What is so special about the center of curvature of a parabola (since a parabola can't actually have a center) ? I mean, it must be something unique for it to be defined. Why is it twice the focal length? What are the geometrical properties of the center of curvature, in a mathematical sense (like with focus, coefficient of x^2=1/4p) ?
[Physics] What are the properties of center of curvature of a parabolic mirror
reflection
Related Solutions
Let's pare away the reflexion part of this problem and just say we have a Gaussian field launched with a phase perturbation on it: i.e. let's for simplicity say that we have a scalar field (one of the EM field's Cartesian components, or one of the Lorenz gauged four-potential components). Then on a transverse cross section through the beam we have a variation in this scalar field given by:
\begin{equation} \begin{split} \Psi(x,y,z) &= E_0 \frac{w_0}{w(z)} \exp \bigg(-(x^2+y^2)\left(\frac{1}{w(z)^2}+i\frac{\pi}{\lambda \,R(z)}\right) + \\ &\hphantom{=E_0 \frac{w_0}{w(z)} \exp \bigg(} i \arctan\left(\frac{z}{z_R}\right) + i\,\phi(x,y)\bigg)\\ &= \Psi_{0,0}(x,y,z) \exp(i\,\phi(x,y)) \end{split} \tag{1} \end{equation}
where $w(z) = w_0\, \sqrt{1 + \frac{z^2}{z_R^2}}$ is the beam waist width in the current transverse plane, $w_0$ the waist width in the focal plane, $R(z) = z + \frac{z_R}{z}$ the beam radius of curvature (i.e. distance from focus), $z_R = \pi w_0^2 / \lambda$ is the so called Rayleigh range defining the shape of the ellipsoidal wavefronts (which converge to spherical ones for $z\gg z_R$) and $\phi(x,y)$ the stochastic phase perturbation wrought by the surface roughness (related in this case to the surface roughness $s(x,y)$ by $\phi(x,y) = 4\pi s(x,y) / (\lambda\sin\theta)$, in your original notation. So your question now is, what effect on the beam as it propagates is wrought by the perturbation $\phi(x,y)$?
The detailed behavior naturally depends on the statistics of $\phi(x,y)$. In my other answer to you for computing wavefront deviations for off-axis mirrors I studied a very similar problem, but there I assumed that the spatial frequencies present were low, so that a $\phi(x,y)$ could be represented by series of Zernike functions with lowish radial and azimuthal orders. Let me "finish" this former answer before shifting on to the other important case, which is where the surface aberration is more often called "surface roughness" or "texture" (as in ISO-10110-8) and the spatial frequencies present in $\phi(x,y)$ are mostly much higher than the reciprocal of the mirror's wavelength. This high frequency case is what I assume for this question - I think the words "surface RMS" without talk of Zernike functions implied this to me - let me know whether I am reading you right. The ISO10110-8 words are "surface roughness" and "surface texture" when one deals with this high frequency case. ISO10110-5 talks of "surface irregularity" to describe the kind of low spatial frequency aberration wontedly resolved into a Zernike function superposition.
The Low Spatial Frequency Aberration Case
In the low spatial frequency case, the Strehl ratio computed for $\phi(x,y)$ with the tilt and defocus terms stripped away will foretell the peak intensity that you can focus your Gaussian beam to as I showed you in the other answer. However, in general, the focal plane variation may be quite different from Gaussian owing to the aberration $\phi(x,y)$. Another question to ask in this case which has a slightly different answer is the one you ask when you say:
what portion of the reflected beam will have the same divergence in the far-field?
Here one imagines the propagating field expanded in a series of orthonormal functions and then finding the component of the aberrated field “in the direction of“ the Gaussian mode through an inner product in the appropriate Hilbert space. Here a good reference is:
and the appropriate orthonormal functions are either Hermite-Gaussian or Hermite-Laguerre modes, depending on the symmetry of your mirror. Both are orthonormal solution sets to the paraxial Helmholtz equation $(\nabla^2 -2\,i\,k\,\partial_z)\psi = 0$, both will do equally well in theory to describe your problem, and both have the same fundamental mode, to wit, that described by equation (1). Hermite-Gauss functions are better for things like resonant cavities with rectangular symmetries in their cross-section; Laguerre-Gauss functions befit an axisymmetric system better. This reference does not explain the surface whereover inner products are to be taken and orthogonality defined, but it actually doesn’t matter. Energy conservation means that the concept of orthogonality is the same when the inner product is computed over any surface homotopically equivalent to a transverse plane (one that can be continuously deformed into a plane and contrawise), i.e. any smooth surface that sunders all of $\mathbb{R}^3$ into two, half infinite, simply connected regions. Define the inner product between two electromagnetic fields $\Psi$ and $\Phi$ over such a surface $S$ by:
$$ \left<\Psi,\,\Phi\right>_S = \int_S (\vec{E}_\Psi \wedge \vec{H}_\Phi) \cdot \hat{\vec{n}}\,\mathrm{d} S \tag{2} $$
and then apply the Lorentz’s Lemma at the particular wavelength in question to the volume enclosed between any two such homotopically equivalent surfaces. With Lorentz’s Lemma (see the [Wikipedia page on the electromagnetic reciprocity] (http://en.wikipedia.org/wiki/Reciprocity_(electromagnetism)) ) one can readily show that two electromagnetic fields $\Psi$ and $\Phi$ are orthogonal over a surface $S$ sundering $\mathbb{R}^3$ into two simply connected half infinite regions if and only if these fields are orthogonal over any other such surface homotopically equivalent to $S$. So, in particular, you can compute overlap integrals on transverse planes or over fundamental mode wavefronts. On doing such an overlap integral, one would find that the component of the aberrated field “in the direction of” the fundamental mode of equation (1) is:
$$ \psi_{0,0} = \frac{\int_S \Psi_{0,0}\, \Psi_{0,0}^*\, \exp(i\,\phi(\mathbf{r}))\,\mathrm{d} S}{\int_S |\Psi_{0,0}|^2\,\mathrm{d} S } \tag{3} $$
This amplitude is a little different from the (complex) Strehl ratio:
$$ \Gamma = \frac{\int_S |\Psi_{0,0}| \exp(i\,\tilde{\phi}(\mathbf{r}))\,\mathrm{d} S}{\int_S |\Psi_{0,0}|\,\mathrm{d} S } \tag{4} $$
where $\tilde{\phi}$ is the aberration with the tilt and defocus terms stripped off as shown in my other answer. The two equations (3) and (4) clearly define different ratios, but their behaviour and indeed their values tend to be very alike for almost any aberration. You could make an argument that either of them is a good answer to your question of “what portion of the reflected beam will have the same divergence in the far-field?”. But neither of them are very physically meaningful answers until either you talk about the focal plane (where the Strehl ratio in (4) is meaningful) or you get out into the extreme far field and, at last, the fundamental mode is slightly lower divergences means that the central farfield is dominated by the fundamental mode and the "vector component amplitude" in (3) holds sway. But over a huge range of lengths the other lower order Hermite-Gauss or Laguerre-Gauss modes will be significant and they will beat with the fundamental mode, so that the total field can seem quite non-Gaussian. In general you will have to do this problem numerically: an example algorithm is one that can:
Take Zernike weights as inputs and imposes the implied aberration function on your Gaussian beam;
Resolves the aberrated field into a set of Laguerre-Gaussian or Hermite-Gaussian beams: the simplest and most reliable way to do this will be through numerical linear regression minimising the least squared difference between the finite orthogonal beam superposition and the aberrated beam on the "lauch plane" i.e. over a plane just in front of the mirror.
Propagates the superposition out to your target propagation distance by separately applying the axial propagation formula applicable to each HG or LG modes (the formulas are given both in the Pampoloni and Enderlein paper and the Wikipedia Page for Gaussian Beam ) and thus checks what the beam looks like after this propagation to see how much the higher order modes disturb your image.
The Strehl ratio would seem to be the better definition if you are ultimately interested in peak focal plane intensities.
The High Spatial Frequency Aberration Case
Things are quite different when the spatial frequency components of the aberration are much higher than the reciprocal of the mirror’s diameter. The Strehl ratio still plays almost the same role as above, but now the splitting of the reflected field into the low spatial frequency Gaussian beam and the high spatial frequency noise is much better defined. So both (3) and (4) will have easy interpretations at almost any plane. I shall now show analytically as well as by posting some numerical simulation results that a good rule of thumb answer to your question is the following:
After propagating a significant distance (i.e. so that Fraunhofer diffraction relates the field on the two planes), the field looks like the ideal field attenuated by the square root of the Strehl ratio (which takes a particularly simple form for this stochastic roughness problem) together with a widely spread, roughly uniform intensity speckle pattern (the numerical results will help make the meaning of this clearer).
This result intuitively makes sense: presumably the stochastic perturbation field has spatial frequency content that is uniformly distributed in a wide spatial frequency range, whereas the main field is a low spatial frequency (i.e. paraxially propagating), Gaussian beam. When these two are Fourier transformed in the Fraunhofer diffraction integral, the stochastic part sprays in all directions: it comprises plane waves mostly skewed at high angles relative to the axial direction. Therefore the perturbation field swiftly sunders itself from the main beam, leading to a roughly uniform speckle pattern over a region much bigger than the beam waist.
Analytical Demonstration of the High Spatial Frequency Rule of Thumb
So now we compute the focal plane intensity of your reflected beam; with standard Fourier optics assumptions holding. The response at the lateral position $(x, y)$ on the focal plane is approximately given by (here f is the focal length):
$$ \Phi(x,y) = \int_{\mathbb{R}^2} \exp\left(i\,\frac{2\,\pi\,(x\,X+y\,Y)}{\lambda\,f} + i\,\tilde{\phi}(X,Y)\right)\Psi_{0,0}(X,Y) \,\mathrm{d}X\,\mathrm{d}Y \tag{5} $$
Now, for a given constant $(x, y)$ we make the following coordinate rotation in the $(X, Y)$ plane:
$$ U = \frac{x\,X+y\,Y}{\sqrt{x^2+y^2}};\quad V= \frac{-y\,X+x\,Y}{\sqrt{x^2+y^2}} \tag{6} $$
so that equation (5) becomes:
$$ \Phi(x,y) = \int_{\mathbb{R}^2} \exp\left(i\,\frac{2\,\pi\,\sqrt{x^2+y^2}}{\lambda\,f} + i\,\tilde{\phi}^\prime(U,V)\right)\Psi_{0,0}^\prime (U,V) \,\mathrm{d}U\,\mathrm{d}V \tag{7} $$
where the primed functions show that their argument coordinates have been rotated. But note that, given reasonable assumptions about the statistics of the surface roughness, the moments and other important statistics of the rotated coordinate aberration function $\tilde{\phi}^\prime(U,V)$ do not change. Now we imagine integrating equation (7) first with respect to $V$ along a line of constant $U$. In this integration the only variation comes from the term $\exp(i\tilde{\phi}^\prime(U,V))$ and thus we need to do the integral:
$$ \int_{V=V_1}^{V=V_2} \exp(i\tilde{\phi}^\prime(U,V))\,\mathrm{d}V \tag{8} $$
where $V=V_1$ and $ V=V_2$ define the exit pupil boundaries at this particular constant value of $U$. Now we assume that $\tilde{\phi}^\prime(U,V))$ is a true random variable and make suitable ergodic assumption about the random process it belongs to so that we can now interpret the integral as approximately the length of the line between $V=V_1$ and $ V=V_2$ times the expected value of $\exp(i\tilde{\phi}^\prime(U,V))$ given whatever probability density function $p(\phi)$ defines the probability distribution of $\phi$. Thus:
$$ \int_{V=V_1}^{V=V_2} \exp(i\tilde{\phi}^\prime(U,V))\,\mathrm{d}V\approx (V_2-V_1)\int_{-\infty}^\infty \exp(i\,\phi)\,p(\phi)\,\mathrm{d}\,\phi \tag{9} $$
Thus we see the original equation (5) can be written:
$$ \Phi(x,y) \approx \int_{\mathbb{R}^2} \exp\left(i\,\frac{2\,\pi\,(x\,X+y\,Y)}{\lambda\,f} \right)\Psi_{0,0}(X,Y) \,\mathrm{d}X\,\mathrm{d}Y \times )\int_{-\infty}^\infty \exp(i\,\phi)\,p(\phi)\,\mathrm{d}\,\phi \tag{10} $$
In other words, this is simply the focal plane field distribution that would arise from a lens with no surface roughness times a constant attenuating factor:
$$ \Gamma=\int_{-\infty}^\infty \exp(i\,\phi)\,p(\phi)\,\mathrm{d}\,\phi \tag{11} $$
set by the surface roughness’s probability distribution (i.e. It is the distribution’s characteristic function evaluated at unity).
If the surface roughness is such that the optical path error imprinted on the field belongs to a Gaussian distribution with zero mean phase and with RMS phase error $\sigma_w$ wavelengths (i.e. in radians the RMS value is $\sigma = 2\,\pi\,\sigma_w$) then:
$$ \Gamma=\frac{1}{\sqrt{2\,\pi}\,\sigma}\,\int_{-\infty}^\infty \exp\left(i\,\phi - \frac{\phi^2}{2\,\sigma^2}\right)\,\mathrm{d}\,\phi = \exp\left(-\frac{\sigma^2}{2}\right) \tag{12} $$
and so we recover Mahajan’s Formula $\Gamma^2 = \exp(-(2\,\pi\,\sigma_w)^2)$ for the Strehl ratio. The original Mahajan reference here is:
V. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance”, J. Opt. Soc. Am. 73 (6): 860–861
but Mahajan’s formula was a semi-empirical fit found to apply for low spatial frequency aberrations represented by Zernike polynomials. Here we see it has a theoretically “exact” interpretation for stochastic surface roughness. In this new interpretation, the Mahajan formula would not seem to be greatly sensitive to the surface roughness’s probability density function. For example, consider uniformly distributed surface roughness; equation (12) then becomes:
$$ \Gamma=\frac{1}{2\,\sqrt{3}\,\sigma}\,\int_{-\sqrt{3}\,\sigma}^{\sqrt{3}\,\sigma }\exp(i\,\phi)\,\mathrm{d}\,\phi = \operatorname{sinc}(\sqrt{3}\,\sigma) \tag{13}$$
which, by plotting these functions, can be shown to be very like the Gaussian case for $\sigma$ up to about 0.2 waves RMS.
Optical power is conserved (i.e. the power output from the lens is the same as that going through the focal plane) and the diffraction integral transform is unitary, so the attenuation at first seems weird. The question naturally arises as to where the rest of the power has gone to. However, recall that Fourier optics used above is grounded on the assumption that the focal plane field is tightly confined around the focus. Therefore, the analysis above has simply found the field that is near the focus. The rest of the power, a fraction $1-|\Gamma|^2$ of the incident power, must therefore go through the focal plane well away from the focus. That is, this power is sprayed randomly over the focal plane.
Numerical Study of the High Spatial Frequency Case
With the numerical procedure I’m about to show, I have found that the Mahajan formula works for Gaussian surface roughnesses begetting RMS aberrations up to 0.5 waves RMS extremely well. This corresponds to a Strehl ratio of about $5\times10^{-5}$, or a central peak loss of more than 40dB! This is far beyond the extent to which Mahajan experimentally found his rule of thumb to fit behaviours for Strehl ratios greater than about 0.1. My own experience (by doing full Maxwell equation simulations of lens systems and comparing the results to those from the Mahajan formula) is that the Mahajan formula is only reliable in the low spatial frequency case down to about 0.3 Strehl and is much rougher that when it is applied to the present, high spatial frequency case. My drawing below shows the numerically simulated system.
Here the full vector Maxwell's equations are integrated numerically by split step Fourier-leapfrog algorithm of my own devising and whose accuracy is limited only by the transverse grid spacing and the axial step size. In both cases, the system input was a plane wave of 488nm freespace wavelength: first a linearly polarised wave, then a circular polarised one.
The surface roughness model was a Gaussian random process with exponential autocorrelation function:
$$\mathcal{R}(\mathbf{x}_1,\,\mathbf{x}_2) = \sigma^2 \exp\left(-\frac{|\mathbf{x}_1-\mathbf{x}_2|}{L}\right)$$
with a correlation length $L$ equal to $1.5\mu\mathrm{m}$. The simulation region's sideways breadth was $100\mu\mathrm{m}\times100\mu\mathrm{m}$ and an example surface roughness topographic map with the above statistics is shown below where the RMS surface roughness is 150nm.
Some results of the numerical simulations are shown below, and the central peak, with the same shape as the theoretical ideal (i.e. zero surface roughness) case, can clearly be seen, as can be the widely spread speckle power. In the linearly polarised case, electric field vector sets a preferred direction for the speckle, and so we see that it is not isotropic, whereas, for the circularly polarised case, there is no such preferred direction and the speckle is clearly isotropic.
In general, reflection and refraction happen when light passes from one medium to another. You can see this if you see your own reflection in a window. Now, as a light ray approaches the critical angle, not only does the refracted ray get closer to the surface, but the amount of light transmitted gets less and less. At the critical angle, the refracted ray would be along the surface, but the amount of light is zero.
The graph below shows how much light is reflected at various angles of incidence. The graph on the right shows your situation of a ray of light moving from a dense medium to a lighter medium (glass to air, for example). At the critical angle, 100% of light is reflected, leaving none to propagate along the surface.
Best Answer
A parabola is a conic section, like circles and ellipses, and all three types of curve can be defined by a focus (or in the case of the ellipse two foci). In the case of a parabola we draw a straight line (the directrix) and choose a point (the focus) and the parabola is the set of points that are an equal distance from the directix and the focus:
(image is from the Wikipedia article linked above).
It should be obvious from the diagram that the focus is at the point where parallel rays hitting a parabolic mirror will converge, so it is both the focus in a mathematical sense and the focal point in an optical sense.
Response to comment:
The centre of curvature is the centre of the osculating circle. We can draw this because near the vertex the parabola looks like part of a circle.
Take the unit circle centred at $(0, 1)$:
$$ (y-1)^2 + x^2 = 1 $$
or expanding this:
$$ y^2 - 2y + 1 + x^2 = 1 $$
If we are very near the origin $y^2 \ll y$ so we can approximate the above expression by:
$$ -2y + 1 + x^2 = 1 $$
which rearranges into the equation for a parabola:
$$ y = \tfrac{1}{2} x^2 $$
And the focus of this parabola is at $(0, \tfrac{1}{2})$. I've used a special case to make the working simple, but you can generalise this to any circle passing centred on the $y$ axis and passing through the origin.