Have you been trying to register on Illuminati brotherhood before and you are unlucky? This site uses Akismet to reduce spam. Learn how your comment data is processed. Follow Six9ja. Search for:. Home » Instrumental. The day of the powerful computer has spawned many consulting optical designers. The design of a lens system for a prototype should be done with a prototype shop in mind.

A formal tolerance analysis may be suitable for a production task but is not appropriate for prototype development. It also does not do any good to specify tight tolerances, unless the buyer is prepared to check to see if the shop can meet the tolerances.

A close relationship between the designer and the shop is highly desirable. It becomes costly to procure components without trust in the prototype shop. Experienced designers find that the most common problems they encounter are with customers who believe that they know precisely what they want and present a list of specifications with unrealistic requirements.

Often the customer does not appreciate the complexity of the design which is to use an ill-defined light source with an inexpensive condensor, illuminating a precise object imaged onto a special detector surface, then to be sampled, digitized, and image processed. The transfer function from each component to its neighbor is usually not known with the precision comparable to the precision one can calculate by using mathematically convenient assumptions.

In a preliminary design phase, the need for first-hand experience is needed. The optical designer must be able to discuss trade-offs with the electrical and mechanical designers, who in turn should also have some understanding of what the optical trade-offs incur. One of the major requirements of good design is to match the methods of design, fabrication, and testing with the economics of the problem.

In this treatment of optics, the reader will be given some idea of what to expect when searching for an optical solution to a technical problem. The Future of Optics The future of optics is without any doubt very bright. New optical devices and techniques appear every day. They are even replacing and complementing electronic devices with advantage.

Lasers and holograms find new applications every day, not only in the laboratory, but also in telecommunications, industry, administration, and even in domestic instruments, like the digital compact disc players. Some day in the near future, the hybrid optical-electronic computer may be a reality. References 1. Kingslake, ed. Kingslake and B. Thompson, eds.

Shannon and J. Wyant, eds. Hardy and F. Twyman, Prism and Lens Making, 2d ed. Rogers and R. Fischer, eds. Types of Optical Systems Optical systems are used to collect light from a variety of sources, transport information, and direct it to a large assortment of detectors. Figure 1 shows a photographic lens which is made up of five elements. All the surfaces are spherical, and the centers of curvature lie on a single axis, i.

The drawing shows the intersection of the spherical surfaces and the plane of the paper. Any plane containing the optical axis is called the rneridional plane. The lens shown has rotational axial symmetry about the optical axis. Figure 2 shows a lens which has a rotational symmetry, but one surface is aspheric.

Elements of this type are used in optical systems, but they cost from three to ten times as much as elements with spherical surfaces when they are made of glass. It is difficult to make aspheric surfaces out of glass with the surface smoothness found in spherical lenses. The aspheric surface also has a single axis which, in theory, must pass through the centers of the spherical surfaces in the complete lens. The aspheric element must be carefully mounted.

When there is sufficient production, they can be made of plastic at low cost after the molds have been paid for. A word of warning: there are many more optical shops that are willing to make an aspheric surface than can actually do so. Figure 3 shows a lens system with a decentered element.

The axial symmetric lenses shown in Figs. During the manufacturing process, the lens elements are usually decentered to some degree. The effect of these manufacturing errors must be evaluated by designers when the lens is toleranced. There are occasional modern optical systems which are designed to use tilted elements to provide asymmetries in some of the other components.

Users beware! If there is no alternative, expect mechanical and testing problems which must be handled with care and at extra expense. Photographic lens of five elements. Optical systems often employ the use of cylindrical and toric surfaces. Spectacle lenses and anamorphic projection lenses are examples. Figure 4 shows a lens system with a prism which bends the optical axis. In the design phase, the lens is considered as a lens system of four elements on a single axis, with a block of glass between them.

In the construction of the system, it is necessary to mount the components on a common axis. This is not a trivial problem for the mirror surface of the prism must be positioned accurately with respect to the optical axis of the two sets of optical components. Folding an optical system like this is hazardous, unless FIG.

An aspheric lens element. A decentered lens. A system with a prism. Figure 5 shows a combination of refractive and reflective components. In the design, the elements are considered as rotationally symmetric. The prism is used to provide an erect image and to direct viewing from the object to the observer.

Those who have had some optical experience may wish to explain why the prism is a pentaprism. If one has difficulty, one can refer to the literature.

It is necessary to decide whether to provide adjustments to align the components or to use precision during the manufacturing process. Adjustments can move, and fixed machining requires close tolerances on both the optical FIG. A metascope. The optics of an infrared viewing telescope. Binoculars, for example, are usually made with some adjustments to keep the prisms aligned; and as a result many binoculars are soon out of alignment.

This is particularly true for the less expensive ones. It amounts to false economy, because a pair of misaligned binoculars will not be used for long, and repairing them costs nearly as much as a new pair.

Many systems use gratings or dispersive prisms. Rotating polygons or holograms for laser scanners are becoming common in modern instruments. Quite often the systems are used over a wide spectral range, so mirrors are used rather than refractive elements. Aspheric surfaces are common in mirror systems. Mirror systems are usually something to avoid unless the problem requires large optics or a wide spectral range.

Mirror systems have many mounting and obscuration problems. The single surface of a mirror is usually asked to do too much, and therefore it becomes sensitive to mounting. In spite of the wide variety of lens systems, the vast majority of them turn out to be axially symmetrical systems.

This chapter deals with the basic tools needed to evaluate and to design such systems. Basic Laws In this section, the basic laws of geometrical optics and image formation will be described. The Law of Refraction The backbone of geometrical optics is ray tracing.

The rays of light travel in straight lines as long as the material in which they travel has a homogeneous index of refraction. When the ray encounters an interface between two different materials, refraction of the ray takes place. The equation governing the refraction is written in vector form in Eq. The derivation is clear from observing the illustration in Fig.

Surfoar The vector law of refraction. In the calculation of optical systems, the object is considered to be made up of a set of point sources of light. The design objective is to shape the lens elements so that each object point is imaged as a perfect point. This means that all the rays passing through the lens from an object point should converge to a single geometrical image point. The rays are normal to the wave fronts of light.

The wave front diverging from a source point is spherical, but the rays remain normal to the wave fronts providing the material is not birefringent. If the designer of the optical system is successful in uniting the rays to a single image point, then the emerging wave front is also spherical and the optical path OP , from the object to the image point, is equal for every ray.

The O P of a ray is the geometrical distance along the ray, multiplied by the index of the material through which the ray is traveling.

This does not mean, however, that the optical paths from all object points to their corresponding image points are equal to each other. To repeat: all the ray paths from a single object point to its perfect image are equal. In most practical systems, the emerging wave front departs from a perfect sphere by an optical path difference OPD. The OPD represents aberration. When the OPD is less than a wavelength, the distribution of light in the image point is determined by diffraction.

The OPD is measured with respect to a central chief ray which is often called the principal ray. When a design calls for a near-diffraction-limited lens, the designer concentrates on reducing the optical-path-length differences to the lowest possible values for several image points over the field of the object. When the images are several waves of OPD, the designer usually concentrates on the geometrical transverse deviations from a perfect point.

Most of the modern lens computer programs can calculate both the diffraction integral and what is called the modulation transfer function MTF. It should be pointed out that most of the design time goes into simply manipulating the design to reduce the geometrical ray deviations or the OPD to small values. The diffraction calculations are only needed near the end of the design to evaluate the precise light distribution in the image.

This is why it is stated that ray tracing is the backbone of geometrical optics and optical-instrument design. The Battle of Notation A lens system consisting of several components and attributes has many parameters which must be labeled with a systematic convention of signs.

The convention used here is widely used, but not universally. The coordinate system is shown in Fig. Figure 8 shows the notation used for the surface parameters: r represents the radius of curvature, t the thickness between surfaces, n the index of refraction, Y the height of a ray on a surface, and U the angle of a ray with respect to the axis. The following rules are used for the notation and the sign of the parameters. The coordinate system to be used. Notation used to specify surface parameters.

The object is assumed to be at one’s left, and the light travels from left to right. The object surface the 0 surface is usually considered to be a plane surface. The modern programs, however, will accept a curved object surface. Surface 1 is the entrance pupil plane. Surface j is a general surface in a lens. The subscript j on the j surface is often omitted.

For example, rj is written r while rip, is written r – I. The radii are positive when the center of curvature of the surface of the lens lies to the right of the pole of the surface, and are negative when the converse is true. The angle U that the ray forms with the axis is positive, in agreement with the convention in analytic geometry.

In Fig. Thickness t , is positive when the front surface of the lens lies to the right of the entrance pupil as in Fig. Often t, is infinite. When the light travels from right to left, the index of refraction is negative, and the signs of the thicknesses are reversed.

Another common notation uses primed quantities to refer to data on the image side of a surface and unprimed quantities for data on the object side of the surface. A comparison between the two notations is given in Fig. Whenxeferring to data pertaining entirely to a single surface, it is convenient to use the prime notation for the data on the image side of the surface.

This notation will be used here when it is clear that reference is being made to a single surface. Comparison between two commonly used conventions for labeling surface data. The lower method will be used in this chapter when it is clearly data pertaining to a single surface. When deriving formulae for optical calculations, one must develop a systematic way to use the sign convention. An incorrect sign indicates a completely different problem than intended. One must be careful to assign the correct sign for input data, because the computer has no way of knowing what one intends.

An incorrect sign can make thousands of calculations and printing worthless. The following procedures are recommended for deriving optics formulae. See Longhurst. Draw the diagram accurately. Label all quantities without regard to sign convention. Assume all the data are absolute values. Derive the geometric relations. Reduce the algebra to the equations needed for the calculations, and then insert negative signs in front of any quantity in the diagram which is negative to the agreed-upon convention.

It is not necessary to be concerned about the signs of quantities which do not appear in the final results. Figure 10 shows a doublet with properly assigned input data. Meridional Ray Tracing The procedure for deriving equations is illustrated by the derivation of the equations for a meridional ray refracted at a single surface, as in Fig. I50 0. A doublet objective with properly assigned signs for the curvatures and thicknesses.

Notice that all the double arrows on lines indicate lengths and angles. This means that they are labeled as an absolute value. Signs for these quantities must then be inserted in the above equations.

Equations 2. They will be used later to explain the derivation of the paraxial-ray equations. They are simple and easy to use, but they lose accuracy for long radii and small angles of U, because it becomes necessary to subtract two numbers which are nearly equal. This is satisfactory when doing the calculations by hand, and the loss of accuracy can be observed as it happens. When it does happen, a more cumbersome formula can be used. If the program is written for an electronic calculator, a differentform of equations should be used.

Diagram illustrating the geometry of the surface-to-surfacetransfer equation. Skew rays are rays that do not remain in the meridional plane as they pass through the system.

Further discussion of the equations is not appropriate for this chapter other than to say that extremely sophisticated tracing programs are now deeply imbedded in several commercially available programs. Since this is the innermost loop of an automatic correction program, they are extremely general and fast.

On a large-scale computer, thousands of ray surfaces can be traced in a second. The meridional rays are the most volatile, so if they and the close skew rays are controlled, the lens is well on the road to good correction. The equation shows that for an object point there is one image point. The angle U does not enter into the equation.

It is useful, however, to think of the paraxial rays as passing through the lens at finite aperture. To do this, one can use the concept of paraxial construction lines PCL to represent the paraxial rays. The following substitutions are made in Eq. The paraxial rays appear to pass through the lens at finite angles, but they refract at the tangent planes of the optical surfaces.

The representation of paraxial rays which refract on the surface tangent plane. In order to simplify the paraxial equations, it is customary to replace the tan u with just u. The paraxial equations for refraction and transfer are then 2.

Graphical Ray Tracing There is a convenient graphical ray-tracing procedure which is illustrated in Fig. The paraxial rays are traced in exactly the same manner as the real meridional rays by using the tangent planes. One can see that the graphical tracing of paraxial rays is an exact method for locating the paraxial images.

Figures 15 a and 15 b show the graphical tracing of real and paraxial rays through a plano-convex lens. They show that when the lens is turned with the strongly curved surface towards the long conjugate, there FIG. Graphical construction for tracing paraxial and meridional rays. Notice also how closely the paraxial construction lines and the real rays stay together as they pass through the lens.

In a well-designed lens, this is nearly true for all the rays. Only when the relative apertures become large do they separate by noticeable amounts. This then is a guide to how the lenses should be shaped and positioned. It also is an explanation of why the paraxial-ray tracing is such a valuable tool for the engineer interested in the mechanical aspects of the problem.

The paraxial-ray heights quite accurately indicate the locations of the real rays. A Paraxial Computation System Paraxial rays are traced through an optical system using the format shown in Fig.

The method is called the ynu method. In the example shown, two rays are traced through a two-surface lens in order to locate the focal planes and the cardinal points. This method permits easy calculations for paraxial-ray tracing. The computation sheet used to locate the cardinal points in a lens. Sometimes the rays are traced forwards and sometimes backwards to solve for the curvatures and thickness. It is not necessary to use formulae for the focal lengths of combinations of lenses or equations for the positions of the cardinal points, since all the information lies in the table when the rays have been traced.

Mirror Systems Mirror systems with reflecting surfaces are handled by substituting -n for n’ on the surfaces where reflection takes place. The system is folded out and ray-traced exactly in the same way as a refracting system.

Figure 17 shows the calculation of a double reflection in a single lens. This calculation locates the position of a ghost image. The problem is set up as follows.

Draw a picture of the lens and sketch the path of the ray to be traced. One should always do this before tracing a ray. It is worthwhile to develop the ability to accurately sketch the passage of a ray through 0. It also helps to reveal stupid input mistakes, which computers enjoy grinding on just as much as on the correct problem.

Insert the curvature in the order they are encountered by the ray to be traced. Use the sign convention for curvature; do not change the sign on curvatures after a reflection. Insert thicknesses in the order that the ray encounters them.

When the ray travels from right to left, insert the thickness as negative. Do the same with the indices as with the thicknesses. Gaussian Optics There is sufficient information from tracing two paraxial rays through an optical system to predict the path of any other ray with a minimum amount of calculation, because there are simple relations between the two rays. The first ray, called the a ray, is labeled using y and u. The second ray, the b ray, is the oblique ray and referred to by j j and ii.

The H is called the optical invariant. The b ray is traced from the edge of the object and is directed through the center of the entrance pupil. See Section 2. These two rays are illustrated in Fig. A Lambertian source emits uniform flux into a given solid angle, independent of the angle between the normal to the surface and the direction of viewing. A diagram illustrating the CI and b rays which are traced through a lens system in order to compute the third-order aberrations.

The diameter of the diffraction image of a point source is inversely proportional to n u k. Table I lists the number of image points that some common types of lenses can image. This large number of image points indicates the remarkable ability of optical systems and shows why they are so useful as informationgathering tools. The value of H is then a direct measure of the magnitude of the design problem. It may be simple enough to trace paraxial rays through a lens system with over 10 million image points, but if the lens is to be built, it must have precision surfaces and centering of the elements.

One should not be too greedy for a large H without being prepared to face the manufacturing problems. Many optical systems have H greater than unity, but the image quality degrades near the edges of the field, and there are not as many image points as given by Eq. For example, H for a pair of binoculars is around 1.

The diff raction-limited performance is needed over only a small central region. The remainder of the image field is used for search, detection of motion, and the reduction of a tunnel-vision appearance.

If a lens system is to be corrected for spherical and coma aberrations, they must have the same magnification for the marginal rays M and for the paraxial rays m. Then, the following expression, called the optical sine condition, may be obtained as follows 2. The NA is often given in the specification of a lens system. When the system works at finite conjugates, this NA value also specifies the nosin Uo on the object side, from Eq.

When the object point is at infinity, Eq. The Cardinal Points of a Lens By tracing two paraxial rays, the six cardinal points for a lens may be found. This subject is treated in every textbook on optics. It is presented here for completeness. Figure 19 shows the six cardinal points of a lens when the object index of refraction is no and the image side index of refraction is n,. The points F, , F 2 , P I , P2are located by tracing a paraxial ray parallel to the optical axis at a height of unity.

A second ray is traced parallel to the optical axis but from right to left. This is why they are called principalplanes. These planes are also planes of unit magnification. The nodal points are located by tracing a ray back from C parallel to the ray direction FIA. If this later ray is extended back towards the F2 plane, it intersects the optical axis at N,. The location of the cardinal points in a lens system. In the diagram, a third ray is traced through the lens, and it intersects the image plane at the point D.

The tracing of these two rays provides enough information to locate all the cardinal points. An excellent exercise is to graphically locate the cardinal points when all of the information one has is the ray diagram shown in Fig. When the object plane is at a finite distance, the a and 6 rays are not traced in parallel to the optical axis. Then the two rays do not directly locate the focal points and the principal planes. It is however possible to locate the cardinal planes from the two rays traced from the finite object by using Eqs.

A diagram showing the a and b rays entering and exiting an optical system. One can locate all the cardinal points from this diagram. The following equations can thus be derived for the cardinal point locations. Conjugate relations for object and image distances. When you click on the green button above, your email address will be added to our contacts list and you will receive an email with free sound effects as well as monthly emails containing new product releases and Sound Dogs news.

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We travel the world to bring you authentic, coveted and royalty-free sounds that make your projects come to life. Peter king 1 year ago wow i really like dis d song was so nice Loading Lucky Ebemen 1 year ago Have you been trying to register on Illuminati brotherhood before and you are unlucky? Marvelous olaniyi 1 year ago Wow nice music I which to see u then I will join you nd sing.

Mx mhobad 10 months ago Pele macheal jackson, mx mhobad Loading Decent oluwa boy 10 months ago Wow I like this mhobad so feel good. These planes are also planes of unit magnification. The nodal points are located by tracing a ray back from C parallel to the ray direction FIA. If this later ray is extended back towards the F2 plane, it intersects the optical axis at N,.

The location of the cardinal points in a lens system. In the diagram, a third ray is traced through the lens, and it intersects the image plane at the point D. The tracing of these two rays provides enough information to locate all the cardinal points. An excellent exercise is to graphically locate the cardinal points when all of the information one has is the ray diagram shown in Fig.

When the object plane is at a finite distance, the a and 6 rays are not traced in parallel to the optical axis. Then the two rays do not directly locate the focal points and the principal planes.

It is however possible to locate the cardinal planes from the two rays traced from the finite object by using Eqs. A diagram showing the a and b rays entering and exiting an optical system. One can locate all the cardinal points from this diagram. The following equations can thus be derived for the cardinal point locations. Conjugate relations for object and image distances.

One of the easiest ways to locate the cardinal points in the laboratory is to image a distant source at Fl and measure the distance to some reference point on the lens.

Then turn the lens around and image the distant object on the point F2 and measure from the same reference point on the lens. Place an object a distance x from Fl and measure the distance xr from F2. Thin Lenses In planning lens systems, it is convenient to use the concept of thin lenses. The thickness of the lens is considered to be zero.

Table I1 shows the computation sheet for analyzing a system of three thin lenses. The y nu method to solve imaging problem 1. Sample Calculation Sheets for Thin Lenses The paraxial calculation of optical systems will be illustrated by means of three examples. Design an optical system to relay an image with a magnification rn equal to -2, a numerical aperture NA on the image side equal to A value of NA equal to 0. Then, tan 5. Now, nkUk may be calculated by using Eq. A sketch of the thin lens with its calculations is illustrated in Fig.

The diagram in Fig. It is easy to see that there is an infinite number of solutions for the positions of the lenses. The y nu method to solve imaging problem 2. The object-to-image distance is then The calculation sheet is shown in Fig. Lens 1 could be made out of two doublet objectives of the same focal length and with parallel light between them. The real half field of view is 3. The system has an objective, a field lens, and an eye lens, as shown in Fig.

The calculation sheet for this lens system is shown in the same figure. Before the powers of lenses 2 and 3 are known, the a and b rays can be traced through the system. The powers may be solved for when the ray data is complete. Chromatic Aberrations The paraxial-ray tracing and the subsequent location of the cardinal points in a lens depend upon the index of refraction of the glass. The index is wavelength-dependent and should be written as n,. An achromatic lens is a lens that is corrected so that the cardinal points are located in the same positions for a band of wavelengths.

Therefore, the lens has to focus a band of wavelengths at the same focal position and have the same focal length. OL53 0. The y nu method to solve imaging problem 3. Figure 25 shows an achromatic lens made up of three spaced lenses. The positive lenses have low dispersion, and the negative lens has high dispersion.

It is clear that the axial ray and the oblique ray must separate for the different wavelengths upon entering the lens. To be achromatic, it is necessary for the color rays to emerge from the same position on the rear surface. The spreading of the rays in the diagram has been exaggerated in order to see the separation, but they are in the correct relative positions. By differentiating Eq. The surface contribution for the j surface is. A diagram illustrating ray paths for F and C light when the lens has been corrected for Tach and Tch.

A l is a wavelength approximately midway between X2 and A,. It is the ray displacement between A2 and A3 as measured in the image plane A ,. When Tach hz – A3 is positive, the A3 ray intersects the image plane above the h2 ray, and the lens is said to be overcorrected for color. The oblique chief ray contributes to the chromatic aberration of the image height.

The formula is the same as for the axial Eq. Table is a calculation sheet of the chromatic calculations of the axial and oblique rays for an air-spaced triplet. The chromatic aberrations of the a and b rays. Calculation of Tach and Tch for a Triplet Objective 0.

Thin-Lens Chromatic Aberrations Further simplification can be achieved by combining the two surfaces of a lens to obtain the chromatic-aberration contributions for a thin lens. The thin-lens contributions to the two chromatic aberrations are given by the equations 2. To assure an accurate value of the index of refraction of the glass, the index is measured at spectral lines. The F light A is F and C light are hydrogen lines, and d light is a helium line.

Glasses are referred to by code numbers using three digits for the V number. Figure 27 shows a plot of the transverse axial color for a visual achromat as a function of the wavelength. Note the following about the curve. This wavelength is not quite midway between the F and C light. This is called secondary color or residual color. In a thin-lens doublet, this is equal to 2. Figure 28 is a plot of P versus V for a large selection of glasses.

A plot of P versus V for a selection of typical glasses. The most outstanding examples are , flourite, and quartz. These materials should be used with caution, for the following reasons. Glass is expensive and tends to tarnish and stain easily. Flourite is expensive and comes in small pieces less than one inch in diameter. It is used primarily in microscope objectives. It does not polish as smoothly as glass and introduces scattered light. If used, it should be cemented between glass elements to reduce the scattering.

The phosphate glasses are also expensive and tarnish easily. The most effective way to reduce the secondary color in optical systems is to keep the Flnumber of the lenses large. When small Flnumbers are essential, the focal lengths should be kept short. These rules are illustrated in the following two cases. The secondary color comes primarily from the objective because its aperture is large. By making the focal length of the eyepiece large, the angular subtense of the secondary color is reduced.

The result is that the telescope should be made as long as possible. For this reason, the microscope objectives have focal lengths ranging from around 2 mm to 32 mm. In the short focal lengths, the working distance becomes extremely small.

The working distance can be increased, but at the expense of increasing the secondary color. In dealing with secondary color in instruments, it is often a mistake to add complexity in order to reduce the secondary color below a detectable amount.

The eye is remarkably capable of visual acuity with secondary color present. This is particulary true for instruments used for pointing. It is more important to form images which are symmetrically round. By adding extra elements with stronger curves, the lens may be manufactured with decentering, which would destroy any gain from secondary color correction.

The chromatic blurs are not additive when the chromatic blurs become large. An example of how the secondary color can be reduced by placing a lens in a position where there is a substantial spreading of the color rays. Figure 29 is an example showing why the simple theory is not correct. The figure shows an exaggerated drawing of an achromat with secondary color.

A lens placed in the focal plane of the d light will have no effect on the d light, however, the F and C light rays will be bent and will reduce the secondary color, adding to the Petzval sum which is field curvature.

This principle is used in microscope objectives. They do not have as much secondary color as their focal lengths suggest, but they have more field curvature. When flat-field microscopes are developed, it is important to expect more secondary color.

The decision to use special glasses then is sometimes necessary, which explains its higher cost. Stops, Pupils, Vignetting, and Telecentricity Most optical systems should have an aperture stop to control the cone angle of flux entering the lens from the object. The diaphragm of a camera or the iris of the human eye are examples of aperture stops. Usually, but not always, the aperture stop is surrounded by lenses, as shown in the simple system illustrated in Fig.

The following should be noted about the diagram. The aperture stop limits the angle uo and u k. The chief ray passes through the center of the aperture stop. The chief ray is directed towards the center of the entrance pupil and emerges from the center of the exit pupil. The field stop limits the angle of obliquity for the chief ray.

The entrance and exit pupil are usually located by tracing the axial a and the chief b ray. Entrmnce P u p i l Plan.. Aparturn Stop Plan..

A diagram showing the pupil planes and field stop for a two-lens system. Figure 30 shows the diameters of the a and b lens and the aperture stop as they project onto the entrance pupil, as viewed along the chief ray. The common area is the shape ofthe beam that passes through the lens for the off-axis point 0.

The lenses a and b vignette the full aperture. This causes a reduction in the flux passing through the lens, and it influences the shape of the image of the point. The field stop in Fig. The field stop should be located in an image plane in order to have a sharp cutoff in the field.

If the film plane had no mechanical limit to the film area, the image brightness would fall off gradually. Optical systems are sometimes designed to image the exit pupil at infinity. The chief ray then emerges parallel to the optical axis; and if there is no vignetting, the perfectly corrected off-axis image is exactly the same as the central image. This avoids an obliquity factor on the receiver and means that slight focus variations do not result in a change of image size.

Lenses of this type are called telecentric on the image side. Lenses can also be telecentric on the object side. The size, shape, and position of the exit pupil has a strong effect on the shape of the image of a point source if the lens is near diffraction-limited.

Too much vignetting can cause an image to appear astigmatic. The exit subtend of the exit pupil also determines the illuminance incident on the image plane. The coordinates of a general ray passing through the paraxial entrance pupil.

The paraxial entrance pupil is used as a reference plane for all the rays traced through the optical system. Figure 31 shows the coordinate system in the entrance pupil of a lens as it is seen from the object plane. The paraxial a ray passes through at the height y,, and the b ray passes through the center of the pupil. The cpordinates of a general ray are Y ,,2,.

There usually is some aberration of the pupil planes. A real ray traced through the center of the paraxial entrance pupil may not pass through the center of the aperture stop. In spite of this, the paraxial pupil plane is usually used as the reference surface for the pupils, and for most lenses the pupil aberrations are small.

In wide-angle systems, however, one must consider the effect of pupil movements. It is pupil aberration that makes it difficult to view through a wide-angle telescope. It is necessary to have an oversize pupil diameter to provide oblique rays that can enter the eye pupil. Paraxial-Ray Summary It may appear odd that optical engineers are so concerned with paraxial rays, when they represent an infinitesimal group of rays surrounding the optical axis.

They are important for the following reasons. Any deviation from the paraxial-ray images represents an image error. The surface contributions to the image errors indicate the source of the aberrations. The deviations are usually less than the tolerance needed to determine the clear apertures of the lenses.

If any hand computing is done, it is concerned with paraxial rays. The ray tracing and image-error reduction is done entirely on the large computer.

It is through thoroughly understanding first-order paraxial optics and the sources of third-order aberration that a designer knows how to specify starting points for the automatic correction of image errors. Third-Order Aberrations. The Aberration Polynomials The time has come to get off the optical axis and note what happens when rays are traced through the lens at finite heights above the optical axis. Figure 32 shows the object plane, the entrance plane, and the image plane as viewed from the object surface and looking towards the image plane.

Third-Order Aberrations In a system which is rotationally symmetric around the optical axis, it is completely general to consider object points along a single radial vector from the optical axis. All object points at a given radius from the optical axis are imaged by the lens in exactly the same way. FOB is the fractional object height.

The general ray passes through the entrance pupil at the coordinates Y l , Zl, and it intersects the image plane at Yk,2,. View of the object plane; entrance pupil as seen looking along the optical axis from the object side and towards the image plane.

Transverse ray plots resulting from a focal shift. The coefficients are 1. The distortion term represents a displacement of the Y position of the chief ray. The Wave-Aberration Polynomial To evaluate a well-corrected optical system, it is necessary to consider the optical paths OP from the object point to the image point. Transverse ray plots resulting from pure third-order spherical aberration.

Transverse ray plots resulting from pure third-order coma. I k 44 FIG. Transverse ray plots resulting from pure third-order astigmatism and field curvature. This means that the wave front converging to the image point is a spherical surface. When there is aberration, the converging wave front is not spherical. In order to specify the path length for an aberrated ray, the path length is measured along the ray from the object point to a line drawn through the image point and normal to the ray.

This is illustrated in Fig. This measure of the path length is justified by the concept that the emerging aberrated wave front can be considered as the superposition of an infinite number of plane waves which are normal to the rays.

The difference between the perfect reference sphere and the aberrated wave front OPD may also be expressed by an aberration polynomial. The term u2 is the coefficient for a transverse shift of the center of the reference sphere from the Gaussian image plane. Plonr – — FIG. The optical pat. This is appropriate when designing optical systems with near diff raction-limited images. There are designs, however, where the OPD values are many wavelengths, and in that case, the size of the image can be computed from the geometrical deviations.

For these problems, it is perhaps more convenient to think of transverse image errors. Prior to large computers, it was difficult to compute accurate optical path lengths.

As a result, designers concentrated on ray deviations in the image plane, and the OPD values were calculated by integrating Eqs. Now that large computers are in use, direct calculation of the optical path is commonly used.

The Calculation and Use of Third-Order Aberration Coefficients The third-order aberration coefficients are used extensively in opticalsystem planning and design. They provide insight as to the source of aberrations and lead to understanding of how to correct image errors and even how to balance out higher-order image errors.

It provides some assurance that the design is located in a configuration that has a chance of success. The third-order coefficients may be computed from the paraxial data by using the following formulae. A diagram showing a stop shift and the data used to compute Q. The following example shows how one can devise concepts to shape lenses and correct aberrations.

The lens is a single plano-convex lens with a focal length of 10mm and an F-number of It illustrates how Eqs. A single lens shaped to minimize spherical aberration, with the stop in contact with it, will have a meridional ray plot for a 20″ off-axis point, as shown in Fig.

The curve is essentially a straight line with a negative slope. This focus is closer to the lens than to the paraxial image plane. The focus will then be at Plotted in the same figure are the terms due to cr, and a,.

Equation 2. In order to reduce the slope of the meridional ray plot, it is necessary to remove the stop from the lens. The sigma a,of a single lens will always be negative, so the term Q’u, will be added to the negative u3and make the slope more negative. However, if the lens is shaped by bending sharply to the left, a, will be positive. A single lens shaped to minimize spherical aberration has a large amount of inward-curving field.

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