Chapter 2 Light Propagation In Optical Fiber 4/25/2017 Chapter 2 Light Propagation In Optical Fiber Course web page contains PowerPoint lecture notes, homework assignments, links to web sites, announcement and other course material. 4/25/2017

Introduction 4/25/2017 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. 4/25/2017

BASIC PRINCIPLE 4/25/2017                                                        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. 4/25/2017

What Makes The Light Stay in Fiber 4/25/2017 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 4/25/2017

The Light is Refracted Higher Refractive index Region 4/25/2017 Lower Refractive index Region This end travels further than the other hand Higher Refractive index Region 4/25/2017

Total Internal Reflection 4/25/2017 Total internal reflection reflects 100% of the light A typical mirror only reflects about 90% Fish tank analogy 4/25/2017

Refraction 4/25/2017 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 4/25/2017

Reflection 4/25/2017 Light inside an optical fiber bounces off the cladding - reflection Reflection LED or LASER Source 4/25/2017

Critical Angle 4/25/2017 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). n1Sin X=n2Sin90o Critical Angle 4/25/2017

Angle of Incidence Also incident angle Measured from perpendicular 4/25/2017 Also incident angle Measured from perpendicular Exercise: Mark two more incident angles Incident Angles 4/25/2017

Angle of Reflection Also reflection angle Measured from perpendicular 4/25/2017 Also reflection angle Measured from perpendicular Exercise: Mark the other reflection angle Reflection Angle 4/25/2017

Reflection 4/25/2017 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”. 4/25/2017

Angle of Refraction Also refraction angle Measured from perpendicular 4/25/2017 Also refraction angle Measured from perpendicular Exercise: Mark the other refraction angle Refraction Angle 4/25/2017

Angle Summary Three important angles 4/25/2017 Three important angles The reflection angle always equals the incident angle Refraction Angle Incident Angles Reflection Angle 4/25/2017

Index of Refraction n = c/v c = velocity of light in a vacuum 4/25/2017 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 away from normal - higher n to lower n Light bends in towards normal - lower n to higher n 4/25/2017

Snell’s Law 4/25/2017 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 4/25/2017

Snell’s Law 4/25/2017 The amount light is bent by refraction is given by Snell’s Law: n1sinq1 = n2sinq2 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 4/25/2017

Snell’s Law 4/25/2017 Normal Refraction Angle(2) Ray of light Lower Refractive index(n2) Ray of light Higher Refractive index(n1) Incidence Angle(1) 4/25/2017

Snell’s Law (Example 1) 4/25/2017 Calculate the angle of refraction at the air/core interface Solution - use Snell’s law: n1sinq1 = n2sinq2 1sin(30°) = 1.47sin(qrefraction) qrefraction = sin-1(sin(30°)/1.47) qrefraction = 19.89° nair = 1 ncore = 1.47 ncladding = 1.45 qincident = 30° 4/25/2017

Snell’s Law (Example 2) 4/25/2017 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(qrefraction) qrefraction = sin-1(1.47sin(70.11°)/1.45) qrefraction = 72.42° nair = 1 ncore = 1.47 ncladding = 1.45 qincident = 30° 4/25/2017

Snell’s Law (Example 3) 4/25/2017 Calculate the angle of refraction at the core/cladding interface for the new data below Solution: 1sin(10°) = 1.45sin(qrefraction(core)) qrefraction(core) = sin-1(sin(10°)/1.45) = 6.88° 1.47sin(90°-6.88°) = 1.45sin(qrefraction(cladding)) qrefraction(cladding) = sin-1(1.47sin(83.12°)/1.45) = sin-1(1.0065) = can’t do light does not refract into cladding, it reflects back into the core (TIR) nair = 1 ncore = 1.47 ncladding = 1.45 qincident = 10° 4/25/2017

Critical Angle Calculation 4/25/2017 The angle of incidence that produces an angle of refraction of 90° is the critical angle n1sin(qc) = n2sin(90°) n1sin(qc) = n2 qc = sin-1(n2 /n1) Light at incident angles greater than the critical angle will reflect back into the core n1 = Refractive index of the core n2 = Refractive index of the cladding Critical Angle, qc 4/25/2017

NA Derivation 4/25/2017

Acceptance Angle and NA 4/25/2017 The angle of light entering a fiber which follows the critical angle is called the acceptance angle, a a = sin-1[(n12-n22)1/2] Numerical Aperature (NA) describes the light- gathering ability of a fiber NA = sina n1 = Refractive index of the core n2 = Refractive index of the cladding Acceptance Angle, a Critical Angle, qc 4/25/2017

Numerical Aperture 4/25/2017 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). 4/25/2017

Acceptance Cone 4/25/2017 There is an imaginary cone of acceptance with an angle a The light that enters the fiber at angles within the acceptance cone are guided down the fiber core Acceptance Angle, a Acceptance Cone 4/25/2017

Acceptance Cone 4/25/2017 4/25/2017

4/25/2017

Formula Summary 4/25/2017 Index of Refraction Snell’s Law Critical Angle Acceptance Angle Numerical Aperture 4/25/2017

4/25/2017 Practice Problems 4/25/2017

Practice Problems (1) 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?

core/cladding interface Practice Problems (2) 4/25/2017 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 nair = 1 ncore = 1.46 ncladding = 1.43 qincident = 12° Answers: r = 8.2° c = 78.4° i = 81.8° light will propagate 4/25/2017

Refractive Indices and Propagation Times 4/25/2017 4/25/2017

Propagation Time Formula 4/25/2017 Metallic cable propagation delay cable dimensions frequency Optical fiber propagation delay related to the fiber material formula t = Ln/c t = propagation delay in seconds L = fiber length in meters n = refractive index of the fiber core c = speed of light (2.998 x 108 meters/second) 4/25/2017

Temperature and Wavelength 4/25/2017 Considerations for detailed analysis Fiber length is slightly dependent on temperature Refractive index is dependent on wavelength 4/25/2017

Classification of Optical Fiber 4/25/2017 Classification of Optical Fiber 4/25/2017

Three common type of fiber in terms of the material used: 4/25/2017 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 4/25/2017

All glass fiber 4/25/2017 The refractive index range of glass is limited which causes the refractive index difference n1-n2 to be small. This small value then reduces the light coupling efficiency of the 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. 4/25/2017

Plastic Clad Silica (PCS) 4/25/2017 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. 4/25/2017

All-plastic fiber 4/25/2017 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. 4/25/2017

Plastic and Silica Fibers

Other fibers Dispersion compensating fibers Dispersion flattened fiber 4/25/2017 Dispersion compensating fibers Dispersion flattened fiber Polarization-maintaining fibers Bend-insensitive and coupling fibers Reduced-cladding fibers Doped fibers for amplifiers and lasers Fiber gratings and photosensitive fibers Holey Fiber 4/25/2017

4/25/2017 Dispersion compensating fibers: fiber with very high negative waveguide dispersion used to cancel the positive chromatic dispersion. Insert a DCF after a normal fiber. 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. 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. 4/25/2017

Holey Fiber: hollow core surrounded by a photonic bandgap cladding 4/25/2017 Reduced-cladding fibers: Has smaller cladding diameter (typically 80 µm) to offer higher packing density and greater flexibility than standard fibers. 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. 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. Holey Fiber: hollow core surrounded by a photonic bandgap cladding 4/25/2017

Step Index Fibers 4/25/2017 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) 4/25/2017

4/25/2017 (a) (b) Figure.2.6 4/25/2017

4/25/2017 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. 4/25/2017

4/25/2017 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. 4/25/2017

Lower tolerance requirements on fiber connectors 4/25/2017 Lower bandwidth applications multimode fibers have several advantages over single mode fibers: The use of spatially incoherent optical sources (e.g. most light emitting diodes which cannot be efficiently coupled to single mode fibers. Larger numerical apertures, as well as core diameters, facilitating easier coupling to optical sources. Lower tolerance requirements on fiber connectors 4/25/2017

Single Mode Step Index fiber 4/25/2017 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. 4/25/2017

Illustration of Mode-Field Diameter (MFD) 4/25/2017 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) 4/25/2017

4/25/2017 4/25/2017

These different beams are called modes. 4/25/2017 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. 4/25/2017

4/25/2017 4/25/2017

4/25/2017 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. 4/25/2017

In an optical fiber, the normalized frequency, V also called the V number, is given by 4/25/2017 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. 4/25/2017

Multimode Step Index Fiber 4/25/2017 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. 4/25/2017

4/25/2017 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. 4/25/2017

Multimode Graded Index Fibers (GRIN) 4/25/2017 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: 4/25/2017

This range of refractive index profile is illustrated in Fig 2.7. 4/25/2017 ∆ 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. 4/25/2017

4/25/2017 4/25/2017

4/25/2017 4/25/2017

4/25/2017 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. 4/25/2017

4/25/2017 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 4/25/2017

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

4/25/2017 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. 4/25/2017

4/25/2017 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: 4/25/2017

For the parabolic profile, the numerical aperture is given by: 4/25/2017 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. 4/25/2017

4/25/2017 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 4/25/2017