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Dispersion

  • Page ID
    75524
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    Implications of the grating equation: angular dispersion

    ALL wavelengths arrive at the same angle, α. We see constructive interference for different wavelengths only as a function of take-off angle, β. The change in constructive interference wavelength with angle. can be found by solving the grating equation for λ and then differentiating with respect to β. The result is:

    \[ \dfrac{d \lambda}{d \beta} =\dfrac{d\cos \beta}{n}\]

    The discriminating reader is hereby authorized to scream at our forebears, who used "d" for groove spacing as well as for "derivative" and "slit width." We are reduced to figuring out which "d" is meant by (frequently unclear) context.

    Implications of the grating equation: linear dispersion

    While angular dispersion is conveniently computed, most detectors are of finite spatial extent and look at light spread out in space. Thus linear dispersion, the separation of light by position instead of angle, is most useful in the lab. For a spectrometer with focal length f, this "spreading of the rainbow" is computed by recognizing that for a small angle θ, the length of an arc at a distance f has length x given by \(x = f θ\). A little work with the chain rule gives:

    \[ \dfrac{d \lambda}{d x} =\dfrac{d\cos \beta}{n\,f}\]

    A convenient approximation for initial design is cos β ~ 1 (exactly true for β = 0, and good within a factor of 2 up to β = 60°. We thus see that in low resolution situations, where we want comparatively wide swaths of wavelength \(\delta\)λ to cover a given detector width \(\delta\)x, we use n = 1 and short focal lengths. For high dispersion, where we want small \(\delta\)λ over a large detector width, higher n and longer f are desirable.


    This page titled Dispersion is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Alexander Scheeline & Thomas M. Spudich via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.