集成光学 第三章

Theory of Optical Waveguides •The electromagnetic wave theory of the physical-optic approach is developed in detail.
•Emphasis is placed on the two basic waveguide geometries that are used most often in optical integrated circuits, the planar waveguide and the rectangular waveguide.
在本章中将学到
•如何解波动方程并导出介质波导模式方程•介质波导中消逝场的特性
•对称波导与非对称波导的截止条件
•如何分析矩形波导
重点掌握波导中消逝场的特性、对称波导与非对称波导的截止条件、2种沟道波导的结构
•rectangular waveguide：矩形波导•standing waves：驻波•permeability：磁导率•transcendental equation：超越方程•confining layer：约束层•waveguiding layer：导引层•evanescent “tail” ：消逝“尾”模•extinction coefficient：消逝系数•symmetric waveguide：对称波导•asymmetric waveguide：非对称波导
•channel waveguide：条形波导
•strip-loaded waveguide：沟道波导
•metal strip-loaded waveguide：金属狭缝沟道波导
•dielectric strip-loaded waveguide：介质条沟道波导
3.1 Planar Waveguide
3.1.1The Basic Three-Layer Planar Waveguide •Considering the basic three-layer waveguide
structure shown in Fig. 3.1. The light confining layers, with indices of refraction n1and n3, are
assumed to extend to infinity in the +x and -x
directions, respectively.
•There are no reflections in the x direction to be
concerned with, except for those occurring at
the n1-n2and n2-n3interfaces.
x=0 x= -t n1
n2
n3
x
y
z
Fig. 3.1 Basic three-layer planar waveguide structure
For the case of TE plane waves traveling in the z direction, with propagation constant β, Maxwell’s wave equation (2.2.1) reduces to with solutions of the form
...3,2,1,22222=∂∂=∇i t E c n E y i y )()(),,(z t i y y e x t z x E βωε-=(3.1.1)
(3.1.2)
•For TE waves, it will be recalled that E x and
E z are zero. Note that in (3.1.2) that
has no y or z dependence because the planar layers are assumed to be infinite in these directions, precluding the possibility of reflections and resultant standing waves.
)(x y
⎪⎩
⎪⎨⎧++-=)](exp[)sin()cos()exp()(t x p D hx C hx B qx A x y ε)()0()0(t x x t x -≤≤-∞≤≤-∞≤≤The transverse function has the general
form
)(x y ε(3.1.3)
where A, B, C, D, q , h and p are all constants that can be determined by matching the boundary conditions, which requires the continuity [3.1]
of and .
)(x y ε()()z y H i x x ωμε=∂∂
Since the permeability μand frequency ωare assumed to be constant, the second condition
translates into a requirement that be
continuous. The constants A, B, C and D can be
determined by making and
continuous at the boundary between Region 1 and Region 2 (x =0), and continuous at x = -t . The procedure provides three equations in four
unknowns, so that the solution for can be
expressed in terms of a single constant C ')(x y ε)(x y εx y ∂∂εx y ∂∂ε
⎪⎩
⎪⎨⎧++'-'-'=)](exp[)]sin()()[cos()]
sin()()[cos()exp(t x p ht h q ht C hx h q hx C qx C y ε)()0()0(t x x t x -≤≤-∞≤≤-∞≤≤c
k k n p k n h k n q ωβββ=-=-=-=212232212222212212)()()(Substitute (3.1.4) into (3.1.2), using the resulting expression for E y (x ,z ,t ) in (3.1.1) for each of the regions, obtaining (3.1.4)
(3.1.5)
By making continuous at x = -t yield the condition
x y ∂∂ε)]
sin()()[cos()cos()()sin(ht h q ht p ht h q h ht h +=----or after simplification )1()tan(2h pq h q p ht -+=(3.1.6)
The transcendental equation (3.1.6), in conjunction with (3.1.5), can be solved either graphically, by plotting right and left sides as a function of βand noting the intersection points, or numerically on a computer. Regardless of the method of solution, the result is a set of discrete allowed values of β, corresponding to the allowed modes. For each βm , the corresponding values of q m , h m and p m can be determined from (3.1.5).
The one remaining unknown constant C 'in (3.1.4) is arbitrary. However, it is convenient to normalize so
that represents a power flow of one Watt per unit which in the y direction. Thus, a mode for
which has a power flow of W/m.)(x y ε)(x A y y εε=2
A Based on normalization condition, one gets
))(11(222
m m m m m m m q h p q t h C +++='βωμ(3.1.7)
For the case of TM modes, the development exactly parallels that which has just been performed for the TE case, except that the non-zero components are H y , E x , and E z rather than E y , H x , and H z .
•场在区域1和区域3是指数式衰减的，衰减的快慢分别由q、p决定。

导波虽然以传输常数 沿波导向z方向传输，但场穿入了光约束层。

q、p值大，即场衰减越快，穿透深度1/q 、1/p就浅。

光约束层中的场称为消逝场，q、p为消逝系数。

•t+1/q+1/p称为有效厚度。

3.1.2 The symmetric Waveguide
•A special case of the basic three-layer planar waveguide that is of particular interest occurs when n1equals n3. The equations developed in
Sect. 3.1.1 apply to this type of waveguide, but a major simplification is possible in the determination of which modes may be supported. In many cases, it is not required to know the ’s for the various modes. The only question being whether the waveguide is capable of guiding a
particular mode or not.
At cutoff (the point at which the field becomes oscillatory in Region 1 and 3) for TE modes, the magnitude of βis given by
3
1kn kn ==β(3.1.8)Substituting (3.1.8) into (3.1.5),one finds 2322212
20
n n k n n k h q p -=-===(3.1.9)
Substituting (3.1.9) into (3.1.6) yields the condition or ,
m s =0,1,2,3,… , that is 0)tan(=ht πs m ht =π
s m t n n k =-2122
Thus, for waveguiding of a given mode to occur, one must have
)(412220212n n t m n n n s +>-=∆λm s =0,1,2,3,...(3.1.10)
The cutoff condition given in (3.1.10) determines which modes can be supported by a waveguide with a given ∆n and ratio of λ0/t . It is interesting to note that the lowest-order mode (m s =0) of the symmetric waveguide is unusual in that it does not exhibit a cutoff as all other modes do. In principle, any
wavelength could be guided in this mode even with an incrementally small ∆n .
If n 2≈n 1, the cutoff condition (3.1.10) becomes
2
2
20
2128n t m n n n s λ>-=∆or, if n 2>>n 1, it is given by
22
20
2124n t m n n n s λ>-=∆m s =0,1,2,3,...
3.1.3 The Asymmetric Waveguide •Another important special case of the three-layer planar waveguide is asymmetric waveguide, in which n3>>n1. It is possible to derive for the case of the asymmetric guide an approximate closed form expression for the cutoff by using a geometrical argument comparing it to the symmetric guide.
Consider an asymmetric waveguide that has a thickness t equal to half the thickness of a corresponding symmetric waveguide. The two lowest order TE modes of the symmetric guide (m s=0, 1) and of the asymmetric guide (m a=0, 1) are both shown in the Fig.3.2.. Note that, for well-confined modes, the lower half of the m s=1 mode of the symmetric guide corresponds closely to the m a=0 mode of the asymmetric guide of half thickness. This fact can be used as a mathematical device to permit one to obtain a closed form expression for the cutoff condition in the case of the asymmetric waveguide.
Symmetric
waveguide Mode correspondence
Asymmetric
waveguide Fig. 3.2. Diagram of the modes in symmetric and asymmetric planar waveguides
Solving (3.1.6) for the case of the symmetric waveguide with thickness equal to 2t in the same fashion as in Sect.
3.1.2 yields the condition
23220232)2)((4t n n m n n n s +>-=∆λm s =0,1,2,3,...However, the asymmetric waveguide supports only modes corresponding to the odd modes of a symmetric guide of twice its thickness. Hence, the cutoff
condition for the asymmetric guide is given by
23220232)(16t n n m n n n a +>-=∆λwhere m a are the elements of the subset consisting of odd values of m s . This can be conveniently expressed by
(3.1.11)
)12(+=m m a m =0,1,2,3,...
Assuming that n 2≈n 3(3.1.11) becomes 222
023232)12(t n m n n n λ+>-=∆(3.1.12)
while the cutoff conditions (3.1.10) and (3.1.12) are valid only for the special cases defined, they offer a convenient means to estimate how many modes can be supported by a particular waveguide. Although (3.1.12) has been derived for the case of TE waves, it can be shown that it also holds for TM waves as long as n 2≈n 3. Thus, the asymmetric waveguide is seen to have a possible cutoff for all modes. This makes the asymmetric waveguide particularly useful as an optical switch.
3.2 Rectangular Waveguides
The planar waveguides discussed in the previous section are useful in many integrated optic applications in spite of the fact that they provide confinement of the optical fields in only one dimension. Some complex optical integrated circuits can be fabricated using planar waveguides. However, other applications require confinement in two dimensions. Sometimes two-dimensional confinement is required merely to guide light from one point on the surface of an OIC to another, to interconnect two circuit elements.
3.2.1 Channel Waveguide
•The basic rectangular waveguide structure consists of a waveguide region of index n1 surrounded on all sides by a confining medium of lesser index n2, as shown in Fig. 3.3. Such waveguides are often called channel guides, strip guides, or 3-dimensional guides.
•It is not necessary that the index in the confining media be the same in all regions.
Fig. 3.3. Basic rectangular dielectric waveguide structure
Fig. 3.4 Cross-section view of rectangular dielectric waveguide bounded by regions of smaller index of refraction
Marcatili has derived an approximate solution to the rectangular channel waveguide problem, by analyzing the structure shown in Fig.3.4.
The key assumption made in Marcatili’s analysis is that the modes are well guided, i.e. , well above cutoff, so that the field decays exponentially in Regions2,3,4, and 5, with most of the power being confined to Region 1. The magnitudes of the fields in the shaded corner regions are small enough to be neglected. Hence, Maxwell’s equations can be solved by assuming relatively simple sinusoidal and exponential field distributions and by matching boundary conditions only along the four sides of Region 1.
Fig. 3. 5 Sketch of a typical mode y E 11
The waveguide is found to support a discrete number of
guided modes that can be grouped into two families, and , where the mode numbers p and q correspond to
the number of peaks in the field distribution in the x and y directions, respectively.
The transverse field components of the modes are
E x and H y , while those of the mode are E y and H x .x pq
E y pq E x pq
E y pq E The (fundamental) mode is sketched in Fig.3.5. Note that the shape of the mode is characterized by extinction coefficients, η2, ξ3, η4, and ξ5in the regions where it is exponential , and by propagation constants k x and k y in Region1.y E
11
The field components in the five regions shown in Fig.3.4 (designated by ν=1,2,3,4,5) have the form
=νy H y x H k i H x z z ∂∂∂-=νν2y x H k n E x z x ∂∂∂-=νννωε2201νννννωεx z y y H k n k
n k E 202
22-=y H n i E x z ∂∂=νννωε20(3.2.1-6)
⎪⎪⎪⎩⎪⎪⎪⎨⎧++-+-++++-=)exp()cos()exp()cos()exp()cos()exp()cos()cos()cos()exp(5544
33221x ik y k M y ik x k M x ik y k M y ik x k M y k x k M t i z ik H x y y x x y y x y x z xv βαβαβαω5
4321=====ννννν
where M νis an amplitude constant,ωis the angular frequency and ε0is the permittivity of free space.The phase constants, αand βlocate the field maxima and minima in Region1, and k x νand k y ν(ν=1,2,3,4,5) are the transverse propagation constants along the x and y directions in the
various media. Matching the boundary conditions requires the assumption that and
x x x x k k k k ===421y
y y y k k k k ===531Also, it can be shown that 222
1y
x z k k k k --=
is the propagation constant of a plane wave with free-space wavelength λ0in a medium of refractive index n 1.Assuming that n 1is only
slightly larger than the other n ν( ),
as is usually the case in an OIC, leads to the condition 1)(15,4,3,21<<-n n n z
y x k k k <<,where
1
0112n kn k λπ
==
Note this expression corresponds, in ray-optics terminology, to a grazing incidence of the ray at the
surfaces of the waveguiding Region1. Calculations show
that the two significant components for the modes
are H x and E y .
y pq E Matching field components at the boundaries of Region 1 yields the transcendental equations
)(tan )(tan )
(tan )(tan 42124122
122
15131ηηπξξπy y y x x x k n n k n n q b k k k p a k ------=--=where the tan -1functions are to be taken in the first
quadrant, and where
(3.2.7)
5
,4,3,2,2)(11)(112210221224,24,24,2225,35,35,3=-=-=-==
-==
νλπ
πηπξνννn n k k A k A k k A k y y x x The transcendental equations (3.2.7) cannot be closed solved exactly in closed form. However, one can assume for well confined modes that most of the power is in Region 1. Hence,(3.2.8)1)(1
)(24,225,3<<<<ππA k A k y x (3.2.9)
Using the assumptions of (3.2.9), approximate solutions of (3.2.7) for k x and k y can be obtained
by expanding the tan -1functions in a power series, keeping only the first two terms. Thus,
121424222153)1()1(--++=++=b n A n A n b q k a
A A a p k y x ππππ(3.2.10)
One can obtain
21221424222424242212535353532214242222253221111111)1()()1()(----⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡⎪⎪⎪⎪⎭⎫ ⎝⎛++-=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎪⎪⎪⎪⎭⎫ ⎝⎛++-=++-++-==b n A n A n b qA A a A A a pA A b n A n A n b q a A A a p
k k z ππηππξππππβ(3.2.11)
The modes are polarized such that E y
is the only
significant component of electric field; E x and E z are
negligibly small. It can be shown for the case of the modes that E x is the only significant electric field
component, with E y and E z being negligible.y pq E x pq
E To develop relationships for the modes corresponding
to those that have been derived from the modes, one can merely change E to H , μ0to -ε0, and vice-verse in the various equations. As long as the assumptions is made
that n 1is only slightly larger than the indices of the surrounding media, then k z , , and are still given by (3.2.11), for the modes just as for the modes.x pq
E y pq
E 53ξ42ηx pq E y pq E
Marcatili’s analysis of the rectangular three-dimensional waveguide is very useful in designing such structures, even though it features an approximate solution to Maxwell’s equations. It must be remembered that the theory assumed well confined modes. When waveguide dimensions a and b are small enough, compared to the wavelength, the theory becomes inaccurate for that mode.
3.2.2 Strip-Loaded Waveguides
•It is possible to make a three-dimensional waveguide, in which there is confinement in both the x and y dimensions, without actually surrounding the waveguide with materials of lesser index. •This is done by forming a strip of dielectric material of lesser index, n3, on top of a planar waveguide, with index n1, as shown in Fig. 3.6. Such a structure is usually called either a strip-loaded waveguide, or an optical stripline.
n1>n2≥n3>n4or
n1>n3≥n2>n4Fig.3.6
n'eff1> n eff1
•The presence of the loading strip on top of the waveguiding layer makes the effective index in the region beneath it, n eff1larger than the effective index, n eff1in the adjacent regions. Thus there can be confinement in the y direction as well as in the x directions, the physical nature of this phenomenon can be visualized best by using the ray optic approach. •Consider a particular mode propagating in the z direction but consisting of a plane wave following the usual zig-zag path in the waveguiding layer.
•Since n 3is larger than n 4, the wave penetrates slightly more at the n 1-n 3interface than it would at the n 1-n 4interface; thus, the effective height of the waveguide is greater under the loading strip than it is in the regions on either side. This means that the zig-zag path of the plane wave would be slightly longer under the loading strip, leading to the result that
k
n k n eff eff β
β=>'='11
Furutae et al. have used the effective index of refraction method to analyze a strip-loaded guide like that of
Fig.3.6, and have shown that its waveguiding properties are equivalent to those of a dielectric waveguiding like that shown in Fig3.7, where the equivalent index in the side confining layers is given by
21
212
1eff eff eq n n n n +'-=The propagation constants of the rectangular
waveguide of Fig.3.7 can be determined by using Marcatili’s method, described in Sec.3.2.1.
A strip-loaded waveguide can also be made with the index of the loading strip, n 3, equal to that of the guide, n 1. That type of waveguide is usually called a ridge , or rib , guide.
Fig.3.7 Cross-sectional view of rectangular dielectric waveguide equivalent to the strip-loaded waveguide of Fig. 3.6
n eff1< n eff1
Fig.3.8. Metal strip-loaded waveguide
Metallic loading strips can also be used to produce optical striplines. In that case, two metal loading strips are placed on the surface of the waveguiding layer, on either side of the region in which confinement is desired, as shown in Fig.3.8.
Since penetration of the guided waves is deeper at the n1-n4 interface than at the n1-n3interface, the desired confinement in the y direction is obtained just as in the case of the dielectric strip-loaded waveguide.
Metal strip-loaded waveguides are particularly useful in applications such as electro-optic modulators, where surface metal electrodes are desired, since these metal stripes can perform the additional function of defining the waveguide.。