## 4.4. Defocus
PAGE HIGHLIGHTS 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 (
The corresponding
P-V wavefront error equals the
differential between two sphere's sagitta, given with the expansion
series Wd=[(1/Rd)-(1/R)]D2/8
+ [(1/Rd3)-(1/R3)]D4/32
+ ... = ΔD2/8RRd
+ Δ(R3+Rd3)D4/32R3Rd3
+ ..., where Wd
= [(1/Rd)-(1/R)]D
with Figure bellow shows the diagram of defocus error (top), as well as the changes in the appearance of otherwise aberration-free diffraction pattern for unobstructed aperture caused by increasing defocus error (bottom) and change in the longitudinal peak intensity as a result of defocus error.
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, " Back to best focus being coinciding
with the Gaussian, defocus
Wd = P with
For very small relative difference between
the two reference radii, such as those for typical defocus values in
telescopes, Wd with
Since this approximation differs from the exact value by a factor of Rd/R, 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
P = -
It is radially symmetrical aberration, with
the geometric
Bd or
Bd=745Δ/F
The peak defocus aberration coefficient
- i.e. P-V wavefront error - and the
RMS wavefront error relate
as
The RMS wavefront error
relates to blur diameter In terms of phase error, the peak value of defocus phase aberration is given as Φd=πΔ/4λF2, in units of the full phase of 2π radians (thus giving the P-V wavefront error in units of wavelength when divided by 2π). The corresponding Strehl ratio is given by S=sinc2(Φd/2)=[sin(Φd/2)/(Φd/2)]2=[sin(180Δ/8λF2)/(πΔ/8λF2)]2.
The peak defocus phase aberration is Φd=1.43
radians, or 1/4.4 of the full phase of 2π.
The corresponding Strehl is S=0.84.
An important property of a telescope is its
Ld = 16WλF2 (26)
where W=Wd/λ
is the defocus P-V wavefront error in units of the wavelength. In terms
of the RMS wavefront error
It is sometimes called
Focus depth in telescopes is also not to be confused with
Taking W=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
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
the P-V wavefront error
Lds
= 16λ(1-16W Taking W=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 0.169mm (note that "defocus range" in this context doesn't equal defocus error; at the best focus location, lower-order spherical aberration is already combined with longitudinal defocus equaling one half of the longitudinal spherical aberration).
For W=0.25, i.e. λ/4 wave of
lower-order spherical aberration in the system, allowed defocus to remain within "diffraction-limited" range
is zero.
In aberration-free systems with central
obstruction, defocus RMS error is smaller by a factor of (1-o2)
than in unobstructed aberration-free system,
S=sinc2(Φd(1-o2)/2)={sin[Φd(1-o2)/2]/[Φd(1-o2)/2]}2= ={sin[(180Δ/8λF2)(1-o2)]/(πΔ/8λF2)(1-o2)]2. If spherical aberration is present, the RMS wavefront error of defocus is reduced by a factor of (1-o2)2, and its defocus error changes similarly to that in clear aperture with spherical aberration. Hence, in terms of error added by defocusing, obstructed system with spherical aberration has wider defocus range than unobstructed system with identical level of spherical aberration. However, its starting point - best focus quality level - is degraded by the effect of obstruction. In order to have any diffraction-limited defocus range, an obstructed aperture needs to have normalized peak diffraction intensity greater than 0.80. This implies the optics Strehl better than 0.8/(1-o2)2. With even moderate linear obstruction ratio o=0.25 (i.e. 0.25D), a system needs to have the optics Strehl better than 0.91 (i.e. better than 1/6.1 wave P-V of spherical aberration level). A 0.95 optics Strehl system, or slightly better than 1/8 wave of lower order spherical level, cannot add more than about 1/24 wave of P-V wavefront error and remain diffraction limited. Unobstructed system with identical correction error can add three times as much of P-V wavefront error (1/8 wave), implicating roughly three times wider diffraction-limited defocus range for given relative aperture. |