## Numerical aperture of fibers

Owing to the requirement of total internal reflection, the only light rays that can propagate losslessly in the core of an optical fiber are those that have a propagation angle smaller than the critical angle for total internal reflection. Light rays for which the propagation angle is too large will consequently not couple into the fiber. Here we consider the coupling of light from an LED light source into an optical fiber. We assume that the fiber end has a polished planar surface normal to the optical axis of the fiber, as shown in Fig. 22.6.

As a consequence of the requirement of total internal reflection, only a range of angles will be “accepted” by the fiber for lossless propagation. Outside the acceptance angle range, light

rays will be refracted into the cladding layer where they will incur losses.

The range of allowed angles can be inferred from Snell’s law. As illustrated in Fig. 22.6, the maximum angle for acceptance in the fiber is given by

(22.9) |

nair sin 9ajr _ n1 sin 9c.

Since the refractive index of air is approximately unity, the maximum acceptance angle in air is given by

(22.10) |

9air _ arcsin (n1 sin 9c).

The maximum acceptance angle defines a cone of allowed angles, as shown in Fig. 22.6. Light rays incident on the core of the optical fiber with propagation angles within the cone can propagate without loss.

Another way to express the acceptance cone is the numerical aperture of the fiber. The numerical aperture (NA) is defined as

9 |

(22.11) |

air |

NA _ n1 sin 9c _ nair sin 9air _ sin 9ajr |

where the approximation sin 9air « 9air is valid for small numerical apertures. Typical NAs for silica single-mode fibers are 0.1 and typical NAs for silica multimode fibers are 0.15-0.25. Plastic optical fibers can have higher NAs, typically 0.2-0.4.

The solid angle corresponding to a certain NA is given by

Solid angle _ Q _ 2n (1 - cos 9air) _ 2n[1 - cos(arcsin NA)] « n NA2 (22.12)

where the small-angle approximations sin 9air « 9air and cos 9air « 1 - (1/2)9air2 have been used. The power emitted by an LED is proportional to the solid angle. Thus the power coupled to an LED is proportional to the NA squared of the fiber in the small-angle approximation.

Exercise: Coupling efficiency of a fiber butt-coupled to an LED. Consider an LED with a point-like emission region that emits an optical power of 1 mW into the hemisphere. For simplicity, assume that the intensity emitted by the LED is independent of the emission angle. What is the maximum acceptance angle of a single-mode fiber with NA = 0.1 and multimode fiber with NA = 0.25? What is the power that can be coupled into the two fibers?

Solution: The maximum acceptance angles of the single-mode and multimode fibers in air are 9air = 5.7° and 14.5°, respectively. The solid angle defined by an acceptance angle 9air is given by Q = 0.031 and 0.20 for the single-mode and multimode fiber, respectively. Since the entire hemisphere has a solid angle of 2n, the power coupled into the single-mode and multimode fibers is given by 0.0049 and 0.032 mW, respectively.

*22.1 *Coupling with lenses The low coupling efficiency of LEDs to optical fibers can be improved with convex lenses, if the light-emitting region of the LED is smaller than the optical fiber core. In this case, the light - emitting region can be imaged on the fiber core, thereby reducing the angle of incidence. The light source is adapted to the NA of the fiber (“NA-matching”).

A convex lens can produce an image with height I of a light-emitting object with height O. If the image is larger than the object, the angles of the light incident from the lens on the image are less divergent than the light emanating from the object towards the lens. The smaller divergence obtained for magnified images allows one to increase the coupling efficiency to fibers. The principle of coupling with a convex lens is shown in Fig. 22.7.

The condition for a focused image (minimum image size) is given by the lens equation

1 + — _ — (22.13)

dO dI f

where dO and dI are the distances of the object and the image from the lens, respectively, and f is the focal length of the lens.

The magnification of the image of the LED light source on the core of the fiber is given by

M = — = . (22.14) O do |

Fig. 22.7. Schematic illustration of coupling with a lens by imaging the light-emitting region of an LED onto the core of an optical fiber. The LED has a circular emission region with diameter О (Object). The emission region is imaged onto the fiber core with diameter / (Image) using a convex lens with focal length/ |

Core Cladding |

As shown in Fig. 22.7, it is

-2 O + do tan ©led = 11 + di tan 9ajr. (22.15)

If the LED and the core of the optical fiber are much smaller than the diameter of the lens and if the angles are relatively small, then Eq. (22.15) can be approximated by

9LED = ~d^9air = 779air. (22.16)

do O

Since dI is larger than dO, the acceptance angle for light emanating from the LED is larger than that of the fiber, implying increased coupling efficiency. Thus, we can define the numerical aperture of the LED, NALED, which defines the angle of light emanating from the LED that is accepted by the fiber. Using Eq. (22.14) and the small-angle approximation for NA, NALED is given by

NAled = ONA. (22.17)

Since the coupling efficiency is proportional to NA2 (see Eq. 22.12), the coupling efficiency is

increased to

The result shows that high coupling efficiencies are obtained for LEDs with small-diameter light-emitting regions, large fiber-core diameters, and large-NA fibers.

Fig. 22.8. (a) Commercial communication LED chip with integrated lens, (b) Detailed picture of the lens etched by a photochemical process into the GaAs substrate (AT&T ODL product line, 1995). |

Lensed LEDs are frequently used in communication applications. A micrograph of an LED with a monolithically integrated lens is shown in Fig. 22.8. The light-emitting region of the LED is 20 |im and the lens shown in Fig. 22.8 has a diameter of about 80 |im.

Exercise: Coupling efficiency of a fiber coupled to an LED with a lens. Consider an LED circular emission region with diameter 20 |jm coupled to a silica multimode fiber with NA = 0.2 and a core diameter of 62.5 |jm. The LED emits a power of 1 mW into the hemisphere lying above the planar LED surface. For simplicity, assume that the LED emission intensity is independent of the emission angle. What is the maximum power that can be coupled into the multimode fiber?

Solution: Improved coupling can be obtained by imaging the LED emission region on to the core of the optical fiber. For maximum coupled power, a convex lens with magnification M = 62.5 ^.m / 20 ^.m = 3.125 can be used. Using the lens, the acceptance angle of the fiber is increased from 9air = 11.5° to 9LED = 35.9°. The solid angle defined by the LED acceptance angle 9LED is given by Q = 1.19. Since the LED emits 1 mW into the entire hemisphere (with solid angle Q = 2n), the power coupled into the fiber is given by 0.189 mW.