore, it radiates. A weakly-guided WG will suffer more from bending lossbecause the range of allowable TIR angles is much smaller.

Fig. 15: Explanation of bending loss by ray analysis (blue/green) and wave analysis (red). R is the radius of curvature of thebent waveguide. SOK Fig. 2-32.


Bending Loss

Bending Loss contA wave analysis can also be used to explain bending loss. In a straight waveguide,the phasefronts of the mode all propagate with a velocity c/N, which by definition isbetween c/nf and c/ncl. In a curved waveguide, the portions of the mode that lietoward the inside edge of the curve will move slower, while the portions that lietoward the outside edge must move faster; this increase and decrease depends onthe distance from the core. Because the mode extends an infinite distance from thecore, there must be a portion of the mode that is being asked to propagatefaster than the speed of light in the cladding. As this is impossible, that portion ofthe mode radiates. Again, for weakly-guided WGs, the evanescent tail extendsfarther into the cladding, where it is more susceptible to bending losses.In fact, when a waveguide is curved, its propagation constant and mode profile areperturbed as well. Even when the waveguide is symmetric, the mode will shift towardthe outside edge of the curve. We will return to this concept in the next chapter.Solving the eigenproblem for a curved waveguide generally involves solving for thestraight waveguide but in a linearly distorted spatial domain. This wont be attemptedin PC421.In the next chapter, we will see that curved waveguides are often required. In thiscase, we must balance the problem of bending loss (for small-radius bends) with thatof propagation loss and device size (for large-radius bends).


Bending Loss

Bending Loss contBending loss is a highly nonlinear function of the bend radius, in that as the bendradius is decreased, the bending loss remains low up until a critical radius, Rc , atwhich point it increases dramatically. The loss coefficient (in units of inverse length,as with absorption) is given by

B = CeR/Rc , (3.60)where C is a constant. As B is large when R < Rc , it is imperative for designers ofphotonic devices to know the critical radius for a particular waveguide, and to ensurethat all curves exceed this radius.Rc depends strongly on the index difference between the core and cladding, as wellas the core thickness. For weak WGs (such as doped glass), it can be on the orderof several millimeters, while for strong WGs (such as semiconductor ribs), it can be afew tens of microns.

0 0.5 1 1.5 2 2.5 320

15

10

5

0

R/Rc

loss

[dB]

Fig. 16: Bending loss for a curve of constant angle, for various bending radii around the critical radius. Note the log scale.


Waveguiding in a Lossy or Gain Medium

Waveguiding in a Lossy or Gain MediumWhen the permittivity of the waveguide materials are complex,

r = rr + iir , (3.61)

(i.e. the material is lossy (ir > 0) or it has gain (ir < 0)), the propagation constantbecomes k = k r + ik i - or, using a Taylor expansion,

k = 0(rr + iir)

0rr

(1 + i

ir

2rr

)= k r i g2 , (3.62)

where k r = 0rr = k0n is the wave vector in the absence of gain or loss, and

g = k r(ir/rr) is the gain coefficient (which is negative, representing loss, if ir > 0).With semiconductor lasers, it is a common situation to have gain in the core and aslight bit of loss in the cladding. In that case, it can be shown that

(0) ig2 + i(1 )

2 . (3.63)In other words, the real part of the propagation constant is unchanged by thepresence of gain or loss, while the imaginary part depends on how much of themode overlaps the lossy and gainy regions. It is most important to note that a modepropagating along a waveguide with a gain coefficient g in the core only experiencesa gain coefficient of g.


Multilayer Waveguides

Multilayer Waveguides TN 3.2The methods that we have seen for finding the guided modes of a slab waveguidecan be extended to the analysis of a multilayer waveguide - one which consists ofan arbitrary number of layers. The details of the analysis are beyond the scope ofPC421, but they can be found on pp. 50-51 of Numai. Essentially, we expresstangential field continuity at the boundary between the ith and (i 1)th layers as a2 2 matrix, then we take the product of each of these matrices to define a matrixfor the waveguide as a whole, from which a characteristic equation is derived.This is only important to us because active (and some passive) photonic devicesnecessarily require multiple layers.


Channel Waveguides

Channel Waveguides TN 3.3Nearly all useful photonic devices use channel waveguides rather than slabwaveguides. Here, light is confined in two transverse dimensions while it propagatesin the third dimension (which is z, by convention).If you found the numerical procedure required to solve slab guides difficult, then Imsorry to tell you that the situation here is much worse. Whereas the numericalsolutions for slab guides were exact, there is no equivalent procedure to exactlysolve for the modes and propagation constants of even the simplest channelwaveguides other than cylindrical fibers. The main problem is that there is no suchthing as a purely TE or TM mode; all channel modes must be of mixed polarization(a cylindrical fiber supports pure radially or azimuthally polarized modes, but theseare in fact equal mixtures of TE and TM). The polarization mixing increases as thestrength of the waveguiding increases and as the aspect ratio becomes moresymmetric (that is, square waveguides have a higher degree of polarization mixingthan do rectangular waveguides).In the next few slides, we will examine two approximate methods for solving channelwaveguides. The first method is somewhat intuitive, but it is limited in scope. Thesecond method is an extension of the slab technique that we have already covered.


Channel Waveguides

Marcatilis Method TN 3.3Consider the channel waveguide shown in Fig. 17. We assume that w > h. Therefractive indices in the corners are not shown because they are irrelevant inMarcatilis method. The rectangular shape of the core ensures that certainpolarization components are more dominant than others. The two types of mode arecalled:

HEpq modes Hx and Ey are the dominant components,EHpq modes Ex and Hy are the dominant components.

The indices p and q indicate the number of nulls in the field in the x andydirection.

Fig. 17: Geometry used in Marcatilis method.


Channel Waveguides

Marcatilis Method cont TN 3.3To solve for the HEpq modes, we write equations for the Hx field component in the 5regions of the waveguide, and assume that Hx = 0 in the corner regions. Along bothdirections, the field is oscillatory in the middle portion of the guide andevanescently decaying in the outer portions. In other words, we assume that:

Hx = e iz

C1 cos(xx + x) cos(yy + y) region 1,C2e2(xw) cos(yy + y) region 2,C3 cos(xx + x)e3(yh) region 3,C4e4x cos(yy + y) region 4,C5 cos(xx + x)e5y region 5,

(3.64)

where 2x + 2y + 2 = (k0n1)2, 2x 22 + 2 = (k0n2)2, and so on.Then we get to apply boundary conditions!Then we get to use Maxwells curl equations to find all of the other field components!Then we do it all over again starting with Ex to find the EHpq modes!

Lets not attempt this any further.


Channel Waveguides

Effective Index Method (EIM) TN 3.3In the effective index method, we convert the 2D problem into a succession of 1Dproblems (which weve already learned how to solve). Here, instead of finding HEand EH modes, we seek quasi-TE and quasi-TM modes. That is, we assume thatthe aspect ratio is large enough - and that the guiding is weak enough - such that thepolarization is primarily of one type.The geometry is shown in Fig. 18. It is identical to that of Marcatilis method, exceptthat we must specify the refractive indices in the corners, since we do not assumethat the field is zero in these regions.

Fig. 18: Geometry used in the Effective Index Method.


Channel Waveguides

Effective Index Method (EIM) cont TN 3.3To find the guided modes, as a first step we separate the 9-section waveguide intocolumns, each of which can be solved individually as a slab guide to find its owneffective index. Then, in a second step, we treat these three effective indices as thecore and claddings of a slab waveguide, which is solved to give the overall effectiveindex. This is best explained using the figure on the next slide.It is important to realize that you must flip your polarization between the twosteps. This is because a polarization that is parallel to the boundaries in step 1 isperpendicular to the boundaries in step 2, and vice versa.The EIM isnt used just for rectangular guides. It is also quite useful for finding theeffective indices of rib or ridge guides, which we will encounter later on. In fact, it caneven be used to analyze waveguides that have a smoothly varying index distribution.Instead of treating them as a 3 - by - 3 problem as in the figure on the previous slide,we turn it into an N - by - N problem. The slab methods we learned earlier in thechapter can be extended to the case where there are multiple layers of core.The EIM can be used to find effective indices, but it is poor at determining modeprofiles. In class, we will see an example to illustrate why this must be the case.


Channel Waveguides

Fig. 19: Effective index method. (top) slab waveguides corresponding to the left, middle, and right columns of the channelguide. (bottom) the corresponding horizontal slab waveguide.


Channel Waveguides

ExamplesRealistically, all waveguide designers use computational techniques such as thefinite difference method, the finite element method, or the imaginary-distancebeam propagation method. Here, the waveguide cross-section is discretized into agrid containing thousands of small elements, and variations of Maxwells equationsare solved in each element, pieced together by boundary conditions. The followingfigures show some mode intensity profiles obtained using these methods. For allfigures, nf = 1.55,ncl = 1.50, = 1.55 , with core dimensions of 5 10 m.

Fig. 20: Intensity profile of EH00 mode. Fig. 21: Intensity profile of EH10 mode.


Channel Waveguides




Channel Waveguides

Examples contIn actual fact, relatively few channel waveguides have rectangular cross-sections.Several other geometries are shown in the following figures (in the figures, n1 = n2).The mode is confined to the shaded regions).Fig. 26 is a strip-loaded waveguide. Transverse confinement is produced in a slabgeometry, while lateral confinement is achieved by a narrow, low-index strip abovethe core. Even though the core layer has no transverse structure, the evanescent tailof the mode sees enough of the strip that lateral confinement is achieved. Thisshould be obvious if you analyze this WG using the EIM.Fig. 27 is a rib waveguide. Mmmmnn, ribs... ... Sorry about that...its like a ridgewaveguide except that the strip is made from the core material (we etch away thecore material outside of the core region). The confinement isnt as strong as with theridge or channel WGs, but there is less propagation loss, as the mode doesntinteract very strongly with material boundaries (which produce scattering losses).Fig. 28 is a ridge waveguide. It is similar to a channel waveguide but with a lowercladding index (usually air) above and beside the core. This configuration is usedwhen we want very strong confinement (small core, tight bending radius, etc.)


Channel Waveguides

Fig. 26: Strip-loaded waveguide. Source: JML

Fig. 27: Rib waveguide. Source: JML

Fig. 28: Ridge waveguide. Source: JML Fig. 29: Diffused waveguide. Source: JML


Channel Waveguides

Examples contFig. 29 is a diffused waveguide. A dopant material is diffused into the substratethrough a patterned mask, resulting in a graded-index profile n(x , y) rather than adiscrete core/cladding boundary. Examples include Ti:LiNbO3 (titanium-diffusedlithium niobate) and ion-exchanged glass (in which network modifiers such assodium are replaced by other ions such as silver).Obviously, the methods described in this chapter can not be used to analyzediffused waveguides since there is an index gradient. Slab diffused WGs (wheren = n(x)) can be solved using analytical methods for some index profiles, but ingeneral they must be solved numerically.Later in this course, we will look at semiconductor lasers and light-emitting diodesthat incorporate optical waveguides. We will see that their geometries are muchmore complicated than those shown on the previous slide. This is because we mustconfine electric current and charge carriers in addition to the optical mode.

Symmetric Slab WaveguidesRay AnalysisWave AnalysisGraphical Solution for the Characteristic EquationsNumerical Solution for the Characteristic EquationsMode CutoffLow- and High-Frequency LimitsNormalized ParametersNormalization ConstantMode OrthogonalityConfinement FactorConfinement Limits - Mode Field DiameterTM Modes

Asymmetric Slab WaveguidesTE ModesCutoff Condition

Bending LossWaveguiding in a Lossy or Gain MediumMultilayer WaveguidesChannel WaveguidesMarcatili MethodEffective Index Method (EIM)Examples

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Optical fibers