Astronomical Optics

Part 1: Basic Optics


Physical & Geometric Optics
Wavelength & Frequency
Refraction
Reflection
Interference
Scattering

Ray Tracing a Lens
Gaussian Concepts
Sign Conventions
Locating the Principal Planes
Image Attributes
Types of Lenses
Image Size & Location (Positive Lens)
Image Size & Location (Negative Lens)

Ray Tracing a Spherical Mirror

Lens Combinations
Thin Lens Formulas
Thick Lens Optical Analysis
Multiple Lens Optical Analysis
Eyepiece Prescription Data

Optical Materials

Optical Coatings


This page describes the optical principles necessary to understand the design and function of telescopes and astronomical eyepieces. Subsequent pages discuss the telescope & eyepiece combined, eyepiece optical aberrations, eyepiece designs and evaluating eyepieces. Included at the end of each page is a list of Further Reading that identifies the sources used to develop the pages and background information available online.

Physical & Geometric Optics

Light propagates in the form of oscillations in an electromagnetic field, which expand from a point light source as evenly spaced and concentric wavefronts. The energy carried in the oscillations is measured in quantum packets known as photons.

The radiation of light through space can be represented in two ways: (1) as actual wavefronts that expand concentrically and radially from the light source (analysis by physical optics), or (2) as imaginary light rays perpendicular to the wavefronts that extend radially from the light source and indicate the direction in which each part of the wavefront is moving (analysis by geometric optics). The basic parameters of optical elements described in this page are developed in terms of geometric optics.

In astronomical applications, light sources are so distant that the concentric wavefronts become a series of equally spaced parallel planes across the width of any practical telescope aperture. To illustrate: across the aperture of a 1 meter (39.4") telescope, light rays from a single point on the Moon, the closest astronomical object at 384,403 kilometers, diverge from parallel by no more than 1/384403000 of a radian or 0.0000026 millimeter, which is 0.0047 or 1/200 a wavelength of "green" light. Since the fabrication limits of the highest quality astronomical optics are around λ/20 wave, or 10 times larger than the wavefront divergence, optical calculations can assume perfectly flat and parallel wavefronts from a distant light source. The optical path of these wavefronts can be summarized as imaginary but analytically useful light rays, defined as parallel to each other and perpendicular to the wavefronts they describe.

Wavelength & Frequency

The distance between identical points on two adjacent wavefronts of light is the wavelength (λ) of the light. The frequency (ν) of light is the number of wavefronts that pass a fixed point in one second, or the cycles per second. The relationship between frequency and wavelength is governed by the speed of light in a vacuum, c:

c = ν·λ = 299,792,458 m·s–1 = 3 x 108 meters per second.

λ = c / ν.

Thus the frequency of "green" light at 550 nm is ν = c/λ550 = 299792458/0.00000055 (wavelength in meters) or 545 trillion cycles per second.

Light wavelengths are commonly measured in angstroms (10-10 meter), nanometers (10-9 meter) or micrometers (10-6 meter). Light that appears green to the human eye has a wavelength of about 550 nanometers — 0.00055 millimeters or 0.55 micrometers (hence submicron). Visible wavelengths of moderately bright light range from about 750 nm ("orange red") to 380 nm ("blue violet"), or 0.00075 to 0.00038 millimeters. Although they are interrelated, note that frequency is used to characterize the energy carried by a photon of light, while wavelength is used to characterize the optical behavior of wavefronts. Longer wavelength "red" light carries less energy, and therefore has a lower frequency, than short wavelength, high energy, high frequency "violet" light.

Refraction

Refraction, or the uniform change in the direction of a wavefront, is the characteristic behavior of light that is incident on the smooth surface of a transparent (transmitting) material.

Wavefronts of light have a uniform speed c in a vacuum. Light can also propagate through various transparent materials, such as air, water or glass, but each material slows the speed of light by a specific value — in some materials, to almost one third its vacuum speed. The speed of light in a vacuum divided by the speed of light in a refracting material (m) is the refractive index (n) of the material:

n = c/m

This is 1.00029 for air; 1.3333 for water; and anywhere from 1.4 to 2.0 for optical glasses.

When light crosses the boundary between materials of different refractive index, the average direction of the wavefront will be deflected in a new direction. The geometrical analysis of this deflection is nicely summarized in a classic illustration made by Christiaan Huygens in 1678 (diagram, left). The physical wavefronts are indicated as parallel white bands, and the geometric light rays as parallel thin white lines perpendicular to the wavefronts.

As the wavefronts of light AB, traveling across distance BC, encounter a refracting boundary AC, the speed of the wavefronts is slowed so that they now travel a shorter distance AB' in the same time. This bends or refracts the wavefronts in a different direction because adjacent points along each wavefront encounter the boundary at different times (t1 to t5) across its width AB.

Geometric rays are always (by definition) at right angles to the wavefronts they describe, so they create the right triangle ABC before refraction and AB'C after refraction, with side AC in common. Inspection of the diagram shows that the angle of incidence (θ1) is equal to the angle BAC, whose sine is equal to BC/AC; and the angle of refraction (θ2) is equal to the angle B'CA, whose sine is equal to B'A/AC. Since AC is a common denominator, the sines differ in the ratio BC/B'A. The diagram shows that this is the ratio of the speed of light in the two media, which is measured as the index of refraction, and therefore the sine ratio is equal to the inverse refraction ratio n2/n1.

This relationship is summarized as Snell's Law or the Law of Refraction, illustrated in the diagram by the yellow arrows and defined mathematically as:

sine(θ1)/sine(θ2) = n2/n1

or

sine(θ1n1 = sine(θ2n2

where n1 and n2 are the refractive indices of the two media that form the refracting boundary, and θ1 and θ2 are the angle of incidence and angle of refraction. These angles are measured from a line normal (perpendicular) to the boundary surface of the two media at the incidence point of a light ray. Both light rays and the line normal must lie in a single plane, and the incident and refracted rays will be on opposite sides of the line normal.

Snell's Law can be derived from the wavefront character of light, as the formation of a single wavefront from the coalescing circular wavefronts expanding from each incidence point along the surface area AC (shown as curved arcs along the line B'C). But for the purposes of describing the refractive effects of optical elements, identical angles of incidence and refraction can be calculated much more easily by applying simple trigonometry to a fictional light ray.

Transparent materials do not refract all wavelengths of light at the same angle. Instead, short wavelength "violet" light is refracted at a greater angle than long wavelength "red" light. This is the reason that prisms (or raindrops in the air) spread "white" light into the characteristic light spectrum. Variation in the refractive index across different wavelengths is called dispersion, discussed below under optical materials.

Reflection

Reflection is the characteristic behavior of light that is incident on a smooth and opaque (reflective) material. In this situation the Law of Refraction must be qualified in three ways:

• For angles of incidence equal to 0°, the ratio of the sines is zero and no refraction occurs: the light is slowed but its direction is not altered.

• If the angle of incidence is greater than a critical angle (θCR), defined as:

θCR = arcsin(n1/n2), n1 < n2

then the ray is reflected from the boundary rather than refracted through it. In a boundary with air, the critical angle is about 49° for water (n2 = 1.33) and 46° to 30° for optical glasses (n2 = 1.4 to 2.0).

• When reflection occurs, either from reflecting surfaces such as mirrors or glass surfaces positioned to the incident light at angles greater than the critical angle, the angle of reflection equals the angle of incidence but on the opposite side of and in the same plane as the line normal:

θ2 = –θ1

For all angles of incidence between the critical angle and 0°, significant reflection also occurs when the difference between the refractive indices of the two media is greater than about 0.25. This light energy is emitted by the surface at the reflected angle of incidence, again with the two angles on opposite sides of the line normal.

Interference

Although the optical behavior of materials can be described in terms of geometric rays, the image forming behavior of light is often better described in terms of physical optics. The most important example is interference or the interaction between light wavefronts from a common light source after they encounter physical edges or surfaces. Interference can arise in two ways.

The first is diffraction or the deflection of wavefronts around an occluding edge of material, such as a telescope tube or lens holder. This was demonstrated by the English naturalist Thomas Young as proof of the wave nature of light (diagram, left).

In this situation light composed of parallel wavefronts from a single light source encounters a thin opaque barrier divided by two parallel, closely spaced and very narrow slits. The slits allow a part of the wavefront to pass, but as it does so the wavefront expands concentrically from each slit aperture.

The slits are positioned so that the oscillations of separate wavefronts must be either coincident or opposed. Where they coincide a wavefront reinforcement occurs and a band of light will be projected onto a screen placed behind the slits. Where they are opposed a wavefront cancellation or destructive interference occurs — the peak of one wavefront is cancelled by the trough of the opposing wavefront — and this produces a dark band on the projection screen.

The interference effects produced by physical obstructions are the origin of the diffraction artifact produced by a star or "point" light source, and the bright and dark pattern of speckles produced in the image of a star disrupted by atmospheric turbulence.

The second way in which interference can arise is through the reflection of the wavefront from thin, parallel layers of transparent media. This was first investigated by Isaac Newton as the interference fringes that appear between two closely spaced sheets of glass; it commonly appears as iridescence in the interior of an abalone shell or in an oil slick on wet pavement.

As just explained, significant light is reflected from an optical boundary between media, such as air and glass, whose refractive indices differ by more than 0.25. This reflection can be minimized by layering the boundary with a third material that has a refractive index between the two materials. This creates two surfaces where reflection can occur, but these reflections can be used to cancel each other.

Interference is governed by the spacing between the first and second reflective surfaces. If the spacing is an even number multiple of the light wavelength divided by 4 (e.g., 2/4 or λ/2) then light reflected from the first surface boundary is one wavelength out of phase with light reflected from the second boundary, and wave reinforcement occurs; if the spacing is equal to an odd number multiple (e.g., 1/4 or λ/4) then light reflected from the first surface boundary is 1/2 wavelength out of phase with light reflected from the second boundary, and cancellation occurs (diagram, above). On this principle, quarter wave optical coatings are commonly used to reduce unwanted reflections from the surfaces of glass lenses.

Scattering

If there are not just two edges or surfaces diffracting light, but hundreds or millions, then the cumulative effect is scattering or a random deflection of some part of the light away from its original path.

If a refracting or reflecting surface is not perfectly smooth at the atomic level, it creates random "error" in the angles of incidence across an incident wavefront. This scattering produces a random variation in the average angle of refraction or reflection that appears as a cloud of diffuse light around the image of a bright light source, such as the star Formalhaut (image, right).

In lenses, scattering is minimized by the absence of scratches, rough spots, pits or small bubbles in the glass, imperfections described by a scratch/dig specification. These imperfections can be somewhat ameliorated by the vacuum application of optical coatings, but can only be minimized by careful glass manufacture, grinding and polishing. In high quality optics scatter is usually due to water condensation (dew), dirt or grease on optical surfaces, or to humidity, dust or exhaust pollution in the atmosphere. In both cases, the behavior of scattering depends on the size of the optical imperfection or atmospheric particles in relation to the dimensions of light wavelengths, so it is also best analyzed by physical optics.

Ray Tracing a Lens

Furnished with the geometrical description of light, the Law of Refraction and information about the refraction and dispersion of optical materials, we can analyze the basic attributes of any optical system. Optics can be divided into two levels of geometrical analysis:

• First order analysis was developed by Carl Friedrich Gauss in his Dioptrische Untersuchungen (1841), expanding on analyses by Isaac Newton, Johannes Kepler and Huygens. It deploys Snell's law and a simplified trigonometric analysis to determine the focal length, magnification and power of an optical system, which yields the location, size and orientation of the image it creates.

• Third order analysis was developed through the combined efforts of several 19th century mathematicians and opticians to describe optical aberrations, or departures of focused light from the optimal image location and size defined by the first order analysis.

First order analysis is developed from the simplified properties of paraxial light rays. These rays are refracted by the surface of a lens very close to its vertex or intersection with the optical axis. In that tiny area the refraction of light by the curved surface of a lens can be diagrammed as the refraction of light by a plane perpendicular to the optical axis. This allows the arithmetic calculation of image size and location without the use of trigonometric functions (sine or tangent).

This paraxial approximation can usefully describe the optical behavior of moderately curved lenses within very small angles, where the quantities of tangent, sine and degree converge to the same value. Thus, an angle of 0.50°, roughly the angular width of the full Moon, is equal to a tangent of 0.546 and a sine of 0.479; an angle of 1/60 or 0.0166667 degree (1 arcminute), roughly the angular diameter of Jupiter at opposition, is equal to a tangent of 0.0166682 and a sine of 0.0166659. Calculations using the paraxial approximation may treat the three quantities as interchangeable.

Exact analysis is possible by calculating the physical properties of light wavefronts, but the paraxial approximation is invaluable for its simplicity and power to describe the basic attributes of an optical system.

Gaussian Concepts

In the Gaussian analysis, the optical system is assumed to provide a perfect (distortion free and precisely focused) image at the optical axis: analysis is only used to define the location, size and orientation of this perfect image.

The analysis builds on the fact that the behavior of an optical system can be diagrammed in relation to three pairs of cardinal points: the focal points, the principal points and the nodal points. However, the nodal and principal points exactly coincide for lenses or mirrors surrounded by air — the standard situation in astronomical optics — so only the focal and principal points are needed to describe the system optical behavior.

The diagram (below) illustrates the key concepts and terminology in the first order analysis of a schematic biconvex lens surrounded by air. These concepts can also apply to single or compound lenses of any type, treated as a single optical unit or "black box".

A few basic properties of the optical system are assumed to apply. All optical components are constructed as solids of rotation, which means their refracting surfaces are symmetrical around an axis. The axes of rotation for all surfaces are identical with a single optical axis when light is passed through the optical system. The intersection of a refracting surface with its optical axis is the vertex of the surface (green dots in the diagram).

Lens surfaces are assumed to be (and in most commercial eyepieces and refractor objectives are) manufactured as sections of a sphere, defined by a radius of curvature originating from a center of curvature located on the optical axis. A two sided lens has two centers of curvature (denoted r1 and r2) and two radii measured along the optical axis from the corresponding vertex. If one side of the lens is a flat (plane) surface, the radius of curvature is zero.

Light rays arise from an object or object space (e.g., area on the celestial sphere) intersected by the optical axis and conventionally diagrammed to the left of the lens. These rays pass through the lens from left to right and terminate in an image plane perpendicular to the optical axis and intersecting the optical axis at a focal point located on the right of the lens. (Note that all real optical images are in fact focused onto a surface that is more or less spherical, with its own radius of curvature; the image plane is the paraxial simplification.) The image receptor (observer's eye, CCD chip, photographic film) is therefore diagrammed at the right of the lens oriented toward the left. The object and image points, and the matching rays connected with them, are termed conjugate.

The focal point can be located by means of collimated rays that are parallel to the optical axis and to each other. If a collimated ray from a point on the object is extended through the lens, and the corresponding oblique image ray is extended back from the conjugate image point, they will intersect in a principal plane perpendicular to the optical axis and intersecting the optical axis at a principal point. All object rays and conjugate refracted image rays will intersect in the same principal plane.

Finally, all refracting optical systems are reversible: they can refract light passing through them from left to right or from right to left. This creates a focal point on each side of the lens. In a thick or compound element (consisting of two or more lenses) there are also two principal points and corresponding principal planes (diagram, above). The first principal plane, first principal point and first focal point are assigned to the surface where light enters the lens; the second principal plane, second principal point and second focal point are assigned to the surface where light exits the lens.

This basic layout defines a number of related concepts, specific labels and symbols, also illustrated in the diagram (above):

• Collimated light is a beam of light originating from a single point on an infinitely distant source; all rays in the beam are parallel to each other and all wavefronts in the beam are parallel planes.

• An oblique ray is an object ray at an angle to the optical axis.

• A chief ray or principal ray is an oblique ray that passes through the first principal point, exits from the second principal point at the same angle to the optical axis, and intersects the image plane at the conjugate image point. Because the ray is not refracted by the optics, it is equivalent to a ray passed through a pinhole at the principal point.

• An axial ray or marginal ray originates from the point where the optical axis intersects the object or object space, enters the first principal plane at the outer edge or aperture radius of the lens, then exits from the second principal plane to intersect the optical axis at the focal point. For objects at infinite distance, the incident marginal rays form a collimated beam parallel to the optical axis and perpendicular to the first principal plane.

• For elements notated by the same letter symbol before and after an optical change, all the elements on the image side — the heights or angles of rays after refraction or reflection by an optical surface — are denoted by an apostrophe (e.g., ƒ and ƒ').

• An object ray intersects the first principal plane at a specific incidence aperture height (y) or distance from the optical axis, and exits from the second principal plane at the refraction aperture height (y'). In the standard Gaussian analysis y = y' (diagram, above).

• The perpendicular distance from the optical axis to the most extreme object point is the object height (h), and the distance from the optical axis to the most extreme (conjugate) image point is the image height (h'). If collimated rays originate in an infinitely distant object then the relation of h to the object height is undefined.

• The effective focal length (ƒEFL) is the distance from the second principal point to the second focal point (ƒ'). The back focal length (ƒBFL) is the distance from the back vertex of the lens to the second focal point. The front focal length (ƒFFL) is the distance from the front vertex of the lens to the first focal point (ƒ).

• In most Gaussian equations the front and back focal lengths are assumed to be equal, and multiples of the focal length can be used to specify the location of images behind the lens and/or objects in front of the lens.

In first order analysis all angles are measured in radians; for the very small slope angles produced by paraxial rays, the tangent and sine of an angle are equal to the angle itself.

In astronomical optics most converging mirrors and a few wide angle eyepieces utilize aspheric surfaces such as ellipsoids, paraboloids or hyperboloids (respectively solids of rotation generated by an ellipse, parabola or hyperbola rotated around its major axis). In first order analysis these are approximated as a spherical surface with a radius that produces the equivalent focal length.

Sign Conventions

In order for Gaussian equations to produce correct numerical values in all variations, the algebraic sign of measured quantities used in the equations must follow arbitrary but specific sign conventions:

• As explained above, light is represented as radiating left to right; the object is always to the left of the lens and the real image is always to the right.

• A measurement reference point, usually a principal point of the optical system, is used as the origin of a Cartesian coordinate system. Lengths measured along the optical axis from left to right indicate the direction of incident light and are positive in sign; lengths measured right to left against the original direction of light are negative. Lengths measured upward from the optical axis are positive; lengths measured downward are negative.

• Important exception: both focal lengths of a converging lens are positive, and both focal lengths of a diverging lens are negative. The single focal length ƒ of a mirror is positive for a concave (converging) mirror and negative for a convex (diverging) mirror.

• The angle of an oblique ray measured at its intersection with the image plane or optical axis, or its originating point in the object space, is positive if the oblique ray must be rotated clockwise to be made parallel with the optical axis, and negative if it must be rotated counterclockwise. (Rotation is always through the smaller angle to the optical axis.)

• Angles of incidence and refraction are positive if they must be rotated clockwise to reach the line normal of the refracting surface, and negative otherwise. (Rotation is always through the smaller angle to the line normal.)

• A radius of curvature and center of curvature are positive if they lie to the right of the surface described, and negative if they lie to the left.

Application of these conventions is illustrated in the optical calculations below.

Locating the Principal Planes

Before developing the Gaussian formulas for optical systems, it will be helpful to illustrate how Snell's Law and basic trigonometry are applied to locate the second principal plane, second principal point and effective focal length of a symmetrical biconvex lens (diagram, below).

In this example, the lens has a refractive index of nL = 1.6, surrounded by air with a refractive index of nA = 1.0. We follow the path of a single collimated light ray (parallel to the optical axis of the lens) that is incident on the front surface of the lens at an aperture height y from the optical axis.

The lines normal to the front surface of the lens are produced as lines radiating from the second center of curvature (r2, shown in the diagram on the far right). Since the lens is symmetrical, the first center of curvature (r1) is of the same length and outside the diagram on the left.

The angle of the line normal to the optical axis at the entrance point i is the arcsine of y/r2. Because we assume the lens produces a perfect focus, the aperture height y can be set to any reasonable value for the analysis. In the example, I've chosen y = 10 and r2 = 30.9: the incidence point then defines a sine of 0.309 to the line normal, giving an entrance incidence angle of θ1 = 18°. (Note the application of the sign conventions to the angle quantities.)

The ratio between the indices of refraction between the first and second materials (air and glass) is 1.0/1.6 = 0.625, so the sine of the refracted ray is 0.309·0.625 = 0.193, for an angle of refraction θ2 = 11° on the opposite side of the line normal, or –7° in relation to the optical axis. A small amount of light is reflected from the lens surface at an angle –θ1 to the line normal (this permits us to see the silhouette of our head in an eyepiece lens). The rest of the light ray continues along the refracted path inside the lens.

The next step is to calculate the thickness of the lens along the refracted light path. This is a function of the relative curvature of the front and back surfaces, the separation t between the two vertices at the optical axis, and the refractive index n1 of the glass. That calculation is explained below; for now, we identify the exit point i' by construction in order to find the focal point and second principal plane.

When the light ray reaches the back surface of the lens at point i', the exit aperture height is y' = 9.2 and the line normal drawn from r1 is arcsin(9.2/30.9) or 17° to the optical axis. Then θ3 = –24° from the interior ray to the line normal, for a sine of –0.407. The ratio of refraction indices is now inverted as 1.6/1.0 = 1.6, so the sine of the refracted angle is –0.407·1.6 = –0.651, which is θ4 = –41° to the line normal outside the lens or –24° to the optical axis.

On this oblique path the image ray continues until it intersects the optical axis at the effective focal point ƒ' with an image ray slope of u' = –24°. (To avoid misunderstanding, note that only a collimated ray parallel to the optical axis will intersect the optical axis at the effective focal point.)

Diagrammatically, the original light ray can be continued from point i as a straight line parallel to the optical axis, and the image ray can be extended backward from point i', until the two rays intersect at point I at aperture height y. The angle of the image ray slope is then defined as u' = –y/ƒ' radians. Since y is given and u' has been calculated, the focal length is calculated as ƒ' = –y/u'.

A plane through the point I and perpendicular to the optical axis is the principal plane of the lens, and the intersection of this plane with the optical axis is the principal point of the lens. The effective focal length (ƒ') is simply the distance along the optical axis between the second principal point and the focal point.

Image Attributes

At this point it is useful to introduce the terminology for the four attributes of an optical image.

(1) An image is real if the image is formed at a location where light rays have actually converged: the sun concentrated by a magnifying glass or light focused by a photographic lens. Real images can affect photosensitive media and can be projected onto a surface. An image is virtual if the image appears at a location where no light is focused: for example, the image in a mirror. Virtual images occur when light rays form a focus when extended backward; they are formed by negative lenses or by placing an object inside the focal length of a positive lens. A virtual image has no location in physical space and therefore no photosensitive media can be placed at their apparent location.

(2) An image is erect if it is oriented vertically in the same way as the object, or inverted if the image is reversed top to bottom.

(3) An image is normal if it is oriented left to right in the same way as the object appears, or reverted if the image is reversed left to right. (Images rotated 180° in relation to the object are both inverted and reverted.)

(4) Finally, an image can be enlarged, actual size or reduced in comparison to the physical size of the object or to the angular width of the object as it appears to the unaided eye.

As the diagram shows, nearly all astronomical telescopes rotate the object image; none actually invert the image, despite the common use of inverting telescope to describe them. Note also that an inverting eyepiece actually produces an erect, normal image: the image is rotated by the objective, not the eyepiece. The exception is the Gregorian telescope, because it reflects light from a secondary placed beyond the focal point of the primary mirror. This rotates the image a second time.

Types of Lenses

The possible combinations of spherical and plane surfaces that can be used to make a lens fall into six generic types of positive or negative lenses, shown in the diagram (below) with two indications of their relative refracting power: the effective focal length and a schematic tracing of two collimated rays. These illustrations show lenses of a typical optical glass (nL = 1.6) in air; nominal radius for the strong curvature (in the meniscus lenses) is r = 50mm and for the weak curvature (all other lenses) is r = 100mm in an aperture diameter Do = 70mm.

The biconvex, positive meniscus and plano convex designs are positive lenses used to concentrate parallel rays of light to a single focus located behind the lens or to reduce the focal length of an optical system. The biconcave, plano concave and negative meniscus designs are negative lenses used to expand parallel rays of light away from a negative focal point (located in front of the lens), typically to extend the focal length and increase the relative aperture of an optical system.

The two principal points are shown by the red dots: even when symmetrical, "thick" lenses have two principal points for the refractive effect in opposite directions. In general, the principal points in a biconvex lens are both internal and spaced about 1/3 the distance from the front to back vertices. In a lens with a plane surface, one principal point is the plane vertex of the lens. In a meniscus lens, one or both principal points can be external to the lens.

The biconvex and biconcave lenses have roughly twice the refractive power of the plane and meniscus forms. The meniscus lenses are most sensitive to changes in the lens thickness, while the plano convex and plano concave lenses are unaffected by variations in thickness. In asymmetrical positive lenses, the power of the lens is greater when the surface with the shorter radius (or the curved surface in plano lenses) is oriented toward the object. In asymmetrical negative lenses, the reverse holds: the power is greater when the greater power or curved surface is oriented away from the object.

Used as single lenses, positive lenses always produce real, inverted, reverted images, while negative lenses produce virtual, erect, normal images.

Positive meniscus lenses are commonly used in eyeglasses to correct for farsightedness or difficulty focusing on objects at short distances from the eye (hyperopia). This occurs because the lens to retina focal distance of the eye is too short, or the cornea and lens of the eye lack optical power, or the lens has hardened with age and cannot adjust to focus on near objects (presbyopia). Negative meniscus lenses are used to correct nearsightedness or difficulty focusing on objects at long distances (myopia), which occurs because the cornea and lens are too strongly curved, or the lens to retina distance in the eye is too large.

Image Size & Location (Positive Lens)

Now that the underlying logic of first order analysis has been illustrated and the concepts defined, we can turn to the basic formulas that result — first for positive or converging lenses, then for diverging or negative lenses.

In the Gaussian model, the optical effect of a lens can be analyzed through the use of three analysis rays. The diagram below shows this analysis applied with two principal planes, which is done by disregarding the space between them.

If it is acceptable to assume that the optical effect of the lens thickness (the distance between the front and back incidence points of a light ray) is inconsequential to the slope of the exiting image ray, then the lens can be modeled by a single principal plane located at the center of the lens, in what is called a thin lens model of the optics. This directly yields the effective focal length (measured from the single centered principal plane) as:

1/ƒ' = (nL–1)·(c1c2).

where c = 1/r. Note that c1 is always numerically negative (by the sign conventions) so the term (c1c2) is never zero; also reversing the lens (direction of light) produces the same focal length but with opposite sign:

1/ƒ = (nL–1)·(c2c1) = –(1/ƒ')

which becomes 1/ƒ' (positive), again by the sign conventions. For a symmetrical biconvex lens where r1 = 10 cm, r2 = –10 cm and nL = 1.6:

1/ƒ' = (1–1.6)·((1/–10)–(1/10)) = 1/(–0.6·–2/10) = 1/0.12, or ƒ' = 8.33 cm and ƒ = 8.33 cm.

The "thin lens" analysis originated in 18th century optics in which the main applications were low power, long focal length spyglass lenses, simple eyepieces and very thin eyeglass lenses. It is applicable to any lens where the focal length is much larger than its maximum thickness. That criterion is ambiguous and only suggests how well a physical lens might be described by the idealized thin lens model: in particular, it implies that wide angle or short focal length (strongly curved) lenses cannot be analyzed in this way.

Given that we have already located the principal points and focal points from the formula above, we are interested to find the location, size and orientation of the image formed by a specific object at a distance s from the lens.

From a single object point off the optical axis at object height h, construct:

A – A collimated light ray from the object point to the first principal plane of the lens, and exiting from an equal height in the second principal plane as an oblique ray through the second focal point ƒ'.

B – An oblique ray from the object point through the first principal point p1 in the first principal plane, and exiting from the second principal point p2 at the same angle to the optical axis.

C – An oblique ray from the object point through the first focal point ƒ to the first principal plane, and from an equal height in the second principal plane as a collimated light ray (parallel to the optical axis). For an object at a finite distance, only two rays, A and C, are necessary to define the image; ray B is shown as a dashed line because it is optional. Rays A and C only require the location of the two focal points, given by the formula above.

For an object at an infinite optical distance — the distance at which large changes in the object distance do not affect the location of the focal point — the use of ray C becomes optional. Instead the location of the image (the focal point) is determined by ray A and the size of the image (h') is determined by the focal distance ƒ multiplied by the field angle (in radians) between the optical axis and ray B.

For an object at finite optical distance, all three rays will intersect at a single point that is not on the optical axis. Given the focal distances ƒ and ƒ' and object height h, this construction defines:

• the image height h'
• the object distance s
• the image focal distance s'
• the ƒ to object distance x, and
• the ƒ' to image distance x' (all labeled in the diagram). The image location is the distance from the second principal point to a line through the ray intersection that is perpendicular to the optical axis; the image size is determined by the distance of the intersection from the optical axis, and the image orientation is shown as the intersection lying above or below the optical axis (indicated by a positive or negative sign in the calculation of the image height or the lens magnification).

Note the explicit symmetry in the procedure: an object height h at distance –s produces an image height –h' at distance s', but an object height –h' at distance s' would produce an image height h at distance –s. This symmetry is characteristic of conjugate points (joined in a reciprocal relationship) in an inverting optical system.

Reasoning from the various congruent triangles created by the three rays, and given the object distance s and focal length ƒ, one can derive the thin lens formula:

1/s' = 1/s + 1/ƒ'

and the other parameters:

[ƒ to object distance] x = s+ƒ

[ƒ' to image distance] x' = –(ƒ'2)/x

[object distance] s = s'·ƒ'/(ƒ's')

[focal distance] s' = s·ƒ'/(s+ƒ')

[object size] h = –(h'·x)/ƒ'

[image size] h' = –(h·x')/ƒ'

and several equations for the image magnification (m), which in the Gaussian analysis refers to the ratio of the linear size of the image over the physical size of the object:

m = h'/h = ƒ'/x = –x'/ƒ' = ƒ'/(s+ƒ')

The curvature (c) of the lens is the reciprocal of the radius of curvature, and increases with shorter radius; the power (φ) of the lens is the reciprocal of the effective focal length, and increases with shorter focal length:

c = 1/r

φ = 1/ƒ'

Refractive power can be expressed in a standardized measure, the diopter, which is power measured in meters (1/ƒm). The shorter the focal length, the more the objective or eyepiece has refracted (or reflected) light, and the higher the diopter. Thus the diopter for an objective with ƒo = 500mm is 1/0.5m or 2; the diopter of an eyepiece with focal length ƒe = 20mm is 1/0.02m or 50.

As an illustration and check on your practice calculations, the relationships among these different measures are illustrated in the table below for a constant object size of 2.5 cm and a symmetrical biconvex lens with an effective focal length of 25 cm.


s/ƒ object
distance (cm)

s
ƒ to object
distance (cm)

x
object
size (cm)
h
focal
distance (cm)

s'
ƒ' to image
distance (cm)

x'
image
size (cm)

h'
magnifi-
cation

m

–0.02–0.5024.502.5–0.51–25.512.551.02
–0.10–2.5022.502.5–2.78–27.782.781.11
–0.20–5.0020.002.5–6.25–31.253.131.25
–0.50–12.5012.502.5ƒ' = –25.00–505.002.00
–0.67–16.758.252.5–50.76–75.767.583.03
–0.80–20.005.002.5–100–12512.55.0
–0.90–22.502.502.5–225–2502510
–0.99–24.750.252.5–2475–2500250100
–1.00–ƒ' = –25.0002.5....
–1.01–25.25–0.252.525252500–250–100
–1.10–27.50–2.502.5275250–25–10
–1.25–31.25–6.252.5125100–10–4.0
–1.5–37.50–12.52.57550–5.0–2.0
–2–50.0–252.55025–2.5–1.0
–5–125–1002.531.256.25–0.625–0.25
–10–250–2252.527.782.78–0.278–0.111
–50–1250–12252.525.510.51–0.051–0.020
–500–12500–124752.525.050.05–0.005–0.002
–5000–125000–1249752.525.010.005–0.0005–0.0002
–50000–1250000–12499752.5ƒ' = 25.000.0005–0.00005–0.00002

Again, note carefully the sign conventions necessary for consistent calculations:

—for a converging lens both focal lengths are positive, for a diverging lens both focal lengths are negative;
s is measured from the first principal plane and is always negative;
x is measured from the first focal point ƒ and is positive if the object is between the lens and the focal point and negative if the object is farther from the lens than the first focal point;
s' is negative if measured from the first principal plane (in front of the lens) and positive if measured from the second principal plane (behind the lens);
x' is measured from the second focal point ƒ' and is negative for a virtual image (in front of the lens) and positive for a real image (behind the lens);
—when x' is negative, the image location measured from the first principal plane is equal to x'+ƒ'.

If only one principal plane is used in the ray tracing (the "thin lens" model), then all distances are measured from it.

This analysis method indicates the changes in the image location, size and orientation produced by objects at different distances from the lens (diagram, above; note that ƒ is negative because it is actually units of –s):

Objects that are far distant enough to make the principal rays A, B and C effectively parallel (collimated) are said to be at optical infinity, and will form an inverted image at the focal point ƒ'. If we take the minimum discernible image divergence from the focal point as equal to a wavelength of light (~0.0005 mm), then an object distance of roughly 50,000ƒ' (e.g., 12.5 kilometers for an ƒ'=25cm lens) is "at infinity" for calculation purposes, with a magnification of no more than –ƒ/50,000 or –0.00002.

(a) When the object is between –50,000ƒ to –2ƒ in front of the lens, the image will appear inverted between 1ƒ and 2ƒ behind lens with a magnification between –ƒ'/50,000 and –1.

(b) When the object is located exactly –2ƒ in front of the lens, the image will appear inverted 2ƒ behind the lens (s' = s) and actual size (h' = –h, m = –1).

(c) If the object is located between –1ƒ and –2ƒ in front of the lens, the image will appear inverted at more than 2ƒ behind the lens, and larger than actual size (–m > 1).

(d) If the object is located exactly at –1ƒ no image can form (the ray C through the front focal point is undefined).

If the object is located at a distance less than –1ƒ then the analysis rays must be extended backward from their intersection points with the principal plane. They intersect in front of the first principal plane to form an erect (h'>0) and enlarged (m > 1.0) virtual image. Near the limit, where s approaches zero, the virtual image forms at the apparent location of the object itself and is actual size.

Note that it is not possible for an object at any distance to form an image between the lens and the second focal point (at a distance less than the effective focal length).

Image Size & Location (Negative Lens)

The diagram below shows the "thin lens" analysis for a negative (diverging) lens, again with two principal planes; the analysis is the same if the two planes are replaced by a single plane.

The major difference from the positive lens analysis is that the effective focal length is measured in front of the lens, and both focal points have negative distances.

From a single object point off the optical axis at object height h, construct:

A – A collimated light ray (parallel to the optical axis) from the object point to the first principal plane of the lens, and exiting from an equal height in the second principal plane as an oblique ray (yellow line) that when extended backward as a virtual ray (magenta line) passes through the first (effective) focal point –ƒ'.

B – An oblique ray from the object point through the first principal point p1 in the first principal plane, and exiting from the second principal point p2 at the same angle to the optical axis.

C – An oblique ray from the object point to the first principal plane that when extended forward from the second principal plane intersects the second focal point –ƒ, but is extended forward from the second principal plane as a collimated light ray (parallel to the optical axis) and is extended backward as a collimated virtual ray.

Given the first and second focal distances –ƒ' and –ƒ and object height h, this construction defines the object distance s (always negative), the image focal distance s' (always negative), the ƒ to object distance x (always negative), the –ƒ' to image distance x' (always positive), and the image height h' (always erect) as shown in the diagram (above).

A real image cannot be produced by the diverging refracted light rays (yellow lines): instead a virtual image is formed at the intersection of the virtual rays (magenta lines) extended backward from the emergent light rays. For an object at finite optical distance, all three virtual rays will intersect at a single point that is not on the optical axis. For an object at infinite optical distance the image is again located by means of only rays B and C from an off axis object point.

The focal points both have negative distances from their respective principal planes, which means (according to the sign conventions and the thin lens formulas) that the image focal distance s' is always negative and the ƒ' to image distance x', the image size h' and the magnification m are always positive. (A virtual image cannot form farther from the lens than the effective focal length –ƒ'.)

This analysis shows that a negative lens always produces an erect but virtual image when viewed from the side opposite the object. For an object at infinity, analysis ray B shows that the virtual image forms at the first focal point ƒ' with a height equal to h' = radians(αƒ', where α is angle of the chief ray B to the optical axis. As the object is brought closer to the lens, the virtual image becomes larger and also approaches the lens; if the object is located at the first focal point then the virtual image is located at s' = ƒ/2 and is half actual size (m = 0.5).

Ray Tracing a Spherical Mirror

In a mirror there is only one optical surface, so analysis is simplified. The major variations are for objects at finite or optically infinite distances, for concave or convex mirrors, and for real and virtual images.

The diagram (below) shows the analysis for a concave and a convex mirror with the object at a finite distance, and for a concave mirror with the object at infinite optical distance (the astronomical case).

The sign conventions for a concave (converging) mirror are that the single focal point and image focal distance are positive, because they are measured in the direction that the light is traveling; the object distance is measured opposite the direction of the light and is therefore negative. For a convex mirror the focal point and image focal distance are negative, because they are opposite the direction of the reflected light. As before, object and image heights and image angles are negative when below the optical axis.

The image size and location are defined as with lenses, by means of three analysis rays from a single off axis object point:

A – A ray from the object point, parallel to the optical axis and reflected by the mirror.

B – A ray from the object point through the mirror focal point and reflected by the mirror.

C – A ray from the object point through the mirror center of curvature (r) and reflected by the mirror; by definition this ray is reflected back through the center of curvature.

D – A fourth ray, from the object point to the mirror vertex, can be used instead of ray C for aspheric (ellipsoid, paraboloid, hyperboloid) mirrors that do not have a center of curvature.

The first diagram (above) shows the ray tracing for a concave (positive, converging) mirror with an object point of height h at finite distance o from the mirror vertex. The tracing shows that all three rays converge at a real image in front of the mirror, shifted from the focal point ƒ toward the object at a focal distance i with a reduced image size –h'. A line through this point and perpendicular to the optical axis will indicate the location of the real image.

In contrast, an object placed inside the focal point of a concave mirror will produce an enlarged, erect and virtual image behind the mirror. This is the standard design of a magnifying vanity mirror, which is constructed so that the focal point is farther from the mirror than the typical viewing distance (behind the viewer's head).

The middle diagram (above) shows the ray tracing for a convex (negative, diverging) mirror, again with an object point at height h and at finite distance o from the mirror vertex. The tracing shows that rays B and C are reflected before they pass through their target points behind the mirror, so all three reflected rays are extended in the opposite direction as virtual rays behind the mirror, where they converge at the location of a reduced, erect virtual image at focal distance i which again is shifted from the focal point ƒ toward the object. Since the object cannot be placed between the mirror and the focal point, no real image is formed. (The exception is when the convex mirror reflects rays that are convergent, for example when used as the secondary mirror in a two mirror telescope.)

The relationships between object distance, image distance and focal length are:

1/ƒ = 1/o+1/i

1/i = 1/o–1/ƒ.

where 1/o is effectively zero for an object at infinite optical distance and 1/ƒ = 1/i. The center of curvature and focal point are related as

ƒ = r/2

which means ray A scales the image size to the object size as

h' = h·i/o

with magnification

M = i/o = h'/h.

and for objects at infinite optical distance the magnification is infinitely small, all objects are reduced to points (h' = ~0), and the optical system only images the angular separation between objects (illustrated by the two "stars", one on axis and the other off axis, in the bottom diagram).

The last diagram (above) shows the ray tracing for a concave mirror with an object at infinite distance — the usual astronomical situation. In this case all rays from the object are parallel, so rays B and D are sufficient to define the image location and size. For off axis objects, both rays define the angular width or field height α measured from the optical axis or center of the image field. Any collimated ray A will identify the focal point, which for a spherical mirror is found as ƒ = r/2.

The image forms at the focal point ƒ, and the field height h' of any part of the image is equal to the focal length times the incidence angle α (in radians) from the object height to the mirror vertex or through the focal point. This angle is opposite in sign to the original angle α because it is reflected on the opposite side of the optical axis:

h' = –radians(αƒ

The average diameter of the Moon is approximately 30 arcminutes or 0.0087 radians; its disk will be 21.8 mm wide at the focal plane of a 250 mm ƒ/10 mirror.

The diagram also shows that as the aperture height y increases, the location of the mirror surface shifts by a distance e from the vertex location due to the surface curvature. The surface of rotation that focuses all rays parallel to the optical axis at the focal point is determined by a parabolic function, so this shift is:

y2 = –4ƒe and e = –y2/4ƒ.

For a 250mm (10 inch) mirror figured to ƒ/4 (ƒ = 1000mm), the depth of curvature required from mirror edge to center (y = 125mm) is –3.9mm (0.15").

Single mirrors, like lenses, produce a curved focal surface whose radius ρ depends on the focal length, roughly as:

ρ = ƒ

Lens Combinations

Astronomical eyepieces are designed as two or more lenses or components mounted along a common optical axis. Their design and evaluation requires optical formulas that extend and generalize the analytical framework described above for a single lens. This section first outlines the traditional thin lens formulas, then more general analytical formulas.

Thin Lens Formulas

A number of optical formulas were derived in the 18th century to characterize the behavior of "thin lens" combinations calculated in relation to a single principal plane for each optical element. We start with this framework for its simplicity and to illustrate basic relationships between two lenses and their spacing as an optical system.

The diagram (below) shows a simple two element eyepiece with an effective focal length of 30mm, analyzed by means of a single principal plane through the center of two positive lenses with focal lengths of 57mm (for the front or field lens) and 35mm (for the back or eye lens), at a separation of 25mm and with a field stop diameter (do) of 23.5mm; the lens indices of refraction and surface curvatures are given as well. This illustrates the application of the thin lens formulas for front and back focal length (ƒFFL and ƒBFL), effective focal length (ƒEFL) and apparent field radius (β), which yields an apparent field of view (AFOV) for this simple eyepiece of 43°.

The design procedure is first to select two lenses of the appropriate aperture and power (φ1 = 1/ƒ1 and φ2 = 1/ƒ2). The lens spacing (the distance between their principal planes) is determined by trial and error or by analytical formula, for example to minimize chromatic and spherical aberration or to optimize eye relief.

Once the interlens spacing is fixed, the front focal length (ƒFFL) is calculated to determine the location of the eyepiece focal plane and field stop. From this point the effective focal length (ƒEFL) is calculated and measured off to locate the eyepiece principal plane. The back focal length (ƒBFL) is calculated and measured from the eye lens principal plane to determine the location of the eye point and the amount of eye relief. Finally, the radius of the eyepiece apparent field of view is determined by the angle β, which is constructed from the eye point to the field stop diameter projected to the eyepiece second principal plane, which is defined by measuring the effective focal length forward from the eye point.

The lens combination can be traced with a pair of collimated rays at the edges of the field stop, called marginal rays. These are sufficient to define the eye point and apparent field of view. Instead, divergent rays (as would be created by a telescope objective) are shown, to illustrate how the lenses both bend the light to a focal point and focus the divergent rays into parallel beams or "pencils" of light that intersect to form the diameter of the exit pupil.

Note the following:

• In the thin lens (Gaussian) model, the focal length of a single spherical lens can be calculated from its index of refraction (nL) and the front and back curvatures (1/r) with the lensmaker's equation as:

1/ƒ = (nL – 1)·[1/r1 – 1/r2]

and the effective focal length of a compound lens (where d = 0) can be found as:

ƒEFL = (ƒ1·ƒ2)/(ƒ1+ƒ2).

Recall that the focal length of a negative (diverging) element is negative.

• The front focal length ƒFFL determines the focal plane of the lens combination used as a magnifier. It defines the optimal location of the field stop.

• The back focal length ƒBFL determines the location of the eyepiece focus point (eye point), and is measured from the eye lens principal plane.

• The two effective focal lengths ƒEFL determine the principal planes of the lens combination. Measured forward from the eye point, the back ƒEFL locates the intersection of the marginal rays with the field stop diameter, which determines the marginal ray angle β.

• The apparent field of view (the angular diameter of the field stop viewed from the eye point) is derived from the radius of the field stop (do) as double the exit angle β of a marginal ray:

AFOV = 2·arctan(0.5do/ƒEFL) = 2β

• To maximize eye relief, the lens with the higher power is usually used as the eye lens. (To verify this, note that ƒFFL is shorter than ƒBFL.)

• A point discovered by Huygens, and implemented in the eyepiece design that bears his name, is that two lenses of the same refractive index but different powers produce the most achromatic image when

d = (ƒ1 + ƒ2)/2.

The Kellner modification of the Huygenian design handles chromatic aberration by using an achromatic doublet for the eye lens.

The graph (below) shows the optical effect on this specific eyepiece when the focal length of either the front or back lens, or the distance between them, is reduced up to 25mm, while holding the other elements constant. (A field stop diameter of 19.5 mm is used to calculate AFOV.)

As the focal length of either the field lens or eye lens is made shorter, the effect on eye relief (red line) is the same: it is reduced. Higher power lenses yield less eye relief, but not as a constant proportion of the focal length. Thus, the field lens focal length is reduced by 43% (from 57 mm to 32 mm), and the eye lens focal length by 71% (from 35 mm to 10 mm), but each change reduces eye relief by a comparable amount, from 17mm to about 7 mm. In contrast, reducing the lens spacing from 25mm to 0 increases eye relief proportionately less, by about 5 mm.

Reducing the focal length of the field lens has a relatively small effect on the system effective focal length (green line) and the system apparent field of view (blue line). This is in contrast to the effect of reducing the focal length of the eye lens, which sharply reduces the focal length and increases the apparent field of the system, an effect that is augmented by a closer lens spacing. As a result, modern wide field eyepieces typically have a high powered eye lens or densely spaced assembly of lenses.

The direction of these changes is the same in both lenses, and differs from the effect of the lens spacing only in the change in eye relief, so the eyepiece focal length decreases and the apparent field of view increases, and by a much greater proportion than the eye relief is reduced, when both lenses are made to a shorter focal length and are placed closer together. This also implies more noticeable aberrations, and a greater emphasis on the dispersion attributes of glasses used for the lenses.

These examples reveal the underlying design principles of traditional astronomical eyepieces:

• The function of the field lens is primarily to "stage" or prepare the image by partially correcting the divergence in the light rays (caused by the relative aperture or ƒ ratio of the objective) and by partially collimating the "pencils" of light after the rays have passed through the focal plane of the eyepiece.

• The function of the eye lens is to bring these corrected rays into a much shorter focus at a much steeper angle to the optical axis, which produces a wider apparent field of view and greater magnification. At the same time, the eye lens eliminates any remaining divergence in the image rays so that they exit the eyepiece as parallel bundles, termed afocal because the image is not formed at a single focal point but remains in focus when projected to any distance.

The significance of the exit pupil is that it is the point where the angular width of the image is at maximum and the projected light has the smallest diameter (obvious in the diagram). This compressed, wide angle beam can most easily pass the small entrance pupil aperture of the observer's eye and fill the wide area of the observer's retina. Both the field lens power and the interlens spacing can be manipulated to compensate for the very short eye relief that a high power eye lens will produce, to place the eye point and exit pupil where it can be comfortably examined by the observer.

The final issue concerns apparent field of view (blue line), which in traditional eyepiece designs rarely exceeds ~45°. A high power eye lens or a very wide field design (when β > ~25°) produces significant optical aberrations, so before c.1850 optical designs focused on minimizing chromatic and spherical aberration. Advances in optical theory and glass manufacture in the late 19th and 20th centuries gradually brought other aberrations under control and made wider field eyepieces practicable.

Thick Lens Optical Analysis

The previous sections have explained how lenses can be analyzed using a paraxial approximation: the front and back surfaces of the lens within an extremely small radius of the optical axis are approximated as a single refractive principal plane, and this plane is extended over the entire diameter of the lens to calculate focal distances (power), image location and orientation, and magnification. This approximation assumes that surface curvature of the lens is relatively small, the power of the lens is weak, and therefore the thickness of the lens, like the thickness in a sheet of window glass, does not have a significant effect on image formation.

For strongly curved (more powerful) lenses the paraxial approximation breaks down, and the analysis must be constructed on either two principal planes within each lens or optical element, or a strict trigonometric analysis of optical surfaces and distances.

The earlier example omitted consideration of the lens thickness in order to illustrate how Snell's Law and simple trigonometry is applied to locate a principal plane. This section introduces calculations that include lens thickness (t), which still rely on the equivalence between the tangent or sine of an angle and the angle itself; the next section introduces calculations based on twin principal planes within and distances between each optical element.

Again, the sign conventions are positive for measurements in the direction of light (left to right) and for objects above the optical axis; negative against the light and below the optical axis. Focal lengths are positive for converging lenses and negative for diverging lenses. Angles are always expressed in radians, which is approximately the tangent for angles less than 20°.

First, let's revisit the procedure outlined above for locating the focal point and second principal plane of a single converging lens. This can be generalized by analyzing an axial rather than collimated incident ray (diagram, below).

The refracting power of lens in air results from the radius of curvature of the two faces of the lens, the thickness of the lens, and the index of refraction of the glass. Therefore measurement of the two surface curvatures (c1 = 1/r2 for the front surface, c2 = –1/r1 for the back surface) and the distance between the two lens vertices (t) is sufficient to identify the second principal plane analytically, using any feasible values for the distance s of an axial point on the optical axis and for the aperture height y of an axial ray from that point to the lens (diagram, above).

For a single lens (where n0 = n2 = the index of refraction for air), the optical angles u1, u'1 = u2 and u'2 are measured in radians — but now in relation to the optical axis or lines parallel to it rather than to the lines normal defined by the lens surfaces.

The step by step calculations proceed as follows:

Given:
c1, c2 – front, back curvatures of lens (=1/r1, 1/r2; r2 and c2 are negative by sign convention)
t – vertex to vertex thickness of lens (positive, by sign convention)
n0, n2 – index of refraction for air ( = 1.0)
n1 – index of refraction for lens material
s – distance of axial point (negative, by sign convention)
y – aperture height of incident axial ray (positive when above the optical axis, by sign convention)

Calculate:

[1. entry incidence slope] u1 = –y/s

[2. entry refraction slope] u'1 = [–y·(n1n0c1+(n0·u1)]/n1

[3. exit (image ray) aperture height] y' = y+[t·(n1·u'1)/n1]

[4. exit (image ray) slope, given u2 = u'1] u'2 = [–y'·(n2n1c2+(n1·u2)]/n2

[5. back focal length] ƒBFL = y'/–u'2

[6. effective focal length] ƒEFL = y/–u'2

[7. power of the lens] φ = 1/ƒEFL.

Once all calculations are completed, first the focal point is located by measuring the back focal length from the back vertex as before. Then the principal plane of the compound lens is located by measuring the effective focal length forward from the focal point. The power of the compound lens is determined as before from the effective focal length.

This procedure can be repeated for a cemented compound lens consisting of two or three elements. Simply make the aperture height and incidence slope of the entry ray into the second (or subsequent) lenses equal to the exit ray aperture height and slope from the previous lens, and calculate steps 2-4 with the front and back curvatures of the second lens. (If the two lenses are separated by air, then a different calculation is necessary, as described below.)

For example, in step 2 n1 would replace n0 and n2 (the index of refraction for the second lens) would replace n1; c2 would be the front curvature of lens 2 and y' the aperture height of the incident ray, and t would be the thickness of the second lens. The exit calculations 3-4 will use the second curvature of the second lens and retain the index of refraction for air. Note again that the angles (in radians) are assumed equal to their tangents or sines, which is the essential condition for the paraxial approximation.

To identify the opposite focal point, the two radii of curvature are reversed in the formulas and the sign conventions will reverse the signs of the radii of curvature.

Multiple Lens Optical Analysis

Next we generalize the "thin lens" formulas for two air spaced lenses or multiple element components treated as single lenses. This procedure departs from the trigonometric analysis given above in that the thickness of the optical components [single or compound lenses] is replaced by a pair of principal planes, located by a prior analysis of each of the components. These are specifically planes of unit magnification, which means, given an object placed at the first focal point, that:

• Front and back focal lengths are equal: ƒ = ƒ'.

• All rays entering the first principal plane at aperture height y will exit the second principal plane at aperture height y' = y. (It is always assumed that the object and image rays lie in the same longitudinal or lengthwise plane, called the meridional plane, which also includes the optical axis.)

• An axial object ray angle u will produce an image ray angle –u', and an oblique object ray angle –u through the first principal point will produce an oblique image ray angle –u' exiting from the second principal point.

• An object point at ƒ will generate collimated rays exiting the element, and collimated rays entering the element will focus exactly at ƒ'; an object located at 2ƒ will form an image at 2ƒ' with image magnification of 1.0, (h = h').

The ray tracing requires only two analysis rays in a common meridional plane: (1) an axial (marginal) ray from the object at distance s from the first principal plane of the field lens of the system that passes through the principal plane at aperture height y (the radius of the system aperture), and (2) a chief ray from object height h that passes through the first principal point of the field lens. Note that h is not necessarily equal to y. Then the object ray relationships will be as diagrammed (below):

u = –y/s and up = h/s (note that s and up as diagrammed are negative by the sign conventions)

and the corresponding image ray relationships for the system will be:

u' = y'/s' and u'p = (h'–y'p)/–s (note that u' and u'p as diagrammed are negative by the sign conventions)

As the diagram makes clear, the image angles have been transformed by the optical system such that the simple equalities y' = y and –u' = u no longer apply. So the generalized procedure must get us from the object to the image values, using only the refractive indices n1 and n2 of the two optical components, the spacing d between the facing principal planes of the two components, and (for finite object distances) the total distance T between object and image.

First, we can get the generalized form of the "thin lens" equations of the previous section, which will then apply to both thin or thick components. Given the unit magnification of the principal planes for a single component yields the basic relationships:

u = –y/s and u' = –y/s'

Substituting these into the thin lens formula (above) by replacing object and focal distances by ray angles (e.g., 1/s = –u/y) and reciprocal focal length with power (1/ƒ = φ) yields as the refraction of the first component:

u'1 = u1y1φ1

the transfer equations into the second component:

y2 = y1 +du'1 and u2 = u'1

the exit ray from the second component:

u'2 = u2 – y2φ2

and the effective focal length of the total system (components 1 and 2 combined at distance d):

ƒEFL = u'2 /y1

Alternately, assume that φ1, φ2 and d are known or fixed by design, and we want to find the focal length or power of the system:

φsys = φ1 + φ21φ2

which is identical to the reciprocal of the focal length formula given above in the "thin lens" case:

φsys = 1/ƒ1 + 1/ƒ2d/ƒ1ƒ2

If the effective focal distance, back focal distance and component spacing are known, the separate effective focal distances of the components can be found as:

ƒ1 = (EFL)/(ƒEFLƒBFL) and ƒ2 = (–BFL)/(ƒEFLƒBFLd)

These equations permit fast layout and approximate analysis of optical systems without the incremental calculation of ray traces through all refracting surfaces.

Finally, all focal optical systems conform to four systematic proportionalities that equivalently define a quantity known as the optical invariant or Lagrange invariant (L). Given an axial ray with a height y at the first principal plane and y' at the back principal plane, and an oblique ray with height yp at the first principal plane and y'p at the back principal plane (diagram, above), then the angles (in radians) of these rays to the optical axis will be u, u', up and u'p respectively. In that case:

L = hn1u = h'nku'.

This quantity states the relationships between refractive power, object distance and image focal distance purely in terms of angles and relative heights. It can be reduced to an alternative statement of system magnification:

M = h'/h = n1u/nku'.

The invariant applies equally to single or compound lenses or multi element optical systems, which are treated as a "black box" and analytically bracketed by the entrance pupil and back principal plane.

In analyses related to optical aberrations, the first principal plane is usually coincident with the entrance pupil of the system, which is the opening in the system that determines its aperture and that intersects the first principal point of the system. This is normally the mounting for the objective lens in a refracting telescope, the circumference of the first mirror in a reflecting telescope, or the mounting of the corrector lens in a catadioptric telescope.)

Eyepiece Prescription Data

In most eyepieces, the thin lens formulas are insufficient to design an eyepiece or guide its manufacture. The total information required to specify an eyepiece optical design is called the prescription data, which includes the vertex to radius of curvature distance for all lens surfaces, the vertex to vertex distance between lens surfaces, as measured along the optical axis, and the specific glasses that provide the refractive indices and Abbe coefficients assumed to calculate the light paths. A full example is shown below for a modern version of the symmetrical Plössl design.

The prescription data are given in tabular form (diagram, lower left): measurements begin at the objective focal plane and end at the eyepiece exit pupil. Note that the distance measurements are strictly sequential — d1 is measured from d0, d2 is measured from d1, etc. — while each radius of curvature is measured from its corresponding vertex. Negative numbers indicate measurements toward the light source (conventionally, to the left), plane surfaces have a radius of 0, and air spaces are shown as a distance with no glass indicated. (This single radius format applies to spherical lenses only; aspheric lenses require a polynomial prescription not shown here.)

Given the prescription data (available in eyepiece patent documents or optical references), an assigned system focal ratio and field diameter (equivalent to the interior diameter of the eyepiece field stop), ray tracing software can calculate the path of light from the objective focal plane through the eyepiece for any field height on the object focal plane. These rays are often shown in different colors for field heights equal to 0%, 70% and 100% of the field radius; the colors do not denote spectral frequencies but help to interpret the diagram visually. Conventionally, three rays are calculated as radiating from each field point (originating from opposite sides and the center of the objective aperture) so that their convergence exactly defines the exit pupil.

Ray tracing allows calculation of the eyepiece principal planes, principal points, effective focal length, front and back focal length, eye relief, the angle of the apparent field of view, and the Petzval radius, which describes the field curvature in the focal plane of the eyepiece. (Note that a negative Petzval radius indicates a positive field curvature — that is, the center of the focal surface is closer to the field lens than the perimeter of the focal surface; a large Petzval radius describes a relatively small field curvature.)

A single spectral frequency is necessary to compute these standard optical parameters; this is usually λ = 550 nm or "yellow green" [called green] light. Analysis of eyepiece chromatic aberrations requires ray tracing at different wavelengths, typically 475 nm ("blue"), 512 nm ("green"), 550 nm ("yellow green"), 587 nm ("orange yellow") and 625 nm ("red orange").

Optical Materials

Glass is the generic material used to refract light in astronomical instruments. Glasses are amorphous (not crystalline) mixtures of fused silica (silicon oxide, SiO2) and oxides of several metals — including sodium, calcium, magnesium and aluminum — added to improve the hardness, durability and water resistance. Glasses are relatively stable under normal temperature changes, readily fabricated and economical.

Glasses, plastics and other transparent materials bend all wavelengths of light through refraction, but bend low energy "red" light less strongly than high energy "violet" light, which spreads out the refractive indices for different spectral hues in an optical effect called dispersion. And dispersion does not increase by an equal proportion across equal wavelength intervals: the increase in the angle of refraction between red and green light is always less than that between green and violet light, and these relative proportions also vary across different types of glass as characteristic partial dispersions.

Optical materials are therefore identified according to those two attributes: (1) the index of refraction (n), (2) the Abbe number (V) or measure of dispersion as the ratio of refraction between two selected wavelengths at the long and short wavelength ends of the visible spectrum. In addition one or more (3) indices of partial dispersion (P) are calculated for limited sections of the spectrum. In glass catalogs the first two attributes are reported as a single six digit code — the first three digits are the index of refraction (after the decimal point), followed by the three digit V number (omitting the decimal point); as many as six indices of partial dispersion may also be included.

Across the entire electromagnetic spectrum, most glasses and plastics have periodic absorption bands of zero transmission. In optical glasses these bands are not normally significant, as they are located above 2200 nm in the infrared (although a few extremely dense flints are transparent up to 4000 or 5000 nm) and below 420 nm to 320 nm in the ultraviolet. This extreme "red" and "blue" absorption causes window glass, sometimes used for large corrector plates, to appear slightly greenish.

Because each wavelength of light is refracted at a slightly different angle by the same lens, the index of refraction must be standardized on a specific wavelength of light, defined as one of the emission lines of a chemical element. By convention this is either the d wavelength of helium "orange yellow" [called yellow] light at 587 nm (nd) or the e wavelength of mercury "yellow green" [called green] light at 546 nm (ne). Then the Abbe numbers Vd and Ve are calculated as:

Vd = [nd–1]/[nCnF]

Ve = [ne–1]/[nC'nF']

and an important index of partial dispersion (PgF) as:

PgF = [ngnF]/[nCnF]

where the comparison wavelengths used (image, left) are the emission lines of hydrogen at C = 656 nm (Hα) and F = 486 nm (Hβ) for d yellow, and the emission lines of cadmium at C' = 644 nm and F' = 480 nm for e green; and the emission line of mercury at g = 436 nm for the partial dispersion. Note that V numbers increase as the difference in refraction between the red and blue wavelengths gets smaller.

The diagram (above) illustrates the relative effect of the refractive index and V number in two equilateral prisms; the actual dispersion has been exaggerated in the diagram for clarity. At the highest dispersion (lowest Abbe number), the indices of refraction for "red" and "blue" wavelengths differ by less than 0.08. When the V number is large, the spread between the "orange red" and "blue" indices of refraction is small, compared to the amount of mid spectrum refraction. Note however that dispersion and refraction are roughly proportional: as the power of refraction increases, so also does the amount of dispersion produced.

But the two metrics are not exactly proportional. When glasses are charted on n and V, they populate an area between diffraction indices of about 1.4 to 2.0, and V numbers between about 90 (low dispersion) to 20 (high dispersion). Although dispersion increases as refraction increases, the two attributes are not perfectly related; so it is possible to choose glasses that have different properties of refraction and dispersion, which, in combination with the shape and spacing of lenses, can produce a wide variety of optical systems.

Glasses on the left (dispersion of 55 or more at refractive index below 1.60 and 50 or more at refractive index above 1.60) are called crowns, and glasses on the right are flints. Curving across the lower right of the diagram is the glass line of flints produced by combining crown glass with lead oxide. The variation of glasses above the glass line is caused by differences in the partial dispersion. The characteristics produced by different chemical additives are shown as different background colors; these produce differences in optical effect denoted by glass categories such as hard, soft, dense and light.

The dots indicate specific glasses available from Schott, one of many major glass suppliers. They illustrate that the distribution of glasses is not random, but is strongly clustered around the righthand diagonal. Most of the variations occur within middle values of the indices of refraction and dispersion, straddling the boundary between crowns and flints. This permits the manufacture of compound lenses made of two or more glasses that produce the same refraction but different dispersions, or the same dispersion but different refractions, which is necessary to eliminate chromatic aberration in the focused image.

Traditionally, crowns are "hard" glasses with higher melting temperatures, made with small quantities of potassium oxide, combined with oxides of other metals such as phosphorus, zinc, lanthanum or barium. Crowns made with boric oxide (borosilicate glasses, including Pyrex™ glass) or fused quartz have been especially important in the manufacture low expansion telescope mirrors. Glasses marketed as ED (for extra low dispersion) are nominally crowns (including fused fluorite or calcium fluoride, CaF2) with a V number above 80 and a partial dispersion far from the Abbe line.

Flints are "soft" glasses with lower melting temperatures, historically made with varying quantities of lead oxide. One of the highest index flints (nd = 1.96) contains 18% silica and 82% lead oxide by weight; some flint glasses used in military applications have even been made with heavier metals such as thorium or uranium. Flints are especially susceptible to clouding under prolonged contact with moisture, which leaches lead from the glass. For environmental reasons, lead oxide is now frequently replaced with oxides of lanthanum, titanium or zirconium.

(Historical note: the term crown arose because in 17th century England the manufacture of glass from sand and lime was protected by a license from the Crown; but at the end of the century that restriction was circumvented by a new type of glass manufactured from nodules of flint, a sedimentary form of quartz, and red lead.)

In the late 20th century, a few hard plastics (acrylics, polycarbonates, urethane monomers) have been successfully used as optical materials in many applications, and lenses incorporating diffractive (ribbed or grooved) surfaces have been used as well. Plastics can be injection molded but are also relatively soft and easily scratched, and can deform under moderately high temperatures, which makes optical coatings impractical. However certain plastics have replaced Canadian balsam as cements commonly used to bond together the separate elements of a compound lens.

Optical Coatings

As explained above, a small fraction of the light directed through an optical system is not refracted by an optical surface but is reflected from it, and additional light is absorbed by the material and scattered by internal reflections at the second (exit) surface.

In general, the transmission of a light ray through both surfaces of a single uncoated optical element of refractive index ni and surrounded by air is:

T = 2ni / (ni2 + 1)

For an angle of incidence of between 0° to ~30°, the two surfaces of a single optical glass element in air transmit about 92% (ni = 1.5) to 81% (ni = 1.95) of unpolarized light, assuming there is zero light absorption by the glass itself. This reflection obviously reduces image illuminance and also often causes scatter, glare or ghosts within the image itself. In addition, the transmission of a multi element optical system is no greater than the product of the separate transmission values of all elements: two lenses of 92% and 81% transmittances combined as an eyepiece yields a total transmission no greater than 75% (and possibly less). Obviously, methods that can reduce surface reflections are highly desirable.

Optical coatings are very thin layers of material vacuum deposited on an optical element to minimize or control the reflection or transmission properties of its surfaces. A variety of materials used for this purpose include fluorides and oxides of various metals, but the most commonly used material is magnesium fluoride (MgF2), which provides both a protective coating to a glass surface and a refractive index that is optimal for controlling reflections.

Coatings manage unwanted reflections by virtue of their optical thickness, defined as the physical thickness times the index of refraction. The material is deposited as a single layer approximately 1/4 or 1/2 wavelength thick, using vacuum processes monitored photoelectrically with monochromatic light.

Light reflected by the internal surface of a 1/4 wavelength layer will be exactly 1/2 wavelength out of phase with light reflected by the external layer, resulting in destructive (wave canceling) interference of all reflected light. This effect is maximized when the refractive index of the layer (in air) is equal to the square root of the refractive index of the glass. Magnesium fluoride (n = 1.38, n2 = 1.90) is optimal for very high index flint glasses; it is preferred for crowns as well due to the durable and protective coating it provides to the optical surface.

Due to its fixed thickness and refractive index, the wave canceling effect of a single layer optical coating becomes less effective at wavelengths longer or shorter than the design wavelength (usually around 550nm to 560nm). This permits some reflected light at "red" and "blue" wavelengths, which combine to produce the characteristic purple tint of a single layer coating.

In multicoated optics, three or more layers of different thicknesses can be designed to eliminate reflected light within a bandpass matching the visible spectrum (~400nm to ~750nm). These lenses generally have a much darker greenish surface tint or lack surface reflections entirely. The "super multicoating" (SMC) used on Pentax optics consists of seven layers and reduces the design wavelength reflection to 99.8%. Coatings of 50 layers or more have been designed to produce a very narrow bandpass (for use as a filter); this bandpass can be shifted to longer or shorter wavelengths simply by increasing or decreasing the thicknesses of all layers by a constant factor.

The commercial designations coated or multicoated indicate that one or more (usually only the external) air/glass surfaces are coated. Fully coated or fully multicoated indicates that all air/glass surfaces (including those inside the instrument) have been coated.

Aluminum mirrors are also coated with very thin layers of magnesium fluoride or silicon monoxide (SiO) to provide protection against oxidation and cleaning. Certain multilayer coatings can also increase the reflectivity of an aluminum coating from 88% up to 99% at certain wavelengths, but not more than ~95% as an average across the entire visible spectrum.

Further Reading

Astronomical Optics, Part 2: Telescope & Eyepiece Combined - the design parameters of astronomical telescopes and eyepieces, separately and combined as a system.

Astronomical Optics, Part 3: The Astronomical Image - analysis of the image produced by a telescope and the eye that receives it.

Astronomical Optics, Part 4: Optical Aberrations - an in depth review of optical aberrations in astronomical optics.

Astronomical Optics, Part 5: Eyepiece Designs - an illustrated overview of historically important eyepiece designs.

Astronomical Optics, Part 6: Evaluating Eyepieces - methods to test eyepieces, and results from my collection.

Modern Optical Engineering by Warren Smith - a classic, authoritative and clearly written survey of optical principles and applications.

Basic Optics and Optical Instruments by Fred A. Carson - a simplified but useful exposition of geometrical analysis.

Ray Tracing of Thin Lenses by Darryl Meister - Clear and well organized explanation of thin lens ray tracing.

Nature and Properties of Light by Linda Vandergriff - overview of modern optical techniques for the detection and manipulation of light.

Basic Geometrical Optics by Leno Pedrotti - the basics of light reflection and refraction and the use of simple optical elements, such as mirrors, prisms, lenses, and fibers.

Basic Physical Optics by Leno Pedrotti - the phenomena of light wave interference, diffraction, and polarization.

 

Last revised 11/26/13 • ©2014 Bruce MacEvoy