7. guided-wave opticsoptics.hanyang.ac.kr/~shsong/p7-guided-wave optics.pdf · microsoft powerpoint...
TRANSCRIPT
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7. Guided-wave optics
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Optical waveguides
slab strip fiber
strip embedded strip rib or ridge strip loaded
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Symmetric & Asymmetric waveguides
Cladding (or, cover) : nc
Core (or, film) : nf
Cladding (or, substrate) : ns
x
y
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Guided waves
A waveguide
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Electromagnetic modes of waveguides
A mode of a waveguide is a stable, propagating pattern of electric and magneticfields that is periodic along the axis of the waveguide, apart from attenuation
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7.1 Planar perfect-mirror waveguides
Waveguide modes
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Condition of self-consistency
2 2cos 2 (1 2sin ) - 2 sin 2 sinAB AC AC AC AB AC d
Bounce angles
(Transverse Component)
: The ray-optics picture of light guidance by multiple reflections
2 2ymd k m
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Propagation constants
Bounce angle Propagation constant
z2 2 2 2 2( / )m zm ymk k k k m d
0k nkymk m
d
m
sin2m md
0 sinym mk nk
2 2 2( / )m k m d
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Field distributions : TE modes
The complex amplitude of the total field in the waveguide is the superposition of the two bouncing TEM plane waves :
upward wave + downward wave
+=
: symmetric modes, odd modes
: antisymmetric modes, even modes
are normalized
are orthogonal in [-d/2. d/2] interval
Assume that the bouncing TEM plane wave is polarized in the x direction, the guided wave is a transverse-electric (TE) wave.
TE mode
( ,0,0); (0, , )x y zE E H H H H
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Each mode can be view as a standing waves in the y direction, traveling in the z direction.
Modes of large m vary in the transverse plane at a greater rate ky, and travel with a smaller propagation constant .
The field vanishes at y = +d/2 for all modes, so that the boundary conditions at the surface of the mirrors are always satisfied.
[ TE guided waves ]
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Number of modes
( d < /2 )
( /2 < d < ) single-mode waveguide
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Group velocities
Group velocity of mode m :
More oblique modes travel with a smaller group velocity since they are delayed by the longer path of the zigzaging process.
Geometrically,
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Field distributions : TM modes
upward wave + downward wave
Magnetic field is in the x direction, the guided wave is a transverse-magnetic (TM) wave.
Since the z component of the electric field is parallel to the mirror, it must behave like the x component of the TE mode :
y components of the electric field:
m
Ez
EyEz components of the electric field:
E
(0, , ); ( 0,0)y z xE E E H H H
TM mode
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Multimode fields
( m = 1 )
( m = 2 )
( m = 1 & 2 )
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7.2 Planar dielectric waveguides
Cladding (or, cover) : nc
Core (or, film) : nf
Cladding (or, substrate) : ns
x
y
n1
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Let’s first consider a symmetric waveguide.
Transverse Component of the wavevector
: The ray-optics picture of light guidance by multiple reflections
Self-consistency condition
2 2 sin 2d m
(Perfect mirror)
2 2ymd k m (Perfect mirror)
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Self-consistency condition : TE modes
Self-consistency condition (TE mode)
From the TIR of TE modes,
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Self-consistency condition : TE modes
(open circles)
0,1, 2,m
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Number of modes : TE modes
Single-mode waveguide (TE mode)
In a dielectric waveguide, there is at least one TE mode,since the fundamental mode m = 0 is always allowed.
No cutt-off frequency
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Propagation constants : TE modes
: propagation constant ( the z-component of wavevector)
n112
M
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Field distributions : TE modes
: Extinction coefficient
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Confinement factor
the ratio of power in the slab to the total power
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Dispersion relation
2 2 2 21 2 1 2
2 22 211
2 tanc
n n n nmn nn n
0 0/N c Nk
0/ 2 /(2 )c c d NA (N : effective index)
Single-mode waveguide (TE mode)
0Nk
3 modes
1 mode
2 modes
2c
1c
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Group velocities
The group velocities lie between c1 and c2(the phase velocities in the slab and substrate).
At a given , the lowest-order mode (the least oblique mode, m = 0)
travels with a group velocity closest to c1. The most oblique mode (m = M) has a group velocity ~ c2.
The higher mode is faster than the lower!
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Group velocities
1
cosGHcz
More oblique modes travel this lateral distance at a fast speed than less oblique modes
증명 !!!
The higher mode is faster than the lower!
Lateral shift
Time delay
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Goose-Hanchen shift
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Goose-Hanchen effect Evanescent field
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Now, consider asymmetric waveguides.
( Slab )
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For TE modes :
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Guided TE modes :
Evanescent waves (x > d, x < 0)
Penetration distances
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Guided waves ( d > x > 0)
Transverse wavevector ( d > x > 0)
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Effective thickness of the waveguide
Dispersion relation for TE modes in a planar dielectric waveguide
Dispersion relation & Effective thickness
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Normalized frequency : V
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Dispersion relation for TE modes in terms of V , b and a:
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Dispersion relation for TM modes in terms of V , b and a:
(TE)
(TE)
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Summary of planar dielectric waveguides
TE :
TM :
has two modes .
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7.3 Two-dimensional waveguides
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Comparison : the number of modes
0
2dM NAl
æ ö÷ç ÷» ç ÷ç ÷÷çè ø
0 0
2/ 2
d dMl l
= =
2 2
0 0
2 14 4 / 2
d dM pp
l l
æ ö æ ö÷ ÷ç ç÷ ÷» =ç ç÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø
22
0
24
dM NApl
æ ö÷ç ÷» ç ÷ç ÷÷çè ø
1-d Mirror Guide
1-d Dielectric Guide
2-d Mirror Guide
2-d Dielectric Guide
( )00
2 dV NA k d NApl
= =
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 fitbut now limited by the angular cutoff characterized by the NA of the guide
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7.4 Optical coupling in waveguides
A. Mode excitation
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Input couplersCoupling by focusing beam
End butt Coupling
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Prism coupling
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B. Coupling between waveguides
In the n1 slab waveguide,
In the n2 slab waveguide,
When a is very large (no-coupling)
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Derivation of coupled wave equations
n1
n2
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: coupling coefficient.
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Coupled wave equations
Coupling coefficients
Phase mismatch per unit length
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Exchange of power
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Exchange of power when the guides are identical (phase-matched)
: transfer distance
: 3-dB coupling
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Power transfer ratio in a small phase mismatched case
EO, TO, MO waveguide switches