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▪ ** **CONTENTS
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1. Rays and waves
▐
1.1.(cont'd) Point spread function** **►
#
1.1.** DIFFRACTION**
According to the
Huygens' principle,
every wavefront point is a source of secondary spherical wavelets, which spread
out in the direction of propagation (FIG. 1,
top/middle). This constitutes a
micro-structure of energy field propagation, with the energy advancing in
the direction of the wavefront, but also spreading out in other directions (Fresnel's modification to this principle is that intensity of wavelets drops as the angle of deviation from the direction of principal rays increases).
Principal waves, or wavefronts, form in the direction determined by
extending straight lines from the point source. Waves moving in other directions generate phase difference,
preventing them from forming another effective
wavefront (**FIG**.** 1**,
top right). However, these diffracted waves do interfere with both,
principal waves and among themselves.
As a consequence of the existence of diffracted wave
energy, placing obstruction of some form in the light path will result
in the "emergence" of this energy in the space behind obstruction. But
the obstruction did not change anything in the way the light propagates
- it merely took out energy of the blocked out principal waves, with the
remaining diffracted field creating some form of intensity distribution
in the space behind obstruction - the diffraction pattern. Likewise, the geometrically unobstructed field in the proximity of obstruction will change due to the absence of contribution from the blocked out portion of the field. It is this second aspect that is relevant for image formation in a telescope.
Wave amplitude at point **P** is a sum of all wave amplitude contributions from the wavefront containing point **M**, which are determined by the ray path length from each of these points. In terms of phase difference at **P**, varying cyclically from 0 to 1, the wavefront can be seen as consisting of concentric zones - Fresnel zones - between the circles for which the phase difference is λ/2, i.e. wave interference produces zero amplitude. Inset top right illustrates the (diffraction) effect of placing obstruction in the light path.
By limiting energy field to an aperture,
the portion passing through it is separated from the rest of the field,
and its energy - this time consisting from both, aperture-shaped
principal waves and diffracted waves from within - will create a pattern of
energy distribution behind the aperture. Again, there is no actual
change in propagation for the light passing the aperture, including
those close to the edge of obstruction (light does not "bend around the
edge"); whatever the form of energy distribution behind the
aperture, it is caused by the interference of the primary and diffracted waves
inherent to the energy field (FIG. 1, middle and bottom). It
is due to the missing portion of the field - the one left out of the
aperture - that the field after passing it changes, with the diffracted
field having different spatial amplitude distribution than the
incident field.
It turns out, complex amplitude diffracted to infinity
is proportional to the Fourier transform
of the field (i.e. its complex amplitude) in the aperture. For the
circular aperture with even transmission, this transform takes the form
of the Amplitude Transfer
Function, the square of which is the
Point Spread Function (PSF) for circular aperture. The intensity
pattern created at infinity by the diffracted field can be brought to a
finite distance by an optical objective that converges diffracted field
to the image plane.
**
Fraunhofer and** **
Fresnel diffraction**
In calculating diffraction effect, the basis is the path length of
diffracted waves, which determines their phase at a point of
interference, thus also the complex amplitude and the wave energy at
this point. Fraunhofer and Fresnel diffraction are terms referring
to the approximations used for calculating path length of the
diffracted wave. They can be based on either (complete) Rayleigh-Sommerfeld,
or (less complete) Fresnel-Kirchhoff diffraction integral. The basic
concept is that of plane wave propagation in free space, illustrated
on **FIG.1** (middle). Since the path length (**PL**) of a diffracted wave
is the hypotenuses of a right-angle triangle whose two sides are the
height of the point of origin relative to the point in the plane of
interference (**h**) and the distance between the planes of
origin and interference (**s**), it is given by (h2+s2)1/2.
This can be expanded into a series PL=s+r2/2s-r4/8s3+...,
with the second term being referred to as *defocus term* and
the third as (primary) *spherical aberration term* (these are reffered to as "aberrations of the diffracted wave"; they are
not the conic surface aberrations, and the terminology is based on the
superficial form identities).
The simplest approximation of the complete diffraction integral -
the Fraunhofer integral - accounts only the first term, hence
applies only for very small angles (cos→1)
- i.e. large distances - for which the difference between PL and **
s** is negligible. Fresnel
integral also accounts for the second, defocus term. Thus, the
difference between Fresnel and Fraunhofer approximations is in the
distance between diffracting aperture and plane of observation to
which one can be applied; the former can be applied to much smaller
distances (*near-field diffraction*) than the latter (*far-field
diffraction*). For very small distances, Fresnel integral also
becomes inaccurate, and a complete integral like Rayleigh-Sommerfeld
is required. Such integral is also applicable to larger distances (Fresnel
and Fraunhofer regions), and Fresnel integral is applicable to the
Fraunhofer region, but more accurate integrals are more complex, and
only used when necessary.
The far field distance D2/λ
is somewhat arbitrary minimum distance at which the defocus term of
the diffracted wave is acceptably small. When pupil-to-plane-of-observation distance is s=D2/λ,
this defocus term, given as OPD by D2/8s,
is 1/8 wave in units of the wavelength, or
π/4
in units of full phase, with **D** being the diameter of a clear
circular pupil (formal condition for the far-field
integral is that the pupil-to-observation plane distance **s** is
significantly larger than πD2/4λ,
which implies that it requires significantly smaller than 1/8 wave
error for high accuracy, but the actual criterion is fairly
arbitrary). At this distance, the
difference between central intensities of a collimated beam (plane
wave propagation) and a
beam focused at that same distance drops to 5%, and the intensity
distribution of a collimated beam is closely approximated by that
for the focused beam. The larger distance, the more accurate Fraunhofer approximation for collimated beam.
Unlike the far-field (Fraunhofer) integral, the near-field (Fresnel)
integral includes defocus term for plane wave propagation, which
makes it applicable to much smaller distances. It is limited by the
next significant term of the diffracted wave, spherical
aberration term, given as OPD by D4/128s3.
For 1/8 p-v wave error at paraxial focus of the focused beam, it
translates into a minimum pupil-to-plane-of-observation distance s3=D4/16λ,
or smaller by a factor of (λ/4D)2/3
than the minimum far-field distance.
With focused beams, defocus term of the diffracted wave cancels out
for the plane of focus, making the far-field Fraunhofer integral
applicable. For defocused patterns, the defocus term re-emerges,
requiring near-field (Fresnel) integral.
**Diffraction image**
In* *order to form
the image of a
point-source, portion of the energy field emitted by it needs to be brought
together, creating a patch of highest energy concentration - a point-source
image. For this, the
primary waves need to meet in a single point in the same state of propagation -
or phase - which, in turn, requires that optical paths from all
wavefront points are identical. The more difference in optical paths,
the less efficient wave interference, resulting in deterioration and,
ultimately, disintegration of the point-image. Obviously, this ideal
wavefront shape for the purpose of optical imaging is a sphere, with
every point on it at identical separation from the center of
curvature. Waves from spherical wavefront
arriving at its center of curvature - or *focus* - all meet in phase, for
the maximally
efficient energy concentration into a point-image.
However, this point-image is not a point of light, but an extended
pattern of intensity distribution (Inset **A**).
The reason are residual wave interactions around the point of convergence,
the concept introduced by Fresnel. This effect is known as diffraction
of light, i.e. interference effect of the diffracted waves. As a result, light energy directed toward focal point is spread into a
cone of converging energy focusing into a 3-D pattern of energy
distribution that sets the limit to image contrast and
resolution.
**
**
A:
Energy converging from the spherical
circular wavefront **W** forms diffraction pattern - or
**Airy pattern**, in honor of
Sir George Airy, who defined it mathematically in 1834 - rather than
a point-like image. The reason is evident from the illustration:
only the wavelets arriving to the center of curvature **C** of
the wavefront - the focal point - have identical paths lengths (OPL)
- equal to the radius of curvature **R** of the wavefront - and
meet in phase, producing the point of maximum intensity. Wavelets
arriving at other points in the image plane have different path
lengths. Consequently, they meet more or less out of phase,
producing field points of generally lower intensity. The resulting
pattern of wave interference for clear, aberration-free circular
aperture consists of the bright central disc
surrounded by a number of rapidly fading concentric rings. This
intensity distribution, normalized to 1 at its peak, is described by
the* *
**Point Spread Function**
(PSF), whose characteristic form is illustrated below the visual of
the diffraction pattern above, and shown in its exact form (for the
first three bright rings) in more detail below (green). The actual
intensity - or *illuminance* (i.e. incident light energy) -
distribution is a product of PSF and the energy in the pupil.
Similarly to the single pair of emitters, this complex superposition of
waves onto the image of a point-source forms a series of subsequent
minimas and maximas, which in a circular aperture appears as a pattern
of concentric bright rings of rapidly descending intensity. Since the
spatial limitation of the aperture effectively creates obstruction to
propagation of waves outside of it, the waves entering the aperture
become diffracted waves, and their interference pattern becomes
diffraction pattern.
In another analogy to a single pair of emitters, where the angular
separation between subsequent minimas and maximas is in inverse
proportion to their separation, angular size of the pattern in
aberration-free circular aperture is inversely proportional to the
aperture diameter. Unlike the simple two-wave interference, as
mentioned, complex superposition of waves in a telescope results in the
constructive interference rapidly diminishing
with the increase in pattern's angular radius, the
consequence of most points in the pupil being at relatively wide
separations.
Physical size
of the diffraction pattern in the plane of best focus is inversely proportional to
the telescope's relative aperture
1/F, with the first minima (Airy disc) radius given by rAD=1.22λF, **λ** being the wavelength of light, and **F** the focal
ratio F=*f*/D, *f*
and **D** being the telescope focal length and diameter,
respectively.
However, the effect on image quality
is not directly related to the physical size of the pattern, rather to
its *angular size*, as subtended on the sky, and the object itself,
through the pupil (aperture) center.
Its radius is given by αAD=1.22λ/D,
in radians. Angular size of diffraction pattern (i.e. of the
point-source image) in a telescope sets direct limit to its theoretical
resolution and maximum useful magnification.
Follows more detailed description of how diffraction shapes
the point-source image formed by a telescope.
#
Diffraction in a
telescope
Optically, any astronomical object is composed of a countless number of
point-sources of light. The telescope forms object's image by imaging
each and every of these point sources in its focal plane. The
point-image itself is created by wave interference around focal point,
due to the
phenomenon known as *diffraction of light*. It is often thought of
as being caused by an obstruction placed in the light path. In fact,
diffracted energy is inherent to the propagation of energy fields, and
the presence of obstructions in their path merely changes the field
properties by excluding a portion of it. This produces various
interference - or diffraction - effects in the image space.
In the case of a telescope, the obstruction in the light path is
effectively in the plane surrounding the aperture. Most reflecting
systems also generate an additional diffraction effect from the
obstruction by a smaller, secondary mirror.
Diffraction image of a point-source in a telescope is a bright central
disc surrounded by rapidly fainting concentric rings. As already stated, this pattern is
created by the interference of light waves. Constructive
interference is at its peak in the center of the pattern, which is the
center of curvature of near-spherical wavefront formed by telescope's
objective. Farther away from the center point, constructive interference
quickly subsides, resulting in the first bright ring much fainter than
the disc, and every successive bright ring much fainter than the
preceding ring. Size of diffraction pattern in a telescope is
proportional to the wavelength **λ**; given wavelength, its physical
size is proportional to telescope's F-number (focal ratio, F=D/*f*), while its angular size is
inversely proportional to the aperture size (**FIG. 2**).
**FIGURE 2**: Diffraction pattern formed as the
image of a point object is a 3-D concentration of energy. In the plane
perpendicular to the axis, at focus, it appears as a bright central disc
surrounded by rapidly fainting rings. Along the axis, the pattern
extends on either side of the focus. Being created by wave interference,
it directly depends on the optical path difference and corresponding
phase difference between the waves from different portions of the
aperture. The greater beam convergence - which is proportional to the
F-number - the faster this phase difference accumulates, and the smaller
resulting linear pattern. Given aperture size, it changes with the telescope's
focal ratio **F**=f/D, i.e. focal length. Angular radius of the
diffraction pattern, given by dividing its linear size with the focal
length (**f**), is inversely proportional to the aperture diameter **
D**; it is constant for given aperture and wavelength. Given F-number, physical size of diffraction pattern is
constant, but since focal length changes in proportion to the
aperture, its angular size changes in the inverse proportion to it. Popular conception of diffraction being caused by light
"bending" around the edges of telescope aperture is somewhat misleading.
It is not the presence of the aperture edge itself, rather edge-to-edge
separation that determines how wide will be angular spread of
light due to diffraction. Central intensity of the diffraction
pattern is the tip of the bell-shaped intensity curve described by its
function (PSF), in units of the flux per unit area (the plot is rotated 90°
counterclockwise about horizontal axis orthogonal to the optical axis,
in order to coincide with the intensity distribution pattern above). For
given wavelength, peak intensity is proportional to (D/F)2,
which means that it changes in proportion to the square of aperture
diameter, and in inverse proportion to the square of focal ratio; doubling
the focal ratio at given aperture results in a fourfold decrease in the peak
physical intensity of the diffraction pattern (point, which has no
physical reality, is not a useful concept for calculating energy
deposited by the light waves; since image radius encircling any given
portion of the energy doubles with the focal ratio, the corresponding
intensity drops fourfold).
The basics of diffraction can be illustrated with interference of
light emitted by an arbitrary pair of points on the wavefront formed by
a telescope objective.
Energy unit of an actual wave is *photon*
- quanta of energy defined by the product of wave frequency (number of
wave cycles per unit of time) and Plank's constant, h=6.6256x10-34
in joules (J). In the following text, wave interference and resulting
energy are described in terms of normalized unit amplitude **
A** - with *wave amplitude* defined as the maximum value of its
oscillation - and resulting intensity I=A2
(electromagnetic wave oscillates in two
perpendicular planes, with the field energy proportional to a product of
their equal amplitudes) of the light wave.
**Optical path difference** (**OPD**)
for any pair of emitters on the wavefront in the pupil of a telescope is
closely approximated by:
OPD = Ssinα**
**
(a)
with **S**
being their linear separation in the pupil, and
**α** the angular radius of a
point in the image space (Inset A). The angle
**α** at which wave
interference becomes destructive is directly related to the linear point
separation (**S**, Inset A), which defines
optical path difference, as given above, and the resulting **
phase difference** in radians as
**ΔΦ**
= 2πOPD/λ
= 2πSsinα/λ
(a')
The angular image radius **α**
at which any given phase difference will be generated is, therefore,
dependant on the point separation **S**. Taking, for instance, phase
difference ∆Φ=π
(which, with the full phase spanning the wavelength, or 2π=λ,
corresponds to λ/2 OPD),
for a pair of wavefront point-emitters separated by S=λ/2 in telescope
pupil, gives the corresponding angular image radius (i.e. radius of the
first minima), rather obvious, as
α=90° (from sinα=OPD/S=∆Φλ/2πS=1).
Given OPD, the efficacy of wave interference depends on their degree of
**coherence**. Two waves are coherent if
their phase differential is constant. Strictly talking, light
is coherent if monochromatic, originating from a point, and has a
constant rate of emission; this
ensures that the energy field has perfectly uniform time-independent
propagation pattern. Such wave is spatially and temporally coherent. As the spatial extension of light source increases,
at some point the separation between individual emitters becomes large
enough to cause a significant phase differential between their fields at
some distant point, resulting in some degree of *spatial incoherence*.
Also, different points radiate independently and the waves they emits become
less coherent, with their
*coherence time*, or *temporal coherence* - defined as the
time interval **t** within which the field has nearly identical phase
continuum - diminishing. So instead of having long trains of nearly
uniformed field oscillation pattern, light consists of many smaller wave
trains with varying phase properties. Spatial period corresponding to
the coherence time, *spatial coherence* or *coherence length* **
l** is
l
= ct, **c** being the speed of light.
Also, as the frequency range of
light **Δν** in Hertz increases, its
temporal coherence diminishes as t~1/Δν. For
white light, with the frequency range of about 320 trillion Hz (with
frequency given as ν=c/λ), temporal
coherence - assuming near uniform intensity over the range - is about
3.1x10-15 seconds, with the corresponding coherence length
l~0.00094mm.
This incredibly fast pace of variation in the configuration of wave
trains contained within the continuum of temporal/spatial coherence
intervals results in suppression of the fringe pattern, as a consequence
of wave interference in low-coherence light (Inset
H, would have been between the top two patterns, with Δλ~0.55λm).
In a different context, polychromatic light with all the
wavelengths emitted simultaneously and continuously from a point-source
is temporally coherent in vacuum only within narrow wavelength range, because
specific phase at any given point in time varies with the wavelengths. But it is spatially
coherent, because all waves come from the same point.
The degree of light coherence for near-monochromatic
light is expressed by its complex degree of coherence value **ɤ**, ranging from 1 (coherence
limit, or complete coherence) to zero (incoherence limit, or complete
incoherence), with the intermediate values that significantly differ
from 1 or 0 indicating partial coherence.
Following page expands on the properties of the building block of any
telescope image, the diffraction image of a point source of light, described by Point Spread Function, or
PSF.
◄
1. Rays and waves
▐
1.1.(cont'd) Point spread function** **►
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