C3 Planar Optical Waveguide

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Slides on Optical Waveguide from BITS Pilani. Course - Fiber optics and Optoelectronics

Transcript of C3 Planar Optical Waveguide

  • BITS Pilani Pilani Campus

    Wave Propagation In Planar Waveguides

    RAHUL SINGHAL

  • Plane TEM Wave

    polarized is xE

    polarized is yH

    Transverse means E & H are perpendicular

    to direction of propagation

  • The net electric flux through any closed

    surface is equal to 1 times the net electric charge enclosed within that closed surface.

    It states that the magnetic field B has

    divergence equal to zero, in other

    words, it is equivalent to the

    statement that magnetic monopoles

    do not exist.

    Ampre's law with Maxwell's addition states that

    magnetic fields can be generated in two ways:

    by electrical current (this was the original

    "Ampre's law") and by changing electric fields

    (this was "Maxwell's addition").

    The induced electromotive force in

    any closed circuit is equal to the

    negative of the time rate of change of

    the magnetic flux through the circuit.

  • 22

    2

    A/mdensity current Electric;J

    A/mIntensity Field Magnetic

    )Sec./m-Vor (Telsa TDensity Flux Magnetic ;

    C/mDensity Flux Electric;

    V/mIntensity Field Electric

    E

    H

    HB

    ED

    E

    0.

    .

    B

    D

    t

    DJH

    t

    BE

    MAXWELL EQUATIONS

    zk

    yj

    xi

    Del or Nabla Operator

  • 0.

    0.

    B

    D

    t

    DH

    t

    BE

    t

    B

    t

    BE

    )(

    2

    2)(

    t

    D

    t

    D

    tt

    H

    27

    00 /104; ANr

    2212

    00 /10854.8; NmCr

    Inside an ideal dielectric, = 0; = 0.

    From Eq based on Gauss law,

  • EEE 2).(

    But,

    2

    2

    2

    2

    2

    22

    zyx

    where,

    0)/.(. DE&,

    2

    2

    2

    2

    2

    22

    t

    E

    t

    E

    t

    DE

    02

    22

    t

    EE

    02

    22

    t

    HH

    Similarly,

  • 02

    22

    t

    1pv

    01

    2

    2

    2

    2

    tvp

    02

    2

    2

    22

    tc

    n

    )(exp0 zti

    pv

    m/s 1031

    that so

    ,1 space, freeor In vacuum

    8

    cvp

    rr

    ncv

    n

    p

    rr

    /Therefore,

    1 with /

    medium, isotropican For

    0

    Solution

  • 0..).( 0 EED r

    0.).(0 EE rr EE rr

    ).(1

    .

    2

    2

    002

    2

    t

    E

    t

    DE r

    2

    2

    00

    2).(t

    EEE r

    0).(2

    2

    00

    2

    t

    EEE r

    0)).(1

    (2

    2

    00

    2

    t

    EEE rr

    r

    0)()(1

    (2

    2

    00

    2

    t

    HHH rr

    r

    Similarly,

    SOLUTION IN INHOMOGENEOUS MEDIUM

    For an isotropic, linear, non-conducting, non-magnetic, but

    inhomogenoeous (or heterogeneous) medium,

    From curl of Gauss Law,

    Rearranging,

    Substituting,

    0. as medium shomogeneoufor ,0 E

    EED

    r

    rr

    0

    00 1

  • )(exp)(E ztixE jj

    )(exp)(H ztixH jj

    t

    H

    t

    BE

    0

    y

    E

    x

    Ek

    x

    E

    z

    Ej

    z

    E

    y

    Ei x

    yzxyz

    zyx HkHjHi

    t

    0

    t

    En

    t

    E

    t

    DH r

    200

    y

    H

    x

    Hk

    x

    H

    z

    Hj

    z

    H

    y

    Hi x

    yzxyz

    zyx EkEjEi

    tn 20

    )(22 xnnr Assume n do not vary

    in y and z directions

    y- & z- dependence of fields,

    will be, in general, of form, )(exp zyi

    zoryxj ,,,

    =0 without any loss of generality

  • )()()()( ztiyztizztixztiy eEx

    keEx

    jeEz

    jeEz

    i

    zero are terms all ,ith not vary w do & As,y

    yHE

    )()()(

    0 zti

    z

    zti

    y

    zti

    x eHt

    keHt

    jeHt

    i

    x

    Ek

    x

    EEijEii

    yzxy

    zyx HikHijHii )(

    )()( 000

    Comparing on both sides,

    xy

    xy

    HE

    HiEi

    0

    0

    or,

    )(

    yz

    x Hix

    EEi )(0

    z

    yHi

    x

    E)(0

  • )()()()( ztiyztizztixztiy eHx

    keHx

    jeHz

    jeHz

    i

    zero are terms all Again,y

    )(2

    0

    )(2

    0

    )(2

    0 zti

    z

    zti

    y

    zti

    x eEt

    kneEt

    jneEt

    in

    x

    Hk

    x

    HHijHii

    yzxy

    zyx EnikEnijEnii )()()( 20

    2

    0

    2

    0

    Comparing on both sides,

    xy

    xy

    EnH

    EniHi

    2

    0

    2

    0

    or,

    )(

    yz

    x Enix

    HHi )( 20

    z

    yEni

    x

    H 20

  • xy HE 0

    yz

    x Hix

    EEi 0

    z

    yHi

    x

    E0

    xy ExnH )( 2

    0

    yz

    x Exnix

    HHi )(20

    z

    yExni

    x

    H )( 20

    TE Modes, Ey component

    zxyyzx HHEHEE ,&,,only involves and zero are ,&,,

    TM Modes, Hy component

    zxyyzx EEHEHH ,&,,only involves and zero are ,&,,

  • PLANAR OPTICAL WAVEGUIDE

  • Planar Optical Waveguide

    z

    y

    x

    x

    z

    sin1x

    cos10 z-a

    +a

    x = 0

    kn11

    n1

    n2

    knn

    m

    11

    1

    22

    1nm

    2k

  • sinsin 11 knx

    cos1 z

    knn

    m

    11

    1

    22

    1

    2cossinn

    nmc

    22

    1

    211min cos kn

    n

    nm

    k

    2

    sin

    2

    m

    x

    1nm

  • mi ia sin4

    2/122211max4sin4sin4

    nnaana

    iM m

    m

    m

    2/122214

    nna

    i

    ia x 24

    z

    sin1x

    cos10 z-a

    +a

    x = 0

    kn11

    n1

    n2 a + a 2a +

    sin

    2

    m

    x

    ... ,3 ,2 ,1 ,0iEach value of i, corresponds to a

    particular mode

    with its own i,

    in z-direction.

    Mode corresponding imax is refracted at interface,

    & may propagate in cladding is called a radiation mode

  • TE Modes of a Symmetric

    Step-Index Planar Waveguide

    )()( 22 xnxn

    yyy ExniExix

    Ei )(1 2

    0

    00

    )(xEE yy

    0)(22002

    2

    2

    yyy

    ExnEdx

    Ed

    kcc

    Exnkdx

    Edy

    y

    ;&1

    where,

    0)(

    200

    222

    2

    2

  • 0221222

    yy

    Enkdx

    Ed

    2

    1)(

    n

    nxn

    ax for

    ax for

    ax for

    0222222

    yy

    Enkdx

    Ed ax for

    Let, 221

    22

    1

    22 nku2

    2

    22

    2

    222 nkw02

    2

    2

    yy

    Eudx

    Edax for

    022

    2

    yy

    Ewdx

    Edax for

    &,

    22222221221 nknk

  • )()(

    )()(

    xExE

    xExE

    yy

    yy

    Anti-symmetric Modes

    Symmetric Modes

    xwyCe

    uxAxE

    cos)(

    ax for

    ax for

    waCeuaA )cos(

    wawCeuauA )sin(

    wuau )tan(

    wauaua )tan(

    222122222221222 nnknknkwu

    222122

    2

    2

    2

    1

    2222 2 nnannakwaua

    222122 2

    nna

    wauaV

    2/122)tan( uaVuaua

    Symmetric Modes

    The continuity of Ey(x) and dEy/dx at x=a gives,

  • Anti-symmetric Modes

    xwy De

    x

    xuxB

    xE

    sin

    )(

    ax for

    ax for

    wauaua )cot(

    2/122)cot( uaVuaua

    From Fig. in next slide,

    2

    122

    2

    mVmm Anti-symmetric Modes

    (m+1)Symmetric Modes

    2

    222

    12

    mVm(m+1) Anti-symmetric

    Modes

    (m+1) Symmetric Modes

    ... ,2 ,1 ,0m

    VM

    2

  • wauahere ; ,

  • 22

    2

    2

    2

    1

    2

    2

    2

    1

    V

    wa

    V

    uab

    m = 0

    m = 1 m = 2

  • 2mVua c

    22212

    nna

    V

    22212

    2nn

    am

    222122

    nn

    ma

  • Problem 3.9 What should be the maximum thickness of the

    guide slab of a symmetrical SI planar waveguide so that it

    supports only the first 10 modes? Take n1 = 3.6, n2 = 3.58,

    and = 0.90 m. Calculate the maximum and minimum values of the propagation constant , .

  • Power Distribution And Confinement Factor

    The power flow is given by Poynting vector defined by

    S = E H where E and H are expressed in complex form but the actual fields are the real part of the complex form. Thus taking Time Average of the

    Poynting vector,

    *Re2

    1ReRe HEHES

    where H* is complex conjugate of H. Thus, time average of S along z-

    direction will be given by

    )(2

    1yxyxz EHHES

  • For TE modes

    yx EH

    0

    therefore, 2

    02

    1yz ES

    For a particular mode, the power associated per unit area per unit

    length in the y-direction will thus be given by

    dxEPx

    y

    2

    02

    1

  • The power inside the guide layer (core)

    and the power inside the cladding (or outside the guide layer)

    dxEP

    a

    ax

    yin

    2

    02

    1

    dxEdxEPax

    y

    a

    x

    yout

    22

    02

    1

  • For symmetric TE modes,

    ,

    ,cos)(

    axCe

    axuxAxE xwy

    Thus