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▪ CONTENTS ◄ 1. TELESCOPE IMAGE ▐ 1.2. Reflection and refraction ► 1.1. DIFFRACTION IN A TELESCOPEOptically, 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, a phenomenon known as diffraction of light. Diffraction image of a point-source in a telescope is a bright central disc surrounded by rapidly fainting concentric rings. What causes the appearance of this pattern is 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, while its angular size is inversely proportional to the aperture size (FIG. 2).
The basics of this phenomenon 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 DΦ=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=DΦλ/2πS=1). Given OPD, the efficacy of wave interference depends on their degree of coherence. Strictly talking, light wave is coherent if monochromatic, and originating from a point; this ensures that the energy field has perfectly uniform time-independent propagation pattern. As the spatial extension of light source increases, 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 E, would have been between the top two patterns, with Δλ~0.55λm). In a different context, polychromatic light with all the wavelengths emitted simultaneously from a point-source can be called temporally coherent in vacuum, because different wavelengths have identical phase at any given point in time. But it is not spatially coherent, because their phases in the plane transverse to propagation at any given distance from the source do not coincide - the more polychromatic light, the more so. 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. The maximum combined intensity for two separated wave sources of intensities I1 and I2 can be written as IC=I1+I2+2(I1I2)0.5cos(OPDπ/λ)ɤ. Thus, for near-equal intensities and near-zero OPD, the maximum combined intensity doubles for incoherent, and quadruples for coherent light. With the maximum combined intensity for incoherent light normalized to 1, the combined intensity of two point sources in the pupil, as a function of OPD, in units of wavelength, is given by:
which for OPD=λ/2 gives I=cos2(π/2)=0. The wider emitter separation, the smaller field angle at which λ/2 OPD is generated and the combined intensity drops to zero. For two wavefront point-emitters at the opposite ends of 100mm diameter pupil (S=182,000λ for λ=0.00055mm), this mutual cancellation will take place at the field angle α=0.57 arc seconds. Since the intensity for incoherent light is the amplitude squared, the combined amplitude of two interfering waves is given by AC=cos(OPDπ/λ). It can be presented as the resultant vector of two unit phase vectors, as illustrated on Inset C. Since the OPD between waves w1 and w2 is λ/4, their combined amplitude is given by AC=cos(π/4)=√0.5; the resulting combined intensity is I1+2=AC2=0.5, as obtained by applying Eq. (b) directly. Likewise, for waves w1 and w3, phase difference is 1.25π, giving OPD=1.25/2=0.625, with the resulting amplitude given by cos(0.625π)=-0.3827, and the combined intensity I1+3=cos2(0.625π)=0.1464. From Eq. (b), the phase differential in terms of combined normalized intensity is given by DΦ=2cos-1√I, and the corresponding field angle sine is sinα=(cos-1√I)λ/πS (cos-1 is the inverse cosine function, i.e. the angle corresponding to a given cosine value). The intensity plot for λ/2 point separation (Inset D, left) shows that the normalized intensity I of two combined waves drops to 0.5 at the phase difference of DΦ=π/2 (α=30°), and that there is little of constructive interference taking place for phase difference DΦ>π/1.31 (α>60°). Intensity drops to zero for α=90° and α=270°, since the two emitters are effectively located on the horizontal axis, centered around zero point and separated by S=λ/2. A plot showing dependence of combined intensity on the angular radius α in the image plane retains this form as long as the S/(λ/2) quotient is a whole number, but the angular radius within which most of constructive interference takes place diminishes. As the point separation increases, the central lobe becomes smaller angularly; within those same coordinates, the plot for S=1000λ separation - which is still only 0.55mm for λ=0.00055mm - would be practically a straight vertical line, but with a number of radially distributed subsidiary maximas whenever the net OPD difference reaches a whole number of waves (for S giving integer when divided with other values than λ/2 the combined intensity at 90° and 270° is non-zero, and it forms full maxima when S consists of a whole number of waves). Of course, energy generated at every point of the image is a sum of wave contributions not only from a pair of wave emitters, but from all wave emitters combined. The complexity of wave interactions is beyond visualization; an attempt of illustration shows the change in oscillation (phase) along the image radius for four pairs of emitters of different inter-separations (Inset A, bottom). A slightly separated pair (1) gives maximum contribution to all field points within the radius shown. Contribution of somewhat less closely positioned pair (2) decreases more rapidly with the increase in field radius, and much more rapidly for more widely separated (3) and the widest (4) pair. As can be seen on the plot for the pair 4, contribution from every pair varies periodically between the maximum and zero, as a function of the field angle (radius). At the field point A, the two more widely separated pairs' phase contribution is zero, but the combined contribution of narrowly separated pairs of emitters (1 and 2) is still close to a maximum; at double the field radius, the two more widely separated pairs' contribution will be at the maximum, but it will be lower for the pair 2. Similarly to a 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. 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.
with t=πr/2
(to simplify the relation), r being the point radius in the
image plane, in units of
λF. The relation is derived from the general PSF relation
for circular aperture
I(r)=[2J1(πr)/πr]2,
where J1(πr)=[1-(t2/1!2!)+(t4/2!3!)-(t6/3!4!)+...]t
is the first order Bessel function (of
the first kind) of
πr, thus 2J1(πr)/πr=2[1-(t2/1!2!)+...]t/πr=[1-(t2/1!2!)+...].[2]
Alternately, point radius r in the image plane can be expressed in terms of radians of phase difference corresponding to it, as (2π/λ)Dsinα/2, where the consecutive PSF minimas fall at 3.83, 5.14, 7.02... With the full phase being 2π radians, the appropriate values in units of phase are 0.61, 0.818, 1.117... (2π/λ, usually denoted by k, is the propagation number of a progressive harmonic wave, corresponding to its single full phase). The logarithmic PSF above more closely resembles visual appearance of diffraction pattern, due to the logarithmic intensity response of the eye. However, since logarithm (exponent) to the base 10 is different than the eye response logarithm base (1000.2~2.512), it compresses nominal differences more, as illustrated on FIG. 9B. Diffraction pattern of a point source, as it appears to the eye is better represented with the plot to the left ("apparent PSF"), modified according to eye's intensity response. It should approximate well appearance of the pattern when both, central maxima and 1st bright ring are well within eye's detection threshold, and the pattern intensity is not too high. As the pattern intensity lowers, the ring appears increasingly fainter than central maxima, and eventually disappears. On the other hand, as pattern's intensity increases, the ring becomes nearly as bright as central disc, probably due to saturation of retinal photoreceptors.
While Eq. (c) implies that the linear, or
transverse size of diffraction pattern changes in proportion to the
telescope's
with a=πDsinα/2, in units of λ, where D is the aperture diameter and α the angular point height in the image plane. Obviously, a is numerically identical to t, so for the first minima (Airy disc radius), the corresponding function value is 2a=πDsinα/λ=1.22π, with sinα=1.22λ/D (which is for very small angles identical to the angle in radians), changing in inverse proportion with the aperture diameter. Or, quite simply, α=rλF/ƒ=rλ/D in radians. For D=100mm and λ=0.00055mm, angular radius of the first minima (r=1.22) is α=1.22∙0.00055/100=0.00000671 in radians or, multiplied by 206,265 (for 180/π degrees in 1 radian x 60 arc minutes in 1 degree x 60 arc seconds in 1 arc minute), 1.384 arc seconds. The diameter of the first PSF minima, given by 2.44λF linearly and 2.44λ/D angularly (in radians) - λ being the wavelength of light, and F the ratio of focal length vs. aperture D of the optical system - is called Airy disc diameter. With the normalized (to 1) encircled energy (EE) within pattern radius r in units of λF given by
(sum in the brackets being zero-order
Bessel function, J0(πr),
and t=πr/2,
as before), it encircles 0.838 of the total energy contained in the
diffraction pattern. Note that
I(r)t2
= [J1(πr)]2,
or the As the PSF and EE plots illustrate, the value of normalized encircled energy within 1st maxima is nearly inverse to the normalized intensity for r<0.8. Thus the encircled energy within 1st maxima, as a relative fraction of the total energy encircled within it, is well approximated as EE~1-I for r smaller than ~0.8. Taking Gaussian approximation of I, which is also good for r<0.8 gives EE~1-2-P, with P=3.77r2.
Following table gives numerical presentation of
the intensity distribution up to 10th dark ring within diffraction
pattern in near-monochromatic light by aberration-free unobstructed
aperture, for linear radius r in units of λF, normalized
intensity (I) and encircled energy (EE)
(source: Optical Imaging and Aberrations 2, Mahajan).
Conventionally, limit to diffraction resolution of two point-object images is set at ~λ/D, nearly identical to the full width at half-maximum (FWHM) of the PSF, 1.03λ/D radians in diameter. Wave interference doesn't only occur radially; wavelets meeting before and after the focal point also interfere, extending the pattern of intensity longitudinally, generally decreasing as the interference takes place farther from the focal point. As a result, diffraction pattern is a 3-dimensional phenomenon. While relative intensities of the central disc vs. rings remain constant, as given above, visual appearance of diffraction pattern - the visibility of its segments, as well as their apparent size and relative apparent brightness - varies with the brightness of the point image.
The above refers to diffraction at a circular
aperture. Aperture forms other than circular will produce different
diffraction patterns. Two examples are square and triangular
aperture, with their PSF, MTF and diffraction patterns (A,
bottom) shown bellow in comparison with the limiting circular
aperture. With the intensity distribution of rectangular aperture
for central intensity normalized to 1 given by Ir=(sinc2α)(sinc2β),
with sincα=sinα/α, α=πax/ƒ
and β=πby/ƒ,
for x and y the Cartesian coordinates in the image
plane with the origin (zero) at the diffraction peak, a and
b the lengths of the two sides of a rectangle, and ƒ
the focal length, normalized intensity distribution for square
aperture is obtained by setting a=b. First minima, along either x
or y axis, is at
λƒ/a, with every next
minima falling at this exact separation (the minimas follow straight
lines running parallel with one or the other axis). The sinc value
is undefined for zero angle, but since it approaches 1 as the angle
approaches 0, 1 is its limit value. Fraunhofer and Fresnel diffraction These terms refer to the two different calculation modes of diffraction effect. Both are approximations that can be based on either complete approach of Rayleigh-Sommerfeld, or more accurate Fresnel-Kirchhoff approximation. The basic concept is that of plane wave propagation in free space, where insertion of a planar obstruction to wave propagation interrupts the pattern of propagation, with the obstructed waves excluded, and those unobstructed and relatively close to the edge of obstruction filling the space behind obstruction. It can be illustrated with Huygens' wavelets, emitted by every point on the wavefront, mainly into the space ahead of wave. However, there is no actual change in propagation for unobstructed points of the wavefront, including those close to the edge of obstruction (points 2 and 3, inset I); whatever diffraction effect results from the presence of obstruction is caused by the absence of energy from the blocked out portion of the wavefront (such as the wavelet from point 1).
(1)Actual
astronomical point-sources - stars - are spatially extended, with the
wave emission phase varying from one point to another, and also in time.
Thus, strictly talking, they emit incoherent
light. However, according to Van Cittert-Zernike Theorem -
stating that the degree of spatial coherence of near-monochromatic light
equals the normalized (to 1) PSF of the source - the degree of coherence
between two wave trains from a star will diminish from 1 to 0 as their
angular separation in the image plane increases from zero to 1.22λ/S,
where S is their separation in the aperture. In other words, the
complex degree of coherence
ɤ
for star of angular size A in radians is 1 for two points in
the aperture nearly coinciding, diminishing to zero as their linear
separation increases to 1.22λ/A.
Hence even the largest star angularly, Betelgeuse (slightly less than
0.045 arc seconds in diameter @550nm), would have its
ɤ
value change only from 1 to over 0.99 for two rays from the opposite
ends of S=D=200mm aperture (1.38 arc seconds Airy disc diameter). On the
other hand, Jupiter with its average 40 arc seconds angular diameter
would have
ɤ
falling from 1 to 0 already at nearly 1/60 of its radius. Since
boundary waves become incoherent when angular radius of the object
equals that of the Airy disc - assuming, for simplicity, circular object
- the corresponding aperture (so called "coherence radius") is given by
Dc=1.22λ/A
for the object's angular diameter A in radians, or Dc=251,643λ/A,
for A in arc seconds, with Dc
being in the same units as λ (when two slits are placed at
separation equaling
D, the fringes in their interference pattern disappear, which is
how Michelson had determined first angular diameter of a star -
Betelgeuse, from
Dc=3070mm
for λ=0.00057mm). At this point, the average coherence, as the averaged
ɤ
value, equals intensity averaged over the Airy disc (as the flux
divided by the area, normalized to the central intensity), which is 0.23
for perfect aperture. Hence, it is partly coherent, low-coherence light.
Note that
Van Cittert-Zernike Theorem strictly
applies to near-monochromatic light, so the fact that stars are
polychromatic sources also needs to be taken into account. As
INSET E shows, widening of the spectral range lowers light
coherency, until it becomes incoherent as the range approaches the mean
wavelength.(2)Value in the brackets, the normalized point intensity, or irradiance, when 1, corresponds to a total field power in the exit pupil, given by a sum of the squared individual wave amplitudes (the amplitude is squared to obtain intensity because an electromagnetic wave has dual identical amplitude, one of its electric field, and the other of its magnetic field). The Bessel function, which can be used to express the sum of phase (i.e. amplitude as sine/cosine function) contributions from the points on the wavefront in this form, replaces much more involved direct integration. Note that the PSF expression with Bessel function is not defined for r=0 (thus t=0), but it approaches 1 as r and t approach zero. Obviously, diffraction of light in a perfect aperture is merely the limiting case of light interference from a circular area of wave emitters (which can be assumed as filling the telescope aperture), with wavefront aberrations and pupil obstructions approaching zero. Presence of wavefront aberrations, or pupil obstruction, alters wave interference and thus the intensity distribution and visual appearance of diffraction pattern as well. Size of diffraction pattern defines a point-source as an object - be it as small as an atom, or as enormously large as a star - whose image in a telescope is smaller than the central diffraction disc. More specifically, according to the optical theory, a point-source image has to be less than 1/4 of the Airy disc in diameter; larger image enlarges the central disc, and alters energy distribution in the area of rings (at the image size of ~0.25 Airy disc diameter, the FWHM is enlarged ~2%, at twice that size it is about 8% larger, and with the image equaling the Airy disc in diameter the FWHM is nearly doubled, and the ring structure greatly supressed).
Physical optics calculates the specifics of wave interactions resulting in the image formation using point spread function (PSF), whose two characteristic forms - actual diffraction pattern and the graph - are illustrated in Inset A. Conventionally accepted limit to point-image resolution set by diffraction equals the full width at half-maximum (FWHM) of the diffraction PSF, given angularly by ~λ/D in radians (or 180λ/Dπ in degrees), D being the aperture diameter, and λ the wavelength of light.
Geometric optics, on the other hand,
limits its scope to the geometry of rays, rays being, as mentioned, wave
paths - or simply straight lines - orthogonal to the wavefront. Since
any wavefront deformation results in disturbance of rays, thus
scattering the rays around the center point of a perfect reference
sphere, it indicates whether an optical system is perfect, or not. To
some extent, ray disturbance indicates the severity of wavefront error,
which makes it a convenient tool for the initial assessment of
wavefront/image quality. Also, it is useful for determining geometric
relations between optical elements and images they form. However, for
the specifics about actual energy distribution around the focal point we
need physical optics.
◄ 1. TELESCOPE IMAGE ▐ 1.2. Reflection and refraction ►
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Eq.
(c) places the first diffraction minima at r=1.22 (slightly
rounded off, from 1.219668), second at r=2.23, third at 3.24,
and so on. Plots to the left ("Optical Imaging and Aberrations
2", Mahajan) show PSF intensity distribution (I, normalized
to 1 for near-monochromatic light) on logarithmic (log
Main
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). The far field distance D