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| '''Ray transfer matrix analysis''' (also known as '''ABCD matrix analysis''') is a type of [[Ray tracing (physics)|ray tracing]] technique used in the design of some [[optics|optical]] systems, particularly [[laser]]s. It involves the construction of a ''ray transfer [[matrix (math)|matrix]]'' which describes the optical system; tracing of a light path through the system can then be performed by multiplying this matrix with a [[vector space|vector]] representing the [[light ray]]. The same analysis is also used in [[accelerator physics]] to track particles through the magnet installations of a [[particle accelerator]], see [[Beam optics]].
| | Parents are rather enthusiastic plus excited to recognize regarding the future height of their kids. Some parents curiously ask themselves - is there any tool which could predict height of the kid whenever he becomes an adult? Is it possible to roughly calculate the future height of the child? Well, many folks do rely on various child height calculators inside order to recognize how tall or how short their young kids will be when they grow up.<br><br>Number 6. Visit the venue before the marriage date, if possible. May it be from the easy backyard of the groom or the cliff of an island, it really is highly suggested to visit the place where the ceremony can be held. This might assist you tremendously for it you'll know what to expect plus tackle. Just watch a step, assuming you'll shoot at the mountains.<br><br>I have constantly loved cycling even because a child. I got my husband a cycle for Valentine's Day. He never waist to height ratio utilized it plus six months later I decided to put it to wise use. So every morning following I loaded my kid into his school bus I would go cycling. I usually did between 3 to 5 kms each morning plus came home.<br><br>Body sort comprises the bone frame plus structure. It is the simple structure which sculpts your figure. They are classified because wide, curvy, plus narrow bone frames. Depending on a bone frame, a body sort is determined. Body kind plus body shape are closely associated, because body kind determines the form of the body. Whether thin or fat, the bone frame chooses a form. So what should you are broad overall? If you are broad total with hardly any curves, you're rectangular fit. Let us see these shapes in detail.<br><br>These types of results are not distinctive to MacMasters. Dr David Heber, Ph.D., from UCLA's Centre for Human Nutrition reports that distribution of body fat is a more important predictor of heart attack risk than the conventional measuring of Body Mass Index (BMI), that is a measuring based on the ratio between your height and fat.<br><br>If you have a bigger booty then stay away from jeans with back pockets that [http://safedietplansforwomen.com/waist-to-height-ratio waist to height ratio] call attention to a behind. Do not buy jeans with flap pockets or with pockets which have a lot of design on them. Carhartt jeans will suit the fanny of any woman, as the pocket shape is easy plus not ostentatious.<br><br>This puts me at the high end of regular. If I were to gain only 10 pounds I would, according to BMI, be overweight. Then most individuals tell me that I am skinny, that I think is kind of silly. I am not skinny yet I absolutely never believe which I will be obese at 185 pounds either. But, I do not like myself at that weight and I will not enable it to result, but which is my own personal problem. Of more value, both medically plus to me personally, is that my percentage body fat runs about 14%, which is lower then most persons of the same height plus fat.<br><br>Percentage body fat is something which is performed at the doctors or by experts inside the area. It is performed by measuring fat under folds of skin. Usually at the waist, hips plus thighs. This will not be certainly accurate because persons age as the fat distribution inside the body changes as you receive older. Its more accurate for people below 40 incredibly nevertheless it still functions OK till age of 55. The accuracy begins to decrease within the age of 40 onwards. |
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| The technique that is described below uses the ''[[paraxial approximation]]'' of ray optics, which means that all rays are assumed to be at a small angle (θ) and a small distance (''x'') relative to the [[optical axis]] of the system.<ref>An exact method for tracing meridional rays is available [http://spie.org/Documents/ETOP/1991/389_1.pdf here].</ref>
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| == Definition of the ray transfer matrix ==
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| The ray tracing technique is based on two reference planes, called the ''input'' and ''output'' planes, each perpendicular to the optical axis of the system. Without loss of generality, we will define the optical axis so that it coincides with the ''z''-axis of a fixed coordinate system. A light ray enters the system when the ray crosses the input plane at a distance ''x''<sub>1</sub> from the optical axis while traveling in a direction that makes an angle θ<sub>1</sub> with the optical axis. Some distance further along, the ray crosses the output plane, this time at a distance ''x''<sub>2</sub> from the optical axis and making an angle θ<sub>2</sub>. ''n''<sub>1</sub> and ''n''<sub>2</sub> are the [[index of refraction|indices of refraction]] of the medium in the input and output plane, respectively.
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| These quantities are related by the expression
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| :<math> {x_2 \choose \theta_2} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}{x_1 \choose \theta_1}, </math>
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| where
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| :<math>A = {x_2 \over x_1 } \bigg|_{\theta_1 = 0} \qquad B = {x_2 \over \theta_1 } \bigg|_{x_1 = 0},</math>
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| and
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| :<math>C = {\theta_2 \over x_1 } \bigg|_{\theta_1 = 0} \qquad D = {\theta_2 \over \theta_1 } \bigg|_{x_1 = 0}.</math>
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| This relates the ''ray vectors'' at the input and output planes by the ''ray transfer matrix'' (RTM) '''M''', which represents the optical system between the two reference planes. A [[thermodynamics]] argument based on the [[blackbody]] radiation can be used to show that the [[determinant]] of a RTM is the ratio of the indices of refraction:
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| :<math>\det(\mathbf{M}) = AD - BC = { n_1 \over n_2 }. </math>
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| As a result, if the input and output planes are located within the same medium, or within two different media which happen to have identical indices of refraction, then the determinant of '''M''' is simply equal to 1.
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| A similar technique can be used to analyze [[electrical circuits]]. ''See'' [[Two-port network]]s.
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| == Some examples ==
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| * For example, if there is free space between the two planes, the ray transfer matrix is given by:
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| :<math> \mathbf{S} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} </math>,
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| where ''d'' is the separation distance (measured along the optical axis) between the two reference planes. The ray transfer equation thus becomes:
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| :<math> {x_2 \choose \theta_2} = \mathbf{S}{x_1 \choose \theta_1} </math>,
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| and this relates the parameters of the two rays as:
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| :<math> \begin{matrix} x_2 & = & x_1 + d\theta_1 \\
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| \theta_2 & = & \theta_1 \end{matrix} </math>
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| * Another simple example is that of a [[thin lens]]. Its RTM is given by:
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| :<math> \mathbf{L} = \begin{pmatrix} 1 & 0 \\ \frac{-1}{f} & 1 \end{pmatrix} </math>,
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| where ''f'' is the [[focal length]] of the lens. To describe combinations of optical components, ray transfer matrices may be multiplied together to obtain an overall RTM for the compound optical system. For the example of free space of length ''d'' followed by a lens of focal length ''f'':
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| :<math>\mathbf{L}\mathbf{S} = \begin{pmatrix} 1 & 0 \\ \frac{-1}{f} & 1\end{pmatrix}
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| \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}
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| = \begin{pmatrix} 1 & d \\ \frac{-1}{f} & 1-\frac{d}{f} \end{pmatrix} </math>.
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| Note that, since the multiplication of matrices is non-[[commutative]], this is not the same RTM as that for a lens followed by free space:
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| :<math> \mathbf{SL} =
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| \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}
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| \begin{pmatrix} 1 & 0 \\ \frac{-1}{f} & 1\end{pmatrix}
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| = \begin{pmatrix} 1-\frac{d}{f} & d \\ \frac{-1}{f} & 1 \end{pmatrix} </math>.
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| Thus the matrices must be ordered appropriately, with the last matrix premultiplying the second last, and so on until the first matrix is premultiplied by the second. Other matrices can be constructed to represent interfaces with media of different [[refractive index|refractive indices]], reflection from [[mirror]]s, etc.
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| == Table of ray transfer matrices ==
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| '''for simple optical components'''
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| {| border="1" cellspacing="0" cellpadding="4"
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| |- style="background-color: #AAFFCC"
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| ! Element
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| ! Matrix
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| ! Remarks
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| |-
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| | Propagation in free space or in a medium of constant refractive index
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| | align="center" |<math>\begin{pmatrix} 1 & d\\ 0 & 1 \end{pmatrix} </math>
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| | ''d'' = distance<br/>
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| |-
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| | Refraction at a flat interface
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| | align="center" | <math>\begin{pmatrix} 1 & 0 \\ 0 & \frac{n_1}{n_2} \end{pmatrix} </math>
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| | ''n''<sub>1</sub> = initial refractive index<br/>
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| ''n''<sub>2</sub> = final refractive index.
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| |-
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| | Refraction at a curved interface
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| | align="center" | <math>\begin{pmatrix} 1 & 0 \\ \frac{n_1-n_2}{R \cdot n_2} & \frac{n_1}{n_2} \end{pmatrix} </math>
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| | ''R'' = radius of curvature, ''R'' > 0 for convex (centre of curvature after interface)<br/>
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| ''n''<sub>1</sub> = initial refractive index<br/>
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| ''n''<sub>2</sub> = final refractive index.
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| |-
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| | Reflection from a flat mirror
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| | align="center" | <math> \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} </math>
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| |-
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| | Reflection from a curved mirror
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| | align="center" | <math> \begin{pmatrix} 1 & 0 \\ -\frac{2}{R} & 1 \end{pmatrix} </math>
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| | ''R'' = radius of curvature, R > 0 for concave, valid in the paraxial approximation
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| |-
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| | Thin lens
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| | align="center" | <math> \begin{pmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{pmatrix} </math>
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| | ''f'' = focal length of lens where ''f'' > 0 for convex/positive (converging) lens.
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| Only valid if the focal length is much greater than the thickness of the lens.
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| |-
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| | Thick lens
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| | align="center" | <math>\begin{pmatrix} 1 & 0 \\ \frac{n_2-n_1}{R_2n_1} & \frac{n_2}{n_1} \end{pmatrix} \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \frac{n_1-n_2}{R_1n_2} & \frac{n_1}{n_2} \end{pmatrix}</math>
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| | ''n''<sub>1</sub> = refractive index outside of the lens. <br/>
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| ''n''<sub>2</sub> = refractive index of the lens itself (inside the lens). <br/>
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| ''R''<sub>1</sub> = Radius of curvature of First surface. <br/>
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| ''R''<sub>2</sub> = Radius of curvature of Second surface.<br/>
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| ''t'' = center thickness of lens.
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| |-
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| | Single right angle prism
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| | align="center" | <math> \begin{pmatrix} k & \frac{d}{nk} \\ 0 & \frac{1}{k} \end{pmatrix} </math>
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| | ''k'' = (cos<math>\psi</math>/cos<math>\phi</math>) is the [[beam expander|beam expansion]] factor, where <math>\phi</math> is the angle of incidence, <math>\psi</math> is the angle of refraction, ''d'' = prism path length, ''n'' = refractive index of the prism material. This matrix applies for orthogonal beam exit.
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| |}
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| == Resonator stability ==
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| RTM analysis is particularly useful when modeling the behaviour of light in [[optical resonator]]s, such as those used in lasers. At its simplest, an optical resonator consists of two identical facing mirrors of 100% [[reflectivity]] and radius of [[curvature]] ''R'', separated by some distance ''d''. For the purposes of ray tracing, this is equivalent to a series of identical thin lenses of focal length ''f''=''R''/2, each separated from the next by length ''d''. This construction is known as a ''lens equivalent duct'' or ''lens equivalent [[waveguide]]''. The RTM of each section of the waveguide is, as above,
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| :<math>\mathbf{M} =\mathbf{L}\mathbf{S} = \begin{pmatrix} 1 & d \\ \frac{-1}{f} & 1-\frac{d}{f} \end{pmatrix} </math>.
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| RTM analysis can now be used to determine the ''stability'' of the waveguide (and equivalently, the resonator). That is, it can be determined under what conditions light travelling down the waveguide will be periodically refocussed and stay within the waveguide. To do so, we can find all the "eigenrays" of the system: the input ray vector at each of the mentioned sections of the waveguide times a real or complex factor λ is equal to the output one. This gives:
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| :<math> \mathbf{M}{x_1 \choose \theta_1} = {x_2 \choose \theta_2} = \lambda {x_1 \choose \theta_1} </math>.
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| which is an [[eigenvalue]] equation:
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| :<math> \left[ \mathbf{M} - \lambda\mathbf{I} \right] {x_1 \choose \theta_1} = 0 </math>,
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| where '''I''' is the 2x2 [[identity matrix]].
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| We proceed to calculate the eigenvalues of the transfer matrix:
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| :<math>\operatorname{det} \left[ \mathbf{M} - \lambda\mathbf{I} \right] = 0 </math>,
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| leading to the [[Characteristic polynomial#Characteristic equation|characteristic equation]]
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| :<math> \lambda^2 - \operatorname{tr}(\mathbf{M}) \lambda + \operatorname{det}( \mathbf{M}) = 0 </math>,
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| where
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| :<math> \operatorname{tr} ( \mathbf{M} ) = A + D = 2 - { d \over f } </math>
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| is the [[trace (linear algebra)|trace]] of the RTM, and
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| :<math>\operatorname{det}(\mathbf{M}) = AD - BC = 1 </math>
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| is the [[determinant]] of the RTM. After one common substitution we have:
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| :<math> \lambda^2 - 2g \lambda + 1 = 0 </math>,
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| where
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| :<math> g \ \stackrel{\mathrm{def}}{=}\ { \operatorname{tr}(\mathbf{M}) \over 2 } = 1 - { d \over 2 f } </math>
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| is the ''stability parameter''. The eigenvalues are the solutions of the characteristic equation. From the [[Quadratic equation#Quadratic formula|quadratic formula]] we find | |
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| :<math> \lambda_{\pm} = g \pm \sqrt{g^2 - 1} \, </math>
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| Now, consider a ray after ''N'' passes through the system:
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| :<math> {x_N \choose \theta_N} = \lambda^N {x_1 \choose \theta_1} </math>.
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| If the waveguide is stable, no ray should stray arbitrarily far from the main axis, that is, λ<sup>N</sup> must not grow without limit. Suppose <math> g^2 > 1</math>. Then both eigenvalues are real. Since <math> \lambda_+ \lambda_- = 1</math>, one of them has to be bigger than 1 (in absolute value), which implies that the ray which corresponds to this eigenvector would not converge. Therefore in a stable waveguide, <math> g^2 </math> ≤ 1, and the eigenvalues can be represented by complex numbers:
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| :<math> \lambda_{\pm} = g \pm i \sqrt{1 - g^2} = \cos(\phi) \pm i \sin(\phi) = e^{\pm i \phi} </math>,
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| with the substitution g = cos(ϕ).
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| For <math> g^2 < 1 </math> let <math> r_+ </math> and <math> r_- </math> be the eigenvectors with respect to the eigenvalues <math> \lambda_+ </math> and <math> \lambda_- </math> respectively, which span all the vector space because they are orthogonal, the latter due to <math>\lambda_+</math> ≠ <math>\lambda_- </math>. The input vector can therefore be written as
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| :<math> c_+ r_+ + c_- r_- </math>,
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| for some constants <math> c_+ </math> and <math> c_- </math>.
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| After ''N'' waveguide sectors, the output reads
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| :<math> \mathbf{M}^N (c_+ r_+ + c_- r_-) = \lambda_+^N c_+ r_+ + \lambda_-^N c_- r_- = e^{i N \phi} c_+ r_+ + e^{- i N \phi} c_- r_- </math>,
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| what represents a periodic function.
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| == Ray transfer matrices for Gaussian beams ==
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| The matrix formalism is also useful to describe [[Gaussian beam]]s. If we have a Gaussian beam of wavelength <math>\lambda_0</math>, radius of curvature ''R'', beam spot size ''w'' and refractive index ''n'', it is possible to define a [[complex beam parameter]] ''q'' by:
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| :<math> \frac{1}{q} = \frac{1}{R} - \frac{i\lambda_0}{\pi n w^2} </math>.
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| This beam can be propagated through an optical system with a given ray transfer matrix by using the equation:
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| :<math> {q_2 \choose 1} = k \begin{pmatrix} A & B \\ C & D \end{pmatrix} {q_1 \choose 1} </math>,
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| where ''k'' is a normalisation constant chosen to keep the second component of the ray vector equal to 1. Using [[matrix multiplication]], this equation expands as
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| :<math> q_2 = k(Aq_1 + B) \,</math>
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| and
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| :<math> 1 = k(Cq_1 + D) \, </math>
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| Dividing the first equation by the second eliminates the normalisation constant:
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| :<math> q_2 =\frac{Aq_1+B}{Cq_1+D}</math>,
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| It is often convenient to express this last equation in reciprocal form: | |
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| :<math> { 1 \over q_2 } = { C + D/q_1 \over A + B/q_1 } . </math>
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| == See also ==
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| * [[Transfer-matrix method (optics)]]
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| * [[Two-port network]]
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| * [[Linear canonical transformation]]
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| == References ==
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| {{reflist}}
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| {{refbegin}}
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| *:{{cite book | title = Fundamentals of Photonics | author = [[Bahaa E. A. Saleh]] and Malvin Carl Teich | publisher = John Wiley & Sons | location = New York | year = 1991 }} Section 1.4, pp. 26 – 36.
| |
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| *:{{cite book
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| |title= Introduction to matrix methods in optics
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| |last1= Gerrard |first1= Anthony
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| |last2= Burch |first2 = James M.
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| |year=1994
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| |publisher=Courier Dover
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| |location=
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| |isbn= 9780486680446
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| |pages=
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| |url=http://books.google.de/books?id=naUSNojPwOgC
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| }}
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| *:{{cite book | title = Tunable Laser Optics | author = [[F. J. Duarte]] | publisher = Elsevier-Academic | location = New York | year = 2003 }} Chapter 6.
| |
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| {{refend}}
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| == External links ==
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| * [http://physics.tamuk.edu/~suson/html/4323/thick.html#Matrix Thick lenses (Matrix methods)]
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| * [http://www.photonics.byu.edu/ABCD_Matrix_tut.phtml ABCD Matrices Tutorial] Provides an example for a system matrix of an entire system.
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| * [http://www.photonics.byu.edu/ABCD_Calc.phtml ABCD Calculator] An interactive calculator to help solve ABCD matrices.
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| * [http://play.google.com/store/apps/details?id=com.dmt195.simple.abcd.optical.designer Simple Optical Designer (Android App)] An application to explore optical systems using the ABCD matrix method.
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| [[Category:Geometrical optics]]
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| [[Category:Accelerator physics]]
| |
Parents are rather enthusiastic plus excited to recognize regarding the future height of their kids. Some parents curiously ask themselves - is there any tool which could predict height of the kid whenever he becomes an adult? Is it possible to roughly calculate the future height of the child? Well, many folks do rely on various child height calculators inside order to recognize how tall or how short their young kids will be when they grow up.
Number 6. Visit the venue before the marriage date, if possible. May it be from the easy backyard of the groom or the cliff of an island, it really is highly suggested to visit the place where the ceremony can be held. This might assist you tremendously for it you'll know what to expect plus tackle. Just watch a step, assuming you'll shoot at the mountains.
I have constantly loved cycling even because a child. I got my husband a cycle for Valentine's Day. He never waist to height ratio utilized it plus six months later I decided to put it to wise use. So every morning following I loaded my kid into his school bus I would go cycling. I usually did between 3 to 5 kms each morning plus came home.
Body sort comprises the bone frame plus structure. It is the simple structure which sculpts your figure. They are classified because wide, curvy, plus narrow bone frames. Depending on a bone frame, a body sort is determined. Body kind plus body shape are closely associated, because body kind determines the form of the body. Whether thin or fat, the bone frame chooses a form. So what should you are broad overall? If you are broad total with hardly any curves, you're rectangular fit. Let us see these shapes in detail.
These types of results are not distinctive to MacMasters. Dr David Heber, Ph.D., from UCLA's Centre for Human Nutrition reports that distribution of body fat is a more important predictor of heart attack risk than the conventional measuring of Body Mass Index (BMI), that is a measuring based on the ratio between your height and fat.
If you have a bigger booty then stay away from jeans with back pockets that waist to height ratio call attention to a behind. Do not buy jeans with flap pockets or with pockets which have a lot of design on them. Carhartt jeans will suit the fanny of any woman, as the pocket shape is easy plus not ostentatious.
This puts me at the high end of regular. If I were to gain only 10 pounds I would, according to BMI, be overweight. Then most individuals tell me that I am skinny, that I think is kind of silly. I am not skinny yet I absolutely never believe which I will be obese at 185 pounds either. But, I do not like myself at that weight and I will not enable it to result, but which is my own personal problem. Of more value, both medically plus to me personally, is that my percentage body fat runs about 14%, which is lower then most persons of the same height plus fat.
Percentage body fat is something which is performed at the doctors or by experts inside the area. It is performed by measuring fat under folds of skin. Usually at the waist, hips plus thighs. This will not be certainly accurate because persons age as the fat distribution inside the body changes as you receive older. Its more accurate for people below 40 incredibly nevertheless it still functions OK till age of 55. The accuracy begins to decrease within the age of 40 onwards.