EE 230: Optical Fiber Communication Lecture 3Waveguide/Fiber Modes

From the movieWarriors of the Net

Optical Waveguide mode patterns

Optical Waveguide mode patterns seen in the end faces of small diameter fibers

Optics-Hecht & Zajac Photo by Narinder Kapany

Multimode Propagation

In general many modesare excited in the guideresulting in complicated field and intensity patterns that evolve in a complex way as the lightpropagates down the guide

Fundamentals of Photonics - Saleh and Teich

Planar Mirror Waveguide

The planar mirror waveguide can be solved by starting with Maxwells Equationsand the boundary condition that the parallel component of the E field vanishat the mirror or by considering that plane waves already satisfy Maxwell’s equationsand they can be combined at an angle so thatthe resulting wave duplicates itself

Fundamentals of Photonics - Saleh and Teich

Mode Components Number and Fields

Fundamentals of Photonics - Saleh and Teich

Mode Velocity and Polarization Degeneracy

Group Velocity derivedby considering the modefrom the view of rays and geometrical optics

TE and TM mode polarizations

Fundamentals of Photonics - Saleh and Teich

Planar Dielectric guide

Characteristic Equation and Self-ConsistencyCondition

Propagation Constants

Number of modes vs frequency

Geometry of Planar Dielectric Guide

The m all lie between thatexpected for a plane wave in the core and for a plane wave in thecladding

For a sufficiently lowfrequency only 1 modecan propagate

Fundamentals of Photonics - Saleh and Teich

Planar Dielectric Guide

Field components have transverse variationacross the guide, with more nodes for higher ordermodes. The changed boundary conditions for the dielectric interface result in some evanescent penetration into the cladding

The ray model can be used for dielectric guidesif the additional phase shift due to the evanescentwave is accounted for.

Fundamentals of Photonics - Saleh and Teich

Two Dimensional Rectangular Planar Guide

In two dimensions the transverse field depends on both kx and ky and the number of modes goes as the square of d/

The number of modes is limited by the maximum angle that can propagatec

Fundamentals of Photonics - Saleh and Teich

Modes in cylindrical optical fiber

• Determined by solving Maxwell’s equations in cylindrical coordinates

011 2

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Key parameters• q2 is equal to ω2εμ-β2 = k2 – β2. It is

sometimes called u2. • β is the z component of the wave propagation

constant k, which is also equal to 2π/λ. The equations can be solved only for certain values of β, so only certain modes may exist. A mode may be guided if β lies between nCLk and nCOk.

• V = ka(NA) where a is the radius of the fiber core. This “normalized frequency” determines how many different guided modes a fiber can support.

Solutions to Wave Equations

• The solutions are separable in r, φ, and z. The φ and z functions are exponentials of the form eiθ. The z function oscillates in space, while the φ function must have the same value at (φ+2π) that it does at φ.

• The r function is a combination of Bessel functions of the first and second kinds. The separate solutions for the core and cladding regions must match at the boundary.

Resulting types of modes• Either the electric field component (E) or the

magnetic field component (H) can be completely aligned in the transverse direction: TE and TM modes.

• The two fields can both have components in the transverse direction: HE and EH modes.

• For weakly guiding fibers (small delta), the types of modes listed above become degenerate, and can be combined into linearly polarized LP modes.

• Each mode has a subscript of two numbers, where the first is the order of the Bessel function and the second identifies which of the various roots meets the boundary condition. If the first subscript is 0, the mode is meridional. Otherwise, it is skew.

Mode characteristics

Each mode has a specific

• Propagation constant β

• Spatial field distribution

• Polarization

- Mode Diagram

Straight lines of d/d correspond to the group velocity of the different modesThe group velocities of the guided modes all lie between the phase velocities forplane waves in the core or cladding c/n1 and c/n2

Step Index Cylindrical Guide

2 20 1 2 1

2( - )» 2V k a n n an

pl

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Fundamentals of Photonics - Saleh and Teich

High Order Fiber modes

Fiber Optics Communication Technology-Mynbaev & Scheiner

High Order Fiber Modes 2

Fiber Optics Communication Technology-Mynbaev & Scheiner

The Cutoff

• For each mode, there is some value of V below which it will not be guided because the cladding part of the solution does not go to zero with increasing r.

• Below V=2.405, only one mode (HE11) can be guided; fiber is “single-mode.”

• Based on the definition of V, the number of modes is reduced by decreasing the core radius and by decreasing ∆.

Number of Modes

Graphical Constructionto estimate the total number of Modes

Propagation constant of the lowest mode vs. V number

2 20 1 2 1

0 0

a 2πV=k a(n -n )=2 an 2Δ

λ λNAp

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Fundamentals of Photonics - Saleh and Teich

Number of Modes—Step Index Fiber

• At low V, M4V2/π2+2

• At higher V, MV2/2

Graded-index Fiber

For r between 0 and a.

Number of modes is

a

rnrn 211

212

aknM

Comparison of the number of modes

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1-d Mirror Guide

1-d Dielectric Guide

2-d Mirror Guide

2-d Dielectric Guide

2-d Cylindrical Dielectric Guide 0

aV=2

λNAp

The V parametercharacterizes the number ofwavelengths that can fit acrossthe core guiding region in a fiber.

For the mirror guide the number of modes is just the number of ½ wavelengths that can fit.

For dielectric guides it is the number that can fit but now limited by theangular cutoff characterized by the NA of the guide

Power propagating through core

• For each mode, the shape of the Bessel functions determines how much of the optical power propagates along the core, with the rest going down the cladding.

• The effective index of the fiber is the weighted average of the core and cladding indices, based on how much power propagates in each area.

• For multimode fiber, each mode has a different effective index. This is another way of understanding the different speed that optical signals have in different modes.

Total energy in cladding

The total average power propagating in the cladding is approximately equal to

MP

Pclad3

4

Power Confinement vs V-Number

This shows the fraction of the power that is propagating in the cladding vs the V number for different modes.

V, for constant wavelength, and material indices of refraction is proportional to the core diameter a

As the core diameter is dereased moreand more of each mode propagates in the cladding. Eventually it all propagates in the cladding and the mode is no longerguided

Note: misleading ordinate lable

Macrobending Loss

One thing that the geometrical ray view point cannot calculate is the amount of bending lossencountered by low order modes. Loss goes approximately exponentially with decreasing radiusuntill a discontinuity is reached….when the fiber breaks!