In general,

*coma*wavefront aberration occurs either due to incident wavefront being tilted with respect to the optical surface, or being axially displaced from it. Hence, it is either an aberration affecting off-axis image points, or the result of axial misalignment of optical surfaces, respectively. As illustrated on**FIG. 25**, comatic wavefront deviation has reverse symmetry along the axis of aberration, with one side flatter, and the other more curved with respect to its perfect reference sphere.**FIGURE 25**: Coma wavefront aberration. RIGHT: incident wavefront

**W**

**i**, tilted in regard to the optical surface, losses the symmetry needed to become spherical after reflection; as a result, rays originating at the actual wavefront scatter, forming comatic blur. Reflected wavefront

**W**

**a**deviates from the reference sphere

**W**

**G**centered at the Gaussian image point

**G**; after corrected for tilt, better reference sphere

**W**

**P**is found centered at the best (diffraction) focus

**F**

**b**. LEFT, an exaggerated illustration of the comatic wavefront deformation, and how it causes rays to spread into comatic blur (for clarity, the blur is rotated 90°). The cross-section of the middle (solid blue) and the edge side projection (dashed blue) of the comatic wavefront indicate the form of its deviation from the perfect reference sphere (red dots). It has an element of tilt combined with lobe-like deformations on the opposite sides of wavefront. Only the central spot of the wavefront is nearly spherical, focusing at the point of highest intensity

**F**

**b**(best, or diffraction focus location), which is located at 2/3 of the sagittal coma, along the axis of aberration. RMS spot radius is the smallest when calculated at the PSF centroid

**C**, when it is 0.29 of the tangential coma. It is obvious that as the wavefront error - and the blur size relative to the Airy disc - increase, best focus location shifts farther away from the Gaussian image point.

Mapping the rays scattered by comatic wavefront deformation follows a simple, elegant symmetry. As illustrated on

**FIG.****26**, every zone at the pupil focuses not into a point, but into a circle of the diameter**R = hp**

**2**/2f

**2**(11)

with

**h**being the linear height in the image plane,**p=ρd=ρD/2**the zonal height in the pupil (**D**is the pupil, or aperture diameter and**r**the zonal height normalized to 1) and**ƒ**the focal length. The center of each circle is also shifted by a length equal to the circle diameter from the Gaussian image point, in the image plane, along the axis of aberration (determined by the chief ray and optical axis).**FIGURE 26**: Geometry of the coma ray spot diagram, created by rays projected from the comatic wavefront. The spot consists entirely from progressively expanding circles, with the exception of the very central ray, projecting into the tip of the sagitta. Both, the diameter of the comatic circles formed in the focal zone and their shift from the Gaussian focus increase with the square of the pupil (zonal) height

**p**, expanding around the axis of aberration

**a**, in the direction of the axis. Diameter of the largest comatic circle, formed by the very edge of the aperture, is 2/3 of the blur length. The ray circle formed by the 0.5 (p=D/4) wavefront zone is only 1/4 of the diameter of of the edge circle (p=D/2), and has 1/4 of its shift. Tangential coma (

**T**) is the entire blur length, with the sagittal coma (

**S**=

**T/3**) being the "V" shaped tip oriented toward Gaussian image point. About 80% of all rays are contained within the sagittal part of the comatic blur. It is the sagittal part of the comatic image that is actually visible as a deformation of the point-source image in a telescope.

**wavefront error**as optical path difference with respect to a reference sphere) for coma at best focus is given by:

with

**C**being the coma peak aberration coefficient,**ρ**the pupil radius normalized to 1 and**q**the pupil angle. Note that**Wc**at its maximum represents 1/2 of the P-V wavefront error. The aberration peaks for ρ=1 and θ=0° and 180°, with the sign of aberration changing symmetrically on both sides of the wavefront. The factor in brackets - effectively correction for wavefront tilt - makes the P-V coma error at best focus ("balanced" coma) three times smaller at its maximum than the P-V error at paraxial (Gaussian) focus, referred to as "classical", or Seidel coma, where the P-V wavefront error is given by W=Cρ3cosθ (**FIG. 25**, left). On

**FIG. 25**, the peak error on the top radius (edge point) is positive, being farther to the center of a perfect reference sphere than its perfect reference point, while the peak error on the bottom radius is negative. This is so called "positive" coma, with the top section of the wavefront flattened with respect to the reference sphere, and the bottom portion more curved. Its comatic tail is oriented outward, away from field center. Presence of the pupil angle factor (**cos****q**) indicates that the aberration is not rotationally symmetrical. The**peak aberration coefficient**is given by: C = cαd

**3**(13) with

**c**being the coma aberration coefficient,**α**the field angle (in radians) and**d**the pupil radius. Coma**aberration coefficient****c**for either refractive or reflective surface, for object at infinity, and aperture stop at the surface is given by: with

**ƒ**being the focal length, and**n**and**n'**being the index of incidence and refraction/reflection, respectively. For mirror surface, the coma coefficient for any object distance is given by:

with

**m**being the transverse image magnification (defined following Eq. 9.). For distant objects mg0, n=1 for mirror in air oriented to the left, and the coefficient reduces to:As the aberration function shows, deviation from the perfect reference wavefront is largest for ρ=1 and either cosθ = 1 or cosθ = -1 (i.e. θ=0 or θ=π

**radians, respectively). With****q**being the position angle in the pupil, measured to the plane of the axis of aberration, the maximum deviation occurs at the marginal points of the wavefront in the plane of aberration (**FIG**.**25**, left). The deviation is zero for cosθ=0 (i.e. θ=π/2 or θ=3π/2), which is the wavefront diameter orthogonal to the plane of aberration. For**r**smaller than |~0.8|, the deviation changes the sign, as a result of the two lobe-like deformations, illustrated on**FIG. 25**, left. As mentioned, the above relations for coma aberration are for a mirror with the aperture stop at the surface. For

**displaced stop**, the coma aberration coefficient**c**changes with a factor**c=1-(1+K)σ**, with**K**being the mirror conic and**σ**the stop-to-mirror separation in units of the mirror radius of curvature. Thus, for displaced stop, the aberration coefficient for object at infinity is: Obviously, stop position doesn't influence coma of a paraboloid (K=-1), while cancels coma of any conic K>-1 for the stop separation

**σ=1/(1+K)**. On the other hand, it increases coma for any K<-1 (hyperboliod). This applies for objects at infinity; for

**close objects**for which the mirror object-image magnification**m**significantly differs from zero, the change factor in the coma coefficient caused by the aperture stop displacement is obtained by replacing both unit figures in the "infinity" object distance factor**c**with (m+1)/(m-1), or (1-Ω),**Ω**being the reciprocal of the object distance in units of the radius of curvature. It gives the coefficient as: Note that (1-Ω) is just a different form of (1-2ψ) parameter,

**ψ**being the reciprocal of the object distance**o**in units of the focal length, ψ=ƒ/o. The wavefront error changes with the coefficient. Considering that the magnification

**m**is negative, coma diminishes with object distance, dropping to zero for m=-1 (for object at the mirror center of curvature) and stop at the surface (σ=0), regardless of the conic.Due to the peak wavefront deviations with coma affecting relatively small area, the error averaged over the entire wavefront is smaller for coma than for spherical aberration for any given P-V wavefront error (

**FIG. 27**).**FIGURE 27**: Coma ray aberration as a ray spot (bottom) and the actual diffraction pattern (top).

*Top*: from left to right, perfect diffraction pattern of a relatively bright star, diffraction pattern affected with 0.42 wave P-V wavefront error of coma (corresponding to 0.074 wave RMS, for 0.80 Strehl, thus comparable to 1/4 wave P-V of primary spherical aberration), and diffraction pattern affected by 0.84 wave P-V wavefront error of coma (~0.15 wave wavefront error RMS, comparable to 1/2 wave P-V of spherical aberration). As the error increases, best focus location shifts from the middle of the Airy disc in the direction of intensity shift. Unlike spherical aberration, there is no gain from axial defocus, which is pretty much obvious from the shape of wavefront deformation.

*Bottom*: corresponding ray spot plots for identical amounts of the aberration. The "V" shaped form obvious in the ray spot doesn't become apparent in the actual diffraction pattern until the error exceeds ~1 wave P-V, provided there is sufficient magnification to make it visible to the eye.

The

**RMS wavefront error**for coma at best focus, in terms of the peak aberration coefficient**C**is:**ωc =**C/72

**0.5**(16)

In terms of the P-V wavefront error it is ω

**c**=2W**c**/**√**32, with**Wc**being, as given above, the coma peak wavefront error (half of the P-V error). Shift from the Gaussian image point to the best focus location is transverse, in the image plane and along the axis of aberration, given by

**4FC/3**. That places best focus at 1/4.5 of the blur length away from the Gaussian image point (the tip). The shift effectively corrects for the wavefront tilt element. Unlike spherical aberration, the peak aberration coefficient**C**for coma does not equal the p-v wavefront error. At the location of best focus, the P-V wavefront error is 2C/3, with the peak wavefront error being ±C/3. The error is smaller by a factor of 3 than at the location of Gaussian image point, for which the tilt is not corrected. Coma

**transverse****aberration**also can be expressed in terms of the peak aberration coefficient**C**. Tangential coma is given by**T=6ƒ****iC/n'D**,**ƒi**being the objective-to-image separation and**D**the aperture diameter. For mirror oriented to the left, n'=-1 and, for distant objects ƒigƒ, resulting in**T=6FC**, with**F**being the focal ratio number, F=f/D. With the**peak aberration coma coefficient****C=**α**D/32F2**for distant objects, tangential coma can be expressed directly as:as transverse aberration (h=αƒ, the linear height in the image plane), and

**T****a=T/****ƒ****= 3**α**/16F2**as angular tangential coma (in radians). From**Eq. 11**, tangential coma changes in proportion to the square of the pupil (zonal) height, or with**ρ****2**for the pupil radius normalized to 1. As already mentioned, sagittal coma**S=T/3**; it is larger than the P-V wavefront error by a factor of 3F. Geometric coma blur (tangential coma) can also be expressed in terms of the RMS wavefront error**ω**, as**t=****ω****√2592****/2.44**, in Airy disc diameters; also, in terms of the peak aberration coefficient as**t=6C/2.44**(in Airy disc diameters), for the RMS wavefront error**ω**and peak aberration coefficient**C**in units of the wavelength. For any given RMS wavefront error, tangential coma is smaller than spherical aberration blur at best focus by a factor (81/360)1/2. Finally, the RMS ray spot radius for balanced coma is given by rRMS=2

**√**7/9FC, which in terms of the tangential coma becomes**√**7/9T/3. In other words, it is smaller than tangential coma blur by a factor 0.294. The coma RMS blur radius is smallest when measured around coma's PSF centroid (at 1/3 of the tangential coma, along the axis of aberration;**C**on**Fig. 25**left). The smallest RMS blur radius is given by r'RMS=**√**8/3FC, or smaller by a factor of 0.926 than the RMS blur radius centered at the diffraction focus. However, due to the increase in wavefront tilt error component, the P-V error of coma for the wavefront centered at the PSF centroid is at the minimum for ρ=1/**√**3, and for the aberration function Wc= C(ρ3-ρ)cosθ. This gives the peak wavefront error Wc=-0.3849C, greater by a factor of 1.15 than the peak wavefront error - and RMS wavefront error - at best focus. In units of Airy disc diameter, the RMS blur diameter at best focus is RRMS=2

**√**7/9C/1.22, with the peak aberration coefficient**C**in units of the wavelength.**EXAMPLE**: For a 200mm ƒ/5 paraboloid, thus d=100 and R=-2000, the peak wavefront error of coma at 1.4mm off-axis (with the field angle α=1.4/1000=0.0014), setting ρ=1 and θ=0 in Eq. 12, comes to C/3. With the peak aberration coefficient

**C**=cαd3=αd3/R2=0.00035mm, the

**peak wavefront error**of coma is

**W**

**c**=0.0001167mm. The

**P-V wavefront error**is twice as much, or 0.000233mm. In units of 550nm (0.00055mm) wavelength, it is 0.424 (1/2.36 wave). The

**RMS wavefront error**is

**ω**=C/

**√**72=0.000041mm or, in units of 550nm wavelength, 0.075 (1/13.33 wave). The RMS wavefront error can also be obtained from the P-V wavefront error as ω=2W

**c**/

**√**32. Either way, the aberration is at the conventional "diffraction-limited" level.

The

**transverse tangential coma****T**=6FC=0.0105mm, or 1.56 Airy disc diameters. Therefore,**angular tangential coma****Ta**=T/ƒ**is 0.0000105 in radians, or 206,265x0.0000105=2.17 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 rRMS=2

**√**7/9FC=0.000617mm, and the RMS blur diameter in units of the Airy disc diameter RRMS=2**√**7/9C/1.22=0.92, for**C**in units of the wavelength. A single thin

Analogous to higher-order spherical aberration, strongly curved surfaces generate **lens**aberration coefficient is given by Eq. 98. It us coma-free if its shape factor**q**and position factor**p**relate as q=-(2n+1)(n-1)p/(n+1). Lens doublet can have both, coma and spherical aberration cancelled. In regard to the aperture stop position, it does affect coma only with objectives not corrected for spherical aberration. With an aplanatic (corrected for spherical aberration and coma for infinity) lens system, stop position doesn't affect neither coma, nor astigmatism and field curvature.**higher-order coma**, similar in appearance to its lower-order form. For given P-V wavefront error, the corresponding RMS error for higher-order coma is larger by a factor of 1.2. After cancelling lower-order coma in the 3rd order (transverse ray, 4th order on the wavefront) approximation, residual higher-order (5th transverse, 6th on the wavefront, or

*secondary*) coma is minimized by reintroducing lower-order coma of opposite sign, larger by a factor of 1.2. The combined RMS error is smaller by a factor of 0.2.

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