Technical and Application Notes

A Reference Guide to Optical Fibers and Light Guides

Coupling Considerations

Entire chapters have been written about the use of the LaGrange or Smith-Helmholtz Invariant in coupling optical fibers to other instruments. We will reduce those chapters to some simple prescriptions.

1. The etendue, or light gathering power, of any optical system can be determined by multiplying the diameter or width of the limiting aperture by the angle of divergence of that aperture. It is a useful method of comparing relative optical performance, not absolute values. When comparing etendue, consistent units must be maintained. We will use degrees and millimeters in this example.

The etendue of a system is two-dimensional. In a system employing non-symmetrical apertures (like slits), there is a different etendue for each axis (x & y). For a monochromator, the etendue are calculated by multiplying both the width and height of the smallest slit by the convergent cone (f/#) angle. For a fiber bundle, the etendue is calculated by multiplying the core area by the acceptance angle (NA).

No system of external optics can increase the etendue of an optical system. This principle is called conservation of etendue.

When coupling a larger etendue to a smaller etendue you will always lose light. When coupling a smaller etendue to a larger etendue, it is possible to design so that you will not lose light.

The effect of etendue when coupling light guides to monochromators is explained below.

We start with a fiber with a diameter of .50 mm, and an acceptance angle (as determined by its NA) of 28.07°. The etendue of the fiber bundle is calculated as:

.50 * 28.77 = 14.39

We are coupling to a monochromator whose slits are set at .2 mm wide and are 4.0 mm high, and an acceptance angle (as determined by its f/#) of 14.250°. The etendue of the slit width (x) is calculated as:

0.2 * 14.25 = 2.85

The etendue of the slit height is calculated as:

4.0 * 14.25 = 57.00

In this arrangement, the etendue of the fiber is greater than that of the monochromator slit width, but smaller than that of the monochromator slit height. With this arrangement, you are always going to lose light when trying to send it from the fiber into the monochromator.

If you try to place the fiber directly at the slit, with the fiber being larger that the slit, you will block a portion of the fiber with the slit. The light that does make it through the slit will be diverging at an angle greater than the acceptance angle of the monochromator. This will further reduce the efficiency of the coupling.

Using additional optics, we can reduce the image size to fit within the slit by collecting the light from the light guide and focusing it onto the slit using a convergence angle that is greater than the divergence angle of the light coming from the light guide. That is demagnification. Since this new angle will be even wider than the original fiber angle, however, it will overfill the collection mirror in the monochromator even more than the fiber alone did. We will lose in geometric proportions.

Or, we can use external optics to change the angle of the light to match the acceptance angle of the monochromator. This will result in a magnified image of the fiber at the entrance slit. We will still lose light, but now it will be in linear proportions.

That is the conservation of etendue.

Reducing the image by a factor of 2 (down to ~ 2.5 mm) requires speeding up the divergence to ~ f/.5. This would reduce f/# matching efficiency to 1.6 %.

(f/.5 +f/4)2

The more efficient method is to stretch the focal cone out to match the monochromator’s acceptance angle using a "slower" lens system. The image will be magnified by the ratio of the fiber f/# to the lens f/#, and more will be blocked by the slit. But the increased image height will fill more of the slit height. Further, the matching focal cone insures that all of the light entering the slit will be diffracted by the grating. Although we are still losing light, this yields a factor of 2 improvement over direct coupling.

That is the Lagrange Invariant. If you want a smaller image, you need a wider angle, which will diverge rapidly. If you want a smaller, less divergent angle, you will get a larger image.

As far as the output of the monochromator is concerned, a monochromator will image the entrance slit onto the exit slit. If an object takes up the full height of the entrance slit, it will also diverge from the full height of the exit slit. Therefore, the exit slit height etendue of a monochromator is somewhat dependant on the object height, with an upward limit being the full slit height. If the image at the exit slit is taller then the entrance to the fiber bundle, yet the acceptance angle of the fiber bundle is larger, the extra etendue of the fiber can be collected and de-magnified by a faster focusing lens, reducing the image size to the bundle diameter.

An in depth explanation of the principle of invariance and of coupling fibers to spectrometers. can be found in "Practical Considerations When Using Fiber Optics with Spectrometers" by C Technologies, Inc., Spectroscopy magazine, vol. 4, no. 6, August, 1989.


There are almost as many factors to be considered when using light guides as there are applications. Some of the more basic considerations are material, wavelength range for the application, core transmission, packing fractions, and coupling. As with all technology, careful attention must be paid to the detail. The trick is in knowing which of the details are relevant and which are not. The attempt here is to bring some of the relevant details to light for those who are inexperienced, and give a reminder to those who have been down this path before.


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