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1. INTRODUCTION AND BASIC PRINCIPLES |

Trigonometric Ray-Tracing |

Sign and Notation Conventions |

Paraxial Ray-Tracing |

2. OPTICAL ABERRATIONS |

Spherical Aberration |

Coma |

Chromatic Aberration |

Other Aberrations |

3. EVALUATING OPTICAL SYSTEM PERFORMANCE |

Optical Tolerances |

Meridional Ray Plots |

Spot Diagrams |

Encircled Energy Plots |

Extended Objects |

APPENDIX |

Aspheric Surfaces |

References |

Optical design and ray-tracing are generally regarded as difficult subjects for the non-professional. Before the advent of mechanical computing machines, optical designers relied on laborious hand calculations with tables of logarithms and trigonometric functions as basic tools of the trade. The work was tedious and subject to errors. Modern high speed digital computers rapidly replaced the mechanical calculator for professionals, but this resource was often not available to the non-professional lens designer. In the interim period before personal computers became generally available much optical design work was done with electronic calculators [1].

Sky Scientific'sdbOpticis intended to aid both the student optical designer as well as science and engineering professionals who may or may not have prior formal training in optical design. Optics are playing an increasingly vital role in complex engineering systems developed today. The text material herein will provide the user with some understanding of the optical principles involved in applying the accompanying software to specific design problems. In this way, the user may gain an overall appreciation of the optical problem rather than merely the numerical prescription that is capable of solving it.

In developing this appreciation, it is intended that the user be spared many of the theoretical details which accompany most optical reference works. The interested reader will find no shortage of suitable material [2],[3].

1.1 Trigonometric Ray-Tracing

The purpose of ray-tracing is to quantitatively determine the quality of the image an optical system will deliver before the system is actually produced. To a great extent, image quality may be determined mathematically: trigonometric ray-trace calculations yield various distances and angles which are utilized in aberration definition formulae. The results provide the designer with the magnitude of aberrations describing image quality of the prescribed system. The aberration magnitudes are compared with pre-established tolerances. The designer then decides whether

(1) the aberration levels are acceptably low and the system may be produced

or

(2) some modification of the design is required.

In the latter case, this may mean changing one or more curvatures or separations or selecting glasses with different refractive properties or all of the above. The correction techniques are applied and the ray-tracing procedure is repeated until all aberrations are within the required tolerances. In some cases, the designer may be faced with the decision of either increasing the acceptable tolerance values or perhaps adding an additional optical element or maybe even aspherizing one or more surfaces. Lens design programs are available which will automatically attempt to determine the design modifications necessary to bring aberration levels within pre-defined tolerances. The approach used indbOpticis to present the user with tools to assist in arriving at the same end result, while providing an appreciation for the process.

The following figure is a representation of a ray striking a spherical surface, used to develop basic ray-tracing equations.

In the above, O is an incoming ray which strikes surface f at point P. If the surface were not present, the unobstructed ray would cross the axis at B. Instead, it crosses at B'. C is the center of curvature of the surface with distance PC equal to the radius r. ZP is the normal to the surface at P. V is the vertex of the surface from which object distance VB ( = L) and image distance VB' ( = L') are measured. n and n' are the indices of refraction in object and image space, respectively.

Angle OBV is the Object Angle ( = U) and angle PB'V is the Image Angle ( = U'). Note that if ray O had struck the surface from an upward angle, point B would be on the optical axis somewhere left of the vertex V. If object point B is a very great distance to the left of the surface, as in the case of an astronomical telescope, then ray O would be parallel to the axis. In that case angle U = 0, distance L is negative infinity and sin I = Y / r where Y is the height at which the ray strikes the lens surface.

Snell's Law, expressed in equation [1-1], establishes the relationship between the angle of incidence OPZ ( = I) and the angle of refraction CPB' ( = I'). The remaining expressions follow from the figure.

sin I' = ( n / n') sin I | [1-1] |

sin I = ( L - r ) sin U / r | [1-2] |

U' = U + I - I' | [1-3] |

L' = r ( 1 + sin I'/ sin U') | [1-4] |

1.2 Sign and Notation Conventions

The following conventions apply:

a. Primed quantities refer to image space (after refraction or reflection). Unprimed quantities refer to object space.

b. Lengths and distances to the right of the surface are positive; to the left they are negative.

c. Curvature radii with centers to the right of the surface are positive. If the center of curvature is to the left of the surface, the radius is negative.

d. For angles between the ray and the axis of the system: mentally rotate the axis into the ray. If the rotation is clockwise, the angle has positive sign. Otherwise, it is negative.

e. Use n' = - 1 for reflective elements.

1.3 Paraxial Ray-tracing

Ray-trace calculations are performed for rays entering the system at various distances from the axis of the system. A simplification of the basic formulae results when rays are traced close to the system axis. This is because the object and image angles become small. When expressed in radians, the sine of a very small angle is equal to the angle. For this reason, equations [1-1] through [1-4] can be rewritten as follows for small angles:

i = u ( l - r ) / r | [1-5] |

i' = i ( n / n' ) | [1-6] |

u' = u + i - i' | [1-7] |

l' = r ( 1 + i'/ u' ) | [1-8] |

Lower case letters are used in the formulae for rays passing near the axis of the optical system. These rays are known as paraxial rays. Rays which pass through the system elements at the largest distance from the axis are called marginal rays. It can be shown that l' is independent of the angles in the formulae. However, it is customary and convenient to select u for the initial paraxial trace to be equal to the value of sin U from the initial marginal trace.

1.4 Parabasal Rays

The usual calculation of image distance breaks down for paraxial rays when the design includes tilted refractive elements. For this special case, paraxial image distance is replaced with the result of tracing using the non-paraxial formulae, but at a very small ray height. These rays are referred to as "Parabasal Rays."

2.1 Spherical Aberration

A point source of light can be thought of as emitting a spherical wavefront represented by a series of expanding concentric spheres with the point source at the center. Those waves which may be intercepted by a converging optical system will be redirected to create an image of the object point. If the original object point is on the optical axis of the system, the image point will also be on-axis. In the absence of spherical aberration and if the light is monochromatic, all of the rays will converge at a single image point. Spherical aberration, given the symbol LA' for "longitudinal aberration,'' is present in an optical system when monochromatic rays from an object point on the axis of the system cross the axis in image space at more than a single point. For a lens, rays passing near the center of the lens cross the axis at a point different from rays which pass near the edge. The lens in the next figure is said to be undercorrected (has a "short" edge).

The ray passing near the edge of the lens at height Y has an image distance VB' equal to L'. The ray passing near the center has a greater image distance Vb' equal to l'. The measure of spherical aberration for the system is simply

LA' = l' - L [2-1]

Generally, L' in the above expression is determined for a marginal ray, although Zonal Spherical Aberration (LZA') where L' is for an intermediate zone is also an important design analysis parameter.

2.2 Coma

Coma, when present in an optical system, manifests itself in flared images of object points which are off the axis of the system. In the figure below, O, P and Q are parallel rays making an angle ø

with the optical axis. Distance g'h' is a measure of coma produced by the system. Since the magnitude of this aberration varies with field width, another optical design parameter which provides a measure of a systems inherent susceptibility to coma is often examined. This parameter is known as the "offense against sine condition.'' When imaging objects at infinity,

OSC = 1 - u'l'/ L'sin U' [3-2]

The above relation holds true only if we follow the customary ray-tracing convention of setting u = sin U for the paraxial trace through the first surface of the system. The expression is also more complicated if the system contains an aperture stop. A short focus Newtonian reflecting telescope is the usual example given for exhibiting comatic star images near the edge of the field, although most systems exhibit some coma at short focal ratios unless specifically designed for wide angle work. References [2] and [3] contain additional information on the use of the parameter OSC.

2.3 Chromatic Aberration

An optical system that is corrected for both spherical aberration and coma is said to be aplanatic. Chromatic aberration, which is present only in systems that contain refractive elements, results from the fact that a refractive medium has a slightly different refractive index for diffeent wavelengths of light. Because white light is made up of a mix of different wavelengths and each must obey Snell's Law, it is clear that the amount of bending will vary with the wavelength. The lens behaves as a prism breaking up the white light into its constituent colors, each crossing the optical axis at a different point. Longitudinal chromatic aberration is the calculated difference between the image distances for two particular wavelengths. The choice of wavelengths depends upon the desired application. Historically, a commonly selected pair for photographic instruments was is the Cadmium F' and C' lines (480 and 643.8 nanometers, respectively). For visual instrument design, the wavelength pair commonly used is one shifted slightly toward the red end of the spectrum: The Hydrogen F and C

lines at 486.1 and 656.3 nm, respectively. Because modern photo-electronic sensors are often designed to operate outside of the visible spectrum, dbOptic implements a more general approach where virtually any pair of spectral lines may be selected for chromatic analysis. However, once the primary design wavelength λ has been selected, the user should choose a 'blue' wavelength shorter than λ and a 'red' wavelength that is longer.

In the figure, distance VC ( = L'_{C}) is the image distance for 'red' light (Hydrogen C-light in this case) and distance VF ( = L'_{F}) is the image distance for blue light (Hydrogen F) . Longitudinal Chromatic Aberration is defined by the following expression:

Chr' = L'_{F}- L'_{C }[2-3]

A lens system which is fully corrected for Longitudinal Chromatic Aberration will usually still suffer from some residual color known as the Secondary Spectrum. This residual is caused by the difference between the now common Blue/Red focus and the slightly different image distance for the other wavelengths of interest. Selection of specialized glasses can minimize the problem in doublets. A three element lens can bring three wavelengths to common focus, creating an apochromatic lens.

2.4 Other Aberrations

While it is true that Spherical and Chromatic Aberration and Coma are often significant sources of image degradation in optical systems, other aberrations are also of concern. Astigmatism, Distortion, Field Curvature, and Lateral Color occur in the imaging of extra-axial object points.

The focused image of a round disk object displaced a distance from the optical axis will appear oval shaped if the lens system suffers from astigmatism. Distortion results in the familiar "barrel'' or "pincushion'' appearance of a square object such as a window frame. Although distortion impacts image shape, it does not lower system resolution. Field Curvature is exactly as implied by the name; the surface of best focus for a flat object is a curved shell rather than being flat as would be desired when imaging on a piece of photographic film or a flat projection screen. Lateral color is the height difference for principal rays of different wavelengths and results in a colored haze around an image at best focus.

The need to predict optical system performance based on a paper design has resulted in the development of a number of different analysis tools with corresponding different levels of usefulness.

3.1 Optical Tolerances

Traditional optical tolerances are based on the 1/4-wavelength criterion established by Lord Rayleigh. That is, that images will not be badly impaired as long as the longest and shortest optical path lengths to a selected focus do not exceed 1/4 wavelength of light.

In addition to calculating the achieved value of most aberrations, thedbOpticsoftware also presents the corresponding tolerance value.

Clearly, the notion of fixed tolerances for performance evaluation implies the existence of a well-defined boundary between acceptable and non-acceptable performance. In fact, image degradation or improvement will be found to be gradual as a design progresses through the tolerance region.

3.2 Meridional Ray Plots

If several rays in a single plane which includes the optical axis originate from an off-axis object point, they will be found to intercept the optical axis in object space at readily determined locations. We can then easilytrace the rays through the optical system using the axis intercepts to define the object distance for each trace. When this is accomplished, it is also possible to determine the height H' in image space at which each ray intercepts a flat plane perpendicular to the optical axis at the paraxial image distance, l'. When the H' value for each trace is plotted against (1) Tan U', the final image angle and (2) L, the object distance, we find two useful graphical indicators of optical system performance.

(1) H' - Tan U' Curves

When final image angle, Tan U', becomes the abscissa then a perfect lens will plot as a straight horizontal line. The two ends of the line represent the first and last rays of the fan or beam to pass completely through the system, called the upper and lower rim rays. The figures below show a variety of outcomes:

(2) H' - L Curves

If the system includes an aperture stop of diameter D at distance Lo from the first lens surface, a wide beam of parallel rays at angle ø to the optical axis are traced as a series of rays from axial object points. The object distances vary in the range Lo ± D / 2 Tan ø .

If we do not know the best location for the stop, it can be determined by tracing a number of oblique rays over a range of object distances L and plotting H' vs L. The stop will allow rays only in the range of Lo ± D / 2 Tan ø to pass. In the figures below, the upper and lower rim rays and the principal ray are noted by u, l and p. The principal ray is defined as the last ray to be cut off as the diameter of the stop is reduced to zero.

In ray-trace calculations the chief ray, midway between the upper and lower rim rays, is often used to represent the principal ray. If there is no vignetting, that is, if the stop is large enough and positioned such that it stops no incident rays, then the chief and principal rays are the same.

Examining the curvature of the graph in the region of the principal ray gives an indication of any coma present. If the curve contains an inflection point, placing a suitable stop at that distance from the lens will minimize the magnitude of coma. Reference [3] Page 210 contains additional information on the interpretation of H' - L curves.

3.3 Spot Diagrams

Skew rays are rays from extra-axial object points that enter the first surface of the optical system outside the meridional plane. Skew rays make up the majority of rays which enter an optical system from an extra-axial point source. If the first surface of a lens is divided into a large number of equal areas and a ray is traced through the center of each area, the ray intersection pattern in the image plane is known as a ray spot diagram. The "tightness'' of the pattern provides an indication of axial as well as off-axis performance of the system.

The image below shows the results of tracing 2,000 rays through a well-corrected 500mm focal length f/8 telephoto lens, as viewed at the paraxial image plane. The image is typically examined at two other locations: (1) at the DLC (Disk of Least Confusion), which is at a point ¼ of the distance from the marginal image plane to the paraxial image plane and (2) Midpoint, which is half way between the paraxial image plane and the DLC. For systems that are well-corrected for spherical aberration there will be little difference between the 3 spot diagrams. The scale shown for the example below is in millimeters.

Spot diagrams are perhaps most useful in determining the quality of off-axis imagery. In the figure below, the same system is examined with rays entering obliquely, at 5 degrees off-axis.

3.4 Encircled Energy Plots

Finally, to provide a more quantitative measure of performance using spot diagrams, a useful tool is the Encircled Energy Plot. This is generated by starting at the center of the spot diagram (using the center point calculated statistically in the off-axis case) and counting the fraction of incident rays that are within any radius R. The figure below is an Encircled Energy Plot for the above lens for the 5 degree off-axis case. In the example, the percentage of incident energy as shown on the Y-axis does not exceed 80% due the fact that not all rays entering the system at 5 degrees off-axis can reach the image plane.

3.5 Extended Objects

Instead of using a single point as the object for a spot diagram, we could consider an array of object points at the same finite or non-finite object distance and create a composite spot diagram at the image plane. The composite image of this "object map" provides a near-realistic view of the image that a candidate optical system might generate.

Original object from file. | Image generated by tracing object array through a well-corrected 76mm f/5 achromat. |

3.6 Modulation Transfer FunctionAPPENDIX

The Modulation Transfer Function (MTF) is a lens evaluation tool designed to characterize the ability of a system to faithfully deliver the contrast present in an object to the image. It depends on the spatial frequency which is simply the number of contrast (light-to-dark) cycles per unit length in the object. MTF is the ratio of image contrast to object contrast, determined for a range of object spatial frequencies. That is,

MTF(µ) = Ci(µ) / Co

where Ci(µ) = Image contrast at frequency µ

The usual object intensity function is a sine wave representing a set of equally spaced lines with the sine wave intensity level cycling between the dark lines (low intensity) and white spaces (high intensity). If I is the intensity, then object contrast is given by

Co = ( Imax - Imin ) / ( Imax + Imin )

When a pair of lines in the object are far apart (low spatial frequency) we could expect that the lens may have little difficulty resolving the lines and the MTF is nearly one. As the spatial frequency increases, a lens will begin to lose its resolving ability. For an imperfect lens, this loss will occur at lower spatial frequencies than for a perfect lens.

References [4] and [5] contain additional information about MTF.

Aspheric Surfaces

The spherical ray-tracing equations presented in Section 2 are a special case of tracing rays through any smooth surface. In order to trace rays through aspheric surfaces, suitable equations must be developed for the

more general case.

The aspheric surface shown above is in general form x = f(y). That is, the displacement of the curve measured as a distance x from the y-axis is a function of some height value y. Note that expressing the surface in this form represents a slight departure from normal mathematical convention where the independent variable is usually selected to be x.

If x and y are the ray intercepts at the surface at P, we can write

y / ( L' - x ) = Tan U' [A-1]

If ZP is the normal to the surface at (x,y), then the angle of incidence I is shown by Angle CPL and the angle of refraction I' is shown by Angle CPL'. Since U' = U + I - I', then

L' = f(y) + y / Tan ( U + I - I') [A-2]

The relationship between I and I' is determined by Snell's Law:

I' = arcsin ( n sin I / n' ) [A-3]

Noting that the slope of the normal to the surface at (x,y) is simply the first derivative of x with respect to y and that this value is exactly equal to Tan ( I + U ); or dx/dy = f '(y) = Tan ( I + U ), then

I = arctan f '(y) - U [A-4]

Equation [A-4] becomes the starting point for the trace through an aspheric surface. The intercept height y at the surface is determined by simultaneous solution of the equations of the surface x = f(y) and the ray y / (L - x) = Tan U. For the paraxial trace, we recall that the sine and tangent (and arcsin and arctan) of small angles are equal to the angle.

References

[1] Mackintosh, A., et al (1986).Advanced Telescope Making Techniques, Vol 1. Willmann-Bell, Inc; Richmond, Virginia.

[2] Conrady, A.E. (1957).Applied Optics and Optical Design, Dover Publications, Inc; New York (Reprint from 1929 Oxford University Press edition).

[3] Kingslake, Rudolf (1978).Lens Design Fundamentals, Academic Press; New York.

[4] Schroeder, Daniel J. (1987).Astronomical Optics, Academic Press; New York.

[5] Mahajan, Virendra N. (1991).Aberration Theory Made Simple, SPIE Optical Engineering Press; PO Box 10, Bellingham, Washington 98227.

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