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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 Δ is the longitudinal defocus (thus Rd=R+Δ). Since the second term already is smaller than the first term by a factor ~3ΔD2/4R2~3Δ/4F2, a full aperture P-V defocus wavefront error, for Δ values of ~λ (wavelength unit) and smaller is accurately (within a small fraction of percentage point) given by:

Wd = [(1/Rd)-(1/R)]D2/8 =  ΔD2/8RRd      (23)

with D being the aperture diameter. The error sign is positive when the OPD from the aberrated to focal point is larger than that from the point at identical zonal height at the reference sphere, and negative for smaller OPD (on the illustration bellow, the wavefront error is numerically negative).

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).

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             (24)

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 Eq. 24. It changes with the square of pupil height.

For very small relative difference between the two reference radii, such as those for typical defocus values in telescopes, Eq. 24 can be written as

Wd ~ (Rd-R)D2/8R2 ~ ΔD2/8R2            (24.1).

with Δ being the longitudinal defocus (effectively, the difference in radii between the two reference spheres).

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 R equaling the system focal length ƒ for object at infinity, (D/R)2=1/F2, where F is the system focal ratio ƒ/D, and the peak aberration coefficient for defocus, equaling the P-V wavefront error, is:

P = -Δ/8F2,           (24.2)

It is radially symmetrical aberration, with blur diameter given simply by Δ/F. In units of the Airy disc diameter (2.44lF) the geometric blur diameter for defocus is:

Bd = 8P/2.44l  = -Δ/2.44lF2          (25)

or Bd=745Δ/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 - i.e. P-V wavefront error - 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.

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.

 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 ω, Ld=32√3ωλlF2.

It is sometimes called focus depth (not to be confused with field depth, a photographic term referring to the range of distance in object space within which object image retains needed quality level; given by Δ=±8Wn(o-ƒ)2/[D2-8(o-ƒ)Wn], where o is the focused distance, ƒ and D the system focal length and aperture diameter, respectively, and Wn the nominal P-V wavefront error of defocus, it is irrelevant in astronomy, where all objects are very distant).

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 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 P-V wavefront error W of lower-order spherical aberration in units of the wavelength - assuming W<0.25 for better than "diffraction-limited" level - with the corresponding longitudinal aberration LAs=64λWF2, allowed ± defocus range for remaining at the conventional "diffraction-limited" level, or better, is given by Lds=xLAs, with x=(1-16W2)0.5/4√15W obtained from Eq. 6. Hence diffraction-limited defocus range in the presence of spherical aberration is given by:

Lds = 16λ(1-16W2)0.5F2/√15             (26.1)

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 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. Eq. 26.1 is accurate for 0.25≥W≥0.0625; for W=1/16, the diffraction limited defocus range becomes identical to that of aberration-free aperture for ±λ/4 wave of defocus, as shown on the illustration to the left, depicting error function over the range of longitudinal spherical aberration, given with Eq. 6 (units on the horizontal axis are Λ, longitudinal aberration normalized to 2 for λ/4 P-V error of spherical aberration; note that the P-V wavefront error of defocus is slightly larger for given RMS error). For W<1/16, Eq. 26.1 gives slightly larger values for diffraction-limited defocus range, than what is obtained for the λ/4 wavefront error of defocus range. This is a consequence of the defocus RMS error being, for given P-V wavefront error, slightly smaller - by a factor of (11.25/12)0.5 - than the RMS error of spherical aberration. For zero wavefront error of spherical aberration (W=0), Eq. 26.1 gives diffraction-limited range larger by a factor of (12/11.25)0.5 than the range for λ/4 wave of defocus. That is the true diffraction limited defocus range, for the RMS error producing 0.80 Strehl at either end; the corresponding P-V wavefront error of defocus is, as mentioned before, 0.258λ.

In aberration-free systems with central obstruction, defocus RMS error is smaller by a factor of (1-o2) than in unobstructed aberration-free system, o being the relative c. obstruction size in units of the aperture diameter. Consequently, the Strehl ratio in terms of peak defocus phase aberration Φd, for linear defocus Δ, is given by 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.

◄ 4.3. Astigmatism   ▐    4.5. Fabrication errors ►

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