Isophase patches and speckles

Consider a distant point-like star. In the ideal case of no atmosphere, the electromagnetic signal would be a plane wave upon its arrival at the telescope (Fig. 8a). Its image in the focal plane would be an Airy disk of angular size ~X/d where d

(aj Plane wavetronl (b) Isoph&se patches

(aj Plane wavetronl (b) Isoph&se patches

Astronomy Celestial Meridian

—Incoming

'T i i x wavefroni

—Incoming

'T i i x wavefroni

Figure 5.8. Isophase patches arriving at lenses and mirrors. (a) Plane-parallel wavefronts yield a small diffraction-limited Airy disk characteristic of the entire lens diameter. (b) Turbulent cells in the atmosphere lead to a non-planar wavefront. The individual isophase patches independently image to large Airy disks, and the interference between the multiple wavefronts leads to speckles in the image plane. Each speckle is a mini-image of the source. (c) Improbable situation wherein phase delays at one instant of time form a tilted planar wavefront and thereby produce a single small speckle (image). (d) Deformable mirror that uses actuators to reshape the reflecting surface every ~1 ms. A non-planar wave becomes planar.

Figure 5.8. Isophase patches arriving at lenses and mirrors. (a) Plane-parallel wavefronts yield a small diffraction-limited Airy disk characteristic of the entire lens diameter. (b) Turbulent cells in the atmosphere lead to a non-planar wavefront. The individual isophase patches independently image to large Airy disks, and the interference between the multiple wavefronts leads to speckles in the image plane. Each speckle is a mini-image of the source. (c) Improbable situation wherein phase delays at one instant of time form a tilted planar wavefront and thereby produce a single small speckle (image). (d) Deformable mirror that uses actuators to reshape the reflecting surface every ~1 ms. A non-planar wave becomes planar.

is the diameter of the telescope mirror. As noted, a 2.4-m telescope would yield a disk of ~0.05". (We use angular units to describe the linear image size.)

In fact, the atmosphere has density enhancements (turbulent cells) of sizes d0 ~ 0.1m that delay portions of the wavefront en route to the telescope. These cells are carried rapidly across the telescope line of sight by high-altitude winds of speeds ~5 m/s. The result is that a distorted (non-planar) wavefront arrives at the telescope (Fig. 8b), and the shape of this varies rapidly with time. The figure shows the front at a given instant. Segments of the wavefront comparable to the ~0.1 m size of the turbulent cells will be nearly planar; they are called isophase patches. There would be about 24 such patches across the diameter of a 2.4-m telescope, and ~500 across the entire two-dimensional aperture.

Each of these isophase patches acts like a plane wave from the distant source and, in passing through the telescope, creates its own slightly offset image of the star. The offset angle will be <X/d0 (Fig. 8b); if it were larger it would not be an "isophase" patch. The overall image thus consists of many interfering plane waves arriving at the focal plane over distances comparable to the angle X/d0 — 1". This overall image size is much larger than the ideal of 0.05" for our 2.4-m telescope.

The overall image size may also be found as follows. Since an isophase patch is only 0.1 m in size, it uses only 0.1 m of the telescope aperture. The image would be the Airy disk expected for a 0.1-m telescope, namely X/d0 — 1", compare to (11). The overall image is thus a summation of 500 slightly displaced overlapping (and interfering) 1" Airy disks. This again leads to an overall image size of 1-2" at any given instant of time.

Inversely, the size of the isophase patches may be obtained from the detected image size (~1"). From the diffraction relation (6),

A00 — X/d0 — 1" (Overall image size) (5.12)

For X = 500 nm, this expression gives the isophase patch size we have already adopted, d0 — 0.1m (Size of isophase patch) (5.13)

The ~500 overlapping Airy disks lead to pronounced interference in the focal plane. This leads to bright and dim patches known as speckles across the 1-2" images, reminiscent of the light patterns seen on the bottom of a swimming pool. The size of the speckles may be obtained by noting that the wavefront of the leftmost isophase patch (dashed line) lags the wavefront of the rightmost patch by d/d0 wavelengths (Fig 8b),

Xd d A60 = d— = — X (Max. path length difference) (5.14)

d0 d0

where d is the diameter of the telescope lens or mirror. This leads to the possibility of as many as d/d0 interference maxima and minima along a given line in the image plane.

The 1" image of size k/d0 is thus broken up into d/d0 speckles (at most) along one dimension, each of which has equivalent angular width of

Ad0 k/d0 k

Telescopes Mastery

Telescopes Mastery

Through this ebook, you are going to learn what you will need to know all about the telescopes that can provide a fun and rewarding hobby for you and your family!

Get My Free Ebook


Post a comment