4.1.2. Lower-order spherical: aberration function, blur size

Wave aberration function for lower-order, or primary spherical aberration, defining the P-V wavefront error as optical path difference at diffraction focus is:

Ws= sd4(r4-r2) = S(r4-r2) (7)

with s being the aberration coefficients for spherical aberration, d the pupil radius and r the ray height at the pupil in units of the pupil radius. The quantity S=sd4 is peak aberration coefficient, equal to the P-V wavefront aberration at the paraxial focus, with s being aberration coefficient of primary spherical aberration. Quantity in the brackets determines the aberration maximum at best (diffraction) focus in terms of the peak aberration coefficient as S/4 for ρ=√0.5.

The sign of P-V wavefront aberration with respect to reference sphere centered at paraxial focus is negative, its numerical value equaling the aberration coefficient, while positive with respect to the reference sphere centered at best focus (based on the aberration being positive when the path length from the aberrated point is longer, and negative when it is shorter than path length from the corresponding point at the reference sphere).

You will notice that this function form for spherical aberration differs from one given for primary aberrations (Eq. 5.1). This is because Eq. 5.1 gives the aberration function for paraxial focus, or so-called classical aberrations. Advance in calculation methods revealed that Gaussian image point is not a best focus location for the three aberrations affecting point-image quality - spherical, coma and astigmatism. Each of them requires shift from the Gaussian image point to their respective best focus location, where the central intensity of diffraction pattern is at its maximum (thus, best focus location is also called diffraction focus). Primary aberrations evaluated at best focus location are called orthogonal or balanced primary aberrations.

For spherical aberration, the amount of defocus from paraxial focus needed for the shift to diffraction focus location is given by P=-S, with P and S being the peak aberration coefficients for defocus and spherical aberration, respectively (FIG. 22). Since for identical longitudinal aberration the P-V wavefront error of defocus is double the P-V error of spherical aberration at paraxial or marginal focus (or P=-2S for the entire longitudinal spherical aberration), best focus for spherical aberration is at the mid point of its longitudinal aberration, as already stated.

Another difference between classical and best focus primary spherical aberration is in the sign of the P-V wavefront error. Since with the classical spherical aberration the point of maximum aberration lies closer to the focus than perfect reference point (that is, wave from the aberrated point has smaller optical path length), it is of negative sign. On the other hand, the point of maximum P-V deviation for balanced (best focus) spherical aberration is farther away than its perfect reference point, giving to the wavefront deviation positive sign. Hence, the aberration coefficient for spherical mirror is negative for the classical aberration form, and positive for the best focus aberration form.

FIGURE 22: Top to bottom: 3-D wavefront deviation plot, wavefront map, ray spot diagrams and actual diffraction patterns for 1/4 wave P-V of spherical aberration at best focus (Λ=1), paraxial (Λ=0), marginal (Λ=2) and 0.866 zone focus (Λ=1.5, circle of least confusion). Here, the perfect pattern has faint, but visible rings(1). They become noticeably brighter as a result of the energy spread caused by 1/4 wave P-V of spherical aberration. As the figure shows, the relation between geometric blur size and the appearance of diffraction pattern is rather loose: while the ray spot size at the circle of least confusion (Λ=1.5) is half that at the best focus location (Λ=1), actual diffraction pattern for the former is significantly more affected. Ray density distribution within the blur gives somewhat better indication of energy distribution than the blur size alone. Note that the 3-D plots on the top are not drawn in proportion; the center number on the wavefront map bellow specifies their actual P-V values.

(1) Bright stars will display more pronounced diffraction rings, especially the first, which may appear nearly as bright as the central disc even with the perfect pattern. This is due to a logarithmic intensity response of the eye: a 56 times lower intensity of the first bright ring may appear to the eye less than twice fainter. Only as the first ring intensity drops close to the threshold of perception, the central disc begin to appear much brighter.

In order to obtain specific wavefront aberration, we need to calculate the aberration coefficient s. General expression for the aberration coefficient of spherical aberration of a thin lens is:

with n being the refractive index, q=(R2+R1)/(R2-R1) the lens shape factor, and p=1-2ƒ/i the lens position factor (ƒ is the lens focal length, and i the image-lens separation).

Suffice to say, single spherical lens cannot be free from spherical aberration, except for the object inside the focal point of a positive or negative meniscus with specific values of p and q (in other words, lens forming a real image cannot have spherical aberration cancelled). The aberration is at its minimum for q=-2p(n2-1)/(n+2). Aspherizing lens surface into conic K<0 (ellipsoid, paraboloid, hyperboloid) for given center thickness increases the in-glass path toward the edge (the conic surface being shallower than sphere). This induces overcorrection, given as the P-V error at best focus for a single surface by W=(n'-n)Kd4/32R3, with d being the lens aperture radius and R the surface radius of curvature. More detailed evaluation of the lens primary spherical aberration is given in section on spherochromatism.

Fortunately, calculating the aberration coefficient for mirror surface is quite simple. Its general form is given by:

with n being the index of refraction of incidence medium, K being the mirror conic, m the magnification, R the radius of curvature and Ω=R/o the inverse of the object distance in units of mirror's radius of curvature, numerically positive (note that the above relation for the coefficient is derived from the general surface coefficient for spherical aberration, using J=(m+1)/(m-1)R and N=-2/nR, valid for mirror surface).

Optical magnification, defined as the ratio of image-to-object size, thus numerically negative when image orientation is opposite to that of the object. For object to the left, and image to the right of the aperture stop it is given by m=(i/o), i and o being for the image (numerically positive) and object distance (numerically negative) from the objective. For the object distance o known it is obtained from m=ƒ/(o+ƒ) when the focal length ƒ is positive, and m=-ƒ/(o-ƒ) for negative focal length. If we denote the reciprocal of the object distance in units of the mirror focal length (numerically negative) as ψ=ƒ/o, then the magnification is m=ψ/(ψ-1), and the aberration coefficient is:

Setting the sum in brackets to zero defines both, zero-aberration conic for given object distance as K=-(1-2ψ)2 and zero-aberration distance for given conic - after expanding (1-2ψ)2 and solving for ψ - as o=2ƒ/(1±√-K), for the object at the far (+), and near focus (-) of the conic. Object distance for K>0 is not defined in this form, because the foci of the oblate ellipse (with respect to horizontal axis) lie off axis.

For distant objects mg0 and the coefficient reduces to:

This gives the peak aberration coefficient for mirror surface and object at infinity as:

with F being the mirror focal ratio, given by -ƒ/D (for mirror in air oriented to the left, ƒ<0 and n=1). The P-V wavefront error of spherical aberration S and the RMS wavefront error ω relate as ω = S/√180. As mentioned, the peak aberration coefficient S equals the P-V wavefront error at the paraxial focus, and is larger than the P-V error at best (diffraction) focus by a factor of 4. Relationship between the P-V and RMS wavefront error in units of the wavelength on one, and the blur diameter at best focus in Airy disc diameters (for the same wavelength) on the other side is given by Bs=8S/2.44λ and Bs= ω√11,520/2.44λ, respectively.

The significance of the medium refractive index n is that it determines the nominal wavelength and, with it, the size of a given wavefront error relative to it. In slower media, such as glass, light waves are in effect compressed, and a given nominal wavefront error results in a proportionally greater phase difference at the focus. From another perspective, the compressed wavelength results in smaller Airy disc, while the size of transverse aberration (blur) remains unchanged, thus making the actual aberration proportionally larger.

Spherical aberration can also be expressed in terms of the peak aberration coefficient S as ray aberrations:

L = 16SF2r2, T = 16SFr3 and Ta = 16Sr3/D (10)

for the longitudinal, transverse and angular aberration (in radians), respectively, with D being the pupil diameter, and 0<r≤1 the relative ray height in the pupil with the radius normalized to 1. Their respective aberration maximums, for ρ=1 and object at infinity, are given with: L=(1+K)D/32F, T=(1+K)D/32F2 and Ta=(1+K)/32F3. The r parameter shows that longitudinal spherical aberration changes with the square of the ray height, while transverse and angular aberration change with the cube of ray height in the pupil. The transverse and angular aberration are for the blur diameter at the paraxial focus; blur at best focus is smaller by a factor of 0.5, and the circle of least confusion by a factor of 0.25.

The sign of peak aberration coefficient determines the sign of ray aberration as negative, indicating that marginal ray from the mirror, determining blur radius, focuses shorter, thus intersecting the blur plane bellow the axis (this also holds for best focus location, but not for the marginal focus location, where the blur boundary is formed by converging rays - that is, rays that focus farther away from the image plane - and all three ray aberrations are positive).

What could be of interest is the RMS blur radius for various locations within the span of longitudinal spherical aberration. It can also be expressed in terms of normalized longitudinal aberration Λ, and peak aberration coefficient S, as:

In units of the paraxial blur radius, the RMS blur radius is just the value in brackets, [0.25-(Λ/3)+(Λ2/8)]1/2. Location of the smallest RMS blur radius is found at Λ=4/3 (defocus location between best focus and circle of least confusion), where it is smaller than (geometric) paraxial blur radius by a factor of 0.167 (in comparison, best focus and circle of least confusion RMS blur radii are smaller than paraxial blur radius by a factor of 0.204 and 0.177, respectively).

The relative wavefront aberration for Λ=4/3 is ŵ=0.408 (Eq. 6), which is smaller than that at the location of the circle of least confusion (ŵ=0.545), but still significantly higher than at the best focus location (ŵ=0.25). It shows that the RMS ray blur size, while generally somewhat more meaningful than the geometric ray blur size, still lacks the accuracy required of a reliable indicator of optical quality.

In units of the Airy disc diameter, the RMS blur diameter is RRMS=8S[0.25-(Λ/3)+(Λ2/8)]1/2/1.22, for the peak aberration coefficient S in units of the wavelength.

EXAMPLE: Running all the numbers for a 6" ƒ/8.15 sphere, thus F=8.15, with d=3" and R=-97.8", gives the aberration coefficient for object at infinity s=(1/4R3)=-0.000000267, the peak aberration coefficient S=sd4=-0.000021647, longitudinal aberration L=16SF2=D/32F=0.02298", and the RMS wavefront error ω=S/√180=0.000001613". Expressing the peak aberration coefficient - which equals the P-V wavefront error at the paraxial focus - in units of 550nm wavelength (0.00002165"), gives the P-V wavefront error at the paraxial focus of 1 wave, the P-V wavefront error at the best focus of 1/4 wave (for ρ=√0.5), and the best focus RMS wavefront error of 1/13.4 wave.

The transverse blur diameter at best focus, T=8SF=D/64F2 or, in Airy disc diameters, is Bs=8S/2.44λ=3.28 is half the blur diameter at the paraxial focus, and twice the smallest blur diameter (circle of least confusion). Angular blur diameter at best focus Ta=T/f=1/64F3=8S/D=0.00002886 radians, or 0.00002886x206,265=5.95 arc seconds. Since both, wavefront error and geometric (ray) aberrations are directly proportional to the aberration coefficient, it implies that they are in a constant proportion themselves. In other words, doubling the wavefront error also doubles the geometric aberration.

The RMS blur radius at best focus (Λ=1) is rRMS=8FS[0.25-(Λ/3)+(Λ2/8)]1/2=0.000288, and the RMS blur diameter in units of the Airy disc diameter RRMS=8S[0.25-(Λ/3)+(Λ2/8)]1/2/1.22=1.338, for S in units of the wavelength.

For object at infinity, spherical aberration in either wavefront or ray form, is independent of the position of aperture stop. That makes it relatively simple to find out the combined spherical aberration coefficient for two or more mirrors. For a pair of mirrors, the combined peak aberration coefficient is given by Sc= S1 + S2, (Eq. 9), with the aperture diameter D for the second surface being determined by the relative height k of marginal ray at it, in units of the primary mirror semi-diameter (equal to the minimum secondary diameter in units of the primary diameter). Since the object for the second surface is the object-image formed by the preceding surface, magnification m for the second surface is greater than zero, given by m=R2/(R2-kR1), R1 and R2 being the radii of curvature of the first and second surface, respectively.

◄ 4.1.1. Primary spherical aberration ▐ 4.1.3. Higher-order spherical aberration ►

Home | Comments