Geometrical optics

Geometrical optics, or ray optics, describes light propagation in terms of "rays". The "ray" in geometric optics is an abstraction, or "instrument", which can be used to approximately model how light will propagate. Light rays are defined to propagate in a rectilinear path as far as they travel in a homogeneous medium. Rays bend (and may split in two) at the between two dissimilar media, may curve in a medium where the refractive index changes, and may be absorbed and reflected. Geometrical optics provides rules, which may depend on the color (wavelength) of the ray, for propagating these rays through an optical system. This is a significant simplification of optics that fails to account for optical effects such as diffraction and interference. It is an excellent approximation, however, when the wavelength is very small compared with the size of structures with which the light interacts. Geometric optics can be used to describe the geometrical aspects of , including optical aberrations.

Explanation

in amplitude. In this image, each maximum amplitude crest is marked with a plane to illustrate the wavefront. The ray is the arrow perpendicular to these parallel surfaces.]] A light ray is a line or curve that is perpendicular to the light's wavefronts (and is therefore with the wave vector).

A slightly more rigorous definition of a light ray follows from Fermat's principle, which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.

Geometrical optics is often simplified by making the paraxial approximation, or "small angle approximation." The mathematical behavior then becomes linear, allowing optical components and systems to be described by simple matrices. This leads to the techniques of Gaussian optics and paraxial ray tracing, which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications.

Reflection

Glossy surfaces such as mirrors reflect light in a simple, predictable way. This allows for production of reflected images that can be associated with an actual (real) or extrapolated (virtual) location in space.

With such surfaces, the direction of the reflected ray is determined by the angle the incident ray makes with the surface normal, a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal. This is known as the Law of Reflection.

For flat mirrors, the law of reflection implies that images of objects are upright and the same distance behind the mirror as the objects are in front of the mirror. The image size is the same as the object size. (The magnification of a flat mirror is unity.) The law also implies that mirror images are parity inverted, which is perceived as a left-right inversion.

Mirrors with curved surfaces can be modeled by ray tracing and using the law of reflection at each point on the surface. For mirrors with parabolic surfaces, parallel rays incident on the mirror produce reflected rays that converge at a common focus. Other curved surfaces may also focus light, but with aberrations due to the diverging shape causing the focus to be smeared out in space. In particular, spherical mirrors exhibit spherical aberration. Curved mirrors can form images with magnification greater than or less than one, and the image can be upright or inverted. An upright image formed by reflection in a mirror is always virtual, while an inverted image is real and can be projected onto a screen.

A device which produces converging or diverging light rays due to refraction is known as a lens. Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation. In general, two types of lenses exist: convex lenses, which cause parallel light rays to converge, and concave lenses, which cause parallel light rays to diverge. The detailed prediction of how images are produced by these lenses can be made using ray-tracing similar to curved mirrors. Similarly to curved mirrors, thin lenses follow a simple equation that determines the location of the images given a particular focal length ($f$) and object distance ($S\_1$):

:$\backslash frac\{1\}\{S\_1\}\; +\; \backslash frac\{1\}\{S\_2\}\; =\; \backslash frac\{1\}\{f\}$

where $S\_2$ is the distance associated with the image and is considered by convention to be negative if on the same side of the lens as the object and positive if on the opposite side of the lens. The focal length f is considered negative for concave lenses.

Incoming parallel rays are focused by a convex lens into an inverted real image one focal length from the lens, on the far side of the lens. Rays from an object at finite distance are focused further from the lens than the focal distance; the closer the object is to the lens, the further the image is from the lens. With concave lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at an upright virtual image one focal length from the lens, on the same side of the lens that the parallel rays are approaching on. Rays from an object at finite distance are associated with a virtual image that is closer to the lens than the focal length, and on the same side of the lens as the object. The closer the object is to the lens, the closer the virtual image is to the lens.

Likewise, the magnification of a lens is given by

:$M\; =\; -\; \backslash frac\{S\_2\}\{S\_1\}\; =\; \backslash frac\{f\}\{f\; -\; S\_1\}$

where the negative sign is given, by convention, to indicate an upright object for positive values and an inverted object for negative values. Similar to mirrors, upright images produced by single lenses are virtual while inverted images are real.

Lenses suffer from aberrations that distort images and focal points. These are due to both to geometrical imperfections and due to the changing index of refraction for different wavelengths of light (chromatic aberration).

Underlying mathematics

As a mathematical study, geometrical optics emerges as a short-wavelength limit for solutions to hyperbolic partial differential equations. In this short-wavelength limit, it is possible to approximate the solution locally by

:$u(t,x)\; \backslash approx\; a(t,x)e^\{i(k\backslash cdot\; x\; -\; \backslash omega\; t)\}$

where $k,\; \backslash omega$ satisfy a dispersion relation, and the amplitude $a(t,x)$ varies slowly. More precisely, the leading order solution takes the form :$a\_0(t,x)\; e^\{i\backslash varphi(t,x)/\backslash varepsilon\}.$ The phase $\backslash varphi(t,x)/\backslash varepsilon$ can be linearized to recover large wavenumber $k:=\; \backslash nabla\_x\; \backslash varphi$, and frequency $\backslash omega\; :=\; -\backslash partial\_t\; \backslash varphi$. The amplitude $a\_0$ satisfies a transport equation. The small parameter $\backslash varepsilon\backslash ,$ enters the scene due to highly oscillatory initial conditions. Thus, when initial conditions oscillate much faster than the coefficients of the differential equation, solutions will be highly oscillatory, and transported along rays. Assuming coefficients in the differential equation are smooth, the rays will be too. In other words, refraction does not take place. The motivation for this technique comes from studying the typical scenario of light propagation where short wavelength light travels along rays that minimize (more or less) its travel time. Its full application requires tools from microlocal analysis.

A simple example

Starting with the wave equation for $(t,x)\; \backslash in\; \backslash mathbb\{R\}\backslash times\backslash mathbb\{R\}^n$

:$L(\backslash partial\_t,\; \backslash nabla\_x)\; u\; :=\; \backslash left(\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\; -\; c(x)^2\; \backslash Delta\; \backslash right)u(t,x)\; =\; 0,\; \backslash ;\backslash ;\; u(0,x)\; =\; u\_0(x),\backslash ;\backslash ;\; u\_t(0,x)\; =\; 0$

assume an asymptotic series solution of the form

:$u(t,x)\; \backslash sim\; a\_\backslash varepsilon(t,x)e^\{i\backslash varphi(t,x)/\backslash varepsilon\}\; =\; \backslash sum\_\{j=0\}^\backslash infty\; i^j\; \backslash varepsilon^j\; a\_j(t,x)\; e^\{i\backslash varphi(t,x)/\backslash varepsilon\}.$ Check that :$L(\backslash partial\_t,\backslash nabla\_x)(e^\{i\backslash varphi(t,x)/\backslash varepsilon\})\; a\_\backslash varepsilon(t,x)\; =\; e^\{i\backslash varphi(t,x)/\backslash varepsilon\}\; \backslash left(\; \backslash left(\backslash frac\{i\}\{\backslash varepsilon\}\; \backslash right)^2\; L(\backslash varphi\_t,\; \backslash nabla\_x\backslash varphi)a\_\backslash varepsilon\; +\; \backslash frac\{2i\}\{\backslash varepsilon\}\; V(\backslash partial\_t,\backslash nabla\_x)a\_\backslash varepsilon\; +\; \backslash frac\{i\}\{\backslash varepsilon\}\; (a\_\backslash varepsilon\; L(\backslash partial\_t,\backslash nabla\_x)\backslash varphi)\; +\; L(\backslash partial\_t,\backslash nabla\_x)a\_\backslash varepsilon\; \backslash right)$ with :$V(\backslash partial\_t,\backslash nabla\_x)\; :=\; \backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; t\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; -\; c^2(x)\backslash sum\_j\; \backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; x\_j\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_j\}$

Plugging the series into this equation, and equating powers of $\backslash varepsilon$, the most singular term $O(\backslash varepsilon^\{-2\})$ satisfies the eikonal equation (in this case called a dispersion relation), :$0\; =\; L(\backslash varphi\_t,\backslash nabla\_x\backslash varphi)\; =\; (\backslash varphi\_t)^2\; -\; c(x)^2(\backslash nabla\_x\; \backslash varphi)^2.$ To order $\backslash varepsilon^\{-1\}$, the leading order amplitude must satisfy a transport equation :$2V\; a\_0\; +\; (L\backslash varphi)a\_0\; =\; 0$

With the definition $k\; :\; =\; \backslash nabla\_x\; \backslash varphi$, $\backslash omega\; :=\; -\backslash varphi\_t$, the eikonal equation is precisely the dispersion relation that results by plugging the plane wave solution $e^\{i(k\backslash cdot\; x\; -\; \backslash omega\; t)\}$ into the wave equation. The value of this more complicated expansion is that plane waves cannot be solutions when the wavespeed $c$ is non-constant. However, it can be shown that the amplitude $a\_0$ and phase $\backslash varphi$ are smooth, so that on a local scale there are plane waves.

To justify this technique, the remaining terms are must be shown to be small in some sense. This can be done using energy estimates, and an assumption of rapidly oscillating initial conditions. It also must be shown that the series converges in some sense.

References

External links

Online book, Hyperbolic Partial Differential Equations and Geometric Optics

Category:Geometrical optics