A compound prism is a set of multiple
triangular prism elements placed in contact, and often cemented together to form a solid assembly. The use of multiple elements gives several advantages to an optical designer:
[Nathan Hagen and Tomasz S. Tkaczyk,]
Compound prism design principles, I
" ''Appl. Opt.'' 50: 4998-5011 (2011).
* One can achieve
spectral dispersion without causing the deviation of the beam at the design wavelength. Thus, light at the design wavelength which enters at an angle
with respect to the optical axis, exits the prism at the same angle with respect to the same axis. This kind of effect is often called "direct vision dispersion" or "nondeviating dispersion".
* One can achieve deviation of the incident beam while also greatly reducing the dispersion introduced into the beam: an achromatic
deflecting prism. This effect is used in
beam steering.
[Zhilin Hu and Andrew M. Rollins, "Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer," ''Opt. Lett.'' 32: 3525-3527 (2007).]
* One can tune the prism dispersion to achieve greater dispersion linearity or to achieve higher-order dispersion effects.
Doublet
The simplest compound prism is a doublet, consisting of two elements in contact, as shown in the figure at right. A ray of light passing through the prism is refracted at the first air-glass interface, again at the interface between the two glasses, and a final time at the exiting glass-air interface. The deviation angle
of the ray is given by the difference in ray angle between the incident ray and the exiting ray:
. While one can produce direct vision dispersion from doublet prisms, there is typically significant displacement of the beam (shown as a separation between the two dashed horizontal lines in the ''y'' direction). Mathematically, one can calculate
by concatenating the Snell's law equations at each interface,
:
so that the deviation angle is a
nonlinear function
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
of the glass refractive indices
and
, the prism elements' apex angles
and
, and the angle of incidence
of the ray. Note that
indicates that the prism is inverted (the apex points downward).
If the angle of incidence
and prism apex angle
are both small, then
and
, so that the nonlinear equation in the deviation angle
can be approximated by the linear form
:
(See also
prism deviation and dispersion.) If we further assume that the wavelength dependence to the refractive index is approximately linear, then the dispersion can be written as
:
where
and
are the dispersion and
Abbe number
In optics and lens design, the Abbe number, also known as the V-number or constringence of a transparent material, is an approximate measure of the material's dispersion (change of refractive index versus wavelength), with high values of ''V'' in ...
of element
within the compound prism,
. The central wavelength of the spectrum is denoted
.
Doublet prisms are often used for direct-vision dispersion. In order to design such a prism, we let
, and simultaneously solving equations
and
gives
:
from which one can obtain the element apex angles
and
from the mean refractive indices of the glasses chosen:
:
Note that this formula is only accurate under the small angle approximation.
Double-Amici
While the doublet prism is the simplest compound prism type, the
double-Amici prism is much more common. This prism is a three-element system (a triplet), in which the first and third elements share both the same glass and the same apex angles. The design layout is thus symmetric about the plane passing through the center of its second element. Due to its symmetry, the linear design equations (under the small angle approximation) for the double-Amici prism differ from those of the doublet prism only by a factor of 2 in front of the first term in each equation:
:
Thus, we can derive the expressions for the prism angles using these linear equations, giving
:
The exact nonlinear equation for the deviation angle
is obtained by concatenating the refraction equations obtained at each interface:
:
The ray deviation angle is given by
.
Triplet
The double-Amici prism is a symmetric form of the more general triplet prism, in which the apex angles and glasses of the two outer elements may differ (see the figure at right). Although triplet prisms are rarely found in optical systems, their added degrees of freedom beyond the double-Amici design allow for improved dispersion linearity. The deviation angle of the triplet prism is obtained by concatenating the refraction equations at each interface:
[Nathan Hagen and Tomasz S. Tkaczyk,]
Compound prism design principles, II: triplet and Janssen prisms
" ''Appl. Opt.'' 50: 5012-5022 (2011).[Nathan Hagen and Tomasz S. Tkaczyk,]
Compound prism design principles, III: linear-in-wavenumber and optical coherence tomography prisms
" ''Appl. Opt.'' 50: 5023-5030 (2011).
:
Here to the ray deviation angle is given by
.
See also
*
Dispersive prism
In optics, a dispersive prism is an optical prism that is used to disperse light, that is, to separate light into its spectral components (the colors of the rainbow). Different wavelengths (colors) of light will be deflected by the prism at di ...
References
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Prisms (optics)