### Source-to-Fiber Power Launching

. Optical output from a source is measured in radians (B). Radiance is defined as the optical power radiated into a solid angle per unit emitting surface area. Radiance is specified in Watts/cm2/Steradian. Radiance is important for defining source to fiber coupling efficiency.
1. Source Output Pattern
. Spatial radiation pattern of source helps to determine the power accepting capability of fiber.
. Fig. 1 shows three dimensional spherical co-ordinate system for characterizing the emission pattern from an optical source. Where the polar axis is normal to the emitting surface and radiance is a function of θ and Φ.

. The Lambartian output by surface emitting LED is equally bright from any direction. The emission pattern of Lambartian output is shown in Fig. 2 and its output is –

Where,

. Both radiations in parallel and normal to the emitting plane are approximated by expression –

Where,
T and L are transverse and lateral power distribution coefficients.
2. Power Coupling Calculation
. To calculate power coupling into the fiber, consider an optical source launched into the fiber as shown in Fig. 3.

. Brightness of source is expressed as B(As , Ωs),
Where, As is area of source.
Ωs is solid emission angle of source.
The coupled power P can be calculated as –

The integral limits are area of source and solid acceptance angle (θ0max).
Here dθs rdr is incremental emitting area.
. Let the radius surface emitting LED is rs, and for Lambartian emitter, B(θ,Φ)  = B0 cosθ, then

Power coupled to step – index fiber
. For step index fiber NA is not dependent on θs and r. Therefore LED power from step index fiber is,

. Consider optical power Ps emitted from source area As into hemisphere (2πSr ).

. When source radius rs ≤ a, the fiber core radius, the LED output power is given from equation 5 .

. When rs ˃ a equation becomes,

Power coupled to graded index fiber
. In graded index fiber, the index of refraction varies radially from fiber axis. Numerical aperture for graded index fiber is given by,

If source radius (rs ) is less than fiber core radius (a) i.e. r< a, the power coupled from surface emitting LED is given as –

. For coupling maximum power to fiber, the refractive index of the medium separating source and fiber must be same, otherwise there will be loss of power. The power coupled is reduced by factor,

Where,
n is the refractive index of medium.
n1 is the refractive index of fiber core.
R is the Fresnel reflection of reflectivity.

3. Power Launching and Wavelength
. Optical power launched into fiber is dependent only on radiance (brightness) of source. It does not depends on the wavelength of the source.
. Considering the expressions for optical power coupled power into the fiber it can be concluded that the power coupled is mainly decided by – source radius, radiance, fiber and numerical aperture.