Lloyd's mirror

Lloyd's mirror is an optics experiment that was first described in 1834 by Humphrey Lloyd in the Transactions of the Royal Irish Academy.^[1] Its original goal was to provide further evidence for the wave nature of light, beyond those provided by Thomas Young and Augustin-Jean Fresnel. In the experiment, light from a monochromatic slit source reflects from a glass surface at a small angle and appears to come from a virtual source as a result. The reflected light interferes with the direct light from the source, forming interference fringes.^[2][3] It is the optical wave analogue to a sea interferometer.^[4]

Setup

 

Figure 1. Lloyd's mirror

 

Figure 2. Young's two-slit experiment displays a single-slit diffraction pattern on top of the two-slit interference fringes.

Lloyd’s Mirror is used to produce two-source interference patterns that have important differences from the interference patterns seen in Young's experiment.

In a modern implementation of Lloyd's mirror, a diverging laser beam strikes a front-surface mirror at a grazing angle, so that some of the light travels directly to the screen (blue lines in Fig. 1), and some of the light reflects off the mirror to the screen (red lines). The reflected light forms a virtual second source that interferes with the direct light.

In Young's experiment, the individual slits display a diffraction pattern on top of which is overlaid interference fringes from the two slits (Fig. 2). In contrast, the Lloyd's mirror experiment does not use slits and displays two-source interference without the complications of an overlaid single-slit diffraction pattern.

In Young's experiment, the central fringe representing equal path length is bright because of constructive interference. In contrast, in Lloyd's mirror, the fringe nearest the mirror representing equal path length is dark rather than bright. This is because the light reflecting off the mirror undergoes a 180° phase shift, and so causes destructive interference when the path lengths are equal or when they differ by an integer number of wavelengths.

Applications

Interference lithography

The most common application of Lloyd's mirror is in UV photolithography and nanopatterning. Lloyd's mirror has important advantages over double-slit interferometers.

If one wishes to create a series of closely spaced interference fringes using a double-slit interferometer, the spacing d between the slits must be increased. Increasing the slit spacing, however, requires that the input beam be broadened to cover both slits. This results in a large loss of power. In contrast, increasing d in the Lloyd's mirror technique does not result in power loss, since the second "slit" is just the reflected virtual image of the source. Hence, Lloyd's mirror enables the generation of finely detailed interference patterns of sufficient brightness for applications such as photolithography.^[5]

Typical uses of Lloyd's mirror photolithography would include fabrication of diffraction gratings for surface encoders^[6] and patterning the surfaces of medical implants for improved biofunctionality.^[7]

Test pattern generation

High visibility cos²-modulated fringes of constant spatial frequency can be generated in a Lloyd's mirror arrangement using parallel collimated monochromatic light rather than a point or slit source. The uniform fringes generated by this arrangement can be used to measure the modulation transfer functions of optical detectors such as CCD arrays to characterize their performance as a function of spatial frequency, wavelength, intensity, and so forth.^[8]

Optical measurement

The output of a Lloyd's mirror was analyzed with a CCD photodiode array to produce a compact, broad range, high accuracy Fourier transform wavemeter that could be used to analyze the spectral output of pulsed lasers.^[9]

Radio astronomy

Main article: Sea interferometry

 

Figure 3. Determining the position of galactic radio sources using Lloyd's mirror

In the late 1940s and early 1950s, CSIRO scientists used a technique based on Lloyd's mirror to make accurate measurements of the position of various galactic radio sources from coastal sites in New Zealand and Australia. As illustrated in Fig. 3, the technique was to observe the sources combining direct and reflected rays from high cliffs overlooking the sea. After correcting for atmospheric refraction, these observations allowed the paths of the sources above the horizon to be plotted and their celestial coordinates to be determined.^[10][11]

Underwater acoustics

An acoustic source just below the water surface generates constructive and destructive interference between the direct path and reflected paths. This can have a major impact on sonar operations.^[12]

The Lloyd mirror effect has been implicated as having an important role in explaining why marine animals such as manatees and whales have been repeatedly hit by boats and ships. Interference due to Lloyd's mirror results in low frequency propeller sounds not being discernible near the surface, where most accidents occur. This is because at the surface, sound reflections are nearly 180 degrees out of phase with the incident waves. Combined with spreading and acoustic shadowing effects, the result is that the marine animal is unable to hear an approaching vessel before it has been run over or entrapped by the hydrodynamic forces of the vessel's passage.^[13]

See also

• Multipath propagation

References

1. Lloyd, Humphrey (1831). "On a New Case of Interference of the Rays of Light". The Transactions of the Royal Irish Academy. 17. Royal Irish Academy: 171–177. ISSN 0790-8113. JSTOR 30078788. Retrieved 2021-05-29.
2. Fresnel's and Lloyd's Mirrors
3. (PDF). University of Arkansas. Archived from the original (PDF) on 7 September 2012. Retrieved 20 May 2012.
4. Bolton, J. G.; Slee, O. B. (1953). "Galactic Radiation at Radio Frequencies V. The Sea Interferometer". Australian Journal of Physics. 6: 420–433. Bibcode:1953AuJPh...6..420B. doi:10.1071/PH530420.
5. "Application Note 49: Theory of Lloyd's Mirror Interferometer" (PDF). Newport Corporation. Retrieved 16 February 2014.
6. Li, X.; Shimizu, Y.; Ito, S.; Gao, W.; Zeng, L. (2013). Lin, Jie (ed.). "Fabrication of diffraction gratings for surface encoders by using a Lloyd's mirror interferometer with a 405 nm laser diode". International Symposium on Precision Engineering Measurement and Instrumentation. Eighth International Symposium on Precision Engineering Measurement and Instrumentation. 8759: 87594Q. Bibcode:2013SPIE.8759E..4QL. doi:10.1117/12.2014467. S2CID 136994909.
7. Domanski, M. (2010). "Novel approach to produce nanopatterned titanium implants by combining nanoimprint lithography and reactive ion etching" (PDF). 14th International Conference on Miniaturized Systems for Chemistry and Life Sciences: 3–7.
8. Hochberg, E. B.; Chrien, N. L. (PDF). Jet Propulsion Laboratory. Archived from the original (PDF) on 22 February 2014. Retrieved 16 February 2014.
9. Kielkopf, J.; Portaro, L. (1992). "Lloyd's mirror as a laser wavemeter". Applied Optics. 31 (33): 7083–7088. Bibcode:1992ApOpt..31.7083K. doi:10.1364/AO.31.007083. PMID 20802569.
10. Bolton, J. G.; Stanley, G. J.; Slee, O. B. (1949). "Positions of Three Discrete Sources of Galactic Radio-Frequency Radiation". Nature. 164 (4159): 101–102. Bibcode:1949Natur.164..101B. doi:10.1038/164101b0. S2CID 4073162.
11. Edwards, Philip. (PDF). National Astronomical Observatory of Japan (NAOJ). Archived from the original (PDF) on 21 February 2014. Retrieved 11 February 2014.
12. Carey, W. M. (2009). "Lloyd's Mirror—Image Interference Effects". Acoustics Today. 5 (2): 14. doi:10.1121/1.3182842.
13. Gerstein, Edmund (2002). "Manatees, Bioacoustics and Boats". American Scientist. 90 (2): 154–163. Bibcode:2002AmSci..90..154G. doi:10.1511/2002.2.154. S2CID 121310274. Retrieved 13 February 2014.

Further reading

• Titchmarsh, P. F. (1941). "Lloyd's single-mirror interference fringes". Proceedings of the Physical Society. 53 (4): 391–402. Bibcode:1941PPS....53..391T. doi:10.1088/0959-5309/53/4/304.

External links

• Lloyd's Mirror @ wolfram