Lab X Single-mode Optical Fibers ECE 476 I. Purpose In this assignment you will study the modal properties and other properties of single mode fibers. You will also make 2D plots of a Gaussian beam. II. Background A. Single-mode Fibers The properties of multimode fibers are easily described in terms of the paths of light rays propagating down the fibers. This ray picture of light propagation is adequate for describing large-core-diameter fibers with many propagating modes, but fails for small-core-diameter fibers with only a few modes or with only a single mode. For fibers of this type, it is necessary to describe the allowed modes of the propagation of light in the fibers. A detailed description of the propagation characteristics of an optical fiber can be obtained by solving Maxwell’s equations for the cylindrical fiber waveguide. This leads to knowledge of the allowed modes which may propagate in the fiber. When the number of allowed modes is very large, the mathematics becomes very complex; this is when the ray picture is used to describe the waveguide properties. An important quantity in characterizing a fiber waveguide is a quantity which is called the V-number of the fiber. V=kfa(NA) (1) where kf is the free-space wavenumber, 2π/λ0 (λ0 is the wavelength of the light in free space), a is the radius of the core, and NA is the numerical aperture of the fiber. The V-number can be used to characterize which guided modes are allowed to propagate in a particular waveguide structure. When V<2.405, only a single mode, the HE11 mode, may propagate in the waveguide. This is the single mode regime. The wavelength at which V is equal to 2.405 is called the “cut-off wave-length” (denoted λc) because that is the wavelength at which the next higher-order mode is cut off and no longer propagates. B. Gaussian Approximation In waveguides in which the diameter of the core is extremely large compared to the wavelength of the light, the lowest order mode has an irradiance pattern, which can be approximated as a Gaussian. That is, the irradiance as a function of radius from the beam axis has the form⎟⎟⎠⎞⎜⎜⎝⎛−=22 2exp)0()(owrIrI (2) where I(0) is the irradiance at the center of the beam and w0 is a measure of the radius of the beam. The LP01 mode of a fiber is very close to a Gaussian mode when the wavelength of the light is near the cut-off wavelength. Fig. 1 shows the shape of a Gaussian mode and the shape of the fundamental LP01 mode near the cutoff with V only slightly less than 2.405, as a function of r/a, where r is the radial position and a is the core radius. The two curves are quite similar and the exact solution near the cut-off wavelength is often approximated by a Gaussian. Figure 1: Gaussian approximation. For a single mode, V = 2.405 is the biggest V number. It can certainly be less and still have single mode transmission as shown in Figure 2. When V is less than V = 2.405, λ is bigger than λcutoff.Figure 3 shows the exact modal distribution along with the Gaussian approximation for a longer wavelength, further from cutoff. It can be seen that the Gaussian approximation is not as good as one gets away from the cut-off wavelength. Figure 3: Gaussian approximation.C. Coupling to a Single-mode Fiber Coupling light into a multimode fiber is relatively easy. However, maximizing the coupling to a single mode fiber is much more difficult. In addition to very precise alignment of the fiber to the incoming beam, it is necessary to match the incident electromagnetic field distribution to that of the mode, which will be propagated by the fiber. The mode profile of the LP01 of a step-index single-mode fiber can be approximated by a Gaussian distribution with a 1/e2 spatial half-width given by ()65.1879.2619.165.0−−++= VVawo (3) where a is the fiber core radius. For example, when V=2.405, the Gaussian spot size is approximately 10% larger than the core diameter. Therefore, in this case, the incident light should be focused to a spot diameter which is 1.1 times the fiber core diameter at the fiber end face. Figure 4 is a plot of the normalized radius of the Gaussian distribution as a function of the V-number. It can be seen that, for a given fiber radius, as V becomes smaller (as λ becomes longer) the spot size increases. You want the beam spot to be a little bigger than the optical fiber diameter but not much. As the wavelength increases, the electromagnetic field of the mode is also less well confined within the waveguide. For this reason, single mode fibers are designed so that the cut-off wave-length is not too far from the wavelength of the light intended for use with the fiber.. Figure 4: Normalized radius of the Gaussian function approximation for the beam spot.III. Assignment A. Optimizing Single-Mode Coupling A-1. For a single mode fiber with core diameter of 4μm and a numerical aperture of 0.11, calculate the V-number of the fiber when light from a HeNe laser (632.8 nm) is coupled to the fiber. Find the spot size of the fiber mode for this V-number (i.e. find the spot diameter 2wo). A-2. When coupling light into a single-mode fiber, one begins by using a microscope objective lens to focus the laser beam to a small spot. The diameter of the spot size of the focused laser beam should match the value calculated in part (1) above. In general, the diameter of the focused laser beam, d1, can be determined from the focal length of the microscope objective lens, and the diameter d, of the laser beam as it enters the focusing lens of focal length f. The relationship of d, d1, and f is given by d1=4λfπd (4) Recall that as the distance z from the front of the laser increases, the beam diameter d increases. Therefore, you need to determine z so that you get the correct d1 for coupling the laser light into the fiber as shown in Figure 5. Perform your calculations for a 40X microscope objective lens, which has a focal length of 4.4 mm. Figure 5. Relationship of d1: the beam spot entering the optical fiber to d: the beam spot entering the lens.B. Gaussian Laser Beams and the Gaussian Approximation for the HE11 Mode B-1. Recall from the Laser Beam Properties lab (Lab 3) that the beam radius at a distance z away from a beam waist w0 is given by: 22001)(⎟⎟⎠⎞⎜⎜⎝⎛+=wzwzwπλ(5) The direction z is parallel to the direction of the laser beam. The intensity of a Gaussian laser beam