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Chapter 2

Light Propagation In Optical Fiber

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Introduction

An optical fiber is a very thin strand of silica glass in geometry quite like a human hair. In reality it is a very narrow, very long glass cylinder with special characteristics. When light enters one end of the fiber it travels (confined within the fiber) until it leaves the fiber at the other end. Two critical factors stand out: Very little light is lost in its journey along the fiber Fiber can bend around corners and the light will stay within it and be

guided around the corners. An optical fiber consists of two parts: the core and the cladding. The

core is a narrow cylindrical strand of glass and the cladding is a tubular jacket surrounding it. The core has a (slightly) higher refractive index than the cladding. This means that the boundary (interface) between the core and the cladding acts as a perfect mirror. Light traveling along the core is confined by the mirror to stay within it - even when the fiber bends around a corner.

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BASIC PRINCIPLE

When a light ray travelling in one material hits a different material and reflects back into the original material without any loss of light, total internal reflection is said to occur. Since the core and cladding are constructed from different compositions of glass, theoretically, light entering the core is confined to the boundaries of the core because it reflects back whenever it hits the cladding. For total internal reflection to occur, the index of refraction of the core must be higher than that of the cladding, and the incidence angle is larger than the critical angle.

                                  

                    

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What Makes The Light Stay in Fiber

Refraction The light waves spread out along its beam. Speed of light depend on the material used called refractive index.

Speed of light in the material = speed of light in the free space/refractive index

Lower refractive index higher speed

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The Light is Refracted

This end travels further than the other hand

Lower Refractive index Region

Higher Refractive index Region

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Total Internal Reflection

Total internal reflection reflects 100% of the light A typical mirror only reflects about 90% Fish tank analogy

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Refraction When a light ray encounters a boundary separating two

different media, part of the ray is reflected back into the first medium and the remainder is bent (or refracted) as it enters the second material. (Light entering an optical fiber bends in towards the center of the fiber – refraction)

Refraction

LED or LASER Source

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Reflection

Light inside an optical fiber bounces off the cladding - reflection

Reflection

LED or LASER Source

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Critical Angle If light inside an optical fiber strikes the cladding too steeply,

the light refracts into the cladding - determined by the critical angle. (There will come a time when, eventually, the angle of refraction reaches 90o and the light is refracted along the boundary between the two materials. The angle of incidence which results in this effect is called the critical angle).

Critical Angle

n1Sin X=n2Sin90o

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Angle of Incidence

Also incident angle Measured from perpendicular Exercise: Mark two more incident angles

Incident Angles

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Angle of Reflection

Also reflection angle Measured from perpendicular Exercise: Mark the other reflection angle

Reflection Angle

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Reflection

Thus light is perfectly reflected at an interface between two materials of different refractive index if:

The light is incident on the interface from the side of higher refractive index.

The angle θ is greater than a specific value called the “critical angle”.

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Angle of Refraction

Also refraction angle Measured from perpendicular Exercise: Mark the other refraction angle

Refraction Angle

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Angle Summary

Refraction Angle

Three important angles The reflection angle always equals the incident angle

Reflection Angle

Incident Angles

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Index of Refraction

n = c/v c = velocity of light in a vacuum v = velocity of light in a specific medium

light bends as it passes from one medium to another with a different index of refraction air, n is about 1 glass, n is about 1.4

Light bends in towards normal - lower n to higher n

Light bends away from normal - higher n to lower n

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Snell’s Law The angles of the rays are measured with respect to the

normal. n1sin 1=n2sin 2

Where n1 and n2 are refractive index of two materials 1and 2 the angle of incident and refraction respectively

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Snell’s Law The amount light is bent by refraction is given by Snell’s Law:

n1sin1 = n2sin2

Light is always refracted into a fiber (although there will be a certain amount of Fresnel reflection)

Light can either bounce off the cladding (TIR) or refract into the cladding

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Snell’s Law

Normal

Incidence Angle(1)

Refraction Angle(2)

Lower Refractive index(n2)

Higher Refractive index(n1)Ray of light

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Snell’s Law (Example 1)

Calculate the angle of refraction at the air/core interface Solution - use Snell’s law: n1sin1 = n2sin2

1sin(30°) = 1.47sin(refraction) refraction = sin-1(sin(30°)/1.47) refraction = 19.89°

nair = 1ncore = 1.47ncladding = 1.45incident = 30°

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Snell’s Law (Example 2)

Calculate the angle of refraction at the core/cladding interface Solution - use Snell’s law and the refraction angle from Example 3.1

1.47sin(90° - 19.89°) = 1.45sin(refraction) refraction = sin-1(1.47sin(70.11°)/1.45) refraction = 72.42°

nair = 1ncore = 1.47ncladding = 1.45incident = 30°

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Snell’s Law (Example 3)

Calculate the angle of refraction at the core/cladding interface for the new data below Solution: 1sin(10°) = 1.45sin(refraction(core))

refraction(core) = sin-1(sin(10°)/1.45) = 6.88° 1.47sin(90°-6.88°) = 1.45sin(refraction(cladding)) refraction(cladding) = sin-1(1.47sin(83.12°)/1.45)

= sin-1(1.0065) = can’t do light does not refract into

cladding, it reflects backinto the core (TIR)

nair = 1ncore = 1.47ncladding = 1.45incident = 10°

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Critical Angle Calculation

The angle of incidence that produces an angle of refraction of 90° is the critical angle n1sin(c) = n2sin(°) n1sin(c) = n2

c = sin-1(n2 /n1) Light at incident angles

greater than the criticalangle will reflect backinto the core

Critical Angle, c

n1 = Refractive index of the coren2 = Refractive index of the cladding

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NA Derivation

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Acceptance Angle and NA

The angle of light entering a fiber which follows the critical angle is called the acceptance angle,

= sin-1[(n12-n2

2)1/2]

Numerical Aperature (NA)describes the light-gathering ability of a fiber

NA = sin Critical Angle, c

n1 = Refractive index of the coren2 = Refractive index of the cladding

Acceptance Angle,

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Numerical Aperture

The Numerical Aperture is the sine of the largest angle contained within the cone of acceptance.

NA is related to a number of important fiber characteristics. It is a measure of the ability of the fiber to gather light at the

input end. The higher the NA the tighter (smaller radius) we can have

bends in the fiber before loss of light becomes a problem. The higher the NA the more modes we have, Rays can bounce

at greater angles and therefore there are more of them. This means that the higher the NA the greater will be the dispersion of this fiber (in the case of MM fiber).

Thus higher the NA of SM fiber the higher will be the attenuation of the fiber

Typical NA for single-mode fiber is 0.1. For multimode, NA is between 0.2 and 0.3 (usually closer to 0.2).

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Acceptance Cone

There is an imaginary cone of acceptance with an angle The light that enters the fiber at angles within the

acceptance cone are guided down the fiber core

Acceptance Cone

Acceptance Angle,

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Acceptance Cone

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Formula Summary

Index of Refraction

Snell’s Law

Critical Angle

Acceptance Angle

Numerical Aperture

vcn

2211 sinsin nn

1

21sinnn

c

22

21

1sin nn

22

21sin nnNA

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Practice Problems

What happens to the light which approaches the fiber outside of the cone of acceptance? The angle of incidence is 30o as in Fig.1 (calculate the angle of refraction at the air/core interface, r/ critical angle, c/ incident angle at the core/cladding interface, i/) does the TIR will occur?

Practice Problems (1)

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Practice Problems (2)

Calculate:angle of refraction at the air/core interface, r

critical angle , c

incident angle at the core/cladding interface , i

Will this light ray propagate down the fiber?

air/core interface

core/cladding interface

Answers:r = 8.2°c = 78.4°i = 81.8°light will propagate

nair = 1ncore = 1.46ncladding = 1.43incident = 12°

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Refractive Indices and Propagation Times

0

1

2

3

4

5

6

Vacuum 1 3.336Air 1.003 3.346Water 1.333 4.446Fused Silica 1.458 4.863Belden Cable (RG-59/U)

N/A 5.551

Refractive Index

Propagation Time (ns/m)

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Propagation Time Formula

Metallic cable propagation delay cable dimensions frequency

Optical fiber propagation delay related to the fiber material

formula

t = Ln/c

t = propagation delay in secondsL = fiber length in metersn = refractive index of the fiber corec = speed of light (2.998 x 108 meters/second)

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Temperature and Wavelength

Considerations for detailed analysis Fiber length is slightly dependent on temperature Refractive index is dependent on wavelength

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Classification of Optical Fiber

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Three common type of fiber in terms of the material used:

• Glass core with glass cladding –all glass or silica fiber

• Class core with plastic cladding –plastic cladded/coated silica (PCS)

• Plastic core with plastic cladding – all plastic or polymer fiber

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All glass fiber

The refractive index range of glass is limited which causesthe refractive index difference n1-n2 to be small.

This small value then reduces the light coupling efficiency ofthe fiber, i.e. large loss of light during coupling.

The attenuation is the lowest compared to the other two fibers making it suitable for long and high capacity.

Typical size: 10/125µm, 62.5/125µm, 50/125µm and 100/140µm.

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Plastic Clad Silica (PCS)

This fiber have higher loss than the all glass fiber and is suitable for shorter links.

Normally, the range of refractive index achievable with plastic fibers are large.

A larger range for the value of refractive index difference.

Light coupling efficiency is better.

Typical size: 62.5/125µm, 50/125µm, 100/140µm 200µm.

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All-plastic fiber

This type has the highest loss during transmission.

Normally used for very short links.

Large core size, therefore light coupling efficiency is high

The core size can be as large as 1mm.

Plastic and Silica Fibers

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Other fibers

1. Dispersion compensating fibers2. Dispersion flattened fiber3. Polarization-maintaining fibers4. Bend-insensitive and coupling fibers5. Reduced-cladding fibers6. Doped fibers for amplifiers and lasers7. Fiber gratings and photosensitive fibers8. Holey Fiber

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1. Dispersion compensating fibers: fiber with very high negative waveguide dispersion used to cancel the positive chromatic dispersion. Insert a DCF after a normal fiber.

2. Polarization-maintaining fibers, also known as polarization preserving fiber: Fiber designed to cope with polarization mode dispersion (PMD). Mainly used in sensors and optical devices that require polarization control. Gyroscope, modulators and couplers.

3. Bend-insensitive and coupling fibers. High coupling efficiency and low bend loss. Used in pigtails, short connection inside optical transmitters, receivers and other devices. Can bend at sharper angle.

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4. Reduced-cladding fibers: Has smaller cladding diameter (typically 80 µm) to offer higher packing density and greater flexibility than standard fibers.

5. Doped fibers for amplifiers and lasers: Fibers that are doped with materials (Erbium, praseodymium, thulium, ytterbium and neodymium) that can be stimulated to emit light. Used as optical amplifiers and fiber lasers.

6. Fiber gratings and photosensitive fibers: Grating are optical filter that reflects certain wavelength and allows transmission of others. Photosensitive fibers are sensitive to UV light and are used to fabricate fiber gratings.

7. Holey Fiber: hollow core surrounded by a photonic bandgap cladding

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Step Index Fibers

The optical fiber with a core of constant refractive index n1 and a cladding of a slightly lower refractive index n2 is known as step index fiber.

This is because the refractive index profile for this type of fiber makes a step change at the core-cladding interface as indicated in Fig which illustrates the two major types of step index fiber.

The refractive index profile may be defined as

n(r) = n1 r < a (core) n2 r ≥ a (cladding)

04/28/23 47Figure.2.6

(a)

(b)

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Fig 2.6(b) shows a multimode step index fiber with a core diameter of around 50µm or greater, which is large enough to allow the propagation of many modes within the fiber core. It illustrates the many different possible ray paths through the fiber.

Fig 2.6(a) shows a single mode or monomode step index fiber which allows the propagation of only one transverse electromagnetic mode and hence the core diameter must be of the order of 2-10µm.

The propagation of a single mode is illustrated in Fig 2.6 (a) as corresponding to a single ray path only (usually shown as the axial ray) through the fiber.

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The single mode step index fiber has the distinct advantage of low intermodal dispersion as only one mode is transmitted.

In multimode step index fiber considerable dispersion may occur due to the differing group velocities of the propagating modes.

This is turn restricts the maximum bandwidth attainable with multimode step index fibers, especially when compared with single mode fibers.

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Lower bandwidth applications multimode fibers have several advantages over single mode fibers:

1.The use of spatially incoherent optical sources (e.g. most light emitting diodes which cannot be efficiently coupled to single mode fibers.2.Larger numerical apertures, as well as core diameters, facilitating easier coupling to optical sources.3.Lower tolerance requirements on fiber connectors

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Single Mode Step Index fiber The advantage of the propagation of a single mode within an optical fiber is the signal dispersion caused by the delay differences between different modes in a multimode fiber may be avoided. Thus achieving a large BW.

In describing SMF, a parameter known as mode-field diameter (MFD) is used. In a SMF light travels mostly within the core and partially within the cladding. MFD is a function of the wavelength.

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Illustration of Mode-Field Diameter (MFD)

Mode-field diameter is a measure of the spot size or beam width of light propagating in a single-mode fiber. Mode-field diameter is a function of source wavelength, fiber core radius, and fiber refractive index profile. The vast majority of the optical power propagates within the fiber core, and a small portion propagates in the cladding near the core (Figure 1)

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• The beams travel at distinct propagating angles ranging from zero to critical value.• These different beams are called modes.• The smaller the propagating angle, the lower the mode.• The mode traveling precisely along the axis is zero- order mode or the fundamental.

Hence for the transmission of a single mode the fiber must be designed to allow propagation of only one mode, whilst all other modes are attenuated by leakage or absorption.

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• This may be achieved through a suitable choice of normalized frequency, V for the fiber. For single mode operation, only the fundamental TE01 mode can exist. The cutoff normalized frequency for the TE01 mode occurs at V=2.405. Thus single mode propagation is possible over the range:

0 ≤ V < 2.405

• For single-mode operation, the normalized frequency should be V ≤ 2.40.

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In an optical fiber, the normalized frequency, V also called the V number, is given by

where a is the core radius, λ is the wavelength in vacuum, n1 is the maximum refractive index of the core, n2 is the refractive index of the homogeneous cladding, and applying the usual definition of the numerical aperture NA. 2

1

22

21

n2nn

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Multimode Step Index Fiber

Multimode step index fibers allow the propagation of a finite number of guided modes along the channel.

The number of guided modes is dependent upon the physical parameters :

1-Relative refractive index difference, 2- core radius (n1) of the fiber 3- The wavelength of the transmitted light.

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it can be shown that the total number of guided modes (or mode volume) Ms , for the step index fiber is related to the v value for the fiber by approximate expression:

Ms ≡ V2

2

Which allows an estimate of the number of guided modes propagating in a particular multimode step index fiber.

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Multimode Graded Index Fibers (GRIN)

GRIN fibers do not have a constant refractive index in the core but a decreasing core index n(r) with a radial distance from a maximum value of n1 at the axis to a constant value n2 beyond the core radius, a in the cladding.

This index variation may b presented as:

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∆ is the relative refractive index difference and α is the profile parameter which gives the refractive index profile of the fiber core.

The equation above is a convenient method of expressing the refractive index profile of the fiber core as a variation of α allows representation of

Step index profile when α = ∞, a parabolic profile when α = 2 and a triangular profile when α = 1.

This range of refractive index profile is illustrated in Fig 2.7.

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→The graded index profiles which at present produce the best results for multimode optical propagation have a near parabolic refractive index profile core with α = 2.

→A multimode graded index fiber with a parabolic index profile core is illustrated in fig 2.8. It may be observed that the meridional rays shown appear to follow curved paths through the fiber core.

→Using the concepts of geometric optics, the gradual decrease in refractive index from the center of the core creates many refractions of the rays as they are effectively incident on a large number of high to low index interfaces.

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Propagation in a Graded index fiber: showing a decreasing number of refractive index changes n1 to n6 for the fiber axis to the cladding. Result in a gradual change in the direction of the ray, rather than the sharp change that occurs in a step index fiber

Figure 2.10 Two types of fiber: (Top) step index fiber; (Bottom) Graded index fiber

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♣ This mechanism is illustrated in Fig 2.9 where a ray is shown to be gradually curved, with an ever-increasing angle of incidence, until the conditions for total internal reflection are met, and the ray travels back towards the core axis again being continuously refracted.♣Although many modes are exited into a graded index fiber, the different group velocities of modes tend to be normalized by the index grading.♣Parameter defined for the step index fiber may be applied to graded index fibers and give a comparison between them.♣However, for the graded index fibers the situation is more complicated since the numerical aperture is a function of the radial distance from the fiber axis.

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♣ Graded index fiber therefore accept less light than corresponding step index fibers with the same relative refractive index difference.

♣To support single mode transmission in a graded index fiber, the normalized frequency is:

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For the parabolic profile, the numerical aperture is given by:

This shown that the NA is a function of the radial distance from the fiber axis (r/a)

The NA drops to zero at the edge of the core.

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Therefore, it is possible to determine fiber parameters which will give single mode operation.

For multimode graded index fibers, the total number of the guided modes, Mg is also related to the V value for the fiber by approximate expression