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Fiber Optic Communications( Chapter 2: Optics Review )

presented byProf. Kwang-Chun Ho

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Section 2.4: Numerical Aperture

vConsider an optical receiver:

uwhere the diameter of photodetector surface area is d.

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Numerical Aperture

vConsider an off-axis beam:

u Light incident beyond the acceptance angle θwill not be focused onto the detector surface.

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Numerical Aperture

vFrom the previous slide, we see that

vThen, the numerical aperture is defined as

u no is the index of refraction for the surrounding media.

u It usually has a value of unity.u θ is the maximum acceptance angle to detect

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Numerical Aperture

vExample:uCompute the NA whenuSolution:

l Since , if θ is so small (i.e. d<<f), we have

10 , 1 , 1of cm d cm n= = =

sin tanθ θ≈

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Numerical Aperture (Applications to Fibers)

vFibers have limited acceptance angles and thus a small NA.

u Typical fibers have an NA in the range 0.1 to 0.5.u Coupling to a low-NA fiber is more difficult and less

efficient than that to a high-NA fiber

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Numerical Aperture

vSome NA and the corresponding acceptance angles:

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

Acceptance Angle

NA

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Section 2.5: Diffraction TheoryvSome experiments do not follow ray theory

closely enough.uA more complete theory based on wave nature of

light is needed to explain these ones

vThis is called Diffraction Theory or Physical Optics.vAs an example of the difference between

ray theory and diffraction theory, consider focusing by a lens of a uniform beam of light.

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Diffraction TheoryuF. Grimaldi’s work in the 1600s !

ud is the diameter of the central spot.

Beam passing a lens Does not converge toA point. Instead, it forms a central spot !

Fiber

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Diffraction Theory

vThe diameter of central spot is found from

(2-14)vExample: u If , then

l If coupling light into fiber with diameter less than 4 µm, the coupling efficiency will be low.

2 , 1f D mλ µ= =

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Gaussian BeamvActual light source produce non-uniform beam

pattern (Gaussian intensity distribution).

-15 -10 -5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

Nor

mal

ized

Inte

nsity

(I/I

o) where w=10 µm

13.534%

2w

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Gaussian Beam

(2.5)uwhere w = spot size and r = radial distance from

the center of the beam pattern.vNote that

uBy definition, w is the spot size. uThe spot diameter is 2w.

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Gaussian Beam

vThe Gaussian intensity distribution is useful because most lasers emit in this pattern.

vAnd it is the field pattern inside a single-mode fiber.

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Focusing a Gaussian Beam

u2w0 is the spot size of focusing beam appearing in the focal plane

2

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Focusing a Gaussian BeamvThe field pattern in the focal plane is given

by

vThe spot size in the focal plane is given by

(2.16)

uThe peak intensity will be more intense, because the beam is compressed.

2

22

o

r

woI I e

−

′=

oI ′

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Collimating a Gaussian BeamvNear the lens the collimation is good (in

Fresnel zone), but far from the lens the beam diverges.

Fresnel Zone

2w

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Collimating a Gaussian BeamvFar from the lens the beam diverges

according to

vThe divergence angle is found fromv Thus,

(2.17)

where

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Collimating a Gaussian BeamvExample:uSpot size: at lens, anduFind θ and wo at z = 10 m, 1 km, and 10 km.uSolution:

1w mm= 0.82 mλ µ=

radians

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Collimating a Gaussian Beaml At z = 10 m, 1 km, and 10 km, respectively

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Beam DivergencevConsider beam divergence

(2.17)

uThe beam divergence depends upon λ/w. u Lenses are analogous to antennas, which direct

energy into desired directions.uA small beam divergence requires a large spot

size relative to the wavelength (λ<<w).

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Beam DivergencevConsider an optical communication system

in atmosphere

u For a long path, much of the light is not collected by the receiving lens.

u This limits the length of the atmospheric system. uOther problems with atmospheric systems are

weather and physical interference with the beam.

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Section 2.6: Summary

vReview of Subjects presenteduRefractive indexuSnell’s lawu Lenses to focus, collimate, and imageuNumerical apertureu Limits due to diffraction theoryuGaussian beams

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ProblemsvP2-8:uCompute the divergence angle of Gaussian beam

of andu If the beam is aimed at moon, what is its spot

size on moon’s surface?lDistance between earth and moon:

uSolution: lDivergence angle:l Spot size on moon:

1w mm= 0.8 mλ µ=

83.8 10moonz m= ×

425.09 10

wλ

θπ

−= = × radians

96.72o moonw z kmθ

== =

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ProblemsvP2-9:uA 6000 km undersea glass-fiber telephone line

crosses the Atlantic ocean connecting U.S.A. and France