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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 28

    Waveguide basicsImportant normalized quantities

    n=1

    ncl

    ncocrit

    crit

    z

    x

    a a kz

    a kx

    clna

    0

    2

    cona

    0

    2

    clcoclcocritcocrit nnnNA === 22sinsin

    Numerical aperture of guide measured in air

    Phase shift of critical wave across guide face

    22

    000 sin clcocritcritx nnakaNAkakakV ===

    Ray view of guiding in a slab waveguide. The most

    extreme ray is trapped via total internal reflection at

    the core/cladding boundary.

    Modal analysis of planar slab waveguidesNormalized quantities

    Phase shift of particular mode across guide face22

    0

    22

    0 sin NnakkkaakakU cozcocox === Exponential loss of particular mode in cladding across guide face

    22

    0

    22

    clclz nNakkkaaW ==

    Snyder and Love

    ncl

    222 WUV +=Note that:Cutoff: U=V, W=0

    Light line: W=V, U=0

    V

    UW

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 29

    TE modal analysis of

    symmetric slab waveguides

    P. Yeh, Optical Waves in Layered Media, Chapter 11

    ( )( ) 0,

    2

    2

    2

    22 =

    zxE

    tc

    rn rr

    Substitute a TEx-dependent mode into 3D wave equation:

    ( ) ( ) zjeyxEzxE = ,r

    ( ) ( ) 0222022

    =

    + xExnkdx

    d

    yieldingncl

    ncoz

    ncl

    x

    Ifn is constant in each layer l

    ( ) xnkjxnkjlll eCeCxE

    2220

    2220

    21

    ++=

    A propagating mode must have

    coclcoclnNnnknk

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 30

    Apply boundary conditions

    TE

    ( )

    ( )

    ( )( )WWDUUBUUAWDUBUA

    WWCUUBUUA

    WCUBUA

    =+=+

    =

    =+

    expsincosexpcossin

    expsincos

    expcossin

    Rearranging slightly by adding and subtracting equations:

    ( ) ( )

    ( ) ( )

    ( ) ( )( ) ( )WDCWUUB

    WDCUB

    WDCWUUA

    WDCUA

    +=+=

    =

    =

    expsin2expcos2

    expcos2

    expsin2

    Modal analysis of planar slab waveguidesTE modes

    x

    EjH

    y

    z

    =

    1

    Tangential components of fields (Ey andHz) must be

    continuous. Since

    This is equivalent to field and its slope are continuous. Yieldsfour conditions for five unknowns (A,B,C,D,), leaving peakamplitude as a free parameter. The conditions are:

    E|| continuous atx = +a

    dE||/dx continuous atx = +a

    E||

    continuous atx = -a

    dE||/dx continuous atx = -a

    Then divide equation 2 by 1 and 4 by 3.

    DCBWUU

    DCAWUU

    =

    =

    ,0tan

    ,0cot

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 31

    1 2 3 4 5 6

    u

    1

    2

    3

    4

    5

    6

    v

    TE SolutionsTwo classes (symmetric and anti-symmetric)

    WUUDCA === tan,,0

    WUUDCB === cot,,0

    Can rewrite the anti-sym equation to make it look similar to sym:

    UUW tan=

    ( )2

    tancot +== UUUUW

    ( )22222

    0

    22

    VnnakWU clco ==+

    No cutoff for lowest mode

    New mode at V>m /2, m=0,1..

    Sym/antisym modes alternate

    Observations

    U

    W

    # TE modes = Int(V//2) + 1

    Transcendental eigenvalue equations

    Modal analysis of planar slab waveguidesTE modes

    ( ) WUUUU =+= 2tancot

    Blue lines are TE solutions

    Green lines are TM solutions

    Always TM mode with TE

    V

    cutoff, W=0

    lightline,U=0

    N=ncl

    N=nco

    Graphical solution of slab transcendental eigenvalue equations:

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 32

    TM modes

    We follow the same approach but now for Hy

    ( )

    >

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 33

    Power carried by modeTE

    Modal analysis of planar slab waveguidesPower

    From Faradays law we can findHx fromEy :

    [W/m]*21

    = dxHEPxyz

    y

    yyc

    yyx

    xy

    EN

    ENEN

    ENk

    EH

    HjEj

    t

    0

    0

    0

    00

    0

    0

    0

    =

    ====

    =

    =

    BE

    r

    r

    From the Poynting vector, we can

    calculate the power carried by this mode.

    Since the guide is infinite iny, we will

    calculate the power per unit length in they

    direction:

    Average power per unity inz

    Faradays Law

    From known tandz dependence

    Impedance of mode

    Giving power in the mode as

    o Impedance of free space = sqrt(0/0)

    = [W/m]22 0

    dxEP yN

    z TE

    z

    x

    Er

    Hr

    TE

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 34

    Power carried by modeTM

    Modal analysis of planar slab waveguidesPower

    From Amperes law we can findEx fromHy :

    [W/m]*

    21

    += dxHEP yxz

    ( )

    ( ) ( ) ( ) ( )

    ( ) y

    yyc

    yyx

    xy

    Hx

    N

    Hx

    NH

    x

    NH

    x

    NkH

    xE

    ExjHj

    t

    0

    0

    0

    00

    0

    0

    0

    =

    ====

    =

    =

    DH

    r

    r

    Amperes Law

    From known tandz dependence

    Impedance of mode

    Giving power in the mode as

    o Impedance of free space = sqrt(0/0)

    ( ) ( )

    == [W/m]22

    1212 0

    0 dxExdxHP xNyxN

    z

    TM

    From the Poynting vector, we can

    calculate the power carried by this mode.

    Since the guide is infinite iny, we will

    calculate the power per unit length in they

    direction:

    Average power per unity inz

    z

    xEr

    Hr

    TM

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 35

    Power confinement

    Modal analysis of planar slab waveguidesPower

    What fraction of the power is in the core?

    ( )

    ( )

    =

    ==

    dxxH

    dxxH

    P

    P

    dxE

    dxE

    dxE

    dxE

    P

    P

    y

    a

    a

    y

    Tot

    co

    y

    a

    a

    y

    yN

    a

    a

    yN

    Tot

    co

    2

    2

    2

    2

    2

    2

    2

    2

    0

    0

    TE

    TM

    a/0=1/1.5, nco=3, ncl=1.5

    The trend is that the

    fractional power in the

    core always decreases

    for increasing mode

    number. It approaches

    1 for tightly bound

    modes with high indexcontrast.

    a/0=1/1.5, nco=3, ncl=1.5

    TETM

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado

    Approximate expression forN

    36

    ( )Tot

    coclcocl

    Tot

    clcl

    Tot

    coco

    P

    Pnnn

    P

    Pn

    P

    PnN 222222 +=+

    It can be shown from variational techniques that the effective index can

    be approximately calculated as (first line TE, second TM):

    The derivative term is small for the fundamental modes of weakly-

    confined modes, leading to a useful conceptual formula:

    Modal analysis of planar slab waveguidesPower

    ( ) ( )

    ( )

    ( ) ( )( ) ( )

    ( )Tot

    yclclcoco

    y

    yy

    Tot

    yclclcoco

    y

    yy

    PdxxndxdHkPnPn

    dxxnxH

    dxxndxdHkxH

    P

    dxdxdEkPnPn

    dxxE

    dxdxdEkxExnN

    +==

    +=

    22

    2

    0

    22

    22

    22

    2

    0

    2

    22

    0

    22

    2

    22

    0

    22

    2

    In general Step index slab

    Nthus approaches ncofor the fundamental

    mode and decreases

    towards ncl with

    increasing mode

    number. Nis less than

    ncl for radiation

    modes,N= ncl cos()

    a/0=1/1.5, nco=3, ncl=1.5

    Snyder and Love section 15-1

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 37

    Ray analysis of step-

    index guides

    Ray analysis of waveguidesForm of waves in step index

    z

    2

    clcl n=

    2

    coco n=

    2

    clcl n=

    k

    r kz

    k

    cln0

    2

    con

    0

    2

    crit

    0kNA

    Range of allowed kin core

    ( )

    =

    +

    +

    claddingin the

    corein the,

    220

    220

    zNxNnjk

    zNxNnjk

    cl

    co

    e

    ezxE

    ( )

    [ ]zNxNnkzkxknk

    zkxkk

    zkxkk

    zz

    zz

    zx

    22

    0

    22

    0

    22

    +=

    +=

    +=

    +=r

    N Effective index = kz / k0

    Notes:

    1) Only one free parameter,N

    2) ncl < N < nco are guided rays

    2)N < ncl are not guided

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 38

    Discrete bound modesSymmetric guide (1/2)

    Ray analysis of waveguidesDiscrete bound modes

    Hunsperger, Section 2.2.2

    zx

    2a

    2

    clcl n=

    2

    cocon=

    2

    clcln=

    makak pcoxpcox 22222 =+++

    0 1 2 3 4 5

    z@mmD

    -15

    -10

    -5

    0

    5

    10

    15

    r

    mm

    ( ) ( ) ( )zxjkzkxkj eezxE zx cossin, ++ == Form of field in core

    NOTE MEASURED FROM || TO BOUNDARY!

    cox

    clx

    cl

    co

    cox

    clz

    cl

    co

    co

    clco

    cl

    coTM

    cox

    clx

    cox

    clz

    co

    clco

    TE

    kn

    n

    k

    kk

    n

    n

    n

    nn

    n

    n

    kk

    kk

    n

    nn

    =

    =

    =

    =

    =

    =

    2

    222

    2

    2222

    2

    2

    22222

    sin

    costan

    sin

    costan

    First lets translate the Goos-Hanchen shifts into this coordinatesystem and write them in terms of convenient quantities:

    Then find phase accumulated in round trip of ray from lower to

    upper and back to lower boundary at single plane z:

    p = TE or TM

    Decay in x in cladding

    Wave # in x in core= -

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 39

    Discrete bound modesSymmetric guide (2/2)

    ( )clxpcoxcox

    cox

    clxpcox

    pcox

    makk

    mk

    ak

    mak

    =

    =

    =+

    2

    1

    tan

    tan22

    22

    Ray analysis of waveguidesDiscrete bound modes

    Simplify slightly

    Substitute GH phase shift

    and rewrite.

    =TM

    TE12

    cl

    cop

    n

    nwhere handles both polarizations.

    m Mode number. Quantizes bound spectrum.

    0.2 0.4 0.6 0.8 1

    H2 p a cL w = HaL k0

    1

    2

    3

    4

    5

    cokr

    clkr

    xk

    zk

    cokr

    xk

    zk

    6

    1

    =

    =

    co

    cl

    n

    n

    Not allowed

    Radiation

    Bound

    a / c = a k0

    Same transcendental equation as modal derivation (good!).

    Plot of solutions found numerically

    akz=ak0N

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 40

    Discrete bound modesAsymmetric guide (1/2)

    Ray analysis of waveguidesDiscrete bound modes

    makak pscoxpclcox 22222 ,, =+++ p = TE or TM

    zx

    2a

    2

    clcln=

    2

    cocon=

    2

    ssn=

    mNnaNn

    nN

    Nn

    nN

    makkk

    mkk

    ak

    mak

    co

    co

    s

    p

    co

    cl

    p

    cox

    cox

    sxp

    cox

    clxp

    cox

    sxp

    cox

    clxpcox

    pspclcox

    =

    +

    =

    +

    =

    =++

    22

    22

    22

    1

    22

    22

    1

    11

    11

    ,,

    2tantan

    2tantan

    tantan2

    2

    Common due to fabrication technology. Typical example is air cladding.

    Following previous, track phase from edge to edge of core.

    substrate

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 41

    Discrete bound modesAsymmetric guide (2/2)

    Ray analysis of waveguidesDiscrete bound modes

    cokr

    xk

    zk n(x)

    xN

    ns

    ncl

    nco

    6

    2

    1

    =

    =

    =

    co

    s

    cl

    n

    n

    nNot allowed

    Comparison of symmetric (ns=ncl) and asymmetric fundamental modes.

    Substrate & cladding radiation

    kz= kcl orN = ncl

    kz = ks orN = ns

    m=0 asymmetric

    m=0 symmetric

    a / c = a k0

    akz=ak0N

    Note analogy to

    particle in potential

    well

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 42

    0 1 2 3 4 5

    z@mmD

    -15

    -10

    -5

    0

    5

    10

    15

    0 1 2 3 4 5

    z@mmD

    -15

    -10

    -5

    0

    5

    10

    15

    0 1 2 3 4 5

    z@mmD

    -15

    -10

    -5

    0

    5

    10

    15

    0 1 2 3 4 5

    z@mmD

    -15

    -10

    -5

    0

    5

    10

    15

    Ray analysis of waveguidesForm of waves in step index

    z

    kz

    k

    z

    kz

    k

    z

    kz

    k

    z

    kz

    k

    cok

    r

    cok

    r

    cok

    r

    co

    kr

    cl

    cl

    cl

    cl

    Guided modes

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 43

    Radiation modesSamples of mode continuum

    z

    kz

    k

    Ray analysis of waveguidesForm of waves in step index

    cokr

    clkr

    z

    kz

    kco

    kr

    clkr

    z

    kz

    kco

    kr

    clk

    r

    0 1 2 3 4 5

    z@mmD

    -15

    -10

    -5

    0

    5

    10

    15

    r

    mm

    0 1 2 3 4 5

    z@mmD

    -15

    -10

    -5

    0

    5

    10

    15

    r

    mm

    0 1 2 3 4 5

    z@mmD

    -15

    -10

    -5

    0

    5

    10

    15

    mm

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 44

    BackgroundRay and eikonal equations

    ( ) ( )rnds

    rdrn

    ds

    d rr

    r

    =

    Eikonal

    ( ) ( )

    zs

    nrn

    zzr

    =+=

    rr

    rr

    Cylindrical coordinates

    Waveguide that is invariant alongz

    Paraxial (small angle) approximation

    ( ) ( )

    r

    r

    r

    ndz

    dn =

    2

    2

    Paraxial eikonal for waveguide inz

    UFdt

    dm ==

    r

    r

    2

    2 Newtons law for particle inpotential well U

    Particle in potential wellAnalogy with ray optics

    Approximations

    ( )

    ( ) ( )

    r

    r

    r

    =

    ==

    2

    2

    2

    2

    2

    dz

    d

    nnn Express in terms of, not n

    Paraxial eikonal for waveguide

    inz, in terms of, not n

    Note similarity with F = m A

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 45

    Particle in potential wellApplication to waveguides

    crit

    critz

    kz

    k

    cln

    0

    2

    con

    0

    2

    crit

    clcoclconn

    =

    0

    22

    0

    22

    max

    2222

    sinsin =====

    clcoclcon

    nn

    cocritcocrit nnnnNAco

    clco

    Numerical aperture of guide measured in air

    Ray view of guiding in a slab waveguide. The most

    extreme ray is trapped via total internal reflection at

    the core/cladding boundary.

    ( ) ( ) ( )[ ] ( )[ ]

    ( ) ( )

    ( )

    22

    22

    22

    2

    2

    2

    2

    sin

    sin

    sin

    2

    1

    2

    clco

    clco

    cl

    nn

    nnn

    n

    dz

    d

    UmV

    dz

    d

    =+==

    r

    r

    r

    r

    rrr

    r

    r

    Kinetic energy < potential to remain bound

    2

    clcln=

    2coco n=

    12 == airair n

    Using analogy

    Express d/dz as sin

    By Snells Law at exit of guide

    Ray analysis of waveguidesNumerical aperture

    T.I.R.

    Can drop uniform

    2

    clcln=

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 46

    Parabolic gradient-index guide

    Solution via eikonal (1/2)

    ( )

    =

    m

    co ann

    1

    ( ) ( )

    ( ) ( )

    +=

    =

    zdz

    dn

    dz

    d

    dz

    rdn

    dz

    d

    ds

    rdn

    ds

    dn

    r

    r

    ( ) ( )

    dz

    dn

    dz

    dn

    d

    d

    Power-law radial index distribution

    < a

    Plug into eikonal

    Paraxial approx.

    Simplify with known dependencies

    Ray analysis of waveguidesGrin guides

    -2 -1 0 1 2

    ra

    -1

    -0.8

    -0.6

    -0.4

    -0.2

    0

    d

    nD

    Index distribution vs radius

    ( )( )

    ( )

    =

    m

    an

    d

    d

    nn

    d

    d

    ndz

    d

    1

    1102

    2

    Subst. n()

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 47

    Ray analysis of waveguidesGrin guides

    Parabolic gradient-index guide

    Solution via eikonal (2/2)

    ( )( )

    ( )

    ( )

    11

    2

    2

    1

    111

    =

    =

    mm

    co

    m

    co

    aa

    m

    aa

    mn

    n

    an

    d

    d

    nn

    d

    d

    ndz

    d

    22

    2 2=

    aadz

    d

    Take deriv.,

    assume n in

    denom ~ nco

    Special case m=2

    ( ) ( ) ( )zzz sincos 00

    += Solution for ray trajectory

    from prev.

    z=0 z=/

    z

    -a

    a

    0

    Rays escape when radial coordinate goes beyond =a, which gives NA:

    ( )

    ( ) +==

    =====

    ====

    221

    2sinsin

    22

    sinmax

    22222

    0

    0000

    cocococlco

    cocococritcocrit

    nnnnnNA

    nndz

    dnnNA

    aa

    z

    Ray slope that hits

    edge

    Defines NA

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado

    Dispersion

    Material

    Modal

    Waveguide

    Polarization

    48

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 49

    Origin of waveguide group delay

    from eigenvalue equation

    Modal analysis of planar slab waveguidesGroup velocity

    ( ) ( )[ ] ( ){ }zjktt zetEztE= 0,, 1 FF

    ( ) ( ) ( ) K+

    +

    +=

    00

    02

    22

    002

    1

    zzzzkk

    kk

    ( )zkj

    z zezk

    tEztE 0

    0

    0,,

    =

    Consider how a temporal pulse that excites a single mode of the guide will

    propagate. We found the solution to the scalar wave equation for a

    sinusoidal temporal excitation at frequency :

    ( ) ( ) ( )zktj zeyxEzxtE = ,,r

    whereE(x) satisfied the DE and kz = N k0 =simultaneously satisfied thetranscendental characteristic equation. For an aribitrary temporal signal,

    we must decompose it into its frequency components and apply this

    transfer function to every frequency independently:

    Let us now expand the propagation constant kz as a Taylor series aroundsome central carrier frequency 0:

    The first term causes a frequency-independent phase shift. The second

    term is a linear phase shift which, by the Fourier shift theorem, is

    equivalent to a shift in the time domain:

    Group delay [ ]mz

    g

    ks

    0

    Pollock and Lipson Ch 6

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 50

    Group index& velocity vs. delay

    Modal analysis of planar slab waveguidesGroup velocity

    ( )c

    N

    d

    dNN

    ccN

    d

    d

    d

    dk gzg

    +=

    =

    1

    0

    Write propagation constant in terms of effective index:

    It is sometimes convenient to have the derivative vs. wavelength:

    ( )

    d

    df

    d

    dfc

    d

    df

    d

    d

    d

    df

    d

    df=== 2 2

    0dk

    dk

    d

    dNN

    d

    dNNN zg ==+=

    thus the effective group index is

    I find group velocity to be a) confusing and b) generally useless, so we

    will attempt to avoid this concept.

    Pollock and Lipson, Section 6.3.2 and 6.3.3

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    ECE 4006/5166 Guided Wave Optics

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    Group velocity dispersion

    second and higher terms

    Modal analysis of planar slab waveguidesGroup velocity dispersion

    Consider a pulse with a finite bandwidth / = /. The latency due topropagation delay of the pulse will be

    ( )[ ]s0 L

    c

    Ng =

    The pulse will spread due to the variation of latency across the spectrum:

    ( ) ( )[ ]

    =

    =

    =

    =

    2

    2

    2

    2

    21

    d

    Nd

    c

    L

    dNd

    ddN

    ddN

    cL

    d

    dNN

    d

    d

    c

    L

    d

    dN

    c

    L

    NNc

    L

    g

    gg

    It is useful to define the GVD parameterD as

    =

    kmnmpsor

    ms

    22

    2

    dNd

    cD

    LD

    Note that exactly the same derivation would hold for propagation in a bulk

    material withNreplaced by the normal refractive index, n.

    Typical value:D = 17 ps/(nm km)

    for SMF-28 at = 1.5 m

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    ExampleSlab waveguide

    Modal analysis of planar slab waveguidesGroup velocity dispersion

    Anomalous dispersion

    Radiation

    Not allowed6

    1

    =

    =

    co

    cl

    n

    n

    Bound

    m=0m=1 m=2 m=3

    1=cln

    6=con

    Modes emerge

    from radiation

    cut-off,

    approach light

    line.

    Zero dispersion

    band near

    cutoff, then

    again when

    tightly bound.

    Ng = ncl at cutoff

    and approaches

    nco from above.

    Normal dispersion

    D

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    ECE 4006/5166 Guided Wave Optics

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    Material dispersionHow to include

    2

    0

    22

    0 NakNnakU coco +=

    ( ) += clcl NaknNakW2

    0

    22

    0

    Modal analysis of planar slab waveguidesGroup velocity dispersion

    In many cases, the core and the cladding are made from similar

    materials, for example, the core is a doped version of the cladding or a

    denser version of the cladding. In this case the dispersion of the two

    materials is very nearly the same. Consider how this impacts the

    effective index:

    ( ) WUUUU =+=2

    tancot

    Thus if the effective index is changed like

    +22

    NNThen U and W are unchanged, the characteristic equation is still

    satisfied:

    thus the mode shape does not change.

    Thus to a good approximation in most cases, material and waveguide

    dispersion can be treated as independent, additive quantities.

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 54

    Orthogonality of modes

    Modal analysis of planar slab waveguidesOrothogonality

    It is possible to show directly from the wave equation that, if the

    materials are lossless (no free current and thus no Joule losses),

    Where Pm is the power carried by the mode (this depends on the

    particular amplitude). The delta functions must be carefully defined:

    ( ) ( ) ( )

    ==

    =

    nm

    nm

    afdxaxxf

    nm0

    1,

    -

    Dirac delta function.

    Note (x) has units of 1/x.

    Discrete delta function.Note m is unitless.

    The modes are not only orthogonal, they are complete, allowing us to write

    ( ) ( ) ( ) ( ) ( )

    ( ) ( ) ( ) ( ) ( )

    +=

    +=

    ml

    zmljzj

    mlml

    zmlj

    ml

    zj

    mlml

    dmdlemlyxHmlaeyxHazyxH

    dmdlemlyxEmlaeyxEazyxE

    ml

    ml

    ,

    ,

    ,,

    ,

    ,

    ,,

    ,;,,,,,

    ,;,,,,,

    ,

    ,

    rrr

    rrr

    If we could find the coefficients a, we would know how arbitrary fields

    propagate. Note al,m and a(l,m)dldm are unitless ifEl,m are not normalized.

    Bound modes,

    l,m discrete

    Radiation modes,

    l,m continuous, e.g. kx, ky

    ( )

    ( )

    ( ) ( )modesBound

    modesRadiation

    2

    2

    2

    1

    ,

    *

    ,,,10

    *

    ,,

    0

    *

    =

    =

    =

    nmm

    nymyyx

    nymy

    nm

    P

    nmmP

    dydxHHN

    dydxEEN

    dydxzHE

    rr

    General:

    TE:

    TM:

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    ECE 4006/5166 Guided Wave Optics

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    Expanding an arbitrary

    field as a sum of modes

    Modal analysis of planar slab waveguidesOrothogonality

    ( ) ( ){ }

    ( ) ( ) ( ) ( )

    +

    =

    dydxzyxHdmdlmlyxEmlayxEa

    dydxzyxHyxE

    pn

    ml

    mlml

    pn

    ,,;,,,

    2

    1

    ,0,,2

    1

    *

    ,

    ,

    ,,

    *

    ,

    rrr

    rr

    Start with the assumed expansion

    Since we only need to perform the expansion at onez plane which will

    then determineEin all ofz, choosez = 0.

    Now cross with theH* field from a bound or radiation mode n,p, take the

    dot product inz and integrate overx andy:

    We can swap the order of thexy integral and the sum/integral over l,m:

    ( ) ( ) ( ) ( )( ) ( )

    += mlamlPaPdmdlpmnlmlPmlaPa

    mlml

    ml

    pmnlmlml,,or,,

    ,,

    ,

    ,,,,

    Thexy integrals collapse via the orthogonality relations:

    ( ) ( )[ ]

    ( )( )

    ( ) ( )[ ] dydxzmlyxHyxEmlP

    mla

    dydxzyxHyxEP

    a mlml

    ml

    ,;,0,,,2

    1,

    ,0,,2

    1

    *

    *

    ,

    ,

    ,

    =

    =

    rr

    rr

    ( ) ( )[ ]

    ( ) ( ) ( )[ ]

    +

    =

    dmdldydxzyxHmlyxEmla

    dydxzyxHyxEa

    pn

    ml

    pnmlml

    ,,;,,2

    1

    ,,2

    1

    *

    ,

    ,

    *

    ,,,

    rr

    rr

    ( ) ( ) ( ) ( ) ( ) += dmdlemlyxEmlaeyxEazyxE zmlj

    ml

    zj

    mlmlml ,

    ,

    ,, ,;,,,,,,

    rrr

    ( ) ( ) ( ) ( ) += dmdlmlyxEmlayxEazyxE mlmlml ,;,,,,,

    ,

    ,,

    rrr

    Substitute assumed expansion

    al,m unitless

    a(l,m) dldm unitless

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    ECE 4006/5166 Guided Wave Optics

    Robert R. McLeod, University of Colorado 56

    Application to TE modes

    of a slab waveguide

    Modal analysis of planar slab waveguidesOrothogonality

    z

    x

    Er

    Hr

    TE( ) ( )= dxxHxEP

    a xnyn

    n

    *

    ,0,2

    1

    Start with the mode expansion equation

    and plug in TE mode. Collapse to 1D.

    ( ) ( )= dxxExEPN

    a ynyn

    n

    *

    ,

    0

    0,2

    1

    use the relationship betweenHandEfor a TE mode to get

    First lets assume that the mode n is bound.

    ( ) ( )

    ( )

    ( ) ( )

    ( )

    ==

    dxxE

    dxxExE

    dxxEN

    dxxExEN

    a

    yn

    yny

    yn

    yny

    n 2

    ,

    ,

    2

    ,

    0

    ,

    00,

    2

    0,2

    Now assume n is a radiation mode. Lets write out the orthogonality

    condition. Assume that a mode m in the cladding has amplitude Cm

    ( ) ( ) ( )nmCNdxeCNeCdydxzHE mxjk

    n

    xjk

    mnmxnxm =

    =

    +

    2

    0

    *

    0

    *

    22

    1

    2

    1,,

    rr

    ( )( ) ( )

    ( )2

    ,

    ,

    cladding

    0,

    yn

    yny

    E

    dxxExEna

    =

    P(n)

    Therefore

    ( ) ( ){ } = dydxzyxHyxEPa n

    n

    n,0,,

    2

    1 *rr

    Can drop cc on flat mode,

    but I dont recommend it.

    an is still

    without

    units

    a(n) dn unitless =

    a(n) has units ofx

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    ECE 4006/5166 Guided Wave Optics

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    Application to TM modes

    of a slab waveguide

    Modal analysis of planar slab waveguidesOrothogonality

    ( ) ( )= dxxExHPb xny

    n

    n ,

    * 0,2

    1

    Start with the mode expansion equation

    and plug in TM mode. Collapse to 1D.

    ( )( ) ( )= dxxHxHxP

    Nb yny

    n

    n ,

    *0 0,1

    2

    use the relationship between H and E for a TM mode to get

    First lets assume that the mode n is bound.

    ( )( ) ( )

    ( )( )

    ( )( ) ( )

    ( )( )

    ==

    dxxHx

    dxxHxHx

    dxxHx

    N

    dxxHxHx

    N

    b

    yn

    yny

    yn

    yny

    n2

    ,

    ,

    2

    ,0

    ,0

    *

    1

    0,1

    1

    2

    0,1

    2

    Now assume n is a radiation mode. Lets write out the orthogonality

    condition. Assume that a mode m in the cladding has amplitude Cm

    ( ) ( ) ( )nmCNdxeCeCNdydxzHE mcl

    xjk

    m

    xjk

    n

    cl

    nmxnxm =

    =

    20*0*

    22

    1

    2

    1,,

    rr

    ( )( ) ( ) ( )

    ( )2

    ,1

    ,1

    *

    cladding

    0,

    yn

    ynyx

    H

    dxxHxHnb

    cl

    =Therefore

    z

    Er

    Hr

    TM

    x

    ( ) ( ){ } = dydxzyxHyxEPb n

    n

    n0,,,

    2

    1 *rr

    ( )( ) ( )= dxxHxHxP

    Nb yny

    n

    n

    *

    ,0* 0,

    1

    2

    or

    bn is still

    without

    units

    P(n)

    b(n) dn unitless =

    b(n) has units ofx

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    ECE 4006/5166 Guided Wave Optics

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    Mode normalization

    Modal analysis of planar slab waveguidesOrothogonality

    It is convenient to scale the mode such that we can drop all of the

    constants in the previous expressions. This is our last free parameter of

    the mode shapes as derived from the transcendental equation.

    ( ) ( )

    ( ) ( )

    ( )( )

    ( )

    ( )

    =

    =

    radiationcladding

    bound

    2

    1

    0,2

    1

    ,

    ,

    2

    ,

    ,

    ,

    0

    ,

    ,

    0

    yn

    yn

    yn

    yn

    yn

    n

    yn

    yny

    n

    n

    E

    xE

    dxxE

    xE

    xEP

    NxE

    dxxExEP

    Na

    ( )( ) ( )

    ( ) ( )

    ( )

    ( )( )

    ( )

    ( )

    =

    =

    radiationcladding

    bound1

    2

    0,1

    2

    ,1

    ,

    2

    ,

    ,

    ,0

    ,

    ,0*

    ynn

    yn

    yn

    yn

    yn

    n

    yn

    yny

    n

    n

    H

    xH

    dxxHx

    xH

    xHP

    NxH

    dxxHxHxP

    Nb

    cl

    The hat symbol will be used for normalized modes and modal amp. This normalizes the modes so that the transverse integral of the |field|2 = is

    1 or (n=0) (radiation). In 2D, mode amplitudes al.m and a(l,m)dldm now carry units ofE= V/m In 2D, mode amplitudes bl,m and b(l,m) dldm now carry units ofH= A/m

    TE:

    TM:

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    SummarizingMode orthogonality for the slab waveguide

    Modal analysis of planar slab waveguidesOrothogonality

    ( ) ( ) ( ) ( ) ( )

    ( ) ( )[ ] ( ) ( ) ( )[ ] ( )

    ++++

    +=

    m

    zmj

    zx

    m

    zj

    zmxmm

    m

    zmj

    y

    m

    zj

    ymm

    dmezmxExmxEmbezxExxEb

    ydmemxEmaexEazxE

    TMTMm

    TETEm

    ;;

    ;,

    ,

    ,

    ,,

    ,

    r

    We may calculate the field at any point down the

    guide. Using modes with arbitrary amplitude:

    TE

    TM

    z

    xAssume we have a field incident on our slabwaveguide, given by its tangentialEatz = 0. We

    may calculateH fromE.

    Using normalized modes:

    This is the complete solution to the waveguide boundary value

    problem, containing all the physics with no approximations!

    ( ) ( ) ( )= dxxEdxxExEa ymymym2

    ,,0,And amplitudes (e.g. TE bound):

    ( ) ( ) ( ) ( ) ( )

    ( ) ( )[ ] ( ) ( ) ( )[ ] ( )

    ++++

    +=

    m

    zmj

    zx

    m

    zj

    zmxmm

    m

    zmj

    y

    m

    zj

    ymm

    dmezmxExmxEmbezxExxEb

    ydmemxEmaexEazxE

    TMTMm

    TETEm

    ;;

    ;,

    ,

    ,

    ,,

    ,

    r

    TE

    TM

    ( ) ( )=dxxExEa

    ynyn ,

    0,And amplitudes (e.g. TE bound):

    ( ) ( ) ( ) dxxExExE ymymym2

    ,,,with normalized modes:

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    ECE 4006/5166 Guided Wave Optics

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    Rewrite TM expansionin terms of E, not H

    ( ) ( ){ }

    ( ) ( )

    ( ) ( ) ( )

    =

    =

    =

    dxxExExPN

    dxxHxE

    P

    P

    dydxzyxHyxEb

    xnx

    n

    ynx

    n

    n

    n

    n

    *

    ,

    0

    *

    ,

    *

    21

    0,2

    1

    0,

    2

    1

    ,0,,

    rr

    Modal analysis of planar slab waveguidesOrothogonality

    ( ) yxH

    x

    NE 0

    =

    TM mode

    Modal expansion,

    modes not normalized

    ( ) ( )

    ( )

    ( ) ( )

    ( )

    ( )

    == radiation

    cladding

    bound

    2

    1normalized

    ,

    ,

    2

    ,

    ,

    ,

    0

    ,

    xncl

    xn

    xn

    xn

    xn

    n

    xn

    En

    xE

    dxxEx

    xE

    xEPN

    xE

    ( )

    = [W/m]22

    1

    0dxExP xNz From previous derivation

    ( ) ( ) ( )= dxxExExb xnxn*

    ,0, ( ) ( )= dxxExEa ynyn

    *

    ,0,

    It is useful and instructive to recast the TM expansion in terms of E fields.

    We can normalize as usual

    Now the two coefficients a and b clearly refer to they andx E-fieldpolarizations (which are the TE and TM modes):

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    Units and values

    61

    1D (slab)

    2D (everything else)

    Not normalized Normalized

    Bound

    Radiatio

    n

    W/m

    W/m2

    V/m, A/m

    V/m, A/m

    V/m1/2 or A/m1/2

    1/m1/2

    Units are for quantities in bold face (i.e. P but not delta function).

    Blank indicates no units. Radiation mode labels (l,m) have units of 1/meters.

    Normalized Poynting vector listed is for TE modes. See pg 34 for TM.

    llb,a

    ll HE,

    = dxHE ll 21lP [ ]12 0N dxEl

    2 1

    llb,a

    llHE ,

    = dxHE ll21lP dxEl

    2V2/m

    Modal analysis of planar slab waveguidesOrothogonality

    ( ) ( ) dllbdl,la ( ) ( )lHlE ,

    ( ) ( ) == dxHEl210lP

    ( ) dxlE2

    V2/m

    V/m or A/m( ) ( )

    dllbdl,la( ) ( )lHlE ,

    ( ) ( ) == dxHEl 0 21lP( ) dxlE

    2 (l=0) [m]

    [ ]12 0N

    Not normalized Normalized

    Bound

    Radiation

    W

    W/m2

    V/m, A/m

    V/m, A/m

    V/m or A/m

    1/m

    l,ml,m b,a

    l,ml,m HE,

    dydxHEml,ml, = 21l,mP [ ]12 0N

    dxdyEml,2

    1

    l,ml,m b,a

    l,ml,m HE ,

    dydxHEml,ml, = 21l,mP

    dydxEml,2

    V2

    ( ) ( ) dldmml,bdldm,ml,a

    ( ) ( )ml,Hml,E ,

    ( ) ( )0, =mlml,P

    ( ) dydxml,E2

    V2/m

    V/m or A/m( ) ( ) dldmml,bdldm,ml,a

    ( ) ( )ml,Hml,E ,

    ( ) ( )0, =mlml,P

    ( ) dydxml,E2

    (l,m=0) [m2]

    [ ]12 0N

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    ECE 4006/5166 Guided Wave Optics

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    Power carried by a modeNot normalized

    Modal analysis of planar slab waveguidesOrothogonality

    ( ) ( ) ( ) ( ) ( )

    ( ) ( )[ ] ( ) ( ) ( )[ ] ( )

    ++++

    +=

    m

    zmj

    zx

    m

    zj

    zmxmm

    m

    zmj

    y

    m

    zj

    ymm

    dmezmxExmxEmbezxExxEb

    ydmemxEmaexEazxE

    TMTMm

    TETEm

    ;;

    ;,

    ,

    ,

    ,,

    ,

    r

    TE

    TM

    ( )

    ( ) ( )[ ] ( ) ( ) ( )[ ] ( )

    ( ) ( ) ( ) ( )

    ++

    +++

    =

    m

    zmj

    y

    m

    zj

    ymm

    m

    zmj

    zx

    m

    zj

    zmxmm

    dmeymxHmbeyxHb

    dmezmxHxmxHmaezxHxxHa

    zxH

    TMTMm

    TETEm

    ;

    ;;

    ,

    ,

    ,

    ,

    ,,

    r

    TE

    TM

    Lets calculate the Poynting vector using the modal field expansion. First we need to writedown bothEandH. TheHfields of the TE modes and theEfields of the TM modes must

    be calculated through the curl equations with proper units so that their Poynting vector is

    correct. The b coefficients can be calculated either fromEorHfields .

    ( ) ( ) ( ) ( )

    ( ) ( ) ( ) ( )

    ++

    +

    +

    =

    =

    dxdmemxHmaexHa

    dmemxEmaexEa

    dxHEP

    m

    zmj

    x

    m

    zj

    xmm

    m

    zmj

    y

    m

    zj

    ymm

    xyz

    TETEm

    TETEm

    ;

    ;

    [W/m]

    ***

    ,

    *

    ,

    21

    *

    21

    ,

    ,

    Note first the the cross product in Poyntings theorem will generate zero

    forETE HTM andETM HTE so we can treat the two polarizations

    independently. Then:

    TE (TM very similar)

    This looks terrible, but note that we can reverse the order of integrations,

    giving us our orthogonality relations, so the entire thing becomes

    [ ] ( ) ( ) ( ) ( )[ ]dmmPmbmPmaPbPaPm

    TETE

    m

    TMmmTEmmz +++=22

    ,

    2

    ,

    2

    Bound and rad

    modes orthogonal,

    so no cross terms.

    So |a|2 represents the power in the mode, scaled by the power the mode

    caries via the arbitrary amplitudes of the modes.

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    Modal analysis of planar slab waveguidesOrothogonality

    ( ) ( ) ( ) ( ) ( )

    ( ) ( )[ ] ( ) ( ) ( )[ ] ( )

    ++++

    +=

    m

    zmj

    zx

    m

    zj

    zmxmm

    m

    zmj

    y

    m

    zj

    ymm

    dmezmxExmxEmbezxExxEb

    ydmemxEmaexEazxE

    TMTMm

    TETEm

    ;;

    ;,

    ,

    ,

    ,,

    ,

    r

    TE

    TM

    ( )

    ( ) ( )[ ] ( ) ( ) ( )[ ] ( )

    ( ) ( ) ( ) ( )

    ++

    +++

    =

    m

    zmj

    y

    m

    zj

    ymm

    m

    zmj

    zx

    m

    zj

    zmxmm

    dmeymxHmbeyxHb

    dmezmxHxmxHmaezxHxxHa

    zxH

    TMTMm

    TETEm

    ;

    ;;

    ,

    ,

    ,

    ,

    ,,

    r

    TE

    TM

    In the case of normalized modes, the Poynting vector will equal N/(2 0) times a deltafunction (radiation modes).

    ( ) ( ) ( ) ( )

    ( ) ( ) ( ) ( )

    ++

    +

    +

    =

    =

    dxdmemxHmaexHa

    dmemxEmaexEa

    dxHEP

    m

    zmj

    x

    m

    zj

    xm

    m

    zmj

    y

    m

    zj

    ymm

    xyz

    TETEm

    TETEm

    ;

    ;

    [W/m]

    ****

    ,

    21

    *

    21

    ,

    ,

    Note first the the cross product in Poyntings theorem will generate zero

    forETE HTM andETM HTE so we can treat the two polarizations

    independently. Then:

    TE (TM very similar)

    Again, this generates the orthogonality relationship, simplifying to:

    ( ) ( ) dmmbmabaN

    Pmm

    mmz

    ++

    += 0

    2

    0

    2

    0

    2

    0

    2 11

    2

    Bound and rad

    modes orthogonal,

    so no cross terms.

    Power carried by a modeNormalized

    The TM terms assumes b has been calculated from the normalized H as per

    page 57 and H fields normalized as per page 58.

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    ECE 4006/5166 Guided Wave Optics

    Coupling efficiency

    Modal analysis of planar slab waveguidesOrothogonality

    One of the most common questions asked about a waveguide is the

    efficiency of coupling from an external source such as a focused beam or

    a previous waveguide. We now have the tools to answer this question.

    ( ) ( )

    [ ] ( ) ( ) ( ) ( )[ ]

    +++

    ++

    +

    =

    dmmPmbmPmaPbPa

    dmmbmabaN

    P

    m

    TETE

    m

    TMmmTEmm

    mm

    mm

    z22

    ,

    2

    ,

    2

    0

    2

    0

    2

    0

    2

    0

    2 11

    2

    Normalized

    Not

    External fields can be decomposed intoEy, Hx which couple into the TE

    modes andEx, Hy which couple into the TM and thus the two can be

    considered separately. Assuming the incident field is in a uniform

    material of index ninc, an external TE field will have z-directed power:

    ( )

    = [W/m]0,2

    2, 0dxxEP y

    n

    inczinc

    The expressions after the brace are the normalized mode amplitudes2 soin the normalized case:

    Assuming appropriate index matching or anti-reflection coating is

    employed such that the difference between ninc andNcan be ignored,

    then (for TE), in the non-normalized case, the coupling efficiency is

    ( ) ( ) ( )

    =

    dmEmadxEa

    dxE

    m

    ym

    ymm

    y

    m 2

    ,

    2

    2

    ,

    2

    2

    cladding0

    1

    incidentPower

    modeinPower

    Bound

    Radiation( ) ( ) ( )

    =

    = dmmaa

    dmma

    a

    dxE

    mm

    y

    m 2

    2

    2

    2

    2

    0

    1