2 Injecting a point source

Injecting starlight into a fiber is usually achieved in the focal plane of the telescope, by forming an image of the star onto the fiber head. To compute the efficiency we first follow the classical method of the overlap integral, whose details can be found in e.g. (Neumann 1988) or (Jeunhomme 1993).

A propagating field is created into the waveguide, which can be decomposed
along a set of spatially orthogonal modes. The amount by which a given mode
is excited by the incident field
is
determined by the scalar product of the spatial distribution for that mode
and the incident field. It follows that the coupling efficiency
between
and the field
of the
fundamental (LP_{01}) mode of the circular waveguide is given by the ratio
of the integrals:

where the integration domain extends at infinity in the focal plane and the symbol denotes a complex conjugate. Since the amplitudes and

Equation (1) expresses how well the spatial distributions of
the two fields match in amplitude *and* in phase. The field distribution
across the LP_{01} mode is fixed and determined by the waveguide
characteristics.
In the case of a step-index, circular core fiber, it is given
by (Gloge 1971a).
In a plane transverse to the axis of the waveguide the phase is constant and
the amplitude shows a symmetry of revolution, with a radial profile that
extends into the cladding and can be well approximated by a Gaussian under
most circumstances (Gloge 1971a, 1971b).

The field distribution at the focus of the telescope, on the other hand, is
proportional to the Fourier transform of the distribution of the complex field
diffracted at the entrance
pupil (Born & Wolf 1980):

where is the pupil transmission function, the electric field amplitude received from the source at the pupil, and a possible random phase mask that takes into account optical aberrations and atmospheric turbulence. The conjugated variables in the Fourier transform are the reduced coordinates on the pupil and the diffraction direction corresponding in the focal plane to the position , where

(4) |

2.1 Diffraction limited beams

Without turbulence nor aberrations, the phase mask at the pupil is unity:
.
The diffraction figure of an
unobstructed circular aperture of radius *S*_{0} = *d*/2 is the Airy pattern

(5) |

where is the reduced radial distance to the optical axis:

(6) |

Most often, however, a secondary mirror of radius

(7) |

Figure 1:
Amplitude profile of the electric field at the focus of a
diffraction limited telescope, for different relative central obstructions
(
corresponds to the 3.60m ESO telescope in La
Silla) |

The central obstruction can seriously damage the coupling efficiency, much
more than one would expect from the loss of collecting power only. Indeed,
in the focal plane the main effect of the pupil obstruction is to reinforce
the first ring of the Airy pattern (Fig. 1), which
contributes *negatively* to the overlap integral (Fig. 2). The coupling efficiency also depends on the input
beam *f*-ratio. Figure 3 shows coupling efficiency vs.
*f*-ratio curves at wavelength
m for a typical fluoride
glass fiber (core diameter
m, numerical aperture
and cutoff wavelength
m). For a
full aperture the maximum efficiency is 78% (obtained at
),
but the performance drops to 48% when the relative obstruction is
,
as it is the case for the 3.6m ESO telescope at La Silla
observatory. Computing the curves for other fibers leads to very similar
values of the maximum efficiency; however the optimum *f*-ratio scales like
the numerical aperture of the fiber.

Figure 3:
Coupling efficiency into the same fluoride glass fiber as a function
of the f-ratio of the input beam, for three values of the relative central
obstruction |

2.2 Coupling with a turbulent wavefront

Inserting the value of (Eq. 3) and developing the squared integral leads to

We can operate a change of variables
and recognize in the product
a pupil function apodized by
the Fourier transform of the fiber mode profile. Equation (9)
can thus be rewritten as the integral of an autocorrelation product:

= | (10) | ||

= |

The autocorrelation of is nothing else than the combined MTF of the atmosphere and the apodized pupil.

In adaptive optics (AO), a common criterion to assess image quality is the Strehl
ratio
which is, for a point source, the ratio of the central
intensity of the corrected image to the central intensity of an ideal,
diffraction limited image of a source with the same
magnitude (Born & Wolf 1980; Tyson 1991).
Since the image intensity on the center is also the integral of the
modulation transfer function, one
has (
is the autocorrelation of the apodized pupil alone):

where is the injection efficiency when there is no turbulence.

Thus, the injection efficiency in presence of turbulence is proportional to the Strehl ratio of the apodized pupil. In practice, if one choses a fiber with a sufficiently small core (or equivalently, if the input beam is slow enough), the coupling variations are a good estimator of the Strehl ratio fluctuations. In the image plane, one can also say that a quasi point-like fiber core samples the intensity at the center of the image, i.e. the Strehl ratio.

The combination of a single-mode fiber and a fast photometer makes up a unique tool to measure the fluctuations of the instantaneous Strehl ratio. This might be especially useful for the qualification of adaptive optics systems. An example is given in Sect. 3.

Copyright The European Southern Observatory (ESO)