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A '''compound prism''' is a set of multiple [[triangular prism|triangular prism elements]] placed in contact, and often cemented together to form a solid assembly.<ref>John Browning, "Note on the use of compound prisms," ''MNRAS'' '''31''': 203-205 (1871).</ref> The use of multiple elements gives several advantages to an optical designer:<ref name="Hagen">Nathan Hagen and Tomasz S. Tkaczyk, "[http://dx.doi.org/10.1364/AO.50.004998 Compound prism design principles, I]," ''Appl. Opt.'' '''50''': 4998-5011 (2011).</ref>
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* One can achieve [[Dispersion (optics)|spectral dispersion]] without causing the deviation of the beam at the design wavelength. Thus, light at the design wavelength which enters at an angle <math>\theta_0</math> 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".<ref>Charles G. Abbott and Frederick E. Fowle, Jr., "A prism of uniform dispersion," ''Astrophys. J.'' '''11''': 135-139 (1900).</ref>
* One can achieve deviation of the incident beam while also greatly reducing the dispersion introduced into the beam: an achromatic [[Prism (optics)#Deflecting prisms|deflecting prism]]. This effect is used in [[beam steering]].<ref>Bradley D. Duncan, Philip J. Bos, and Vassili Sergan, "Wide-angle achromatic prism beam steering for infrared countermeasure applications," ''Opt. Eng'' '''42''': 1038-1047 (2003).</ref><ref>Zhilin Hu and Andrew M. Rollins, "Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer," ''Opt. Lett.'' '''32''': 3525-3527 (2007).</ref>
* One can tune the prism dispersion to achieve greater dispersion linearity or to achieve higher-order dispersion effects.


==Doublet==
Legal process can be confusing and complicated when you deficiency a history inside the rules, so make sure to enable your legal representative know when you are puzzled or maybe if you with a better notion of what you should expect through your trial run. The individual need to come back your calls on time.<br><br>Do not really feel compelled to hire a lawyer since you fulfilled several times and got some beneficial assistance. You need to indicator a contract only once you decide on charges and feel relaxed along with your attorney. When you are hesitating because you have heard terrible things about this lawyer or feel the fees are far too high, continue to keep seeking.<br><br>You have to figure out what your financial budget will probably be. It may not be worth the cost cost a good idea to file or overcome a lawsuit. Analysis the kinds of charges you will process when you purchase a specific lawyer or attorney. Schedule a getting together with with them to go about what you are able manage and your anticipations. Know that you might look at price range, at times.<br><br>Ask lots of questions to each lawyer that you simply meet with. They should provide an answer for everything. It really is their job to ensure you that you are currently satisfied with their knowledge and data. When they do not accommodate your requests, basically depart.<br><br>Steer clear of legal professionals that use the word "slam dunk" with regards to any situation or situation. Experienced legal professionals realize that legal requirements is almost never a lower and dry matter. Normally, attorneys might not have to exist whatsoever! One does desire a self-confident lawyer, but not an conceited or ignorant a single.<br><br>An excellent idea to not forget whenever using a lawyer is to try and get all the carried out in achievable in just one getting together with. You're likely to pay far more to schedule different events to talk about numerous points. Exceeding a number of subjects in one conference will save you lots of money.<br><br>Figure out up front how often you should be in hard with the attorney. If you require these people to communicate with you easily as soon as you make contact with them, interact that in their mind in the beginning. Should you face this issue together with your present legal representative, get a new lawyer at your earliest convenience.<br><br>Do not let you to ultimately evaluate a legal professional depending on how excellent their ads are. The truth is that the lawyer or attorney which is reputable doesn't ought to publicize to get new customers. Consider your attorney's background and use everything you get to assist you to choose about if you should work with them.<br><br>I require a major brand legal professional! I want them to shock the evaluate! Exactly like in everyday life, would like are fantastic, but needs are more crucial. You really a summary of your needs and go with a attorney according to that listing, its not all the would like you may have inside your cardiovascular system.<br><br>Make sure you talk about repayment with any lawyer or attorney you are planning on employing. Some are paid for with the hour among others may possibly charge you a set rate. Consider before hand about what you really are most at ease with (or maybe if it even truly matters to you). Speak the attorney relating to your tastes to see if they will be willing to do business with you.<br><br>Prepare a few pre-determined questions for the initially conference with the [http://www.youtube.com/watch?v=s7Bzuq8LIIM Schwartz attorneys at law] attorneys you are thinking about using the services of. Anticipate asking them questions with regards to their encounter and results and get a lot of questions in the things they take into consideration your needs. Do not have confidence in a lawyer who would seem exceedingly confident in remarkable ability to win your circumstance before you even provide them with every detail.<br><br>Regardless of who you are, you can always utilize a great legal professional. The reason why you might need a legal representative can depend on your situations. No matter this, you'll find a legal representative that may focus on any legal problem you might have. Maintain this short article at heart when it's a chance to locate one.
[[File:Prism2.svg|right|A doublet prism, showing the apex angles (<math>\alpha_1</math> and <math>\alpha_2</math>) of the two elements, and the angles of incidence <math>\theta_i</math> and refraction <math>\theta'_i</math> at each interface. The deviation angle of the ray transmitted by the prism is shown as <math>\delta</math>.|400px]]
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 <math>\delta</math> of the ray is given by the difference in ray angle between the incident ray and the exiting ray: <math>\delta = \theta_0 - \theta_4</math>. 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 <math>\delta</math> by concatenating the Snell's law equations at each interface,<ref name="Hagen"/>
:<math>
  \begin{align}
      \theta_1 &= \theta_0 - \beta_1                              &\theta_3 &= \theta'_2 - \alpha_2 \\
      \theta'_1 &= \arcsin (\tfrac{1}{n_1} \, \sin \theta_1) \quad &\theta'_3 &= \arcsin (n_2 \, \sin \theta_3) \\
      \theta_2 &= \theta'_1 - \alpha_1                            &\theta_4 &= \theta'_3 + \tfrac{1}{2} \alpha_2 \\
      \theta'_2 &= \arcsin (\tfrac{n_1}{n_2} \, \sin \theta_2)
  \end{align}
</math>
so that the deviation angle is a [[nonlinear equation|nonlinear function]] of the glass refractive indices <math>n_1 (\lambda)</math> and <math>n_2 (\lambda)</math>, the prism elements' apex angles <math>\alpha_1</math> and <math>\alpha_2</math>, and the angle of incidence <math>\theta_0</math> of the ray. Note that <math>\alpha_i</math> indicates that the prism is inverted (the apex points downward).
 
If the angle of incidence <math>\theta_0</math> and prism apex angle <math>\alpha</math> are both small, then <math>\sin \theta \approx \theta</math> and <math>\text{arcsin} (x) \approx x</math>, so that the nonlinear equation in the deviation angle <math>\delta</math> can be approximated by the linear form
:<math>
  \delta (\lambda) = \big[ n_1 (\lambda - 1 \big] \alpha_1 + \big[ n_2 (\lambda) - 1 \big] \alpha_2 \ .
</math>
(See also [[Prism (optics)#Deviation angle and dispersion|Prism deviation angle and dispersion]].) If we further assume that the wavelength dependence to the refractive index is approximately linear, then the dispersion can be written as
:<math>
  \Delta = \frac{\delta_1 (\bar{\lambda})}{V_1} + \frac{\delta_2 (\bar{\lambda})}{V_2} \ ,
</math>
where <math>\delta_i</math> and <math>V_i</math> are the dispersion and [[Abbe number]] of element <math>i</math> within the compound prism, <math>V_i = (\bar{n} - 1) / (n_F - n_C)</math>. The central wavelength of the spectrum is denoted <math>\bar{\lambda}</math>.
 
Doublet prisms are often used for direct-vision dispersion. In order to design such a prism, we let <math>\bar{\delta} = 0</math>, and simultaneously solving equations <math>\delta</math> and <math>\Delta</math> gives
:<math>
  \delta_1 (\bar{\lambda}) = - \delta_2 (\bar{\lambda}) = -\Delta \Big( \frac{1}{V_2} - \frac{1}{V_1} \Big)^{-1} \ ,
</math>
from which one can obtain the element apex angles <math>\alpha_1</math> and <math>\alpha_2</math> from the mean refractive indices of the glasses chosen:
:<math>
  \begin{align}
  \alpha_1 &= \frac{\Delta}{\bar{n}_1 - 1} \Big( \frac{1}{V_1} - \frac{1}{V_2} \Big)^{-1} \ , \\
  \alpha_2 &= \frac{\Delta}{\bar{n}_2 - 1} \Big( \frac{1}{V_2} - \frac{1}{V_1} \Big)^{-1} \ .
  \end{align}
</math>
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 [[Amici prism|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:<ref name="Hagen"/>
[[File:Double Amici prism with refraction angles.svg|right|A double-Amici prism, showing the apex angles (<math>\alpha_1</math> and <math>\alpha_2</math>) of the three elements, and the angles of incidence <math>\theta_i</math> and refraction <math>\theta'_i</math> at each interface.|500px]]
:<math>
\begin{align}
  \delta (\bar{\lambda}) &= 2 \delta_1 (\bar{\lambda}) + \delta_2 (\bar{\lambda}) = 2 \big( \bar{n}_1 - 1) \alpha_1 + \big( \bar{n}_2 - 1) \alpha_2 \ , \\
  \Delta &= 2 \frac{\delta_1 (\bar{\lambda})}{V_1} + \frac{\delta_2 (\bar{\lambda})}{V_2} \ .
\end{align}
</math>
Thus, we can derive the expressions for the prism angles using these linear equations, giving
:<math>
\begin{align}
  \alpha_1 &= \frac{\Delta}{2 (\bar{n}_1 - 1)} \Big( \frac{1}{V_1} - \frac{1}{V_2} \Big)^{-1} \ , \\
  \alpha_2 &= \frac{\Delta}{\bar{n}_2 - 1} \Big( \frac{1}{V_2} - \frac{1}{V_1} \Big)^{-1} \ .
\end{align}
</math>
 
The exact nonlinear equation for the deviation angle <math>\delta</math> is obtained by concatenating the refraction equations obtained at each interface:
:<math>
  \begin{align}
      \theta_1 &= \theta_0 + \alpha_1 - \tfrac{1}{2} \alpha_2      &\theta'_3 &= \arcsin (\tfrac{n_2}{n_1} \, \sin \theta_3) \\
      \theta'_1 &= \arcsin (\tfrac{1}{n_1} \, \sin \theta_1) \quad &\theta_4 &= \theta'_3 - \alpha_1 \\
      \theta_2 &= \theta'_1 - \alpha_1                            &\theta'_4 &= \arcsin (n_1 \, \sin \theta_4) \\
      \theta'_2 &= \arcsin (\tfrac{n_1}{n_2} \, \sin \theta_2)    &\theta_5 &= \theta'_4 + \alpha_1 - \tfrac{1}{2} \alpha_2 \\
      \theta_3 &= \theta'_2 - \alpha_2
  \end{align}
</math>
The ray deviation angle is given by <math>\delta = \theta_0 - \theta_5</math>.
 
==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:<ref name="Hagen3">Nathan Hagen and Tomasz S. Tkaczyk, "[http://dx.doi.org/10.1364/AO.50.005012 Compound prism design principles, II: triplet and Janssen prisms]," ''Appl. Opt.'' '''50''': 5012-5022 (2011).</ref><ref name="Hagen2">Nathan Hagen and Tomasz S. Tkaczyk, "[http://dx.doi.org/10.1364/AO.50.005023 Compound prism design principles, III: linear-in-wavenumber and optical coherence tomography prisms]," ''Appl. Opt.'' '''50''': 5023-5030 (2011).</ref>
[[File:Triplet prism with refraction angles.svg|right|A triplet prism, showing the apex angles (<math>\alpha_1</math>, <math>\alpha_2</math>, and <math>\alpha_3</math>) of the three elements, and the angles of incidence <math>\theta_i</math> and refraction <math>\theta'_i</math> at each interface.|400px]]
:<math>
  \begin{align}
      \theta_1 &= \theta_0 + \alpha_1 + \tfrac{1}{2} \alpha_2      &\theta'_3 &= \arcsin (\tfrac{n_2}{n_3} \, \sin \theta_3) \\
      \theta'_1 &= \arcsin (\tfrac{1}{n_1} \, \sin \theta_1) \quad &\theta_4 &= \theta'_3 - \alpha_3 \\
      \theta_2 &= \theta'_1 - \alpha_1                            &\theta'_4 &= \arcsin (n_3 \, \sin \theta_4) \\
      \theta'_2 &= \arcsin (\tfrac{n_1}{n_2} \, \sin \theta_2)    &\theta_5 &= \theta'_4 + \alpha_3 + \tfrac{1}{2} \alpha_2 \\
      \theta_3 &= \theta'_2 - \alpha_2
  \end{align}
</math>
Here too the ray deviation angle is given by <math>\delta = \theta_0 - \theta_5</math>.
 
==See also==
*[[Dispersive prism]]
 
==References==
{{reflist}}
 
[[Category:Prisms]]

Latest revision as of 21:47, 19 November 2014

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