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 ${\bf E}_{\mbox{\scriptsize focus}}$ is determined by the scalar product of the spatial distribution for that mode and the incident field. It follows that the coupling efficiency $\rho$ between ${\bf E}_{\mbox{\scriptsize focus}}$ and the field ${\bf E}_{01}$ of the fundamental (LP₀₁) mode of the circular waveguide is given by the ratio of the integrals:

$$\rho = \frac{ \left\vert \int_{A_\infty} {\bf E}_{\mbox{\scr... ...\, \int_{A_\infty} \! \vert{\bf E}_{01}\vert^2 \, {\rm d}A} ,$$ (1)

where the integration domain extends at infinity in the focal plane and the symbol $^\ast$ denotes a complex conjugate. Since the amplitudes $E_{\mbox{\scriptsize focus}}$ and E₀₁ are each squared both in the numerator and the denominator, the ratio does not depend on the overall field intensities and Eq. (1) is more conveniently rewritten using fields whose energies have been normalized to unity, so that

$$\rho = \left\vert \int_{A_\infty} {\bf E}_{\mbox{\scriptsize focus}} \, {\bf E}_{01}^\ast \, {\rm d}A \right\vert^2 .$$ (2)

Equation (1) expresses how well the spatial distributions of the two fields match in amplitude and in phase. The field distribution across the LP₀₁ 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 ${\bf E}_{\mbox{\scriptsize pupil}}$ diffracted at the entrance pupil (Born & Wolf 1980):

$${\bf E}_{\mbox{\scriptsize pupil}}({\vec s}) = {\bf E}_\star \, \mbox{\boldmath$\Psi$ }({\vec s}) \, G({\vec s}) ,$$ (3)

where $G({\vec s})$ is the pupil transmission function, ${\bf E}_\star$ the electric field amplitude received from the source at the pupil, and $\mbox{\boldmath$\Psi$ }({\vec s}) = {\rm e}^{j \phi({\vec s})}$ 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 ${\vec s} / \lambda$ on the pupil and the diffraction direction corresponding in the focal plane to the position ${\vec r} / f$, where f is the focal length of the telescope. A change of variables leads then to (the sign $\tilde{\mbox{ }}\$ denotes a Fourier Transform)

$${\bf E}_{\mbox{\scriptsize focus}}({\vec r}) \propto \tilde{{... ...criptsize pupil}}\left(\frac{{\vec r}}{\lambda f}\right )\cdot$$ (4)

   
2.1 Diffraction limited beams

Without turbulence nor aberrations, the phase mask at the pupil is unity: $\mbox{\boldmath$\Psi$ }({\vec s}) = 1$. The diffraction figure of an unobstructed circular aperture of radius S₀ = d/2 is the Airy pattern

$${\bf E}_{\mbox{\scriptsize focus}} \propto {\bf E}_\star \frac{2 \, J_1(\zeta)}{\zeta} ,$$ (5)

where $\zeta$ is the reduced radial distance to the optical axis:

$$\zeta = 2 \pi \, S_0 \, \frac{r}{\lambda f} \cdot$$ (6)

Most often, however, a secondary mirror of radius s₀ obstructs the beam and the focal field is then expressed as ( $\alpha = s_0 / S_0$ is the relative obstruction of the pupil):

$${\bf E}_{\mbox{\scriptsize focus}} \propto {\bf E}_\star \lef... ...pha^2 \, \frac{2 \, J_1(\alpha \zeta)}{\alpha \zeta} \right] .$$ (7)

Figure 1: Amplitude profile of the electric field at the focus of a diffraction limited telescope, for different relative central obstructions $\alpha$ ( $\alpha = 0.436$ corresponds to the 3.60m ESO telescope in La Silla)

Figure 2: Compared amplitude profiles (at wavelength $\lambda = 2.2\,\mu$m) of the electric field at the focus of a telescope ( $\alpha = 0.436$, f/d=4) and the fundamental mode of a fluoride glass fiber (core diameter $2 a = 8.5\, \mu$m and cutoff wavelength $\lambda_{\mbox{\scriptsize c}}=1.91\,\mu$m)

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 $\lambda = 2.2\,\mu$m for a typical fluoride glass fiber (core diameter $2 a = 8.5\, \mu$m, numerical aperture $N\!A = 0.17$and cutoff wavelength $\lambda_{\mbox{\scriptsize c}}=1.91\,\mu$m). For a full aperture the maximum efficiency is 78% (obtained at $f/d \simeq 3.35$), but the performance drops to 48% when the relative obstruction is $\alpha = 0.436$, 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 $\alpha$

   
2.2 Coupling with a turbulent wavefront

Ground-based telescopes are affected by atmospheric turbulence so that the phase of the incident wavefront is disturbed and the pupil transmission has a imaginary part. This section deals mostly with the coupling fluctuations in presence of turbulence. For detailed simulations of the time averaged coupling efficiency between a single-mode fiber and a turbulent wavefront, the reader is referred to (Shaklan & Roddier 1988; Ruilier 1998). Going back to Eq. (2), one can make use of Parseval's theorem to rewrite the integral in the conjugated Fourier space, i.e. the pupil plane, where we have after normalization of the field energies:

$$\rho = \left\vert \int_{A_\infty} {\bf E}_{\mbox{\scriptsiz... ...pil}} \, {\bf \tilde{E}}_{01}^\ast \, {\rm d}A \right\vert^2 .$$ (8)

Inserting the value of ${\bf E}_{\mbox{\scriptsize pupil}}$ (Eq. 3) and developing the squared integral leads to

 

$$\rho \int\!\!\int E_\star^2 \, \mbox{\boldmath$\Psi$ }({\vec s}) \, \mbox{\boldmath$\Psi$ }^\ast(\vec{s'}) \, G({\vec s}) \, G^\ast(\vec{s'}) \,$$ = (9)

$$\times \, {\bf\tilde{E}}_{01}^\ast\left(\frac{\vec s}{\lambda f}\... ...1}\left(\frac{\vec{s'}}{\lambda f}\right) \, {\rm d}\vec{s} \, {\rm d}\vec{s'}.$$

We can operate a change of variables ${\vec s_1} = \vec{s'} - {\vec s}$and recognize in the product $G_{\rm a}({\vec s}) = G({\vec s}) {\bf \tilde{E}}_{01}(\frac{\vec{s'}}{\lambda f})$ 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:

$$\rho E_\star^2 \!\!\int \!\! {\rm d}{\vec s} \int \!\! \mbox{\boldmath... ...s}) \, G_{\rm a}({\vec s}) G_{\rm a}^\ast(\vec{ s_1}+\vec{s}) {\rm d}{\vec s_1}$$ = (10)

$$E_\star^2 \, \int A_{\mbox{\boldmath$\Psi$ } G_{\rm a}}({\vec s})\, {\rm d}{\vec s}.$$ =

The autocorrelation $A_{\mbox{\boldmath$\Psi$ } G_{\rm a}}$ of $\mbox{\boldmath $\Psi$ } G_{\rm a}$ is nothing else than the combined MTF of the atmosphere and the apodized pupil.

2.2.1 Coupling efficiency and Strehl ratio

In adaptive optics (AO), a common criterion to assess image quality is the Strehl ratio ${\cal S}$ 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 ( $A_{G_{\rm a}}$ is the autocorrelation of the apodized pupil alone):

 

$${\cal S} \frac{\int A_{\mbox{\boldmath$\Psi$ } G_{\rm a}} \, {\rm d}{\vec s}}{\int A_{G_{\rm a}} \, {\rm d}{\vec s}}$$ = (11)

$$\frac{\rho}{\rho_0} ,$$ =

where $\rho_0$ 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.