Coupling Light into a Multimode Fibre

The task of collecting thermally emitted light and reflected light from that same surface is essentially the same procedure. The difference is that the reflection behaviour is determined by how the surface is initially illuminated, but the task

of collecting this reflected light is the same as if it was directly emitted from the surface.

To simplify things, we will begin by assuming we have an extended diffuse light source. We want to collect the emitted light and couple it into a multimode fibre as efficiently as possible. By diffuse, I mean that the source emits light in all directions and is of the same intensity in all directions. This may not be case for all photonic crystal emitters, in fact for the devices in this work it clearly is not, as they have angularly dependent emission behaviour yet assuming diffuse emission is a good starting point.

The goal of the optical system is to maximise the amount of radiated power from the source to the detector, where the detector in our case is replaced with the facet of a multimode fibre, which is connected to a spectrometer.

Firstly, we consider a simple optical imaging system containing one lens and examine how an arbitrary optical ray propagates from the object h1 at s1 to

the image h2 at s2, Fig. 2.8a. The dotted line in the figure shows the ray from

the object that has the maximum angle to the optical axis and is still within the aperture of the lens. This is the maximal ray and is very important for the design of an optical system. Assuming a thin lens and the paraxial approximation (sin(θ)≈θ), and after applying some basic geometry, we get

θ1 = R s1 , θ2 = R s2 (2.1) where θ1 and θ2 are the angles which the ray makes with the optical axis on

the object and image side, respectively. R is the radius of the lens and s1 and

s2 are the image and object distances from the lens. Using the definition of

magnification (h2/h1 =s2/s1) and Eq. 2.1 we have

h2θ2 =h1θ1 (2.2)

where h1 and h2 are the object and image height. This relationship is a funda-

mental law of optics, called the optical invariant, and is true for any ray through the system, not just the maximal as illustrated here. For any optical system of lenses, regardless of the number of lenses, the product of the image size and the ray angle is constant, which is called the ´entendu. The optical invariant is a special case of the more general geometrical extent, where the product of the area and the solid angle is invariant. However, for systems that are rotation- ally symmetric (e.g. lenses and optical fibres), the 1D optical invariant is equally accurate.

In addition to the source (detector) and the entrance (exit) lenses, other elements may limit the optical invariant of a system. For example, internal aper- tures or some other physical restriction will limit the optical invariant. However, it is more often than not limited by the spectrometer or monochromator as these instruments have small entrance pupils and/or small detector areas and limited acceptance angles. Another possible limitation on the optical invariant is the use of fibre optics. The numerical aperture and the size of the fibre core can be the limiting factors and determine the best optical invariant achievable. It is important to understand the part of the system that limits the optical invariant of the system, fixes the maximum possible throughput since no amount of clever optical design can improve on this.

f f s1 s2 h1 h2 θ1 θ2 f f s1 s2 θ1 θ2 h1 h2 Multimode Fibre Diffuse Source Diffuse Source Butt Coupling Multimode Fibre Diffuse Source f f Multimode Fibre h1 h2 (d) (a) (b) (c)

Figure 2.8: Four different optical imaging systems, each with the goal of efficiently coupling light form a diffuse source into a multimode fibre. (a) Simple optical imaging system illustrating the optical invariant by looking at an arbitrary ray propagating through the system. (b) Imaging of a diffuse light source onto the facet of a mulitmode fibre using a single lens with a 1:1 magnification. (c) Illustrating the butt coupling of a multimode fibre to a diffuse source. (d) Two lens coupling setup. Effectively the same coupling as (c) with the added advantage of the separation distance between the two lenses being more flexible which allows for easy integration of other optical components

Returning to our optical system design, Fig. 2.8b shows a possible system to couple light from a diffuse source into a multimode fibre. We must not forget that we are imaging an area of the source on to the facet of the fibre and in this case with a 1:1 magnification ratio. The source is placed at s1 (2f), and its

image is at s2 (2f) at the other side of the lens (indicated with green arrows).

The radius of the fibre core facet is h2, therefore the light is only collected from

an area of radius h1 at the source, (1:1 magnification). The largest possible

angle coupled into the fibre is restricted by its own numerical aperture (NA, NA = nsin(θ), where n is the refractive index (this case n=1) of the environment and θ is the maximum acceptable angle). In this case we letθ2 = NA, according

to the paraxial approximation NA ≈ θ. Therefore, by the optical invariant, the maximum angle sampled from the source is θ2 (θ1 = θ2 = NA). Thus, it is the

fibre and not the lens that limits the coupling of the light assuming the lens has a sufficiently small F/# to overfill the acceptance cone of the collecting fibre. Using the paraxial approximation, the F/# of a lens can be defined as

F/#≈ f

D (2.3)

where f is the focal length and D is the diameter of the lens. The smaller the F/# the greater the radiant flux collected by the lens i.e. the larger the diameter or the smaller the focal length the better. Ideally, for this single lens 1:1 setup, require a F/#<1/(2NA).

Note that if we remove the lens and move the fibre directly over the source, effectively butt-coupling the fibre to the source, we will collect light from an area of h2, the radius of the fibre core, and collect light with an angle of NA

(Fig. 2.8)c. This is what the lens and the 1:1 imaging system achieves, therefore any imaging system of lenses cannot collect more light than can be collected by simply butt coupling the fibre directly to the source.

In fact, this result is entirely general. If we try and use the multiple lenses to collect light from a much larger area and focus it down to the fibre facet, we will not gain anything. For example, by collecting light from a larger area, we will lose in collection angle.

In conclusion, the total light that is collected (area × solid angle× radiated power) is the same in each case. Therefore, the maximum efficiency that any lens system can achieve is to match that retrieved through butt coupling. A lens system makes it possible to place the fibre away from the source and get maximum coupling. To truly maximise the light throughput, a fibre with a large core diameter and a large NA is required.