5.10 Fiber optics
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5.10 Fiber optics

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5.10 Fiber optics 5.10 Fiber optics Presentation Transcript

  • Module 5: Digital Techniques and Electronic Instrument Systems 5.10 Fiber Optics
  • Introduction  Fiber optics are used to create data links.  A fiber optic data link performs 3 functions:  Electric signal is converted to an optical signal.  The optimal signal is send over an optical fiber.  The optical signal is converted again to electrical.  The fiber optic data link is consist of 3 parts:  Transmitter (e.g. LED)  Optical Fiber  Receiver (e.g. Photodiode)
  • Advantages and Disadvantages of Fiber Optics  Advantages:  Huge bandwidth:  Ethernet cable: 1Gbps  Fiber optics: 250Gbps  Immunity to electrical noise  No crosstalk  Reduced size and weight cables  Resistance to corrosion and temperature variations.  Disadvantages:  Expensive in comparison with conventional electrical cables.  Expensive and difficult installation.
  • Frequency and Bandwidth  Bandwidth: Amount of information transmitted at one time. Can be above Tbps.  Wavelength: around 700nm. Fiber optics wavelength
  • 1. Properties of Light
  • Fiber Optic Light Transmission  Light wave is a form of energy moving in wave motion.  When studying the propagation of light (e.g. in a fiber optic cable), we consider it as an electromagnetic wave.  When studying light interaction with semiconductors (e.g. in the photoelectric effect), we consider it a sum of particles (i.e. photons).  When light transmitted from a source encounters a substance can be:  reflected  absorbed  refracted
  • Fiber Optic Light Transmission  Transparent:  Almost all light rays pass through.  Translucent:  Some light rates can pass.  Opaque:  No light rates can pass.  Always, an amount of light rays is absorbed by the material and the rest is reflected (which defines its color and makes them visible) or / and refracted, if it is not opaque.
  • Light Reflection  Reflected rays: Rays that are not absorbed nor refracted. (e.g. in a mirror).  2 rays:  The incident (falling on the relecting surface.  The reflected ray.  The angle of incident and the angle of reflection are the equal. (Law of reflection).
  • Refraction of light  When the light passes through a medium, the direction of rays changes. This is called Refraction.
  • Refraction of Light  View underwater.  Blue lines: the real light rays path from water to the air.  Red lines: the light path as if there was no water.
  • Light Diffusion  In a non smooth material the reflected light is broken into many light beams and scattered through many directions.
  • The index of refraction  Index of refraction (or refractive index) shows how light propagates through a medium.  n: refractive index.  c: speed light in vacuum (3x108 m/sec).  v: speed of light in the material. v c n n 0 λ is the wavelength of light in the medium and λ0 in the vacuum.  v is the frequency of light multiplied with λ. fv
  • Snell’s law  As angle of incidence increases, angle of refraction increases as well.  When angle of refraction exceeds 90 degrees, then no retraction happens. All light is reflected. (Total internal reflection).  The angle of incidence at which the angle of reflection is 90 degrees is called critical angle of incidence (θc). 2211 sinsin nn 1 2 sin n n c
  • 2. Structure of Fiber Optics and Light Transmission
  • Basic structure of the Fiber Optics  Jacket:  Determines the mechanical robustness of the cable.  Usually plastic, PVC etc.  Buffer (or coating):  Protects the fiber optic from physical damage. Fiber identification etc.  Cladding:  Dielectric material, with index of refraction less than the core material. Made of glass or plastic.  Core:  Light propagation medium. Dielectric material (usually glass). Conducts no electricity.
  • How the light propagates in the optical fiber cable?  2 ways to describe the propagation:  The ray theory: to get a clear picture of the propagation of light inside the cable.  The mode theory: to understand the behavior of the light inside the cable (comprehending of the properties of absorption, attenuation and dispersion).
  • The Ray Theory  2 types of rays propagate along the optical fiber cable:  Meridional rays: Rays that pass through the center axis of the core.  Skew rays: Rays that travel through the fiber without passing through the center axis of the core.
  • Meridional Rays  Can be classified as:  Bound: Rays that propagate through the fiber by total internal reflection.  Unbound: Rays that are refracted from the core.  Cladding imperfections prevent total internal reflection.  Part of the bound rays are refracted inside the cladding. (unbound rays).  Unbound rays eventually escape from the cable.
  • How light rays enter the cable?  For avoiding making the rays unbound, the rays must enter the core with incident angle equal or greater than the critical angle.  The range of angles which make the rays bounded is the cone of acceptance.  All rays entering the core with angle less than the half-angle of the core of acceptance (acceptance angle) become unbounded and finally lost.
  • The Numerical Aperture  Illustrates the relation between the cone of acceptance and the indexes of refraction. 22 claddingcore nnNA  Shows the light-gathering ability of the optical fiber  High NA means that the fiber optic “admits” more light.
  • Skew Rays  The acceptance angle of skew rates is larger than the acceptance angle of meridional rays.  Most rays entering the core are skew rays.  The contribute to the amount of light capacity of the fiber optic (especially in case of high NA).  However, they contribute also the amount of light losses of the fiber optic.  A large portion of the number of skew rays becomes leaky rays.  Leaky rays, although they are predicted to be totally reflected, they are partially refracted due the curved shape of the fiber boundary.
  • The Mode Theory  Mode theory considers light to be electromagnetic waves.  Describes light propagation in the optical fiber as a propagation of electromagnetic waves.
  • Plane Waves  A light wave can be represented as a plane wave.  Plane wave’s surfaces are parallel plates, normal to the direction of the propagation.  3 characteristics of a plane wave:  Direction  Amplitude  wavelength
  • Plane Waves  Planes having the same phase are called wavefronts. nf c wavelength )(  c: speed of light in the vacuum (3*108)  f: frequency of light  n: index of refraction  Only wavefronts with angle of incident equal or grater with the critical angle will propagate inside the core.  All wavefronts propagating inside the core must have phase equal to an integer multiplied with 2π in order for light to be transmitted.  If they total phase is not integer x 2π, then destructive interference takes place. (one wavefront eliminates the other).  This is the reason that a finite number of modes can be propagate along the fiber.  A set of electromagnetic field patterns , form a mode.
  • Light inside the fiber optics  The electromagnetic wave is composed by an electric field and a magnetic field.  If the magnetic field is in the direction of the propagation, the mode is called TM.  If the electric field is the direction of the propagation, it is called TE.
  • The modes  Different electromagnetic wave patterns can propagate in a fiber optic. These patterns are called modes.  The number of maxima of each pattern is the order of the mode (e.g. TE0 has one maximum, TE2 has 3, etc.).  Each mode can be realized with different wavelength.
  • Low order and High order modes  Low order modes have high incident angle, while high order modes have low incident angle.
  • Single vs. Multi-mode Optical Fibers  Not every mode can propagate in any fiber optic.  The number of modes than can propagate depend on:  ncore, ncladding, wavelength (λ) of the fiber optic and core diameter (α). 222 claddingcore nn a V  Normalized frequency (V) show the number of modes that a fiber optic can support.  The larger the V, the more modes can be supported.
  • Single vs. Multi-mode Optical Fibers  Each fiber optic has a specific normalized frequency, for a specific wavelength.  Fiber optics with V < 2.405 can support only one mode. They are called single mode fibers.  Fiber optics with V > 2.405 can support more modes. They are called multi- mode fibers.  Y axis shows the propagation constant.  The higher the propagation constant, the less the dispersion of the wave during the propagation.  The higher the mode order, the higher the dispersion.
  • Single vs. Multi-mode Optical Fibers  Single mode fibers:  The core diameter is small: 8 – 10 μm.  Only the lowest mode can be propagated (around 1300nm wavelength).  Normalized frequency less or equal to 2.405.  Lower signal loss in comparison with multi-mode and higher bandwidth. Low fiber dispersion.  They use laser diodes to generate light.  Multi-mode fibers:  Can propagate up to 4 modes (TE0, TE1, TE2, TE3. Also called OM1, OM2, OM3, OM4).  Typical core sizes: 50 – 100 μm.  The light enters easily the fiber and the connections are easier.  They use LED to generate light (cheaper, less complex and last longer).  However, they have higher dispersion than the single mode.  Specifically they exhibit mode dispersion: Modes arrive at the receiver at different times.
  • More on Multi-mode Fibers  Multi-mode fibers are basically because they are cheaper and easier to install.  The limited bandwidth they offer in comparison with the single-mode fibers is sufficient for more applications.  OM1 multi-fiber optics (TE0) are 62.5/125μm fibers and offer 200/500MHz x Km.  OM2 multi-fiber optics (TE0 & TE1) are 50/125μm fibers and offer 500MHz x Km.  OM3 multi-fiber optics (TE0 & TE1 & TE2) are 50/125μm and offer 2GHz x Km. Are designed for 2Gbps transmission.  OM4 multi-fiber optics (TE0 & TE1 & TE2 & TE3) are designed for speeds from 40 to 100Gbps.  Increasing the mode used, we increase the bandwidth. However, intermodal dispersion limits the bandwidth. Transmission Standards 100 Mb Ethernet 1 Gb Ethernet 10 Gb Ethernet 40 Gb Ethernet 100 Gb Ethernet OM1 (62.5/125) up to 2000 meters 275 meters] 33 meters Not supported Not supported OM2 (50/125) up to 2000 meters 550 meters 82 meters] Not supported Not supported OM3 (50/125) up to 2000 meters 550 meters 300 meters 100 meters[ 100 meters[ OM4 (50/125) up to 2000 meters 1000 meters 550 meters 150 meters[ 150 meters[
  • 3. Properties of Fiber Optics
  • Performance of Fiber Optics  Performance is affected by:  Signal loss  Bandwidth  Bandwidth in telecommunications is defined as the difference between the highest and the lowest frequency of the signal.  Bandwidth in data transmission is defined as bits transmitted per second.
  • Attenuation & Dispersion  Fiber optics properties that affect performance are:  Attenuation  Dispersion  Attenuation is a result of:  Light absorption  Light scattering  Bending losses  If the signal strength is reduced below a specific point, the receiver is unable to detect it. Dispersion is the spreading of the signal. The spreading limits how fast data can be transmitted along the fiber. The receiver is unable to distinguish between input pulses caused by the spreading of each pulse.
  • Attenuation  Attenuation is the loss of power as the light travels inside the fiber. It is caused by:  Light absorption  Light scattering  Bending losses o i P P L nattenuatio log 10 L: Fiber’s length in km. Pi: Signal’s power in the input. Po: Signal’s power in the output. attenuation units: dB/Km example: Signal’s input power is 400W and the power at the receiver is 100W. The length of the fiber is 1Km. attenuation = 10/1 * log(400/100) = 6 dB / Km
  • Explaining Decibel scale  dB is a logarithmic scale, describing ratio. 1 2 log10 P P dB  If P2 is twice the P1, the ratio is 10*log(2) = 3dB.  If P2 is half the P1, then the ratio is 10*log(1/2) = -3dB  If P1 is 1,000,000 the P1, then the ratio is 10*log(106) = 60dB.  log(10n) = n*log(10) = n  log (1) = 0
  • Light Absorption  Absorption is the conversion of optical power into another energy form, such as heat.  Factors contributing to light absorption:  Imperfections in the structure of the fiber material.  The intrinsic material properties.  Photons are absorbed by electrons and excites them to higher energy levels.  The extrinsic material properties.  Impurities in the atomic structure of the silicon (e.g. iron, nickel, chromium) results in the absorption of photons by the electrons of these metal and electrons excitation to higher energy levels.
  • Light Scattering  Scattering is the result of density fluctuations in the fiber structure.  During the fabrication process of the fiber, there are areas where the density of the molecular structure is higher or lower.
  • Bending losses  Bending losses:  Macrobends: Too sharp bend.  A part of the mode is converted to a higher order mode, which is lost inside the cladding.  Microbends: Imperfections in the fiber or external force which deforms the cable.
  • Dispersion  2 types of dispersion:  Intramodal dispersion: Occurs in all kinds of optical fibers, where more than one wavelenghts are used (e.g in WDM).  Different wavelengths of signals of the same mode travel at different speeds inside the fiber, so exit the fiber at different times.  Intermodal dispersion: Occurs only in multi-mode fibers.  Each mode travels at different speed inside the fiber, so, they do not exit from the fiber at the same time.  Single mode fibers exhibit less dispersion than multi- mode.
  • Wave Division Multiplexing  Each signal is transmitted in the fiber using a different wavelength.  e.g. the light of LED is split to separate channels, where each channel has a different wavelength.  Can be used in single or multi-mode fibers.  Can increase the bandwidth of the fiber up to 40 times.
  • 4. Optical Fiber Cables
  • Step-index and Graded-index fiber cables  Step-index: The refractive index of the core is uniform.  Graded-index: The refractive index of the core is varies gradually, as a function of radial distance from the fiber center.  Single mode fibers are normally step-index.  Graded-index multimode fibers performance is superior to step-index multimode.
  • Why Graded-index fibers exhibit high performance?  Light travels faster in materials with low refractive index.  Therefore, rays that travel the longer distance, propagate through lower refractive indexes. So, they travel faster.  So, the modal dispersion of the fiber decreases. n c v
  • Graded-index fibers specifications  The relative refractive index difference shows the difference between the core and the cladding refractive index: 2 1 2 2 2 1 2n nn Most common values of graded-index multimode fibers are:  core: 62.5μm  cladding: 125μm  Δ = 0.02  They are the most common fiber optic cables used in aircrafts.  A basic advantage is the low sensitivity to microbend and macrobend. This allows relatively sharp bending during the installation procedure.
  • Fiber Buffers  Tight buffer cables are used for fiber installed inside buildings, aircrafts, etc.  Loose-tube buffer cables are typically used outdoors.
  • Types of cables
  • Zip-core cables  1: Jacket (3mm)  2: Tight buffer (900μm)  3: Fiber optic  Normally used in patch boards  patch boards: panels in telecommunication centers where temporary connections of circuits are made.  Indoor cable.
  • Distribution cable  Tight buffer (900μm)  Low cost cable, suitable for short distances.  Fibers are not individually buffered. Difficult termination.  Indoor cable.
  • Breakout Cable  Several zip-core cables hold together.  Each fiber is individually buffered. Easy termination.  Large and more expensive than distribution cables. However, the easy installation (i.e. termination) can eventually make then more economic.  Indoor cables
  • Gel-filled loose-tube cable  Several fibers inside each gel-filled loose buffer.  Gel is used to repel water.  Designed to endure high temperature and moisture conditions.  Can be burried into the ground.  Used outdoors
  • Dry loose-tube cable  No gel.  High durability and easier installation than gel-filled.  Used outdoors.
  • Armored cables  Steel or aluminum armor.  Extremely high durability in harsh environments.  Used outdoors.
  • Submarine communication cables  69mm diameter  10Kg/m  In 2010 submarine optical cables linked all world’s continents.  7: gel  3: steel bars
  • Cables Coloring Fiber Type Diameter (µm) Buffer Color Multimode 50/125 Orange Multimode 62.5/125 Slate Multimode 85/125 Blue Multimode 100/140 Green Singlemode - Yellow
  • 5. Optical Splices, Connectors and Couplers
  • Fiber Optic Data Links  Fiber optic data links perform 3 basic functions:  Conversion of the input electrical to optical signal.  Transmission of the optical signal through the fiber.  Conversion of the optical signal to output electrical signal.  Fiber optic connections permit the transfer of optical power from one component to another. A connection requires:  a fiber optic splice (permanent join) or  a connector (allows reconfiguration) or  a coupler (combination of optical signals)  Poor connections degrade the system performance (coupling losses).
  • Optical Fiber Coupling Losses  Coupling losses are due to:  Reflection losses.  Fiber misalignment  Poor fiber end preparation  Mismatches between the fiber characteristics
  • Reflection Losses  When optical fibers are connected a part of the source power will be reflected back, to the source fiber.  It is caused by a small air gap that may exists in the interface, causing a discontinuity of the refracting index.  Solution: Index matching gel.
  • Fiber Misalignment  3 main coupling errors:  longitudinal misalignment  Lateral misalignment  Angular misalignment (θ must be less than 1 degree).  Matching gel reduces coupling loss from fiber separation.  However, it does not affect lateral misalignment losses and increases angular misalignment losses.
  • Poor Fiber End Preparation  First we remove the buffer (usually with chemical materials).  Then, using a diamond blade and constant tension between the two ends is used to break the fiber.  Finally the polishing using and abrasive paper and a microscope.
  • Fiber Mismatches  Geometry mismatches  e.g. difference in core diameter.  NA mismatches  different n1, n2.  Relative refracting index difference  different Δ.
  • Fiber Optic Splices  A device that connects a fiber optic with the other permanently.  2 kinds of splices:  Mechanical splice  Aligning takes place mechanically.  Fusion splice  The end of the two fibers is melted by heat to fuse them together.
  • Mechanical Splices  3basic kinds:  A tube, with a diameter slightly larger than the fiber optic’s diameter.  The fiber optics are placed inside the tube and they are aligned.  A groove where the two fibers are placed.  Rotary splices  A transparent adhesive is placed after the alignment, to make the alignment permanent and to reduce the coupling losses.
  • Rotary splices  The fiber is inserted into each half of the splice.  It’s epoxied with an ultraviolet cure epoxy.  Using the alignment sleeve the two halves are connected.  Mechanical stability is added due to the two springs.  They are used extensively in aircraft applications due to their durability in environmental conditions and easy installation.
  • Fusion Splices  The two ends of the fiber optics are melted to form a connection.  Takes longer time than the mechanical connection.  Requires extensive training by the operators and the final result depends greatly on their experience.
  • Fiber Connector characteristics  Connectors allow system reconfiguration. (No permanent connections).  Every connector is defined by its ferrule diameter.  Ferrules (made of ceramic) hold the end of the fibers and keeps them aligned.  There are different connectors for single mode and multi-mode fibers.  There are screw and push- pull connectors.  Most connector types are made in two types:  UPC.  APC.
  • Connector installation  Very fast installation in comparison with splices.  Installation procedure is almost the same, in all connector types:  Buffer removal  Boot insertion  Plug of the fiber optic in the connector and cramp.  Polishing and cleaning of the fiber end. Boot
  • Fiber Connector evaluation  2 parameters define the quality of the connector:  Insertion loss: The power difference between the source and the destination fibers (around 0.5dB).  Return loss: The amount of reflected power (around 40 – 70dB).  A high quality connector has low insertion loss and high absolute return loss value.
  • UPC vs. APC Connectors  In all fiber optic connections (using splices or connectors) a part of the incoming light ray is reflected back to the source fiber.  It is preferable the reflected light to be lost in the cladding, than be reflected back in the core of the source fiber.  Angled polished connectors (APC) are cut in 8 degrees angle to increase the possibility of the reflected light to be lost in the cladding.  Depending on the application requirements, a proper selection between UPC and APC can be made.  APC are usually single mode.
  • UPC vs. APC comparison  UPC has lower insertion loss, but APC has superior return loss, due to its diameter.
  • Types of Connectors  Most common types:  SC connectors (square connector or subscriber connector or standard connector).  A push-pull connector  2.5mm ferrule diameter  Extremely common in datacom and telecom.  LC connectors (lucent connector, little connector).  Half the size of SC.  1.25 mm ferrule diameter
  • Aircraft Connectors (AVIO connectors)  Materials:  Ferrule: Zirconium (Zr).  External parts: Copper – Nickel.  Dimensions:  Ferrule diameter: 900μm  Boot diameter: 3mm  Screw style.
  • AVIO Connectors specifications
  • Fiber Optics in Boeing 787
  • AFDX switches  Boeing 787 and Airbus 380 use a new ARINC specification: 664.  664 is a full duplex bus, using a twisted pair of optical cables.  Transmitting and receiving for each LRU connected in the network is independent.  It is considered rather a network, than a bus. LRUs are connected to the bus through switches.  All connections are 10Mbps or 100Mbps.  The network is called AFDX (Avionics Full-Duplex Switched Ethernet).
  • Fiber Optic Couplers  Some fiber optic links require more than simple point-to- point connections.  Such connections are more complex.  These configurations require devices that can distribute an optical signal from one fiber among two or more fibers. These devices are called couplers.  Example 1x2 couplers distributes equally the input signal among two outputs.  Each output signal has less than half power of the input signal. (losses around 3dB).
  • Types of couplers  Optical splitters: splits the optical power of a signal optical fiber in two or more fibers.  Optical combiners: combines the signal of more than one optical fibers to one fiber.
  • 6. Fiber Optic Measurement Techniques  Laboratory Measurements:  Attenuation  Cut-off Wavelength (in single mode only).  Bandwidth  Dispersion  Fiber geometry  Core diameter  Numerical Aperture  Insertion loss  Return loss
  • Attenuation Measurement  Cut-back method:  Pi and Po measurement in a known length L.  Cut-back to 2m and repeat measurements with the same Pi.  Repeating the measurement in a short length fiber is necessary to ensure correctness. o i P P L nattenuatio log 10
  • Cut-off Wavelength Measurement  The wavelength a which a mode seizes to propagate.  It depends on the fiber length and the bending conditions.  Attenuation is measured with specific bending radius for different wavelengths.  Cut-off wavelength is the longest λ for which attenuation is equal to 0.1dB.
  • Bandwidth Measurement  The fiber bandwidth is defined as the lowest frequency at which the magnitude of the fiber frequency response has decreased to one-half its zero frequency value. )( )( log)( fP fP fH i o  Power frequency response:
  • Intramodal (Chromatic) Dispersion Measurement  Differential group delay τ(λ) is the difference between the arrival time of different wavelengths.  We use a reference fiber and the fiber under testing.  The pulse delay for the reference fiber of length Lref and the delay is tin(λ). The pulse delay for the test fiber of Ls is tout(λ).  The differential group delay for a specific λ per unit of length is: refs outin LL )()( )(
  • Fiber Geometry Measurement  Specific fiber geometry characteristics, such as the cladding diameter are obtained by video camera images, analyzed by a computer.  Core diameter is measured by the range of the intensity of light at the end of the fiber.
  • Numerical Aperture Measurement  It is measured with a similar way with the core diameter. 22 claddingcore nnNA
  • Insertion and Return Loss Measurement  Insertion loss is similar to attenuation, but is used to indicate the attenuation in an optical fiber connector. 1 0 0 1 log10 M M P P P P lossinsertion r i P P lossreturn log10  P0 is power measured without the interconnection.  P1 is the power measured with the interconnection.  Return loss is measured using a 2x2 (or 2x1) coupler. The returning power is measured in one of input ports of the coupler.  Pi is power in the source fiber.  Pr is the power reflected.  return loss may be defined as 10logPr/Pi. Then, it will be negative.
  • Optical Time-Domain Reflectometer  Electronic instrument used to characterize a fiber optic.  Injects a series of optical pulses into the fiber under test.  It estimates attenuation, reflected losses, scattered light etc.  It is used for field measurements (i.e. installed fiber optics, not in laboratory).
  • 7. Fiber Optics Transmitters and Receivers
  • Fiber Optic Transmitter  Converts electrical signals to optical signals and lunches the optical signal into the fiber.  The part of the transmitter that converts electrical energy (current) to optical energy (light) is the optical source.  Optical sources must:  Launch sufficient light power to the fiber to overcome attenuation and coupling losses.  Emit light at proper wavelengths.  Optical sources are LEDs.  Emit infrared light normally at 850, 1300 or 1550nm.  Today, common LEDs have been replaced with VCSELs.
  • VCSEL (Vertical Cavity Surface Emitting Laser)  High speed, low power consumption.  Can used to design 2D Arrays.  Higher light density.  Low temperature sensitivity.
  • Transmitter  850nm transmitter.  Multi-mode SC type.  Can transmit up to 1.65 or 3.5Gbps. Electrical InputOptical Output  4 digital data inputs are multiplexed with slightly different wavelengths.  VCSELs produce light of slightly different wavelength.  Light is forwarded to an SC type fiber optic cable.
  • Fiber Optic Receivers  Photodiodes: Semiconductor devices that convert light into current.
  • Fiber Optic Transceiver  Interfaces between fiber optics and Ethernet. ReceiverTransmitter Electrical Input / Output 10 Gb Ethernet Transceiver SR - 850 nm, for a maximum of 300 m LR - 1310 nm, for distances up to 10 km ER - 1550 nm, for distances up to 40 km ZR - 1550 nm, for distances up to 80 km
  • Formulas Formula Units Explanation Number A number that shows the size of the acceptance cone. dB/Km Pi: input Po: output L: fiber optic length Number Power Frequency Response: Po(f): Power at the output Pi(F): Power at the input nsec/Km. Differential group delay: tin: the group delay of the reference fiber with length Lref Tout: the group delay of the test fiber with length Ls. dB Pi: power at the source fiber. P0: power at the destination fiber. dB Pi: power at the source fiber. Pr: power reflected. 22 claddingcore nnNA o i P P L nattenuatio log 10 )( )( log)( fP fP fH i o refs outin LL )()( )( 0 log10 P P lossinsertion i r i P P lossreturn log10