Wavefront aberration

Atmospheric turbulence is a dynamic process and the wavefront aberrations are constantly changing. A characteristic timescale may be defined as the time over which the mean square change in wavefront error is less than 1 rad. If the turbulence were concentrated in a single layer with Fried parameter ro moving with a horizontal speed of v ms then the characteristic time, ro, is given by... 

One of the attractive features of single mode waveguides is their ability to filter the spatial field distribution. All the wavefront aberrations only result in a photometric fluctuation easy to monitor. It results in a very good calibration of the interferometric data as firstly demonstrated by FLUOR, as seen in Fig. 8 (Coude du Foresto et al., 1998). Nevertheless, care has to be taken to keep in mind that turbulence may have a spectral selectivity in the launching pro-... 

The second problem is how we can obtain a linear relationship between the coefficients describing the wavefront and our measurements. It is how this linear relationship is obtained that differentiates, for example, a Shack-Hartmann and a curvature sensor. In all wavefront sensors to transform a wavefront aberration into a measurable intensity fluctuation it is necessary to propagate the wavefront. As a first approximation this propagation is described by geometric optics, and we discuss the linear relationship between the wavefront slope and the image displacement in Section 24.3. 

According to the wave theory of aberrations the converging wave that propagates toward an image point should in principle have exactly spherical wavefronts. The maximum distance between the wavefront and a true sphere is called the wavefront aberration. If the wavefront aberration exceeds one-quarter wavelength, then the lens is aberration-limited. The point spread function is broader than the Airy disk, and the intensity of the peak is correspondingly reduced. The ratio of the peak intensity of the point spread function to that of the Airy disk is called the... 

Strehl ratio and may be used as a measure of the image quality. A Strehl ratio of 0.8 approximately corresponds to a wavefront aberration of one-quarter wavelength, and a lens with a Strehl ratio greater than 0.8 may be taken to be diffraction-limited. 

If the wavefront aberrations become smaller than 100 nm RMS, the surface gradients have to be analysed and the encircled energy of the light is evaluated. 

Figure 8.1 Adaptive optics in astronomy, (a) The image of a star appears as a fuzzy blob in a telescope due to the wavefront aberrations caused by turbulence in the atmosphere, (b) If a deformable mirror is used to correct the image of the star to make it back into a point of Ught, a nearby galaxy with a more complex structure will also be corrected, (c) The wavefront distortions can be measured with a wavefront sensor. The wavefront distortions from the guide star are fed back through a control system to deform the adaptive mirror to a shape that is conjugate to the distorted wavefront, correcting the reflected wavefront of the guide star back into a plane wave. (Credit Claire Max, Center for Adaptive Optics.) [7],...

In astronomy, we are interested in the optical effects of the turbulence. A wave with complex amplitude U(x) = exp[ irefractive index, resulting in a random phase structure by the time it reaches the telescope pupil. If the turbulence is weak enough, the effect of the aberrations can be approximated by summing their phase along a path (the weak phase screen approximation), then the covariance of the complex amplitude at the telescope can be shown to be... 

Figure 6. Path of aberrated ray on pyramid when circular modulation is applied. The offset of the circle from the pyramid vertex is proportional to the local wavefront slope.

As an example, consider a planar wavefront from an incoherent source passing through an aberration-free circular lens. When the image is diffraction-limited, an Airy disc pattern is observed (Goodman, 1996). For an aperture of radius 1 / 2n) the pdf for photon arrival is given by... 

Figure 9. The Shack-Hartmann sensor with (a) a planar wavefront and (b) an aberrated wave-front. The dashed lines are the perpendicular bisectors of the lenslets.

Since the aberrations of an acoustic lens can be much less than a wavelength, an alternative is not to think of them in geometrical terms, but rather to consider the phase aberrations introduced into the wavefront (Lemons and Quate 1979). In the absence of any aberration the wavefront after passing though the lens would be a sphere whose centre was at the focus. The difference between such a sphere and the actual wavefront is the aberration... 

Mathematically, aherrations are described as wavefront deviations (or errors), i.e., the difference in phase (or path difference) of the actual wavefront emerging from the lens compared to the ideal spherical wavefront. In other words, each primary aberration will produce unique deviations within the lens pupil. This phase difference is a function of position within the lens pupil, which can be described with an aberrated pupil function. An aberrated pupil function is described in terms of wavefront deformation as ... 

Aberration The departure of a ray or a wavefront from the path predicted by the paraxial theory. 

If the wavefront is nonspherical and the displacement of the actual wavefront from a spherical one is, we call it a wave aberration. Ray aberration is also reduced from wave aberration. 

For example, the phase conjugate operation is of particular interest in optical systems." In identifying the wavefront phase distribution y), a phase conjugate operation produces a counterpropagating wave with opposite phase — (x,y). Therefore, it is possible to construct systems where the phase aberrations are automatically cancelled by a double pass through the distorting media via a phase conjugate mirror. 

The second approach to measuring aberrations is by interferometry. The aberrations are, after all, essentially imperfections that appear when the lens focuses the incoming light and causes distortion in the output wavefront. Such deformation would be reflected from the interference patterns between the output wave and a reference wave (usually a plane wave). The measured interference patterns can be used in turn to determine the aberrations, usually using Zemike pol5momials [8-10]. 

P. D. Pulaski, J. P. Roller, D. R. Neal, and K. Ratte, "Measuremenst of aberrations in microlenses using a Shack-Haitmann wavefront sensor," Proceedings ofSPIE, vol. 4767, pp. 1-9,2002. 

Figure 5.12a depicts an example of the measured interferograms. Five interferograms were taken with a phase shift between each interferogram and a wrapped phase map could be generated. The unwrapped phase map representing the actual optical path difference profile generated by the diffractive lens appears in Figure 5.12b. A good spherical wave was obtained with very few higher order aberrations as indicated by an rms wavefront error of only 5.0889A,.

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