Dear Sirs,
The setup for the dispersion measurement is as follows. Hg lamp, collimator, goniometer with triangular prism. We measure the light dispersion using the prism. To do so we determine the minimal deviation angle of the particular color line and then calculate the refraction coefficient (the wavelengths are known from the standard source). To find the minimal deviation angle we rotate the prism.
In the above setup does anybody know whether the smooth minimum of the deviation angle (as a function of incidence angle) can increase the error of the measurement of the minimal deviation angle? Please correct me if there is a mistake.
For the general case of refraction in a prism spectrometer, the angle of deflection depends on the refractive index of the prism, the vertex angle of the prism, and the orientation of the prism with respect to the incident beam.
The angle of deviation is a minimum when the prism is aligned symmetrically with respect to incident and refracted output beam directions. In this orientation the deviation depends only weakly on the rotation of the prism. To calculate the refractive index, we need only the deviation angle φ, and the prism vertex angle, θ.
n = sin( (θ+φ)/2 ) / sin(θ/2)
A small error, ε, in the prism orientation gives rise to a much smaller error in the deflection angle, proportional to the square of the orientation error ε2. In practice, with reasonably careful adjustment, this error can be neglected.
I don't see how the minimum deviation angle could be made more sensitive to prism rotation, but if this were possible, it would increase the errors arising from measurements of the prism alignment.
The deviation angle does become extremely sensitive to incident angle when the internal refracted ray approaches the critical angle at either the entrance or exit face of the prism. The deviation angle is a maximum at the critical angle, where the beam will mostly be reflected rather than refracted through the prism.
For the general case of refraction in a prism spectrometer, the angle of deflection depends on the refractive index of the prism, the vertex angle of the prism, and the orientation of the prism with respect to the incident beam.
The angle of deviation is a minimum when the prism is aligned symmetrically with respect to incident and refracted output beam directions. In this orientation the deviation depends only weakly on the rotation of the prism. To calculate the refractive index, we need only the deviation angle φ, and the prism vertex angle, θ.
n = sin( (θ+φ)/2 ) / sin(θ/2)
A small error, ε, in the prism orientation gives rise to a much smaller error in the deflection angle, proportional to the square of the orientation error ε2. In practice, with reasonably careful adjustment, this error can be neglected.
I don't see how the minimum deviation angle could be made more sensitive to prism rotation, but if this were possible, it would increase the errors arising from measurements of the prism alignment.
The deviation angle does become extremely sensitive to incident angle when the internal refracted ray approaches the critical angle at either the entrance or exit face of the prism. The deviation angle is a maximum at the critical angle, where the beam will mostly be reflected rather than refracted through the prism.
Dear Mr. Alan Robinson,
Thank you very much for your detailed answer.
Dear Mr. Khripov,
the abruptness depends on the material and the vertex angle of the prism.
(In the case of diffraction gratings, the abruptness depends on the grating period).
If you rotate the prism and the moving slit image of a chosen wavelength changes direction at some point, this is exactly where you have minimum deviation.
For a given prism with known material and vertex angle, you can calculate and plot deviation as a function of incidence angle, and see the position of minimum deviation as the minimum of the graph.
Kind regards,
Sascha Grusche
Dear Mr. Sascha Grusche,
Thank you for the reply. Could you clarify whether the abruptness is stronger when the prism refraction index is bigger? And what is about the vertex angle influence on it?
Did anybody try to use converging lenses placed between the collimator and the prism to make the abruptness more stronger?
Anatoly A Khripov asked: "Could you clarify whether the abruptness is stronger when the prism refraction index is bigger? And what is about the vertex angle influence on it?"
If the deflection angle is φ, the prism vertex angle is θ, and the prism rotation from the minimum deflection orientation is ε, then a measure of "abruptness" is the second derivative of deflection with prism rotation angle. At the minimum deviation condition we have:
d2φ / dε2 = 2 n sin(θ/2) (n2 - 1) / (1 - n2sin2(θ/2))3/2
which increases monotonically with both prism refractive index, n, and with prism vertex angle, θ. Note that we require (n sin(θ/2) < 1) to avoid total internal reflection within the prism.
So "abruptness" increases with both prism refractive index and with vertex angle, and is maximised for a vertex angle just less than 2 arcsin(1/n).
Anatoly A Khripov , you can increase precision of your measurements if you will approximate their results in the proximity of minimum by parabolic dependence and find position of minimum from fitted parabola parameters.
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