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http://www.handprint.com/ASTRO/ae1.htmlAstronomical Optics

Part 1: Basic Optics

The Geometry of Light Refraction Reflection

Wavelength & Frequency Interference Ray Tracing the Optics 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

Single Lens Optical Analysis Multiple Lens Optical Analysis Eyepiece Prescription Data Optical Materials Optical Coatings

This page introduces the optical principles necessary to understand the design and function of

telescopes and astronomical eyepieces. Subsequent pages discuss the telescope & eyepiececombined , 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 usedhere and background information available online.

The Geometry of Light

Light propagates in the form of oscillations in an electromagnetic field, which expand from thelight source as evenly spaced and concentric wavefronts whose energy is measured in thequantum packets known as photons. Considered as photons, light travels in a straight line fromthe light source unless or until it is absorbed, reflected or refracted by matter.The radiation of light through space can be represented in two ways: (1) as wavefronts thatexpand concentrically and radially from the light source (analysis by physical optics ), or (2) asstraight lines originating in the light source and perpendicular to the wavefronts that indicate thedirection in which each part of the wavefront is moving (analysis by geometric optics ). Basic

optics is entirely developed in terms of geometric optics.In all astronomical applications, light sources are so distant that the concentric wavefronts areeffectively parallel planes over any practical telescope aperture, and the geometric raysdescribing the direction of the wavefronts are also parallel. To illustrate: across the aperture of a


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254mm (10") telescope, light rays from a single point on the moon, the closest astronomicalobject, diverge from parallel by no more than 0.00000000066 millimeter, or 1/1,000,000 awavelength of light. Therefore, geometric optics can assume perfectly flat and parallelweavefronts from a distant light source, and light rays that are straight, parallel andperpendicular to the wavefronts.

Refraction

Refraction is the characteristic behavior of light that is incident on the smooth surface of atransparent (transmitting) material.Light has a uniform and universal speed in a vacuum of nearly 300,000 kilometers per second.Light can also propagate through various transparent materials, such as air, water or glass, buteach material slows the speed of light by a specific value in some materials, to almost onethird its vacuum speed. The speed of light in a vacuum divided by the speed of light in arefracting material is the refractive index ( n ) of the material: this is 1.00029 for air; 1.3333 forwater; and anywhere from 1.4 to 2.0 for optical glasses.When light crosses the boundary between materials of different refractive index, the averagedirection of the wavefront can be deflected in a new direction. The geometrical analysis of thisbehavior 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 thin white lines.

As the wavefronts of light AB , traveling across distance BC , encounter a refracting boundary AC,the speed of the wavefront is slowed so that it now travels a shorter distance AB' in the sametime. The refracted wavefronts are also turned in a different direction, which occurs because thesame wavefront encounters the boundary at different times ( t 1 to t 5) across its width AB .Because geometric rays are always (by definition) at right angles to the wavefronts theydescribe, they create the right triangles ABC before refraction and AB'C after refraction, withside AC in common. Inspection shows that the angle of incidence ( 1) is equal to the angle BAC ,whose sine is equal to BC / AC ; and that the angle of refraction ( 2) is equal to the angle ACB' ,whose sine is equal to AB' / AC . Since AC is a common denominator, the sines differ in the ratioBC / AB' . This ratio is the proportional slowing in the refracted wave, which we have seen is

defined as the index of refraction, and therefore the sine ratio is equal to the inverse refractionratio n 2 / n 1 .

These relationships are summarized as Snell's Law or the Law of Refraction, illustrated in thediagram by the yellow arrows and defined mathematically as:

sine( 1)/sine( 2) = n 2 / n 1

or

sine( 1) n 1 = sine( 2) n 2

where n 1 and n 2 are the refractive indices of the two media that form the refracting boundary,and 1 and 2 are the angle of incidence and angle of refraction. These angles are measured froma line normal (perpendicular) to the boundary surface of the two media at the incidence point ofa light ray. Both light rays and the line normal must lie in a single plane, and the incident andrefracted rays will be on opposite sides of the line normal.This analysis can be explaned in terms of the wavefront character of light, as the formation ofsingle wavefronts by the coalescing wavefronts expanding from each incidence point (curved arcsalong the line B'C ). However Snell's law equally applies if the wavefront is summarized as asingle light ray parallel to the direction of the light and perpendicular to the wavefront (theyellow arrows). The angles of incidence and refraction within a single plane remain the same.

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: thelight is slowed but its direction is not altered. If the angle of incidence is greater than a critical angle ( CR), defined as:

CR = arcsin( n 1 / n 2), n 2 > n 1


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then the ray is reflected from the boundary rather than refracted through it. In a boundary withair, the critical angle is about 49 for water ( n = 1.33) and 46 to 30 for optical glasses ( n = 1.4to 2.0). When reflection occurs, either from reflecting surfaces such as mirrors or glass surfacespositioned 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 linenormal:

2 =

1

For all angles of incidence between the critical angle and 0, significant reflection also occurswhen 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 twoangles on opposite sides of the line normal.

Wavelength & Frequency

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

Interference

Although the optical behavior of light can be described in terms of geometric rays, the imageforming behavior of light requires the description of wavefronts in terms of physical optics. Themost important behavior is the interaction between light wavefronts that produces interference. Interference can arise in two ways.The first is diffraction or the concentric expansion of wavefronts from an occluding edge. Thiswas demonstrated by the slit experiment of the English naturalist Thomas Young which provedthe 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 apart of the wavefront to pass, but as it does so the wavefront expands concentrically from eachslit aperture.The slits are positioned so that the oscillations of separate wavefronts must be either coincident

The distance between identical points on twoadjacent wavefronts of light is the wavelength ( ) ofthe light. The frequency ( ) of light is the number ofwavefronts that pass a fixed point in one second, orthe cycles per second. The relationship betweenfrequency and wavelength is governed by the speedof light in a vacuum, c :

c = = 299,792,458 ms 1 = 3 x 10 8 m/s.

m = c / .

Light wavelengths are commonly measured inangstroms (10 -11 meter), nanometers (10 -9 meter)or micrometers (10 -6 meter). Light that appearsgreen to the human eye has a wavelength of about550 nanometers 0.55 micrometers or 0.00055millimeters. Visible wavelengths range from about750 nm ("orange red") to 380 nm ("blue violet"), or0.00075 to 0.00038 millimeters. Longer wavelength"red" light carries less energy, and therefore has alower frequency, than shorter wavelength "violet"light.


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or opposed. Where they coincide a wavefront reinforcement occurs and a band of light will beprojected 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 theopposing wavefront and this produces a dark band on the projection screen.The interference effects produced by physical obstructions are the origin of the diffractionartifact produced by a star or "point" light source, and the bright and dark pattern of specklesproduced 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 fromseparate transparent media. This was first investigated by Isaac Newton as the interferencefringes that appear between two closely spaced pieces of glass. In the optimal situation, thespacing between the first and second reflective surfaces is equal to 1/4 wavelength of the light.Light is partially reflected from the first and second boundaries, and these separate wavefrontsare exactly 1/2 wavelength out of phase. The peak of one wavefront mixes with the trough of thesecond, and the result is that the two wavefronts cancel each other and the light is destroyed.This laminar interference is the physical cause of iridescent colors from laminated, pearlescentmaterials such as abalone shells or bird feathers, or from a thin film of oil on the surface ofwater. It is also the basic mechanism exploited in reflection reducing optical coatings on thecomponents of astronomical eyepieces.

Ray Tracing the Optics

Furnished with the geometrical description of light, the Law of Refraction and information aboutthe refraction and dispersion of optical materials , we can analyze the basic attributes of anyoptical 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 deploysSnell'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 itcreates.

Third order analysis was developed through the combined efforts of several 19th centurymathematicians and opticians to describe optical aberrations , or departures of focused lightfrom 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 raysare 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 therefraction of light by a plane perpendicular to the optical axis. This allows the arithmeticcalculation of image size and location without the use of trigonometric functions (sine ortangent).This paraxial approximation can usefully describe the optical behavior of moderately curvedlenses with a field angle of up to 20 if we accept errors of calculation of 1%, or to lenses of 6with errors of 0.1%. Exact analysis is possible with trigonometric methods and finally bycalculating the physical properties of light wavefronts, but the paraxial approximation isinvaluable 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 andprecisely focused) image at the optical axis: analysis is only used to define the location, size andorientation of this perfect image.The analysis builds on the fact that the behavior of an optical system can be diagrammed inrelation 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 surroundedby air the standard situation in astronomical optics so only the focal and principal points areneeded to describe the system optical behavior.The diagram (below) illustrates the key concepts and terminology in the first order analysis of aschematic biconvex lens surrounded by air. These concepts can also apply to single or compoundlenses of any type, treated as a single optical unit or "black box".


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A few basic properties of the optical system are assumed to apply. All optical components areconstructed as solids of rotation, which means their refracting surfaces are symmetrical aroundan axis. The axes of rotation for all surfaces are identical with a single optical axis when light ispassed through the optical system. The intersection of a refracting surface with the optical axis isthe vertex of the surface.Lens surfaces are assumed to be (and in nearly all real situations are) manufactured as sectionsof a sphere, defined by a radius of curvature originating from a center of curvature located onthe optical axis. A two sided lens has two centers of curvature (denoted r 1 and r 2) and two radiimeasured 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 bythe optical axis and conventionally positioned to the left of the lens. These rays pass through thelens from left to right and terminate in an image or image plane located on the right of the lens;the light receptor (the observer's eye, a CCD chip, photographic film) is therefore located at theright of the lens oriented toward the left.

An object ray originates from some point on the object or object space being imaged by theoptical system; an image ray terminates at a corresponding point in the image of the object. Allimage points are located on an image plane perpendicular to the optical axis and intersecting theoptical axis at the focal point . (Note that all real optical images are in fact focused onto a surfacethat is more or less spherical or paraboloid, with its own radius of curvature; the image plane is aparaxial simplification.)Finally, all refracting optical systems are reversible: they can focus light passing through themfrom left to right or from right to left. This creates a focal point on each side of the lens. Theobject and image, and matching distances and points connected with them, are conjugate. This is the basis for a number of related concepts, specific labels and symbols, illustrated in thediagram (above):

Collimated rays are parallel to the optical axis and to each other. Oblique rays or abaxial rays are at an angle to the optical axis (originate from an object point not on the optical axis).

If a collimated ray from a point on the object is extended through the lens, and thecorresponding oblique image ray is extended back from the conjugate image point, they willintersect in a principal plane perpendicular to the optical axis and intersecting the optical axis ata principal point . All rays from the object and refracted to the image will intersect in the sameplane. (Again, aberration free optical systems refract light as if from a spherical or paraboloidsurface; the principal plane is a paraxial simplification.)

In a thick or compound lens there are two principal points and corresponding principal planes.The first principal plane, first principal point and first focal point are assigned to the surfacewhere light enters the lens; the second principal plane, second principal point and second focalpoint are assigned to the surface where light exits the lens.

The effective focal length ( EFL) is the distance from the second principal point to the secondfocal point ( ' ). The back focal length ( BFL) is the distance from the back vertex of the lens tothe second focal point. The front focal length ( FFL) is the distance from the front vertex of thelens to the first focal point ( ). In most Gaussian equations the front and back focal lengths areassumed 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. A chief ray or principal ray arises from any object point not on the optical axis, passes throughthe first principal point, exits from the second principal point at the same angle to the opticalaxis, and intersects the image plane at the conjugate image point.


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An axial ray or marginal ray originates from the point where the optical axis intersects theobject or object space, enters the first principal plane at the outer edge or aperture radius of thelens, then exits from the second principal plane to intersect the optical axis at the focal point.

An object ray intersects the first principal plane at a specific incidence aperture height ( y ) ordistance from the optical axis, and exits from the second principal plane at the refractionaperture height ( y' ).

The perpendicular distance from the optical axis to the most extreme object point is the objectheight ( h ), and the distance from the optical axis to the most extreme (conjugate) image point isthe image height ( h' ). If the ray is collimated then the aperture height of the incident raycorresponds to the object height of the originating point.

For elements notated by the same letter symbol before and after refraction, elements on theimage side, and rays or angles of rays after refraction by any surface, are denoted by anapostrophe (e.g., and ' , y and y' , h and h' ).

In first order analysis all angles are measured in radians; for the very small slope anglesproduced 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 generatedby an ellipse, parabola or hyperbola rotated around its major axis). In first order analysis theseare 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, thealgebraic sign of measured quantities used in the equations must follow arbitrary but specificsign conventions:

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

A measurement reference point, usually the principal point of the optical system, is used as theorigin 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 leftare negative. Lengths measured upward from the optical axis are positive; lengths measureddownward are negative.

Angles measured at the optical axis are positive if the oblique ray must be rotated clockwise toreach the optical axis, and negative otherwise. Angles of incidence and refraction are positive ifthey must be rotated clockwise to reach the line normal of the refracting surface, and negativeotherwise. (Rotation is always through the smaller angle to the optical axis or line normal.)

Both focal lengths of a converging lens are positive, and both focal lengths of a diverging lensare negative. The single focal length is positive for a concave (converging) mirror and negativefor a convex (diverging) mirror.

A radius of curvature is positive if it lies to the right of the surface it describes, negative if it lies

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 howSnell's Law and basic trigonometry are applied to locate the second principal plane, secondprincipal point and effective focal length of a symmetrical biconvex lens (diagram, below).

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

The lines normal to the front surface of the lens are produced as lines radiating from the secondcenter of curvature ( r 2 , shown in the diagram on the far right). Since the lens is symmetrical, thefirst center of curvature ( r 1) 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 / r 2 .


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Because we assume the lens produces a perfect focus, the aperture height y can be set to anyreasonable value for the analysis. In the example, I've chosen y = 10 and r 2 = 30.9: theincidence point then defines a sine of 0.309 to the line normal, giving an angle of entranceincidence of 1 = 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 0.3090.625 = 0.193, for an angle ofrefraction 2 = 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 1 to the line normal (thisis, for example, why we can see reflections in storefront windows); the rest of the light raycontinues 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 afunction of the relative curvature of the front and back surfaces, the separation t between thetwo vertices at the optical axis, and the refractive index n 1 of the glass. That calculation isexplained below ; for now, we identify the exit point i' by construction in order to focus onfinding 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 y' =9.2 and the line normal drawn from r 1 is 17 to the optical axis. Then 3 = 24 from the interiorray to the line normal, for a sine of 0.407. The ratio of refraction indices is now inverted as1.6/1.0 = 1.6, so the sine of the refracted angle is 0.4071.6 = 0.651, which is 4 = 41 tothe 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 effectivefocal point ' with an image ray slope of u' = 24. (To avoid misunderstanding, note that onlya 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 tothe optical axis, and the image ray can be extended backward from point i' , until the two raysintersect at point I at aperture height y . The angle of the image ray slope is then defined asu' = y / ' radians. Since y is given and u' has been calculated, the focal length is calculated as' = 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. Theeffective focal length ( ' ) is simply the distance along the optical axis between the secondprincipal 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 actuallyconverged: the sun concentrated by a magnifying glass or light focused by a photographic lens.


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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, theimage 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 positivelens. A virtual image has no location in physical space and cannot affect photosensitive media.

(2) An image is erect if it is oriented vertically in the same way as the object, or inverted if theimage 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, orreverted if the image is reversed left to right. (Images rotated 180 in relation to the object areboth inverted and reverted.)(4) Finally, an image can be enlarged, actual size or reduced in comparison to the physical size ofthe 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 actuallyinvert the image, despite the common use of inverting telescope to describe them. Note also thatan inverting eyepiece actually produces an erect, normal image: the image is rotated by theobjective, not the eyepiece. The exception is the Gregorian telescope , because it reflects lightfrom a secondary placed beyond the focal point of the primary mirror. This rotates the image asecond time.

Types of Lenses

The possible combinations of spherical and plane surfaces that can be used to make a lens fallinto six generic types of positive or negative lenses, shown in the diagram (below) with twoindications of their relative refracting power: the effective focal length and a schematic raytracing. These illustrations show lenses of a typical optical glass ( n L = 1.6) in air; nominalradius 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 D o = 70mm.

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


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The effective principal point is shown by the red dot. Note that even when symmetrical, "thick"lenses have two principal points for the refractive effect in opposite directions. In general, theprincipal points in a biconvex lens are both internal and spaced about 1/3 the distance from thefront to back vertices. In a lens with one plane surface, the first principal point is the front vertexof 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 andmeniscus forms. The meniscus lenses are most sensitive to changes in the lens thickness, whilethe plano convex and plano concave lenses are unaffected by variations in thickness. Inasymmetrical positive lenses, the power of the lens is greater when the surface with the shorterradius (or the curved surface in plano lenses) is oriented toward the object. In asymmetricalnegative lenses, the reverse holds: the power is greater when the greater power or curvedsurface is oriented away from the object.

Used as single lenses, positive lenses always produce real, inverted, reverted images, whilenegative lenses produce virtual, erect, normal images.Positive meninscus lenses are commonly used in eyeglasses to correct for farsightedness ordifficulty focusing on objects at short distances from the eye ( hyperopia ). This occurs becausethe lens to retina focal distance of the eye is too short, or the cornea and lens of the eye lackoptical power, or the lens has hardened with age and cannot adjust to focus on near objects( presbyopia ). Negative meninscus lenses are used to correct nearsightedness or difficultyfocusing on objects at long distances ( myopia ), which occurs because the cornea and lens are toostrongly 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 conceptsdefined, 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 threeanalysis rays. The diagram below shows this analysis applied with two principal planes, which isdone by disregarding the space between them. This is equivalent to replacing both planes by asingle principal plane midway between them, in what is called the thin lens model of the optics.

If it is acceptable to assume that the optical effect of the lens thickness (the distance betweenthe front and back incidence points of a light ray) is inconsequential to the slope of the exitingimage ray, then the lens can be modeled by a single principal plane located at the center of thelens. This directly yields the effective focal length (measured from the single centered principalplane) as:

1/ ' = ( n L1)( c 1 c 2).

where c = 1/ r . For a symmetrical biconvex lens where r 1 = 10 cm, r 2 = 10 cm and n L = 1.6:

1/ ' = (11.6)((1/10)(1/10)) = 1/(0.62/10) = 1/0.12, or ' = 8.33 cm.

(Note application of the sign conventions.)

The "thin lens" analysis originated in 18th century optics in which the main applications were lowpower, long focal length spyglass lenses, simple eyepieces and very thin eyeglass lenses. It isapplicable to any lens where the focal length is much larger than its maximum thickness. Thatcriterion is ambiguous and only suggests how well a physical lens might be described by theidealized thin lens model: in particular, it implies that wide angle or short focal length (stronglycurved) 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 specificobject 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 (parallel to the optical axis) from the object point to the first principalplane of the lens, and exiting from an equal height in the second principal plane as an oblique raythrough the second focal point ' .B An oblique ray from the object point through the first principal point p 1 in the first principalplane, and exiting from the second principal point p 2 at the same angle to the optical axis.


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C An oblique ray from the object point through the first focal point to the first principal plane,and from an equal height in the second principal plane as a collimated light ray (parallel to theoptical axis).Note that only two rays, A and C, are necessary to define the image; ray B is shown as a dashedline because it is optional. Rays A and C only require the location of the two focal points, givenby the formula above. For an object at an infinite optical distance the distance at which allrays from the object are parallel to the optical axis the use of ray C becomes invalid, becauseto intersect it must be identical to the optical axis. In that case the image is identical with thefocal point intersected by ray A and the size of the image is determined only by the focaldistance multiplied by the field angle (in radians) between the optical axis and the object pointemitting ray B.

For an object at finite optical distance, all three rays will intersect at a single point that is not onthe optical axis. Given the first and second focal distances and ' and object height h , thisconstruction defines the object distance s , the image focal distance s' , the to object distance x ,the ' to image distance x' , and the image height h' .The image location is the distance from the second principal point to a line through the rayintersection that is perpendicular to the optical axis; the image size is determined by the distanceof the intersection from the optical axis, and the image orientation is shown as the interesectionlying above or below the optical axis (indicated by a positive or negative sign in the calculation ofthe image height or the lens magnification).Note the explicit symmetry in the procedure: an object height h at distance s produces animage height h' at distance s' , but an object height h' at distance s' would produce an imageheight h at distance s . This symmetry is characteristic of conjugate points (joined in areciprocal relationship) in an inverting optical system.

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

1/ s' = 1/ s + 1/ '

and the other parameters:[ to object distance] x = s + [ ' to image distance] x' = ( ' 2)/ x [object distance] s = s' ' /( ' s' )

[focal distance] s' = s ' /( s + ' )[object size] h = ( h' x )/ ' [image size] h' = ( h x' )/ '

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

m = h' / h = ' / x = x' / ' = ' /( s + ' )

The curvature ( c ) of the lens is the reciprocal of the radius of curvature; the power ( ) of thelens is the reciprocal of the effective focal length:c = 1/ r

= 1/ '

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


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(b) When the object is located exactly 2 in front of the lens, the image will appear inverted 2 behind the lens ( s' = s ) and actual size ( h' = h , m = 1).(c) If the object is located between 1 and 2 in front of the lens, the image will appearinverted at more than 2 behind the lens, and larger than actual size ( m > 1).(d) If the object is located exactly at 1 no image can form (the ray C through the front focalpoint never reaches the principal plane).(e) If the object is located at a distance less than 1 then the analysis rays must be extendedbackward from their intersection points with the principal plane. They intersect in front of thefirst principal plane to form an erect ( h' >0) and enlarged ( m > 1.0) virtual image. Near the limit,where s approaches zero, the image forms on 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 andthe 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 twoprincipal 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 measuredin 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 principalplane 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 thefirst (effective) focal point ' .B An oblique ray from the object point through the first principal point p 1 in the first principalplane, and exiting from the second principal point p 2 at the same angle to the optical axis.C An oblique ray from the object point to the first principal plane that when extended forwardfrom the second principal plane intersects the second focal point , but is extended forward fromthe second principal plane as a collimated light ray (parallel to the optical axis) and is extendedbackward as a collimated virtual ray.

Given the first and second focal distances ' and and object height h , this construction definesthe object distance s , the image focal distance s' , the to object distance x , the ' to imagedistance x' , and the image height h' as shown in the diagram (above).A real image cannot be produced by the diverging refracted light rays (yellow lines): instead avirtual image is formed at the intersection of the virtual rays (magenta lines) extended backwardfrom the emergent light rays. For an object at finite optical distance, all three virtual rays willintersect at a single point that is not on the optical axis. For an object at infinite optical distancethe 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, whichmeans (according to the sign conventions and the thin lens formulas ) that the focal distances' is always negative and the ' to image distance x' , the image size h' and the magnification m are always positive. (A virtual image cannot form outside the effective focal length ' .)This analysis shows that a negative lens always produces an erect but virtual image when viewedfrom the side opposite the object. For an object at infinity, analysis ray B shows that the virtualimage forms at the first focal point ' with a height equal h' = radians( ) ' . As the object isbrought closer to the lens, the virtual image becomes larger and also approaches the lens; if theobject is located at the first focal point then the virtual image is located at s' = /2 and at half


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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 forobjects at finite or optically infinite distances, for concave or convex mirrors, and for real andvirtual images.The diagram (below) shows the analysis for a concave mirror with the object at a finite distance,and for a concave and convex mirror at infinite optical distance.The sign conventions for a concave mirror are that the single focal point and image focal distanceare positive, because they are measured in the direction that the light is traveling; the objectdistance is measured opposite the direction of the light and is therefore negative. For a convexmirror the focal point and image focal distance are also negative. The object and image heights

and image angles are negative below the optical axis.


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The image size and location are defined as with lenses, by means of three analysis rays from asingle 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 and reflected by themirror; 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 foraspheric (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 withan object point of height h at finite distance o from the mirror vertex. The tracing shows that allthree rays converge at a real image in front of the mirror, shifted from the focal point towardthe object at a focal distance i with a reduced image size h' . A line through this point andperpendicular 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 vanitymirror, where the focal point is farther from the mirror than the typical viewing distance.

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 tracingshows that rays B and C are reflected before they pass through their target points behind themirror, so all three reflected rays are extended in the opposite direction as virtual rays behindthe mirror, where they converge at the location of a reduced, erect virtual image at focaldistance i which again is shifted from the focal point toward the object. Since the object cannotbe placed between the mirror and the focal point, no real image is formed. (The exception iswhen the convex mirror reflects rays that are convergent, for example when used as thesecondary mirror in a two mirror telescope .)

The relationships between object distance, image distance and focal length are:1/ = 1/ o +1/ i 1/ i = 1/ o 1/ .The center of curvature and focal point are related as = r /2which means ray A scales the image size to the object size ash' = h i / o with magnificationM = i / o = h' / h .

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

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

The average diameter of the Moon is approximately 30 arcminutes; its disk will be 21.8 mm wideat the focal plane of a 250 mm /10 mirror.The diagram also shows that as the aperture height y increases, the location of the mirrorsurface shifts by a distance e from the vertex location due to the surface curvature. The surfaceof rotation that focuses all rays parallel to the optical axis at the focal point is determined by aparabolic function, so this shift is:

y2

= 4 e and e = y2

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


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

Lens Combinations

Astronomical eyepieces are designed as two or more lenses or components mounted along acommon optical axis. Their design and evaluation requires optical formulas that extend andgeneralize the analytical framework described above for a single lens . This section first outlinesthe 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 twolenses and their spacing as an optical system.The design procedure is first to select two lenses of the appropriate aperture and power ( 1 =1/ 1 and 2 = 1/ 2). The lens spacing (the distance between their principal planes) isdetermined by trial and error or by analytical formula, for example to minimize chromatic and

spherical aberration and optimize eye relief.

Once the interlens spacing is fixed, the front focal length ( FFL) is calculated to determine thelocation of the eyepiece focal plane and field stop. From this point the effective focal length ( EFL)and location of the eyepiece principal plane are determined. The back focal length ( BFL) iscalculated to determine the location of the exit pupil and the approximate amount of eye relief.Finally, the radius of the eyepiece apparent field of view is determined by the angle , drawnfrom the center of the exit pupil to the edge field stop aperture projected onto the eyepieceprincipal plane.

A worked example for a simple two element eyepiece, using the two second principal planes fortwo components with focal lengths of 115mm (for the front or field lens ) and 70mm (for the back

or eye lens ) at a separation of 50mm, is shown in the diagram (below). This illustrates theapplication of the formulas for front and back focal length ( FFL and BFL) and effective focallength ( EFL). If the "thin lens" approximation is applied, the same formulas can be used with asingle principal plane through the center of each lens.

Note the following:

Assuming the thin lens model is valid, then the focal length of a single spherical lens can be

calculated from its index of refraction ( n L) and the front and back curvatures (the lensmaker'sequation ) as:1/ = ( n L 1)[1/ r 1 1/ r 2]

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


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EFL = ( 1 2)/( 1+ 2).

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

The effective focal length determines the principal plane of the lens combination, and ismeasured forward from the field stop. The EFL will typically be shorter than the focal length ofeither of the lenses separately in positive (converging) lens combinations.

The apparent field of view (the angular diameter of the field stop viewed from the exit pupil) is

derived from the radius of the field stop ( d o) as double the exit angle of a marginal ray:AFOV = 2*arctan( d o /2* EFL) = 2*

To maximize eye relief, the lens with the higher power is usually used as the eye lens. (Toverify this, note that FFL is shorter than BFL.)

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 achromaticimage whend = ( 1 + 2)/2.

The Kellner modification of the Huygenian design handles chromatic aberration by using anachromatic doublet for the eye lens.The graph (below) shows the optical effect when the focal length of either the front or back lens,or the distance between them, is reduced by 50mm while holding the other elements constant.

The effect on eye relief (red line) is the same as the focal length of either the field lens or eyelens is made shorter. Higher power lenses yield less eye relief, but not as a proportion of thefocal length. Thus, the field lens focal length is reduced by 43% (from 115mm to 65mm), andthe eye lens focal length by 71% (from 70mm to 20mm), but each change reduces eye relief bythe same amount, from 34mm to 12mm. In contrast, reducing the lens spacing from 50mm to 0increases eye relief, but only by 10mm.

Reducing the focal length of the field lens has a relatively small effect on the system effectivefocal length (green line) and the system apparent field of view (blue line). This is in contrast tothe effect of a higher power eye lens, which sharply reduces both the focal length and apparentfield of the system, an effect that is augmented by a closer lens spacing.

The direction of these changes is the same in both lenses, and differs from the effect of the lensspacing only in the change in eye relief, so the eyepiece focal length decreases and the apparentfield of view increases, and by a much greater proportion than the eye relief is reduced, whenboth lenses are made to a shorter focal length and are placed closer together.The diagram (below) shows the analysis for a "thick" lens eyepiece, the 28 mm Edmund RKE .Again, the marginal "pencils" or beams of light are traced through the system, but now the

calculation of focal lengths is based on the pair of principal planes for each component; theintervals between the principal planes are ignored.


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These examples reveal the underlying design principle of astronomical eyepieces:

The function of the field lens is primarily to "stage" or prepare the image by partially correctingthe divergence in the light rays (caused by the relative aperture or ratio of the objective) afterthe rays have passed through the focal plane of the eyepiece, refracting the rays from adivergent to an approximately collimated beam. Additionally in the RKE (and all "wide field"eyepieces), the field lens increases the radius of the beam so that a longer eye relief will deliverthe same apparent field.

The function of the eye lens is to bring these corrected rays into a much shorter focus at amuch steeper angle to the optical axis, which produces a wider apparent field of view and greatermagnification. At the same time, the eye lens eliminates any remaining divergence in the imagerays so that they exit the eyepiece as parallel bundles, termed afocal because the image is notformed 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 beam compression of the systemis at maximum and has the smallest diameter (obvious in the diagram). This highly compressedbeam can most easily accommodate the small entrance pupil aperture of the observer's eye.Both the field lens power and the lens spacing can be manipulated to compensate for the veryshort eye relief that a high power eye lens will produce, to make alignment of eye pupil and exitpupil comfortable for the observer.

The main constraint remaining is that a high power eye lens or a very wide field design (when > ~20) produces significant optical aberrations , an optical design problem that was graduallyovercome by advances in optical theory and glass manufacture in the 19th and 20th centuries.

Single Lens Optical Analysis

Optical systems can be analyzed using the paraxial ideal by modifying the analysis based ondistances in relation to a single principal plane to one based on angles and field heights inrelation to multiple principal planes.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.Angles are always expressed in radians, which is approximately the tangent for angles less than20.First, let's revisit the procedure outlined above for locating the focal point and second principalplane of a single converging lens. This can be generalized by analyzing an axial rather thancollimated incident ray (diagram, below).


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The refracting power of lens in air results from the radius of curvature of the two faces of thelens, the thickness of the lens, and the index of refraction of the glass. Therefore measurementof the two surface curvatures ( c 1 = 1/ r 2 for the front surface, c 2 = 1/ r 1 for the back surface) andthe distance between the two lens vertices ( t ) is sufficient to identify the second principal planeanalytically, using any feasible values for the distance s of an axial point on the optical axis andfor the aperture height y of an axial ray from that point to the lens (diagram, above).For a single lens (where n 0 = n 2 = the index of refraction for air), the optical angles u 1 , u' 1 = u 2 and u' 2 are measured in radians but now in relation to the optical axis or lines parallel to itrather than to the lines normal defined by the lens surfaces.

The step by step calculations proceed as follows:

Given:

c 1 , c 2 front, back curvatures of lens (=1/ r 1 , 1/ r 2 ; r 2 and c 2 are negative by sign convention )

t vertex to vertex thickness of lens (positive, by sign convention)

n 0 , n 2 index of refraction for air ( = 1.0)

n 1 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] u 1 = y / s

[2. entry refraction slope] u' 1 = [ y ( n 1 n 0) c 1+( n 0 u 1)]/ n 1

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

[4. exit (image ray) slope, given u 2 = u' 1] u' 2 = [ y' ( n 2 n 1) c 2+( n 1 u 2)]/ n 2

[5. back focal length] BFL = y' / u' 2 [6. effective focal length] EFL = y / u' 2

[7. power of the lens] = 1/ EFL.

This procedure can be repeated for a cemented compound lens consisting of two or threeelements. 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 radii of curvature of the second lens. (If the two lenses areseparated by air, then a different calculation is necessary, as described below .)For example, in step 2 n 1 replaces n 0 , n 2 (the index of refraction for the following lens) replacesn 1 , c 2 is the curvature and y' replaces y , etc. The exit calculations 3-4 will use the final curvatureand the index of refraction for air. Note again that the angles (in radians) are assumed equal totheir tangents or sines, which is the essential condition for the paraxial approximation.Once all calculations are completed, first the focal point is located by measuring the back focallength from the back vertex as before. Then the principal plane of the compound lens is locatedby measuring the effective focal length forward from the focal point. The power of the compound


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lens is determined as before from the effective focal length.To identify the opposite focal point, the two radii of curvature are reversed in the formulas andthe sign conventions will reverse the signs of the radii of curvature and ray angles.

Multiple Lens Optical Analysis

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

Front and back focal lengths are equal: = ' .

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

An axial object ray angle u will produce an image ray angle u' , and an oblique object rayangle u through the first principal point will produce an oblique image ray angle u' exitingfrom the second principal point.

The image magnification will be 1.0, or 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 thesystem that passes through the principal plane at aperture height y (the radius of the systemaperture), and (2) an oblique ray from object height h that passes through the first principalpoint of the field lens. Note that h is not necessarily equal to y . Then the object ray relationshipswill be (diagram, below):

u = y/s and u p = h/s (note that s and u p are negative according to 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 s' , u' and u' p are negative by the signconventions )

As the diagram makes clear, the image angles have been transformed by the optical system suchthat the simple equalities y' = y and u' = u no longer apply. So the generalized procedure mustget us from the object to the image values, using only the refractive indices n 1 and n 2 of the twooptical 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, whichwill then apply to both thin or thick components. Given the unit magnification of the principalplanes for a single component yields the basic relationships:


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u = y / s and u' = y / s'

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

the transfer equations into the second component:

y 2 = y 1 + du' 1 and u 2 = u' 1 the exit ray from the second component:

u' 2 = u 2 y 2 2

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

EFL = u' 2 / y 1

Alternately, assume that 1 , 2 and d are known or fixed by design, and we want to find the

focal length or power of the system:

sys = 1 + 2 d 1 2

which is identical to the reciprocal of the focal length formula given above in the "thin lens" case: sys = 1/ 1 + 1/ 2 d / 1 2

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:

1 = ( d EFL)/( EFL BFL) and 2 = ( d BFL)/( EFL BFL d )

These equations permit fast layout and approximate analysis of optical systems without theincremental calculation of ray traces through all refracting surfaces.Finally, all focal optical systems conform to four systematic proportionalities that equivalentlydefine a quantity known as the optical invariant or Lagrange invariant ( L). Given an axial raywith a height y at the first principal plane and y' at the back principal plane, and an oblique raywith height y p 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' , u p and u' p respectively.

In that case:

L = hn 1u = h'n ku' .

This quantity states the relationships between refractive power, object distance and image focaldistance purely in terms of angles and relative heights. It can be reduced to an alternativestatement of system magnification:M = h' / h = n 1u / n ku' .The invariant applies equally to single or compound lenses or multielement optical systems,which are treated as a "black box" and analytically bracketed by the entrance pupil and backprincipal plane.

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

Eyepiece Prescription Data

In most modern eyepieces, the thin lens formulas are insufficient to design an eyepiece or guideits manufacture. The total information required to specify an eyepiece optical design is called the

prescription data, which includes the radius of curvature for all lens surfaces, the distance

between lens surfaces and centers of curvature as measured along the optical axis, and thespecific glasses that provide the refractive indices and Abbe coefficients assumed to calculate thelight paths. A full example is shown below for a modern version of the symmetrical Plssldesign .


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The prescription data are given in tabular form (diagram, lower left): measurements begin at theobjective focal plane and end at the eyepiece exit pupil, assumed to be coincident with the eyepupil. Note that measurements are relative the radius r 1 is measured from d 1 , distance d 2 ismeasured from d 1 , r 2 is measured from d 2 , and so on. Negative numbers indicate measurementstoward the light source (usually, to the left), plane surfaces have a radius of 0, and air spacesare shown as distances with no glass indicated. (This single radius format applies to sphericallenses only; aspheric lenses require a polynomial prescription not shown here.)

Given the prescription data, available in eyepiece patent documents or optics references, and anassigned diameter for the exit pupil (5 mm in the diagram) and object field diameter (equivalentto the interior diameter of the eyepiece field stop), ray tracing software can calculate the path oflight from the objective focal plane through the eyepiece for any field height on the object focalplane. These rays are often shown in different colors for field heights equal to 0%, 70% and100% of the object field radius; the colors do not denote spectral frequencies but help tointerpret the diagram visually. Conventionally, three rays are calculated as radiating from eachfield point so that they exactly fill the exit pupil.

Ray tracing allows calculation of the eyepiece principal plane, principal point, effective focallength, back focal length, eye relief, the angle of the apparent field of view, and the Petzvalradius, which describes the field curvature in the focal plane of the eyepiece. (Note that anegative Petzval radius indicates a positive field curvature the center of the focal surface iscloser to the field lens than the perimeter of the focal surface; a large Petzval radius describes arelatively small field curvature.)

A single spectral frequency is necessary to compute these standard optical parameters; this isusually = 550 nm or "yellow green" [called green] light. Analysis of eyepiece chromaticaberrations 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


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amorphous (not crystalline) mixtures of fused silica (silicon oxide, SiO 2) and oxides of severalmetals including sodium, calcium, magnesium and aluminum added to improve thehardness, durability and water resistance of the glass. It is relatively stable under normaltemperature 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 outthe refractive indices for different spectral hues in an optical effect called dispersion. Opticalmaterials are therefore identified according to those two attributes: (1) the index of refraction and (2) the Abbe V number or measure of dispersion. In glass catalogs the two attributes arereported as a six digit code the first three digits are the index of refraction (after the decimalpoint), followed by the three digit V number (omitting the decimal point).Across the entire electromagnetic spectrum, most glasses and plastics have periodic absorptionbands of zero transmission. In optical glasses these bands are located above 2200 nm in theinfrared (although A few extremely dense flints are transparent up to 4000 or 5000 nm) andbelow 420 nm to 320 nm in the ultraviolet. "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, theindex of refraction must be standardized on a specific wavelength of light, defined as one of theemission lines of a chemical element. By convention this is either the d wavelength of helium"orange yellow" [called yellow] light at 587 nm ( n d) or the e wavelength of mercury "yellowgreen" [called green] light at 546 nm ( n e ). Then the Abbe numbers Vd and Ve are calculated as:Vd = [ n d1]/[ C F]Ve = [ n e 1]/[ C' F' ]where the comparison wavelengths used (image, left) are the emission lines of hydrogen at C =656 nm (H) and F = 486 nm (H) for d yellow, and the emission lines of cadmium at C' = 644nm and F' = 480 nm for e green. Note that V numbers increase as the difference in refractionbetween the red and blue wavelengths gets smaller.

The diagram (above) illustrates the relative effect of the refractive index and V number in twoequilateral prisms; the actual dispersion has been exaggerated in the diagram for clarity. At thehighest 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 spectrumrefraction. Note however that dispersion and refraction are roughly proportional: as the power ofrefraction increases, so also does the amount of dispersion produced.


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When glasses are charted using thosetwo metrics, they populate an areabetween diffraction indices of about 1.4to 2.0, and V numbers between about90 (low dispersion) to 20 (highdispersion). Although dispersionincreases as refraction increases, thetwo attributes are not perfectly related;so it is possible to choose glasses thathave different properties of refractionand dispersion, which, in combinationwith the shape and spacing of lenses,can produce a wide variety of opticalsystems.Glasses on the left (dispersion of 55 ormore at refractive index below 1.60 and50 or more at refractive index above1.60) are called crowns , and glasses onthe right are flints. Curving across thelower right of the diagram is the glassline of flints produced by combiningcrown glass with lead oxide. The

characteristics produced by different chemical additives are shown as different backgroundcolors; these produce small differences in refractive or dispersive power denoted by terms suchas hard, soft, dense and light.

The dots indicate specific glasses available from Schott, one of many major glass suppliers. Theyillustrate that the distribution of glasses is not even, and is strongly clustered around therighthand diagonal of the diagram, indicating that refraction and dispersion are correlated. Mostof the variations occur within middle values of the indices of refraction and dispersion, straddlingthe 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 thesame 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 smallquantities of potassium oxide, combined with oxides of other metals such as phosphorus, zinc,lantahnum or barium. Crowns made with boric oxide (borosilicate glasses, including Pyrexglass) or fused quartz have been especially important in the manufacture low expansiontelescope mirrors. Glasses marketed as ED (for extra low dispersion ) are nominally crowns(including fused fluorite or calcium fluoride, CaF 2) with a V number above 80 and a partialdisperion far from the Abbe line.

Flints are "soft" glasses with lower melting temperatures, historically made with varyingquantities of lead oxide. One of the highest index flints ( n d = 1.96) contains 18% silica and 82%

lead oxide by weight; some flint glasses used in military applications have even been made withheavier metals such as thorium or uranium. Flints are especially susceptible to clouding underprolonged 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.In the late 20th century, a few hard plastics (acrylics, polycarbonates, urethane monomers) havebeen successfully used as optical materials in many applications, and lenses incorporatingdiffractive (ribbed or grooved) surfaces have been used as well. Plastics can be injection moldedbut are also relatively soft and easily scratched, and can deform under moderately hightemperatures, which makes optical coatings impractical. However certain plastics have replacedCanadian balsam as cements commonly used to bond together the separate elements of acompound lens.

Optical Coatings As explained above , a small fraction of the light directed through an optical system is notrefracted by an optical surface but is reflected from it, and additional light is absorbed by thematerial and scattered by internal reflections at the second (exit) surface.


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In general, the transmission of a light ray through both surfaces of a single uncoated opticalelement of refractive index n i and surrounded by air is:T = 2 n i / ( n i2 + 1)

For an angle of incidence of between 0 to ~30, the two surfaces of a single optical glasselement in air transmit about 92% ( n i = 1.5) to 81% ( n i = 1.95) of unpolarized light, assumingthere is zero light absorption by the glass itself. This reflection obviously reduces imageilluminance and also often causes scatter, glare or ghosts within the image itself. In addition, thetransmission of a multielement optical system is no greater than the product of the separatetransmission values of all elements: two lenses of 92% and 81% transmittances combined as aneyepiece yields a total transmission no greater than 75% (and possibly less). Obviously, methodsthat can reduce surface reflections are highly desirable.

Optical coatings are very thin layers of material vacuum deposited on an optical element tominimize or control the reflection or transmission properties of its surfaces. A variety of materialsused for this purpose include fluorides and oxides of various metals, but the most commonlyused material is magnesium fluoride (MgF 2), which provides both a protective coating to a glasssurface and a refractive index that is optimal for controlling reflections.Coatings manage unwanted reflections by virtue of their optical thickness, defined as the physicalthickness 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 withmonochromatic light.

Light reflected by the internal surface of a 1/4 wavelength layer will be exactly 1/2 wavelengthout 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, n 2 = 1.90) is optimal for very high index flint glasses; it is preferred for crowns as well dueto 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 opticalcoating 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 eliminatereflected light within a bandpass matching the visible spectrum (~400nm to ~750nm). Theselenses 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 thedesign wavelength reflection to 99.8%. Coatings of 50 layers or more have been designed toproduce a very narrow bandpass (for use as a filter); this bandpass can be shifted to longer orshorter wavelengths simply by increasing or decreasing the thicknesses of all layers by aconstant factor.

The commercial designations coated or multicoated indicate that one or more (usually only theexternal) 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 siliconmonoxide (SiO) to provide protection against oxidation and cleaning. Certain multilayer coatingscan also increase the reflectivity of an aluminum coating from 88% up to 99% at certainwavelengths, but not more than ~95% as an average across the entire visible spectrum.

Further Reading Astronomical Optics, Part 2: Telescope & Eyepiece Combined - the optics of astronomical telescopes and eyepieces, separately andcombined as a system.Astronomical Optics, Part 3: Optical Aberrations - an in depth review of optical aberrations in astronomical optics.Astronomical Optics, Part 4: Eyepiece Designs - an i llustrated overview of historically important eyepiece designs.Astronomical Optics, Part 5: Evaluating Eyepieces - how 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 b asics of light reflection and refraction and the use of simple optical elements, such asmirrors, prisms, lenses, and fibers.Basic Physical Optics by Leno Pedrotti - the phenomena of light wave interference, diffraction, and polarization.


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Astronomical Optics Part 2: Telescope & Eyepiece Combined

Telescope & Eyepiece Combined Three Basic Telescope Functions Stops, Pupils, Windows & Baffles

Focal Length & Field of View

Telescope Designs Refractors Refractor Design Principles Reflectors Newtonian Design Principles Cassegrain Design Principles

Refractor/Reflector Hybrids Field of Full Illumination

Eyepiece Design

Telescope Optical Attributes Collimation

Focus Light Grasp

Relative Aperture (Focal Ratio) Exit Pupil Resolution Magnification Magnification Selection Apparent Field of View True Field of View

Effect of Central Obstruction

Optics of the Eye Individual Differences Illuminance & Luminance Luminance Adaptation

Light Grasp Resolution Contrast Sensitivity True Field of View

Focus Diopters

This page introduces the optical principles necessary to understand the design and performanceof astronomical telescope systems the telescope and eyepiece used as a visual instrumentwith the eye included as a third component. It is one of series: the previous page explainedbasic optics ; subsequent pages discuss eyepiece optical aberrations , eyepiece designs andevaluating eyepieces .Included at the end of each page is a list of Further Reading that identifies the sources usedhere and background information available online.

Telescope & Eyepiece Combined

In this section I describe the basic functions, parts and terminology that characterize theKeplerian telescope an "inverting" objective combined with an eyepiece. Diagrams show thetelescope objective schematically as a single refracting lens, but it may be a dioptric system(comprising one or more lenses), a catoptric system (one or more mirrors), or a catadioptric system (mirror and lens combination). The objective is combined with an eyepiece , used as amagnifier to inspect the detailed content of the objective image.

Three Basic Telescope Functions

An astronomical telescope has three basic functions:(1) Light grasp . Our ability to see very faint (low luminance) objects is limited by the area ofthe pupil opening of the eye, which admits only a small amount of light. The telescope admits acolumn of light whose cross sectional area is many times larger than the pupil, increasing thetotal illuminance of the image. Implicit is the complementary function of beam compression ,which means that the light gathered by the large aperture is concentrated into a much smallerdiameter beam that can completely pass through the relatively tiny pupil of the eye.(2) Angular resolution . Our ability to see objects that are small or at great distances is limitedby the eye's nominal angular resolution which can be from about one to over sixarcminutes. A telescope transmits the image of a distant object through a very wide pupil,producing diffraction limited resolution on the order of one arcsecond. Implicit is thecomplementary function of magnification of the telescopic image, which means its arcseconddetails are expanded to match the arcminute visual resolution. This magnified image remainssufficiently illuminated due to the light grasp of the telescope.


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(3) Pointing. The eye as a mobile receptor can look in any direction and remain fixed on anysingle object. The telescope must be able to match this capability so that its optical axis can bealigned in any specific direction and the alignment sustained for extended observation or longexposure photography. For telescopes on the surface of the rotating earth, this implies tracking through a moveable mounting and sidereal motors.Note that light grasp and beam concentration can be entirely analyzed in terms of geometriclight rays , but angular resolution and magnification must be analyzed in terms of physical light

wavefronts.

This cost/benefit calculation divides the range of available aperture into two regimes. The benefitin telescopes below the half aperture point is primarily in their resolution, and any increase inaperture within this range increases resolution proportionately more than light grasp. Above thehalf aperture point, increases in aperture deliver proportionately more light grasp thanresolution. As a result: Double star, lunar and planetary visual astronomers generally prefer high optical qualitytelescopes in the "resolution" half of the regime, despite the comparative lack of light grasp which favors refractors and reflectors with apertures below 300 cm that are less sensitive toatmospheric turbulence and permit a longer focal length (which produces greater objectivemagnification) Deep sky astronomers generally prefer high transmission and small focal ratio instruments inthe "light grasp" half of the regime above 300 cm despite the resolution problems that arisefrom mirror flexure, miscollimation and atmospheric turbulence, and the field curvature andcoma/astigmatism produced by a small focal ratio. Astrophotographers prefer telescopes at or just below the inflection point in aperture and atmoderate focal ratios, maximizing light grasp and field of view without introducing excessiveoptical aberrations or sensitivity to atmospheric turbulence.

For visual use, a key feature of astronomical telescopes is that the combination of telescope

objective and eyepiece creates an afocal optical system . The objective focuses a beam ofparallel light rays from an "infinite" (far distant) object or object space (such as the celestialsphere) as a point on the image plane at the apex of a converging light cone. The eyepiecereceives the light rays diverging from the opposite side of the image plane and focuses them asecond time into a much smaller exiting beam of collimated light rays.As a result, the telescope and eyepiece combination does not have a single focal point or focallength, although the separate objective and eyepiece do. This allows variation in true field, thebasis of resolution, and field width, the basis of luminance.

The afocal output provides a comfortable visual inspection of the telescope image. The eye isstructured so that small muscles can cause the eye lens to bulge, reducing its focal length tofocus the diverging light rays from nearby objects. Infinitely distant objects enter the eye as

parallel light rays, and to focus these the focal eye muscles become completely relaxed. Becausea telescope produces an exit beam of parallel light rays from every object point, the eye canview the telescope image in a completely relaxed state, allowing extended viewing withoutstrain. When the telescope is used with a camera or other recording instrument, the afocal

Resolution and light grasp are both exponential functions ofaperture - as reciprocal aperture (1/D) and aperture area (D 2)- but light grasp increases more than resolution in largerapertures. This leads to the invariant optical relationshipbetween light grasp and resolution shown in the diagram(right). For any arbitrary range of apertures up to a maximumfeasible aperture (defined as 1.0), then 1/2 the total gains inresolution are realized at 1/2 the maximum aperture, whichyields only 1/4 of the total feasible gains in light grasp. Theseproportions apply no matter what aperture we choose as thefeasible maximum. Thus, if we (arbitrarily) take the practicable

limit of amateur telescopes as a 700 mm (28") Dobsonianreflector, then 1/2 the potential gains in resolution are realizedwith an aperture of 350 mm (14"), which delivers only 1/4 thepotential gains in light grasp.


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output is not required and the telescope is typically used without an eyepiece.These generic telescope functions light grasp and beam compression, resolution andmagnification, pointing and tracking may be adapted in different ways for solar astronomy,spectroscopy, wide field photography, transit measurements or field portability, but all areimportant in telescopes used for visual astronomy.

Stops, Pupils, Windows & Baffles

The light grasp and beam compression functions of a telescope are defined by stops, which arephysical obstructions within the system. The optical lenses and mirrors of the system createimages of these obstructions, called pupils and windows, which can be viewed from either theobject or observer end of the telescope (diagram, below). Baffles are used outside the path ofthe light cone in order to minimize internal reflections.

The optical axis is the axis of rotation for all refracting or reflecting surfaces; mirrors, lenses,stops and baffles are centered on the axis and perpendicular to it, with the exception of tilted

optics such diagonal mirrors or prisms. Axial light rays are emitted from a point on the optical axis; if the point is sufficiently far away(at optical infinity ), the light rays reach the telescope as a collimated beam of rays parallel toeach other and to the optical axis.

A stop is a physical obstruction that limits the amount of light entering or leaving the telescopesystem.

The aperture stop is the obstruction that limits axial light rays entering the telescope system:its interior diameter defines the aperture of the telescope. In astronomical telescopes this iseither the mounting of the refracting lens or corrector plate, or the circumference of a reflectingmirror. The aperture stop defines the principal point of the objective; its position affects the

appearance of aberrations in the image created at the objective focal surface. The field stop is the obstruction that limits the area of the image or the amount of light exiting the telescope system. It is usually coincident with the objective focal plane, and is either a barreldiaphragm or field lens lock nut in the eyepiece, or the edges of a camera CCD chip or filmholder. The field stop controls the apparent field of view (the angle of marginal rays passingthrough the eyepiece or onto the CCD sensor), and it limits the appearance in the image ofaberrations that increase with field height.

The linear diameter of the aperture stop defines the clear aperture ( D o), which determines theangular resolution of the objective. For reflecting telescopes with a central obstruction(secondary mirror support), the area of the secondary obstruction ( d s) must be subtracted fromthe area of the aperture stop to compute the effective aperture ( D eff ) that determines the lightgrasp of the system.

A pupil is the image of the aperture stop, viewed from a point on the optical axis at either end ofthe telescope.


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The entrance pupil is the virtual image of the telescope aperture stop, illuminated from theimage space and viewed from the telescope object end. The exit pupil is the virtual image of the telescope aperture stop, illuminated from the objectspace and viewed from the observer end. (The images of the aperture stop with or without theeyepiece in place are

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