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Lecture 4: Optical waveguides. □ Waveguide structures. □ Waveguide modes. □ Field equations. □ Wave equations. □ Guided modes in symmetric slab ...
Typology: Lecture notes
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References: A significant portion of the materials follow “Photonic Devices,” Jia-Ming Liu, Chapter 2
There are two basic types of waveguides: In a nonplanar waveguide of two-dimensional transverse optical confinement, the core is surrounded by cladding in all transverse directions , and n(x, y) is a function of both x and y coordinates. E.g. the channel waveguides and the optical fibers In a planar waveguide that has optical confinement in only one transverse direction , the core is sandwiched between cladding layers in only one direction, say the x direction, with an index profile n(x). The core of a planar waveguide is also called the film , while the upper and lower cladding layers are called the cover and the substrate.
5 n 3 n 1 n 2 Planar (slab) waveguides for integrated photonics (e.g. laser chips) n 2 Cylindrical optical fibers n 1 (x, y, z) (r, φ, z)
(n 1 > n 2 , n 3 ) (n 1 > n 2 )
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Most waveguides used in device applications are nonplanar waveguides. For a nonplanar waveguide, the index profile n(x, y) is a function of both transverse coordinates x and y. There are many different types of nonplanar waveguides that are differentiated by the distinctive features of their index profiles. One very unique group is the circular optical fibers (to be discussed in Lecture 5). Another important group of nonplanar waveguides is the channel waveguides , which include The buried channel waveguides The strip-loaded waveguides The ridge waveguides The rib waveguides The diffused waveguides.
n 1 n 3 n 1 n 2 w n 2 d w d n 1 n 2 w d n 1 n 2 w n 2 w n 1 d rib waveguide Diffused waveguide Buried channel waveguide Strip-loaded waveguide Ridge waveguide x y z d h
A rib waveguide has a structure similar to that of a strip or ridge waveguide, but the strip has the same index as the high- index planar layer beneath it and is part of the waveguiding core. These four types of waveguides are usually termed rectangular waveguides with a thickness d in the x direction and a width w in the y direction, though their shapes are normally not exactly rectangular. A diffused waveguide is formed by creating a high-index region in a substrate through diffusion of dopants, such as LiNbO 3 waveguide with a core formed by Ti diffusion. Because of the diffusion process, the core boundaries in the substrate are not sharply defined. A diffused waveguide also has a thickness d defined by the diffusion depth of the dopant in the x direction and a width w defined by the distribution of the dopant in the y direction.
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For a waveguide of two-dimensional transverse optical confinement , there are two degrees of freedom in the transverse xy plane, and the mode index υ consists of two parameters for characterizing the variations of the mode fields in these two transverse dimensions. E.g. υ represents two mode numbers, υ = mn with integral m and n, for discrete guided modes. For the planar waveguide , the mode fields do not depend on the y coordinate. Thus, the electric and magnetic fields of a mode can be simplified to In this case, υ consists of only one parameter characterizing the field variation in the x dimension.
Consider the qualitative behavior of an optical wave in an asymmetric planar step-index waveguide, where n 1
n 2 n 3
For an optical wave of angular frequency ω and free-space wavelength λ, the media in the three different regions of the waveguide define the following propagation constants: where k 1
k 2 k 3 We can obtain useful intuitive picture from considering the path of an optical ray , or a plane optical wave , in the waveguide.
As the wave is reflected back and forth between the two interfaces, it interferes with itself. A guided mode can exist only when a transverse resonance condition is satisfied s.t. the repeatedly reflected wave has constructive interference with itself. In the core region, the x component of the wavevector is k x
k 1 cos θ for a ray with an angle of incidence θ, while the z component is β = k 1 sin θ. The phase shift in the optical field due to a round-trip transverse passage in the core of thickness d is 2k 1 dcos θ.
k x = (n 1 ω/c) cos θ β = (n 1 ω/c) sin θ k 1 k x β θ core n 1 z x
