4.4. Defocus

Formally, defocus wavefront aberration results from the image formed by a telescope objective being observed not at the location of the Gaussian image point, but at a point longitudinally displaced from it. In more practical sense, defocus error occurs when point of observation is displaced from best, or diffraction focus, a point with peak diffraction intensity embedded into longitudinal extension of the central maxima. Longitudinal shift away from this point, to either side, results in gradual decrease of central intensity, with the energy spreading from the central maxima out to the rings area.

In its pure form, defocus occurs in aberration-free aperture, in which case the longitudinal defocus error equals longitudinal displacement from Gaussian image point, and the corresponding P-V wavefront error is given by the sagitta difference between two reference spheres in the entrance pupil: one centered at Gaussian focus (F on the illustration bellow), and the other at the point of observation (Fd). Hence, longitudinal defocus is given by the difference between the two radii, Ld=R0-Rd, where R0 being the pupil-to-image distance (equaling focal length for object at infinity) and Rd the distance from pupil to the point of defocus. The corresponding P-V wavefront error for full pupil diameter is:

Wd = 0.125[(1/R0)-(1/Rd)]D2 (24)

with D being the aperture diameter.

In such instances when this is not the case - such as the presence of spherical aberration, or astigmatism - certain amount of defocus from the Gaussian image point is necessary to shift to the actual best focus location. In other words, defocus can be balanced with another aberration in order to have the combined aberration minimized (for instance, best focus in the presence of spherical aberration is at the point halfway between paraxial and marginal focus, thus combines spherical aberration and defocus).

In everyday's jargon, "defocus" has somewhat different meaning: it is simply an axial deviation from best focus location, correctable by mere refocusing.

Back to best focus being coinciding with the Gaussian, defocus P-V wavefront error is measured as the optical path difference between a perfect reference sphere for the point other than the Gaussian, and a perfect reference sphere for Gaussian image point. It is expressed as:

Wd = Pr2 (25)

with r being the in-pupil ray height in units of pupil radius, and P the peak aberration coefficient for defocus (equal to the P-V wavefront error) given by P=0.125[(1/R0)-(1/Rd)]D2. It changes with the square of pupil height.

For very small relative difference between the two reference radii, Eq. 24 can be written as

Wd ~ (R0-Rd)D2/8R02 ~ LdD2/8R02 (25.1).

Since this approximation differs from the exact value by a factor of Rd/R0, it is as good as the exact value for all practical purposes (for instance, the relative difference between the true value and approximation for 1mm defocus in a 200mm ƒ/10 system is 0.9995, and still only 0.9975 in a 80mm ƒ/5 system).

With R0 equaling the system focal length ƒ, (D/R0)2=1/F2, where F is the system focal ratio ƒ/D, and the peak aberration coefficient for defocus is:

P = -Ld/8F2, (25.2)

Ld being the longitudinal defocus (effectively, the difference in radii between the two reference spheres). It is radially symmetrical, with the blur diameter given simply by Ld/F. In units of the Airy disc diameter (2.44lF) the geometric blur diameter for defocus is:

Bd = 8P/2.44l = -Ld/2.44lF2 (26)

or Bd=745Ld/F2 for 550nm (0.00055mm) wavelength. Since the peak aberration coefficient P equals the P-V wavefront error Wd for ρ=1, the blur size in Airy disc diameters is, from Eq. 26, also given by Bd=8W'd/2.44, with W'd being the P-V wavefront error of defocus in units of the wavelength.

The peak defocus aberration coefficient and the RMS wavefront error relate as ω=P/√12. This implies that the P-V wavefront error of defocus corresponding to "diffraction-limited" level, (i.e. to ω=1/√180, for 0.80 Strehl) is P=√12/√180=1/√15=0.258, in units of wavelength.

The RMS wavefront error relates to blur diameter Bd as ω=2.44Bd/√768.

EXAMPLE: Thus, an ƒ/10 system defocused by Ld=0.1mm has the P-V wavefront error of defocus given by Wd=P=0.1/800=0.000125 or 1/4.4 wave, with the corresponding geometric blur diameter of Bd=74.5/100=0.745 Airy disc diameters, both for 550nm wavelength. This same linear defocus in an ƒ/5 system will produce Wd=P=0.1/200=0.0005 or 1/1.1 wave P-V wavefront error, with the corresponding geometric blur diameter of Bd=74.5/25=3 Airy disc diameters.

An important property of a telescope is its defocus sensitivity. As Eq. 25 implies, it is proportional to the longitudinal defocus Ld, and inversely proportional to F2. Taking as the maximum allowable P-V error of defocus P=xλ in either direction from the exact focus, the longitudinal range (including both sides of defocus) within which defocus error will not exceed this level is given by Ld=16xλF2. It is sometimes called focus depth (not to be confused with field depth, a photographic term referring to the range of distance in the object space within which the object image retains needed quality level; given by Δ=±8W(o-ƒ)2/[D2-8(o-ƒ)W], where o is the focused distance, ƒ and D the system focal length and aperture diameter, respectively, and W the corresponding PV wavefront error of defocus, it is irrelevant in astronomy, where all objects are very distant).

Taking x=0.258 for 1/3.87 wave of defocus (0.0745 wave RMS) allowable for the conventional "diffraction-limited" level of 0.80 Strehl, gives the corresponding ± range of defocus as Ld=4.13λF2. For an ƒ/10 system and λ=0.00055mm, it gives 0.227mm.

This is, of course, an idealized system. In the presence of spherical aberration - rather common scenario - allowed defocus is significantly smaller. For a system with Ws=xλ P-V of lower-order spherical aberration - assuming x<0.25 for better than "diffraction-limited" level - with the corresponding longitudinal aberration LA=64WsF2, allowed ± defocus range for remaining at the conventional "diffraction-limited" level, or better, is closely approximated by 32λ[√x-2x]F2. Taking x=1/6, for λ/6 wave P-V of spherical aberration, and λ=0.00055mm, allowable defocus range in an ƒ/10 system (so F=10) is 0.133mm (note that "defocus range" in this context doesn't equal defocus error; at best focus, lower-order spherical aberration is already combined with longitudinal defocus equaling one half of the longitudinal spherical aberration).

For x=0.25, allowed defocus to remain within "diffraction-limited" range is zero.

In aberration-free system with central obstruction, defocus RMS error is smaller by a factor of (1-o2), o being the relative c. obstruction size in units of aperture diameter. If spherical aberration is present, its RMS wavefront error is reduced by a factor of (1-o2)2, and its defocus error changes in a similar manner to that of spherical aberration at clear aperture. Hence, in terms of added error by defocusing, obstructed system with spherical aberration has wider defocus range than unobstructed system with identical level of spherical aberration (over the entire area of optical surfaces), but its starting point - best focus quality level - is, for spherical aberration of ~λ/4 P-V wavefront error or smaller, degraded by the effect of obstruction (with the increase in obstruction and spherical aberration level, the overall best focus degradation diminishes, turning into slight improvement at 0.5D c. obstruction and λ/2 P-V wavefront error of lower-order spherical aberration level).

◄ 4.3. Astigmatism ▐ 4.5. Fabrication errors ►

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