e-book Introduction to Geometrical Physics

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In order to understand the optical system of a simple microscope locations of the lens elements in a common laboratory microscope are presented in Figure 1 , the basic properties of a simple thin lens having two light-refracting surfaces and a central optical axis must first be described. Each lens has two principal planes and two focal planes that are defined by the geometry of the lens and the relationship between the lens and the focused image.

Light rays passing through the lens will intersect and are physically united at the focal plane see Figure 2 , while extensions of the rays entering the lens will intersect at the principal plane with extensions of the rays emerging from the lens. The focal length of a lens is defined as the distance between the principal plane and the focal plane, and every lens has a set of these planes on each side front and rear. Traces of light rays passing through a simple bi-convex thin glass lens are presented in Figure 2, along with the other important geometric parameters necessary in forming a focused image by the rays.

An Introduction to Geometrical Physics

The focal points of the lens are denoted by the variable F , and there are two separate focal points, one in front of the lens on the left-hand side of Figure 2 and one behind the lens on the right. The principal planes of the lens P and P' in Figure 2 are denoted by dashed lines, and the distance between each principal plane and its respective focal point represents the focal length f. Because the bi-convex lens illustrated in Figure 2 is symmetrical, the principal planes are located equal distances from the lens surfaces, and the front and rear focal lengths are also equal.

The object or specimen being imaged by the lens is positioned in the object plane , located on the left-hand side of the lens by convention, and is represented by a red arrow that travels upward from the centerline or optical axis , which passes through the center of the lens, perpendicular to the principal planes. Ray traces through the lens yellow arrows emanate from the object and proceed from left to right through the lens to form a magnified real image inverted red arrow in the image plane on the right-hand side of the lens. The distance between the front principal plane of the lens and the specimen is known as the object distance , and is represented by the variable a in Figure 2.

In a similar manner, the distance from the rear principal plane to the image the variable b in Figure 2 is termed the image distance. These parameters are the fundamental elements defining the geometrical optics of a simple lens and can be used to calculate important properties of the lens, including focal length and magnification factor. Lenses can be either positive or negative depending upon whether they cause light rays passing through to converge into a single focal point, or diverge outward from the optical axis and into space.

Positive lenses illustrated in Figures 2 and 3 converge incident light rays that are parallel to the optical axis and focus them at the focal plane to form a real image. As shown in Figure 3, positive lenses have one or two convex surfaces and are thicker in the center than at the edges. A common characteristic of positive lenses is that they magnify objects when they are placed between the object and the human eye. In contrast, negative lenses diverge parallel incident light rays and form a virtual image by extending traces of the light rays passing through the lens to a focal point behind the lens.

Negative lenses have at least one concave surface and are thinner in the center than at the edges see Figure 3. When a negative lens is placed between an object and the eye, it does not form a real image, but reduces or demagnifies the apparent size of the object by forming a virtual image. The distinction between a real and a virtual image is an important concept when imaging specimens through a lens or mirror system, regardless of whether the system consists of a single or multiple components.

In general, images are defined by the regions where light rays and their extensions become convergent as the result of refraction by a lens or reflection by a mirror. In cases where the light rays intersect at a focal point, the image is real and can be viewed on a screen, recorded on film, or projected onto the surface of a sensor such as a CCD or CMOS placed in the image plane.

When the light rays diverge, but project imaginary extensions that converge to a focal point, the image is virtual and cannot be viewed on a screen or recorded on film. In order to be visualized, a real image must be formed on the retina of the eye. When viewing specimens through the eyepieces of a microscope, a real image is formed on the retina, but it is actually perceived by the observer to exist as a virtual image located approximately 10 inches 25 centimeters in front of the eye.

The primary lens geometries for the positive lens elements illustrated in Figure 3 are bi-convex Figure 3 a and plano-convex Figure 3 b ; having one planar or flat surface. In addition, the convex-meniscus Figure 3 c lens has both convex and concave surfaces with similar curvatures, but is thicker in the center than at the edges. Bi-convex lenses are the simplest magnifying lenses, and have a focal point and magnification factor that is dependent upon the curvature angle of the surfaces.

Higher angles of curvature lead to shorter focal lengths due to the fact that light waves are refracted at a greater angle with respect to the optical axis of the lens. The symmetric nature of bi-convex lenses minimizes spherical aberration in applications where the image and object are located symmetrically. When a bi-convex optical system is fully symmetric in effect, a magnification , spherical aberration is at a minimum value and coma and distortion are equally minimized or cancelled.

Generally, bi-convex lenses perform with minimum aberrations at magnification factors between 0. Convex lenses are primarily employed for focusing applications and for image magnification. Typical plano-convex lenses Figure 3 b have one positive convex face and a flat plano face on the opposite side of the lens. These lens elements focus parallel light rays into a focal point that is positive and forms a real image that can be projected or manipulated by spatial filters.

The asymmetry of plano-convex lenses minimizes spherical aberration in applications where the object and image lie at unequal distances from the lens. The optimum case for reduction of aberration occurs when the object is placed at infinity in effect, parallel light rays enter the lens and the image is the final focused point.

However, the plano-convex lens will produce minimum aberration at conjugate ratios up to approximately When the curved surface of a plano-convex lens is oriented toward the object, the sharpest possible focus is achieved. Plano-convex lenses are useful for collimating diverging beams and to apply focus to a more complex optical system. The positive meniscus lens Figure 3 c has an asymmetric structure with one face shaped as a convex radius, while the opposite face is slightly concave.

Meniscus lenses are often employed in conjunction with another lens to produce an optical system having either a longer or shorter focal length than the original lens. As an example, a positive meniscus lens can be positioned after a plano-convex lens to shorten the focal length without decreasing optical system performance.

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Positive meniscus lenses have a greater curvature radius on the concave side of the lens than on the convex side, enabling formation of a real image. Negative lens elements are the bi-concave Figure 3 d , plano-concave Figure 3 e ; with a single planar surface , and concave-meniscus Figure 3 f , which also has concave and convex surfaces, but with the center of the lens being thinner than the edges.

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For both positive and negative meniscus lenses, the distances between the surfaces and their focal planes are unequal, but their focal lengths are equal. The line connecting the centers of the lens curved surfaces in Figure 3 is known as the optical axis of the lens. Simple lenses having a symmetrical shape bi-convex or bi-concave have principal planes, denoted by dotted lines in Figure 3, that are equally spaced with respect to each other and the lens surfaces.

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Curvature radii of the two convex surfaces for a bi-convex lens are indicated by black arrows in Figure 3 a. The lack of symmetry in other lenses, such as the meniscus lenses and the plano negative and positive lenses, causes the locations of the principal planes to vary according to lens geometry. Plano-convex and plano-concave lenses have one principal plane that intersects the optical axis, at the edge of the curved surface, and the other plane buried inside the glass.

The principal planes for meniscus lenses lie outside the lens surfaces. Bi-concave lenses Figure 3 d are primarily utilized for diverging light beams and image size reduction, as well as increasing optical system focal lengths and collimating converging light beams.


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Often termed the double-concave lens, this optical element refracts parallel input rays so that they diverge away from the optical axis on the output side of the lens, forming a negative focal point in front of the lens. Although the output light rays do not actually unite to form a focal point, they do appear to be diverging from a virtual image located on the object side of the lens. Bi-concave lenses can be coupled to other lenses to reduce optical system focal lengths.

The plano-concave lens illustrated in Figure 3 e is a divergent element that has a negative focal point and produces a virtual image.

When a collimated light beam is incident on the curved surface of a plano-concave lens element, the exit side will form a divergent beam. This beam will appear to emerge from a smaller virtual point source than if the planar lens surface had faced the collimated light beam. Even so the incorporation of material on the Ising theory is helpful in this regard.

HC Verma Class 11 Physics Part-1 Solutions for Chapter 18 - Geometrical Optics

I'm going to guess there will be no good introductory texts at that level that will be all that useful. You will probably have to delve into the physics literature particularly, review articles. Although, a lot of lattice QCD stuff is written for experimentalists who know little formal QCD or math, so you should be able to find something reasonably accessible for a beginning step. The text "Differential Geometry, Gauge Theories, and Gravity" is sufficient for the geometric part of the mathematical background of gauge theories in general, and even has a paragraph about lattice QCD!


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  • And applying this formalism to lattice QCD isn't "hard" thus, a paragraph ;. However, the "quantum" part of QCD is not covered by the classical gauge theories described in that text, and to really understand that you have to know QCD. As for the physical intuition of what's going on, I doubt you will get much without a reasonable understanding of quantum field theories in general. You certainly can't get an understanding of these without losing rigor, since there is not yet a rigorous version of QFT there are some very good books out there that do rigorously what we can do rigorously, but that is not much.

    Lattice QCD is on much better footing, but you do lose a lot of the intuition skipping straight to that.

    Introduction to Geometric Optics

    So I would suggest going to the arxiv, and trying to find a good review article that makes you happy, and then check out its references when you get confused. But if you want to really develop a physical intuition for what's going on, I think that would be a lot of work. Offhand, the book mentioned by Steve looks good, but if it is good for you or not depends on your background. And by math standards it does not look particularly mathematical.

    Tony Phillips, a topologist at Stony Brook who taught me differential topology when I was a first-year graduate student, has worked on lattice gauge theory since the mid-to-late s. You can try to take a look at his papers on MathSciNet on this topic. His language is certainly geometrical. These two volumes instantly became my favorite intro to geometric methods in theoretical physics. Convex mirrors reflect incoming plane waves into outward-moving spherical waves with the center of the sphere appearing to be behind the mirror they are diverging mirrors.

    Also the very same relation between the image and object distances applies:. Mirrors have an advantage over lenses in that they do not suffer chromatic aberration. This phenomenon arises due to dispersion, causing the lens to have not just one focal length but a small band of focal lengths corresponding to the different amounts by which it refracts the different colors. This means that it is impossible to focus colored images precisely with a lens.

    Geometrical Methods for Physics

    Mirrors, because they do not rely on refraction, do not suffer this problem. Corrections arising from this and other considerations cause aberrations or deviations from the simple equations developed here for spherical lens and mirror systems. In fact, there are five primary, monochromatic aberrations called spherical aberration, coma, astigmatism, field curvature, and distortion. They are collectively known as the Seidel aberrations. Geometric Optics.

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