1.1. DIFFRACTION IN A TELESCOPE

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).

FIGURE 2: Angular radius of diffraction pattern (α, αAD for the Airy disc) is inversely proportional to the aperture diameter D; it is constant for given aperture. However, linear size of diffraction pattern (ri, rAD for the Airy disc) changes in proportion to the telescope focal length ƒ; at given aperture size, it corresponds to the change in telescope's F-number, F=ƒ/D. Vice versa, for given F-number, physical size of diffraction pattern is constant, but since the focal length changes in proportion with the aperture, its angular size changes in the inverse proportion to it. The reason for the angular size of diffraction pattern being inversely proportional to the aperture diameter is less obvious; it is due to the efficiency of constructive wave interference at a circular aperture being angularly dependant (inversely proportional) on the width of telescope pupil. Popular conception of diffraction being caused by light "bending" around the edges of telescope aperture is somewhat misleading. It is not a presence of the aperture edge itself, rather edge-to-edge separation that determines how wide will be angular spread of light due to diffraction.

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 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; this is not strictly true for the exaggerated geometry shown, but is valid approximation for both, widely separated points on an actual wavefront, due to very small angular span of the field angle α encompassing diffraction pattern, and for close points due to their small linear separation S). 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 DΦ=2πOPD/λ.

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, 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, 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 given 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) 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/1.25 (α>53°). Within those same coordinates, the plot for S=100mm separation would be practically a straight vertical line.

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.

A: Energy converging from the spherical 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 is described by the Point Spread Function (PSF), whose characteristic form is illustrated bellow the pattern. Intensity at any point of the pattern for near-monochromatic point-source[1], normalized to 1 for peak intensity at the center is given by:

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]

Intensity distribution within the 1st PSF maxima (the bell-shaped central portion) is well approximated for r~1 and smaller by a Gaussian function of form I~2-P, with P=(x2+y2)/r'2, where x and y are the point coordinates in horizontal plane (zero at the center), and r' the FWHM radius, both in units of λF. Substituting for FWHM and setting y to zero, with x effectively becoming r as defined with Eq. (c), gives the exponent for the 2-D Gaussian central maxima approximation as P=3.8r2. For r>1 , Gaussian approximation asymptotically approaches horizontal axis, without any hint of the ring structure.

Eq. (c) places 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 (adapted from "Optical Imaging and Aberrations 2", Mahajan) show PSF intensity distribution (I, normalized to 1 for near-monochromatic light) for aberration-free circular aperture, as a function of point radius r in units of λF (for object radiance constant over the range of radiation). The ring structure is most pronounced for near-monochromatic light (spectral range Δλ is 0.0001 of the mean wavelength λm), gradually vanishing as the range widens to Δλ~λm. Most astronomical objects emit in relatively wide spectral ranges, but narrowing toward their peak intensity. Thus, a typical pattern is intermediate between those for near-monochromatic and those for widely polychromatic light with the radiance nearly constant over the entire range; it is roughly similar to the pattern plotted for Δλ=0.1λm.

While Eq. (c) implies that the linear, or transverse size of diffraction pattern changes in proportion to the telescope's F-number, angular size of the pattern changes in inverse proportion to the aperture diameter. As a function of the angular pattern size, equation for the normalized PSF intensity can be written as:

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 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 1st-order Bessel function of πr squared, so EE(r = 1 - [J0(πr)]2 - [J1(πr)]2. With I(r) being practically zero at every minima for near-monochromatic light, as long as well defined pattern of intensity change exists, the encircled energy within any given minima radius reduces to 1-[J0(πr)]2, and the remaining energy outside that radius to [J0(πr)]2. As the diffraction plots to the right indicate, energy spread caused by diffraction extends far beyond the first few bright rings. According to Schroeder's EE approximation for larger r values (quite accurate already for r>2), EE~1-(2/rπ2), about 2% of the total energy is still contained beyond the 10th dark ring. In amateur telescopes, however, this thinly stretched energy is below the threshold of detection in both, visual observing and astrophotography.

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.

Note that the in focus diffraction pattern is usually calculated using Fraunhofer's far-field approximation (for the pattern far enough from the source to assume planar wavefronts at the size scale of the diffraction pattern). The defocused pattern requires applying Fresnel's more accurate, near-field approximation (which can also be used to calculate the far-field in focus pattern).

(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 in 550nm wavelength), 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 the boundary waves become incoherent when the angular radius of the object equals that of the Airy disc radius - assuming, for simplicity, circular object - the corresponding aperture (so called "coherence radius") is given by Dc=251,643λ/A, where A is the object's angular diameter in arc seconds, with D 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 when the range width starts approaching 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 p=0), but it approaches 1 as r (thus also p) approaches 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).

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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