Chapter 5 Power Launching and Coupling 5.1 Source-to-Fiber Power Launching 5.1.2 Power-Coupling Calculation 5.1.3 Power Launching versus Wavelength 5.1.4 Equilibrium Numerical Aperture 5.2 Lensing for Coupling Improvement 5.2.1 Nonimaging Microsphere 5.2.2 Laser Diode-to-Fiber Coupling 5.3 Fiber-to-Fiber Joints

5.1 Source-to-Fiber Power Launching Radiance (or brightness) B at a given diode drive current is the optical power radiated into a unit solid angle per unit emitting surface area and is generally specified in terms of W/cm2.sr. Consider Fig. 5-1, which shows a spherical coordinate system characterized by R, q, and f, with the normal to the emitting surface being the polar axis. The radiance may be a function of both q and f, and can also vary from point to point on the emitting surface.

5.1 SOURCE-TO-FIBER POWER LAUNCHING Surface-emitting LEDs are characterized by their Lambertian output pattern. The power delivered at an angle q, measured relative to a normal to the emitting surface, varies as cosq because the projected area of the emitting surface varies as cosq with viewing direction. The emission pattern for a Lambertian source follows the relationship B(q, f) = Bocosq (5-1) where Bo is the radiance along the normal to the radiating surface. The radiance pattern for this source is shown in Fig. 5-2.

5.1 SOURCE-TO-FIBER POWER LAUNCHING Figure 5-1. Spherical coordinate system for characterizing the emission pattern from an optical source.

5.1 SOURCE-TO-FIBER POWER LAUNCHING Figure 5-2. Radiance patterns for a lambertian source and the lateral output of a highly directional laser diode. Both sources have B0 normalized to unity.

5.1 SOURCE-TO-FIBER POWER LAUNCHING Edge-emitting LEDs and laser diodes have different radiances B(q, 0o) and B(q, 90o) in the planes parallel and normal, respectively, to the emitting-junction plane of the device. The radiances can be approximated by (5-2)

5.1 SOURCE-TO-FIBER POWER LAUNCHING The integers T and L are the transverse and lateral power distribution coefficients, respectively. For edge emitters, L=1 (a Lambertian distribution with a 120o half-power beam width) and T is significantly larger. For laser diodes, L can take on values over 100.

5.1 SOURCE-TO-FIBER POWER LAUNCHING Example 5-1 : Figure 5-2 compares a lambertian pattern with a laser diode that has a lateral (f = 0o) half-power beam width of 2q = 10o. From Eq. (5-2), we have B(q=5o, f = 0o) = Bo(cos5o)L = (1/2)Bo Solving for L, we have L = [log 0.5 / log(cos5o)] = [log 0.5/log 0.9962] = 182 The narrower output beam from a laser diode allows more light to be coupled into an optical fiber.

5.1.2 Power-Coupling Calculation Consider the case shown in Fig. 5-3 for a symmetric source of brightness B(As, Ws), where As and Ws are the area and solid emission angle of the source, respectively. The coupled power can be found using the relationship (5-3) The radiance B(q, f) is integrated over the solid acceptance angle of the fiber. qo,max is the maximum acceptance angle of the fiber, which is related to the NA through Eq. (2-23).

5.1.2 Power-Coupling Calculation The total coupled power is determined by summing up the contributions from each individual emitting-point source of incremental area dqsrdr; that is, integrating over the emitting area. If the source radius rs is less than the fiber-core radius a, then the upper integration limit rm = rs; for source areas larger than the fiber-core area, rm = a.

5.1.2 Power-Coupling Calculation Figure 5-3. Schematic diagram of an optical source coupled to an optical fiber. Light outside of the acceptance angle is lost.

5.1.2 Power-Coupling Calculation Assume a surface-emitting LED of rs < a. This is a Lambertian emitter, Eq. (5-1) applies and Eq. (5-3) becomes (5-4) where the NA is defined by Eq. (2-23). For step-index fibers the NA is independent of qs and r, so that Eq. (5-4) becomes (for rs < a) PLED,step = (prs)2B0(NA)2 = 2(prs)2B0n12D (5-5)

5.1.2 Power-Coupling Calculation Assume Total optical power Ps emitted from the source area As into a hemisphere (2psr) is given by (5-6) Equation (5-5) can be expressed in terms of Ps : PLED,step = Ps(NA)2 for rs < a (5-7) When the radius of the emitting area is larger than the radius a of the fiber-core area, Eq. (5-7) becomes PLED,step = (a/rs)2Ps(NA)2 for rs > a (5-8)

5.1.2 Power-Coupling Calculation Example 5-2 : Consider an LED that has a circular emitting area of radius 35-mm and a Lambertian emission pattern with 150 W/(cm2.sr) axial radiance. Let us compare the optical powers coupled into two step-index fibers, one of which has a core radius of 25-mm with NA = 0.20 and the other has a core radius of 50-mm with NA = 0.20.

5.1.2 Power-Coupling Calculation For the larger core fiber, we use Eqs. (5-6) and (5-7) to get PLED,step = Ps(NA)2 = p2rs2B0(NA)2 = p2(0.0035cm)2[150W/(cm2.sr)](0.20)2 = 0.725 mW When the fiber end-face area is smaller than the emitting surface area, with Eq. (5-8), the coupled power is less than the above case by the ratio of the radii squared: PLED,step = (a/rs)2Ps(NA)2 = (25mm/35mm)2(0.725mW) = 0.37 mW

5.1.2 Power-Coupling Calculation The power coupled from a surface-emitting LED into a graded-index fiber becomes (for rs < a) (5-9) where the last expression was obtained from Eq. (5-6).

5.1.2 Power-Coupling Calculation If the refractive index n of the medium is different from n1, then the power coupled into the fiber reduces by the factor R = (n1-n)2/(n1+n)2 (5-10) where R is the Fresnel reflection or the reflectivity at the fiber-core end face. The reflection coefficient r = (n1-n)/(n1+n) relates the amplitudes of the reflected and the incident wave.

5.1.2 Power-Coupling Calculation Example 5-3 : A GaAs optical source with a refractive index of 3.6 is coupled to a silica fiber that has a refractive index of 1.48. If the fiber end and the source are in close physical contact, then, from Eq. (5-10), the Fresnel reflection at the interface is R = [(n1-n)/(n1+n)]2 = [(3.60-1.48)/(3.60+1.48)]2 = 0.174

5.1.2 Power-Coupling Calculation This value of R corresponds to a reflection of 17.4% of the emitted optical power back into the source. Given that Pcoupled = (1-R)Pemitted the power loss L in decibels is found from L = -10.log[Pcoupled/Pemitted] = -10.log(1-R) = -10log(0.826) = 0.83 dB This number can be reduced by having an index-matching material between the source and the fiber end.

5.1.3 Power Launching versus Wavelength The number of modes that can propagate in a graded-index fiber of core size a and index profile a is M = [a/(a+2)].[2pan1/l]2D (5-11) Twice as many modes propagate in a given fiber at 900-nm than at 1300-nm.

5.1.3 Power Launching versus Wavelength The radiated power per mode, Ps/M, from a source at a particular wavelength is given by the radiance multiplied by the square of the nominal source wavelength, Ps/M = Bol2 (5-12) Twice as much power is launched into a given mode at 1300-nm than at 900-nm. Two identically sized sources operating at different wavelengths but having identical radiances will launch equal amounts of optical power into the same fiber.

5.1.4 Equilibrium Numerical Aperture An example of the excess power loss is shown in Fig. 5-4 in terms of the fiber NA. At the input end of the fiber, the light acceptance is described in terms of the launch numerical aperture NAin. If the light-emitting area of the LED is less than the cross-sectional area of the fiber core, then the power coupled into the fiber is given by Eq. (5-7), where NA = NAin.

5.1.4 Equilibrium Numerical Aperture In long fiber lengths after the launched modes have come to equilibrium (which is often taken to occur at 50-m), the effect of the equilibrium numerical aperture NAeq becomes apparent. The optical power in the fiber scales as Peq = P50 (NAeq / NAin)2 (5-13) where P50 is the power expected in the fiber at the 50-m point based on the launch NA.

5.1.4 Equilibrium Numerical Aperture Figure 5-4. Example of the change in NA as a function of fiber length.

5.2 LENSING FOR COUPLING IMPROVEMENT Possible Lensing Schemes are shown in Fig. 5-5:   A rounded-end fiber; A small glass sphere (nonimaging microsphere) in contact with both the fiber and the source; A larger spherical lens used to image the source on the core area of the fiber end: A cylindrical lens generally formed from a short section of fiber; A system consisting of a spherical-surfaced LED and a spherical-ended fiber; and A taper-ended fiber.

5.2 LENSING FOR COUPLING IMPROVEMENT Figure 5-5. Example of possible lensing schemes used to improve optical source-to-fiber coupling efficiency.

5.2.1 Nonimaging Microsphere On examining nonimaging microsphere for surface emitter shown in Fig. 5-6, we make assumptions that the spherical lens has a refractive index of 2.0, the outside medium is air (n = 1.0), and the emitting area is circular. The focal point can be found from the gaussian lens formula n/s + n’/q = (n’ - n)/r (5-14) where s and q are the object and image distances, respectively, n is the refractive index of the lens, n' is the refractive index of the outside medium, and r is the radius of curvature of the lens surface.

5.2.1 Nonimaging Microsphere To find the focal point for the right-hand surface of the lens shown in Fig. 5-6, we set and solve for s in Eq. (5-14), where s is measured from point B. With n = 2.0, n’ = 1.0, and r = -RL, Eq. (5-14) yields s = f = 2RL The focal point is located on the lens surface at point A.

5.2.1 Nonimaging Microsphere Figure 5-6. Schematic diagram of an LED emitter with a microsphere lens.

5.2.1 Nonimaging Microsphere Figure 5-7. Theoretical coupling efficiency for a surface-emitting LED as a function of the emitting diameter. Coupling is to a fiber with NA = 0.20 and core radius a = 25-mm.

5.2.1 Nonimaging Microsphere Placing LED close to the lens surface results in a magnification M of the emitting area. This is given by the ratio of the cross-sectional area of the lens to that of the emitting area: M = pRL2 / prs2 = (RL/rs)2 (5-15) Using Eq. (5-4), with the lens, the optical power PL coupled into a full aperture angle 2q is given by PL = Ps(RL/rs)2sin2q (5-16) where Ps is the total output power from the LED without the lens.

5.2.1 Nonimaging Microsphere For a fiber of radius a and numerical aperture NA, the maximum coupling efficiency hmax is given by (a/rs)2(NA)2 , for (rs/a)2 > NA hmax = { (5-17) 1 , for (rs/a)2 < NA When the radius of the emitting area is larger than the fiber radius, no improvement in coupling efficiency is possible with a lens. In this case, the best coupling efficiency is achieved by a direct-butt method. Based on Eq. (5-17), the theoretical coupling efficiency as a function of the emitting diameter is shown in Fig. 5-7 for a fiber with an NA of 0.20 and 50-mm core diameter.

5.2.2 Laser Diode-to-Fiber Coupling Edge-emitting laser diodes have an emission pattern that nominally has a FWHM of 30-50o in the plane perpendicular to the active-area junction and an FWHM of 5-10o in the plane parallel to the junction. Since the angular output distribution of the laser is greater than the fiber acceptance angle, and since the laser emitting area is much smaller than the fiber core, spherical or cylindrical lenses or optical fiber tapers can be used to improve the coupling efficiency between edge-emitting laser diodes and optical fibers.

5.2.2 Laser Diode-to-Fiber Coupling Spherical glass lenses with a refractive index of 1.9 and diameters ranging between 50 and 60-µm were epoxied to the ends of 50-mm core-diameter graded-index fibers having an NA of 0.2. The measured FWHM values of the laser output beams were as follows: 1. Between 3 and 9 µm for the near-field parallel to the junction. 2. Between 30 and 60o for the field perpendicular to the junction. 3. Between 15 and 55o for the field parallel to the junction. Coupling efficiencies in these experiments ranged between 50 and 80%.

5.3 FIBER-TO-FIBER JOINTS For a graded-index fiber with a core radius a and a cladding index r2, and with k = 2p/l, the total number of modes is (5-18) where n(r) defines the refractive-index variation of the core. The total number of modes can relate to a local numerical aperture NA(r) through Eq. (2-80) to yield (5-19)

5.3 FIBER-TO-FIBER JOINTS The fraction of energy coupled from one fiber to another is proportional to the common mode volume Mcomm. The fiber-to-fiber coupling efficiency is given by hF = (Mcomm/ME) (5-20) where ME is the number of modes in the emitting fiber. The fiber-to-fiber coupling loss LF is given in terms of hF as LF = -10 log hF . (5-21)

5.3 FIBER-TO-FIBER JOINTS Consider all fiber modes being equally excited, as shown in Fig. 5-8a. If steady-state modal equilibrium has been established in the emitting fiber, the optical power is concentrated near the center of the fiber core, as shown in Fig. 5-8b. The optical power emerging from the fiber then fills only the equilibrium NA (see Fig. 5-4).

5.3 FIBER-TO-FIBER JOINTS Figure 5-8. Different modal distributions of the optical beam emerging from a fiber lead to different degrees of coupling loss. (a) When all modes are equally excited, the output beam fills the entire output NA; (b) For a steady-state modal distribution, only the equilibrium NA is filled by the output beam.