Abbe sine condition - Citizendia

The Abbe sine condition is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects. A lens is an optical device with perfect or approximate Axial symmetry which transmits and refracts Light, converging or diverging It was formulated by Ernst Abbe in the context of microscopes. Ernst Karl Abbe ( January 23, 1840 &ndash January 14, 1905) was a German Physicist and professor at the University A microscope ( Greek: ( micron) = small + ( skopein) = to look or see is an instrument for viewing objects that are

The mathematical condition is as follows:

$\frac{\sin u'}{\sin U'} = \frac{\sin u}{\sin U}$

where the variables u, U are the angles (relative to the optic axis) of any two rays as they leave the object, and u’, U’ are the angles of the same rays where they reach the image plane (say, the film plane of a camera). For example, (u,u’) might represent a paraxial ray (i. In Geometric optics, the paraxial approximation is an Approximation used in ray tracing of light through an optical system (such as a lens) e. a ray nearly parallel with the optic axis), and (U,U’') might represent a marginal ray (i. In Optics, a ray is an idealized narrow Beam of light. Rays are used to model the propagation of Light through an optical system by dividing the real light e a ray with the largest angle admitted by the system aperture); the condition is general, however, and does not only apply to those rays.

Put in words, the sine of the output angle should be proportional to the sine of the input angle.

Magnification and the Abbe Sine Condition

Using the framework of Fourier optics, we may easily explain the significance of the Abbe sine condition. Huygens-Fresnel principle|geometrical optics Fourier optics is the study of classical optics using techniques involving Fourier transforms and can be seen Say an object in the object plane of an optical system has a transmittance function of the form, T(x_o,y_o). We may express this transmittance function in terms of its Fourier transform as:

$T(x_o,y_o) = \int\!\!\!\int T(k_x,k_y) ~ e^{j(k_x x_o + k_y y_o)} ~ dk_x dk_y$

Now, assume for simplicity that the system has no image distortion, so that the image plane coordinates are linearly related to the object plane coordinates via the relation

x i = M x o

y i = M y o

where M is the system magnification. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and Magnification is the process of enlarging something only in appearance not in physical size Let's now re-write the object plane transmittance above in a slightly modified form:

$T(x_o,y_o) = \int\!\!\! \int T(k_x,k_y) ~ e^{j((k_x/M) (Mx_o) + (k_y/M) (My_o))} ~ dk_x dk_y$

where we have simply multiplied and divided the various terms in the exponent by M, the system magnification. Magnification is the process of enlarging something only in appearance not in physical size Now, we may substitute the equations above for image plane coordinates in terms of object plane coordinates, to obtain,

$T(x_i,y_i) = \int\!\!\! \int T(k_x,k_y) ~ e^{j((k_x/M) x_i + (k_y/M) y_i)} ~ dk_x dk_y$

At this point we can propose another coordinate transformation relating the object plane wavenumber spectrum to the image plane wavenumber spectrum as

$k^i_x = k_x / M$

$k^i_y = k_y / M$

to obtain our final equation for the image plane field in terms of image plane coordinates and image plane wavenumbers as:

$T(x_i,y_i) = M^2 \int\int T(M k^i_x,M k^i_y) ~ e^{j(k^i_x x_i + k^i_y y_i)} dk^i_x dk^i_y$

From Fourier optics, we know that the wavenumbers can be expressed in terms of the spherical coordinate system as

k x = k sinθcosφ

k y = k sinθsinφ

If we consider a spectral component for which $φ = 0$ , then the coordinate transformation between object and image plane wavenumbers takes the form

k i sinθ i = k sinθ / M

This is another way of writing the Abbe sine condition, which simply reflects Heisenberg's uncertainty principle for Fourier transform pairs, namely that as the spatial extent of any function is expanded (by the magnification factor, M), the spectral extent contracts by the same factor, M, so that the space-bandwidth product remains constant. Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters Huygens-Fresnel principle|geometrical optics Fourier optics is the study of classical optics using techniques involving Fourier transforms and can be seen In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and

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