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▪ CONTENTS 1.1. Diffraction in a telescope  PSF ► 1. TELESCOPE IMAGE: RAY, WAVEFRONT and DIFFRACTIONIN THE NUTSHELL ● Telescope produces image by transforming plane waves into spherical  i.e. by transforming collimated into focusing beam
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Telescope image has: By making space objects brighter and larger, telescopes greatly expand our ability of their detection and observation. But how do they do it? What influences their performance, and how? This is the subject of this text. There are two main aspects to how telescopes form images of planets, stars and galaxies. One is concerned with the physics of image formation and the other with its geometry. We need the former to determine how light waves behave in the telescope, or optical system in general: where do they go, and how their interactions determine image brightness, contrast and resolution. The latter is an interface based on the simplified directional model of light propagation that provides convenient way of determining image location and magnification, as well as the initial assessment of the size of image aberrations. The former is physical optics, and the latter geometrical optics. The former is real, and the latter  well, more like a drawing. Any optical image  and those formed by telescopes are no exception  is made of light: a form of electromagnetic radiation. More precisely, a telescope image is made by imaging a countless number of lightemitting pointsources from faraway objects. As shown on FIG. 1, light waves emitted by a pointsource spread out in a concentric pattern, propagating as an oscillating energy field. It is convenient to present wave oscillation as a cycle, the full cycle being 360 degrees, or 2π radians. Phase of wave oscillation is, for harmonic sinusoidal wave, defined by o=Asin(2πx/λ), where A is the wave amplitude, defined as the maximum value of wave oscillation, x is the length of wave path from the origin, and λ the wavelength of light (FIG. 1, top left). An imaginary surface connecting wave points of identical oscillatory motion, or phase, is called phasefront. Geometrical approximation of the phasefront, based on the identical ray optical path length (OPL) from the source is called optical wavefront, or simply wavefront. For optical telescopes, phasefront and wavefront are, for all practical purposes, identical as long as the wavefront error remains relatively small. The difference between the two comes from the latter increasing directly with the nominal wavefront deviation, while the former follows the increase nominally, but effectively it oscillates from the maximum constructive interference for wavefront points deviating any whole number of waves  including, of course, zero deviation  decreasing to zero constructive interference from any wavefront point deviating by a whole number of halfwave deviations. Ray, on the other hand, is simply a straight line with the origin at the pointsource, that remains perpendicular to the wavefront. While rays are useful in presenting geometrical aspects of optical phenomena, they represent only a tiny fraction of the total energy propagating through the energy field. Furthermore, it is only their geometric properties that are being considered. Therefore, ray (or geometric) optics has no direct relation with the physical properties of the energy field.
FIGURE 1,
(TOP): Wavefronts emerging from a point source (left). Small section of a
greatly
expanded wavefront is practically flat (middle), which is how it
normally enters telescope aperture. Rays are the wave paths
perpendicular to the wavefront. However, energy does not travel in lines  it
spreads throughout energy field, but waves traveling in other directions generate phase
difference, preventing wavefront formation. Electromagnetic
wave propagates similarly to a mechanical wave moving along a loose cord,
in the transverse motion, or oscillation (o). The magnitude
of wave oscillation for any given point in space and time is a
product of wave amplitude (A) and its wave function; for
harmonic (sinusoidal) wave, this function is a sine of the angle
corresponding to the wave phase. The phase effectively varies
between 0 and 2π
radians, with 0 and 2π
corresponding to the beginning and the end of the full oscillation
cycle, respectively. Full phase is directly related to the
wavelength, which is the distance transversed by a wave between two
consecutive states of identical phase
Geometry of rays is superficial, but useful concept, not only for approximating image location and size, but for the initial assessment of its quality as well. 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. The wavefront, while itself geometric category, is more directly related to the underlying physics. It identifies the location of inphase wave sources, making it the basis for calculations determining the properties of wave interactions at and around focal point  i.e. diffraction calculation. Hence, the significance of the wavefront is in that its form directly determines quality of optical imaging in a telescope. Obviously, form of the wavefront and geometric properties of the rays are directly interrelated, but the ray geometry remains only loosely related to the interactions taking place within the energy field. The most striking example is that of a spherical wavefront, whose rays all meet in a single point. At the same time, the actual physical image formed by waves emerging from the wavefront is a bright spot surrounded by a series of fading rings. How is this taking place? Diffraction According to Huygens' principle, every wavefront point is a source of secondary wavelets, through which spreads in the direction of propagation. This constitutes a microstructure of energy field propagation, with the energy advancing in the direction of the wavefront, but also spreading out in other directions. 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. Similarly, 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, apertureshaped principal waves and diffracted waves 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 primary and diffracted waves inherent to the energy field (FIG. 1, middle and bottom). 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 RayleighSommerfeld, or (less complete) FresnelKirchhoff 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 rightangle 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+r2/2sr4/8s3+..., with the second term being referred to as defocus term and the third as (primary) spherical aberration term (they are not the same as the 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 (nearfield diffraction) than the latter (farfield
diffraction). For very small distances, Fresnel integral also
becomes inaccurate, and a complete integral like RayleighSommerfeld
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. Diffraction image In order to form the image of a pointsource, portion of the energy field emitted by it needs to be brought together, creating a patch of highest energy concentration  a pointsource 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 pointimage. 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 pointimage. However, this pointimage 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 3D pattern of energy distribution that sets the limit to image contrast and resolution. 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=ƒ/D, ƒ 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 pointsource image) in a telescope sets direct limit to its theoretical resolution and maximum useful magnification. Follows more detailed description of how diffraction shapes the pointsource image formed by a telescope.Diffraction in a telescopeOptically, any astronomical object is composed of a countless number of pointsources of light. The telescope forms object's image by imaging each and every of these point sources in its focal plane. The pointimage 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 pointsource 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 nearspherical 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 Fnumber (focal ratio, F=D/ƒ), while its angular size is inversely proportional to the aperture size (FIG. 2).
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=6.6256x1034 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 pointemitters 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 timeindependent 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 two 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 3.1x1015 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 lowcoherence 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 and continuously from a pointsource 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 nearmonochromatic 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. Wave interference, combined intensity With wave energy  or intensity  I defined as the wave amplitude A squared, i.e. I=A2 (general form), the combined intensity of two waves is their combined amplitude squared, or IC=(A1+A2)2. Hence, the maximum combined intensity for two separated wave sources of intensities I1 and I2 can be written as IC = I1 + I2 + 2A1A2 cos(ΔΦ)ɤ, with ɤ, the complex degree of coherence, ranging from 0 for incoherent to 1 for fully coherent light. The third term at right is so called interference term, which vanishes in incoherent light (ɤ=0) so that the combined intensity is simply a sum of the two intensities, ICi=I1+I2. With partly coherent light, value of the interference term varies with both, phase difference ΔΦ and ɤ, while for fully coherent light (ɤ=1) it varies with phase difference; for two waves in phase, i.e. ΔΦ=0 or ΔΦ=2π, cos(ΔΦ)=1 and the combined intensity is at its maximum, ICcmax=I1+I2+2A1A2. For A1=A2=A0, and A02=I0, the combined intensity is ICcmax=4I0, as opposed to ICi=2I0 with incoherent light and A1=A2=A0. Putting it in words, while coherent light in phase interferes in sustained manner, so that the amplitudes add first, and a square of their sum gives the combined intensity, with incoherent light the fields are essentially independent and the combined intensity is given by a sum of individual intensities, i.e. of the individual amplitudes squared. In other words, coherent light is linear in complex amplitude (square of which gives intensity), whereas incoherent light is linear in intensity, given by a sum of the individual wave amplitudes squared. Thus, for nearequal amplitudes and nearzero OPD, the maximum combined intensity doubles with doubled flux 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 pointemitters 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. The combined amplitude of two interfering waves is given by AC=cos(OPDπ/λ), with (OPDπ/λ)=ΔΦ being the phase difference. 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 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 ∆Φ=2cos1√I, and the corresponding field angle sine is sinα=(cos1√I)λ/πS (cos1 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 ∆Φ=π/2 (α=30°), and that there is little of constructive interference taking place for phase difference ∆Φ>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 nonzero, 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 at illustration shows the change in oscillation (phase) along the image radius for four pairs of emitters of different interseparations (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. Diffraction pattern Similarly to the single pair of emitters, this complex superposition of waves onto the image of a pointsource 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 aberrationfree circular aperture is inversely proportional to the aperture diameter. Unlike the simple twowave 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.
Following page expands on the properties of the building block of any telescope image, the point image described by Point Spread Function, or PSF. 1.1. Diffraction in a telescope  PSF ►
