Geometrical optics

From Wikipedia, the free encyclopedia
  (Redirected from Ray optics)
Jump to: navigation, search

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 they travel in a homogeneous medium. Rays bend (and may split in two) at the interface 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 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 imaging, including optical aberrations.



A light ray is a line or curve that is perpendicular to the light's wavefronts (and is therefore collinear 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.[1]



Main article: Reflection (physics)



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 equal to one.) 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.[3]


Main article: Refraction
Illustration of Snell's Law

Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction n_1 and another medium with index of refraction n_2. In such situations, Snell's Law describes the resulting deflection of the light ray:

n_1\sin\theta_1 = n_2\sin\theta_2\


v_1\sin\theta_2\ = v_2\sin\theta_1

where v_1 and v_2 are the wave velocities through the respective media.[3]




A ray tracing diagram for a simple converging lens.



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.[5] The focal length f is considered negative for concave lenses.


Likewise, the magnification of a lens is given by




Underlying mathematics[edit]

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) \approx a(t,x)e^{i(k\cdot x - \omega t)}

where k, \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\varphi(t,x)/\varepsilon}.

The phase \varphi(t,x)/\varepsilon can be linearized to recover large wavenumber k:= \nabla_x \varphi, and frequency \omega := -\partial_t \varphi. The amplitude a_0 satisfies a transport equation. The small parameter \varepsilon\, 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[edit]

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

L(\partial_t, \nabla_x) u := \left( \frac{\partial^2}{\partial t^2} - c(x)^2 \Delta \right)u(t,x) = 0, \;\; u(0,x) = u_0(x),\;\; u_t(0,x) = 0

assume an asymptotic series solution of the form

u(t,x) \sim a_\varepsilon(t,x)e^{i\varphi(t,x)/\varepsilon} = \sum_{j=0}^\infty i^j \varepsilon^j a_j(t,x) e^{i\varphi(t,x)/\varepsilon}.

Check that

L(\partial_t,\nabla_x)(e^{i\varphi(t,x)/\varepsilon}) a_\varepsilon(t,x) = e^{i\varphi(t,x)/\varepsilon} 
\left( \left(\frac{i}{\varepsilon} \right)^2  L(\varphi_t, \nabla_x\varphi)a_\varepsilon + \frac{2i}{\varepsilon} V(\partial_t,\nabla_x)a_\varepsilon + \frac{i}{\varepsilon} (a_\varepsilon L(\partial_t,\nabla_x)\varphi) + L(\partial_t,\nabla_x)a_\varepsilon   \right)


V(\partial_t,\nabla_x) := \frac{\partial \varphi}{\partial t} \frac{\partial}{\partial t} - c^2(x)\sum_j \frac{\partial \varphi}{\partial x_j} \frac{\partial}{\partial x_j}

Plugging the series into this equation, and equating powers of \varepsilon, the most singular term O(\varepsilon^{-2}) satisfies the eikonal equation (in this case called a dispersion relation),

0 = L(\varphi_t,\nabla_x\varphi) = (\varphi_t)^2 - c(x)^2(\nabla_x \varphi)^2.

To order \varepsilon^{-1}, the leading-order amplitude must satisfy a transport equation

2V a_0 + (L\varphi)a_0 = 0

With the definition k : = \nabla_x \varphi, \omega := -\varphi_t, the eikonal equation is precisely the dispersion relation that results by plugging the plane wave solution e^{i(k\cdot x - \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 \varphi are smooth, so that on a local scale there are plane waves.

To justify this technique, the remaining terms 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.

See also[edit]


  2. ^ Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides 1. SPIE. pp. 19–20. ISBN 0-8194-5294-7.  edit
  3. ^ a b c d e f g Hugh D. Young (1992). University Physics 8e. Addison-Wesley. ISBN 0-201-52981-5. Chapter 35
  5. ^ a b Hecht, Eugene (1987). Optics (2nd ed. ed.). Addison Wesley. ISBN 0-201-11609-X.  Chapters 5 & 6.

Further reading[edit]

English translations of some early books and papers:

External links[edit]