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DOUBLE-SLIT INTERFERENCE AND DIFFRACTION COMBINED
7 years ago
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In focusing an image, a lens passes only the light that falls within its circular perimeter. From this point of view, a lens behaves like a circular aperture in an opaque screen. Such an aperture forms a diffraction pattern analogous to that of a single slit. Diffraction effects often limit the ability of telescopes and other optical instruments to form precise images. The image formed by a lens can be distorted by other effects, including chromatic and spherical aberrations. These effects can be substantially reduced or eliminated by suitable shaping of the lens surfaces or by introducing correcting elements into the optical system. However, no amount of clever design can eliminate the effects of diffraction, which are determined only by the size of the aperture (the diameter of the lens) and the wavelength of the light. In diffraction, nature imposes a fundamental limitation on the precision of our instruments. When we used geometrical optics to analyze lenses, we assumed diffraction not to occur. However, geometrical optics is itself an approximation, being the limit of wave optics. If we were to make a rigorous wave-optical analysis of the formation of an image by a lens, we would find that diffraction effects arise in a natural way. Figure 42-13 shows the image of a distant point source of light (a star) formed on photographic film placed in the focal plane of the converging lens of a telescope. It is not a point, as the (approximate) geometrical optics treatment suggests, but a circular disk surrounded by several progressively fainter secondary rings. Comparison with Figure 42-1 leaves little doubt that we are dealing with a diffraction by a circular aperture, which is beyond the level of this text, shows that (under Fraunhofer conditions) the first minimum occurs at an angle from the central axis given by sin q = 1.22 λ/d where d is the diameter of the aperture. This is to be compared with Equation 42-1, sin q = λ/a’ which locates the first minimum of a sit of width a. These expressions differ by the factor 1.22, which arises when we divide the circular aperture into elementary Huygens sources and integrate over the aperture.
In focusing an image, a lens passes only the light that falls within its circular perimeter. From this point of view, a lens behaves like a circular aperture in an opaque screen. Such an aperture forms a diffraction pattern analogous to that of a single slit. Diffraction effects often limit the ability of telescopes and other optical instruments to form precise images.
The image formed by a lens can be distorted by other effects, including chromatic and spherical aberrations. These effects can be substantially reduced or eliminated by suitable shaping of the lens surfaces or by introducing correcting elements into the optical system. However, no amount of clever design can eliminate the effects of diffraction, which are determined only by the size of the aperture (the diameter of the lens) and the wavelength of the light. In diffraction, nature imposes a fundamental limitation on the precision of our instruments.
When we used geometrical optics to analyze lenses, we assumed diffraction not to occur. However, geometrical optics is itself an approximation, being the limit of wave optics. If we were to make a rigorous wave-optical analysis of the formation of an image by a lens, we would find that diffraction effects arise in a natural way.
Figure 42-13 shows the image of a distant point source of light (a star) formed on photographic film placed in the focal plane of the converging lens of a telescope. It is not a point, as the (approximate) geometrical optics treatment suggests, but a circular disk surrounded by several progressively fainter secondary rings. Comparison with Figure 42-1 leaves little doubt that we are dealing with a diffraction by a circular aperture, which is beyond the level of this text, shows that (under Fraunhofer conditions) the first minimum occurs at an angle from the central axis given by
sin q = 1.22 λ/d
where d is the diameter of the aperture. This is to be compared with Equation 42-1,
sin q = λ/a’
which locates the first minimum of a sit of width a. These expressions differ by the factor 1.22, which arises when we divide the circular aperture into elementary Huygens sources and integrate over the aperture.
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