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Proceedings Paper

Further discussion of Huygens-Fresnel principle
Author(s): Zhiqiang Liu; Kiyoshi Uchikawa
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Paper Abstract

Huygens-Fresnel principle is well used in analyzing wavefront propagation in vacuum. However, we should clear if this principle is available in the space including absorptive objection, such as a grating. In this paper we analyze if a grating can diffract a beam when the grating set on the focus point and the grating pitch is larger than the Airy disc, especially the phenomenon in a special interferometer [1]. The special interferometer shown in reference [1] has an extended incoherent light source modulated by a binary grating. The light source is imaged onto a sinusoidal transmission grating by the tested lens. The pitch of the binary grating is half of the period of the sinusoidal grating. Light coming from an arbitrary point of the light source is focused onto the sinusoidal grating. Two kinds of analysis method are considered. One is using Huygens-Fresnel principle as described in reference [1]: the concave beam can be resolved to a pair of waves. Each wave is diffracted by the sinusoidal grating. By using the modulated extended light source, only one color interferogram interfered by -1st and +1st order beams can be observed. By shifting the sinusoidal grating; the phase of the interferogram can be modulated; therefore, the intensity after the grating will change. The intensity variation is the contrast of the interferogram. The other analysis method is to calculate the intensity after the sinusoidal grating as in a normal Ronchi test. If two points on the light source separated by distance of the binary grating's pitch is picked up, the intensity of each point after the grating changes sinusoidally when the sinusoidal grating is shifted. Because the period of the modulation grating is half of the sinusoidal grating's pitch, the intensities of the two points change with "pi" phase difference. The total intensity after the sinusoidal grating does not change even if the sinusoidal grating is shifted. We found that the confusion in reference of complex transmission and amplitude transmission is one reason of the difference. We also reviewed the developing of Huygens-Fresnel principle from Maxwell equations. To get Huygens-Fresnel principle, condition of non absorptive elements is needed. Because transmission grating is an absorptive element, wavelet should be integrated before the grating. We also found that the substrate influences the phase of complex transmission when NA of the test is large. We performed an experiment to show that the contrast is almost zero, not 21% as calculated by the theory shown in the paper of refer.

Paper Details

Date Published: 4 October 2011
PDF: 10 pages
Proc. SPIE 8171, Physical Optics, 817106 (4 October 2011); doi: 10.1117/12.899036
Show Author Affiliations
Zhiqiang Liu, Nikon Corp. (Japan)
Kiyoshi Uchikawa, Nikon Corp. (Japan)

Published in SPIE Proceedings Vol. 8171:
Physical Optics
Daniel G. Smith; Frank Wyrowski; Andreas Erdmann, Editor(s)

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