How many light rays?? Recall that only light rays which enter the core with an angle less than the acceptance angle will propagate Optical Communications Systems There are an infinite number of possible ray angles, all less than acceptance angle In theory then there are an infinite number of light rays? Range of angles over which light will not be transmitted Propagation in Range of angles over which light will be transmitted Observing Modes Experimentally Electromagnetic Modes in Visible light is used as a source, typically HeNe laser (Red, 670 nm) Output from a fibre is projected onto a reflective surface, such as a white card in a darkened room To obtain an improved model for propagation in a fibre, EM wave theory must be used. Ray diagram or Geometric Optics approach remains useful as a way to visualise propagation in a fibre. Basis of EM analysis is a solution to Maxwells equations for a fibre. For ease of analysis a fibre is frequently replaced by a planar optical waveguide, that is a slab of dielectric with a refractive index n1, sandwiched between two regions of lower refractive index n. Output from singlemode fibre, HE11 mode n Output from a fibre supporting two modes Output from a multimode fibre, a socalled speckle pattern n 1 n Planar waveguide
F E A F E A Formation of Modes in a (I) Formation of Modes in a (II) B B θ θ d d D C D C Propagation of an individual ray takes place in a zigzag pattern as shown In practice there is at the fibre input an infinite number of such rays, called more properly plane rays. Each ray is in reality a line drawn normal to a wavefront, for example the wavefront shown by the dotted line FC above. For plane waves all points along the same wavefront must have identical phase. The wavefront intersects two of the upwardly travelling portions of the same ray at A and C. Unless the phase at point C differs from that at point A by a multiple of π then destructive interference takes place and the ray does not propagate. Moving along the ray path between A and C involves a phase change caused by the distance AB and BC and a phase change caused by reflection Combining these two phase changes and setting the result equal to a multiple of π we get a condition for propagation of a "ray", more properly now called a mode. Types of Optical Three distinct types of optical fibre have developed The reasons behind the development of different fibres are explored later Concern here is to examine propagation in the different fibres The three fibre types are: Step Index Step index fibre Graded index fibre Multimode fibres Singlemode fibre (also called monomode fibre)
Step Index Normalised Frequency for a 0 N N1 For an optical fibre we can define the so-called normalised frequency "V" Convenient dimensionless parameter that combines some key fibre variables It is defined thus: πa V =. n 1 - n where a is the fibre radius and is the operating wavelength Simplest and earliest form of fibre The larger the core diameter the more modes propagate With a large core diameter many thousands of modes can exist Refractive index profile for a step index optical fibre V is also very commonly defined using the numerical aperture NA thus: π V = a.na We will use this definition Relative Refractive Index Modes in a MM Step Index It is also possible to define a so called relative refractive index for a fibre Normally the symbol is used is defined thus: = if is << 1 then is given by: The normalised frequency V can be written in terms of : n 1 - n n 1 = V = n 1 - n n 1 π a.n1 In a multimode step index fibre, a finite number of guided modes propagate. Number of modes is dependent on: ƒwavelength, refractive index n 1 ƒrelative refractive index difference, radius a Number of propagating modes (M) is normally expressed in terms of the normalised frequency V for the fibre: M = V Problem: A step index fibre with a core diameter of 80 µm has a relative refractive index difference of 1.5%, a core refractive index of 1.48 and operates at 850 nm. Show (a) that the normalised frequency for the fibre is 75.8 and (b) that the number of modes is 873
Influence of Size and Wavelength As the core diameter increases and with it the normalised frequency, the number of modes increases with a square law dependency on core size As the wavelength increases the number of modes decreases 850 nm 130 nm Graded Index Graded Index Propagation in a Graded Index 0 N N1 Parabolic variation in refractive index Typical core diameter for this fibre type: 50 to 10 µm Different refractive index profiles have developed An expanded ray diagram for a graded index fibre, showing a discrete number of refractive index changes n 1 to n 6 for the fibre axis to the cladding. Result is a gradual change in the direction of the ray, rather than the sharp change which occurs in a step index fibre
Propagation in a Graded Index Graded Index Profiles Axis b a Light ray (a) and (b) are refracted progressively within the fibre. Notice that light ray (a) follows a longer path within the fibre than light ray (b) The index variation n(r) in a graded index fibre may be expressed as a function of the distance (r) from the fibre axis n(r) = n 1 (1- (r/a)) α for r < a (core) n(r) = n 1 (1- ) = n for r > a (cladding) Meridional (axial) rays follow curved paths in the fibre as shown Benefits of using graded index design are considered later Most common value of the profile parameter α is, a so called parabolic profile. An infinite profile parameter implies a step index fibre Refractive index profiles for Graded Index fibres Modes in a Graded Index Calculating the number of modes in a graded index fibre is very involved As an approximation it can be shown that the number of modes is dependent on the normalised frequency V and on the profile parameter α. That is M = α α + where, is again given by: = V n 1 - n n 1 if is << 1 Singlemode Exercise For the most common value of α show that for fibres with similar relative refractive indices, core radii and operating wavelengths, the number of modes propagating in a step index fibre is twice that in a graded index fibre
Singlemode Optical Refractive Index Profiles for SM s Small Multimode step index Multimode graded index 0 N N 1 Small Conventional singlemode fibre (so called matched cladding) Depressed cladding singlemode fibre (less susceptible to bend loss) Refractive index profile Triangular profile singlemode fibre (used in dispersion shifted fibre) Up-and-down profile singlemode fibre (used in dispersion flattened fibre) also called multicladding fibre Normalised Frequency for SM s Energy Distribution in a Singlemode Singlemode fibre exhibits a very large bandwidth and has thus become the fibre of choice in most high speed communications systems. Singlemode operation is best considered with the aid of the fibre normalised frequency V: V = π a. NA Single mode operation takes place where V is less than the so-called cutoff value of Vc =.405. The single mode is the lowest order mode that the waveguide will support, referred to as the HE11 mode. This mode cuts off at V=0. As will be explained practical V values are normally between about to.4 Singlemode operation is achieved by altering the fibre radius, NA or the wavelength in use so that V lies in the range above. The amplitude distribution of the optical energy in a singlemode fibre mode is not uniform, nor is it confined only to the core In multimode fibres if we assume a mode model instead of ray diagram approach then some small percentage of the energy is contained within the cladding close to the core, but typically < 1% so the ray model is still a valid view Ray diagram model does not work for singlemode fibre > 50 µm 7-9 µm Multimode energy distribution is confined to the core Singlemode energy distribution peaks in the centre of the core (Darker shading = higher energy)
Mode Field and Spot Size (I) Mode Field and Spot Size (II) Mode field diameter (MFD) is an important property of SM fibres. The amplitude distribution of the HE11 mode in the transverse plane is not uniform, but is approximately gaussian in shape, as shown below centre The MFD is defined as the width of the amplitude distribution at a level 1/e (37%) from the peak or for power 13.5%from the peak The spot size is the mode field radius w. Its value relative to core radius is given by the expression: w a -3/ -6 = 0.65 + 1.619V +.879V As the V value approaches.4 the spot size approaches the fibre radius. For V < the spot size is significantly larger than the core size. For V < the beam is partially contained within the cladding and loss increases For this reason V should be between about and.4 MFD or spot size is frequently specified as well as core radius or diameter for the fibre Normalised spot size as a function of the fibre V value Cutoff Wavelength SM Summary and Problem Singlemode operation only takes place above a theoretical cutoff wavelength c where V < V c =.405 c = π V c a NA In practice the theoretical cutoff wavelength is difficult to measure. An alternative is EIA (Electronics Industry Association of America) cutoff wavelength, which states that the cutoff wavelength is: The wavelength at which the power in the HE1 mode is 0.1 db of the power in the HE11 (fundamental mode) The EIA cutoff wavelength can be 100 nm less than the theoretical cutoff wavelength size is a useful parameter for multimode fibres, but is not so useful for SM fibres. Telecommunications systems are normally designed to work close to the cutoff wavelength for good power confinement (small spot size), but not close enough to cutoff so that significant power is carried in higher modes. Exercise A singlemode fibre has a core refractive index of 1.465 and a cladding refractive index of 1.46. What is the maximum core size if the fibre is to support only one mode at 1300 nm? Answer: core radius 4.11 microns, 8.3 microns core diameter. If the wavelength is increased to 1550 nm what is the new fibre V value, the spot size and the MFD? Answer: V =.0, Spot size 5.18 microns, MFD 10.4 microns