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MACH-ZEHNDER INTERFEROMETRY

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The Mach-Zehnder interferometer is a classical mirror-interferometer. For a long time it was the most common dual-beam-interferometer used to measure continuously refractive index distributions of transparent objects. Developed by Mach and Zehnder in 1892, it has been frequently used in heat and mass transfer and in gas dynamics, for example combustion, until it was superceded by the holographic interferometer.

Figure 1 shows a Mach-Zehnder interferometer with a specific phase object ("phase object" is used to denote the transparent object being studied which influences the phase of light passing through it) in its measuring beam. It consists of two reflecting and two beam-splitting mirrors in a parallelogram (can also be a rectangle) arrangement. First the interferometer has to be adjusted in such a way that the two beams have equal optical path lengths. When the physical process of interest, for example heat transfer, is introduced into the measuring beam an optical path difference between reference and measuring beam is produced. A superposition of the two beams then generates an interference pattern in which the interference fringes correspond to lines of uniform difference of the refractive index.

Parallelogram arrangement of a Mach-Zehnder lnterferometer, M1, beamsplitting mirrors, M2, reflecting mirrors, L1 and L2 lenses, MS test section with a constant temperature gradient (and so a constant refractive index gradient), tm — tm adjustment plane, ti — ti image plane.

Figure 1. Parallelogram arrangement of a Mach-Zehnder lnterferometer, M1, Parallelogram arrangement of a Mach-Zehnder lnterferometer, M1, beamsplitting mirrors, M2, reflecting mirrors, L1 and L2 lenses, MS test section with a constant temperature gradient (and so a constant refractive index gradient), tm — tm adjustment plane, ti — ti image plane. beamsplitting mirrors, M2, Parallelogram arrangement of a Mach-Zehnder lnterferometer, M1, beamsplitting mirrors, M2, reflecting mirrors, L1 and L2 lenses, MS test section with a constant temperature gradient (and so a constant refractive index gradient), tm — tm adjustment plane, ti — ti image plane. reflecting mirrors, L1 and L2 lenses, MS test section with a constant temperature gradient (and so a constant refractive index gradient), tm — tm adjustment plane, ti — ti image plane.

Test section windows in the measuring beam, and corresponding plates compensating their effect in the reference beam, have to be manufactured with special precision and accuracy. The same goes for mirrors and lenses of all optical components because imperfections influence the beams differently. This makes the Mach-Zehnder interferometer very expensive.

Example

Figure 2 shows the interference pattern of a horizontal annulus filled with air when the inner cylinder is heated isothermally. The outer tube is cooled by water at constant temperature. When evaluating the interferogram the origin, i.e., the position of the inner wall, has to be determined. But the interference line close to the wall cannot be distinguished because of defraction effects. This problem is solved by extrapolating the temperature distribution—known from the interference pattern—from the wall temperature (measured by thermocouples) so the wall position and, as a result, the temperature gradient and the heat transfer coefficient can be determined.

Interferogram of a horizontal annulus with infinite fringe field. di = 40 mm; da = 98 mm; s = 29 mm; = 0, 73;. Angle position 30°: Nus = 4, 82; Grs = 7, 28×104; = 29, 7 K [Photo according to Hauf (1966)].

Figure 2. Interferogram of a horizontal annulus with infinite fringe field. di = 40 mm; da = 98 mm; s = 29 mm; Interferogram of a horizontal annulus with infinite fringe field. di = 40 mm; da = 98 mm; s = 29 mm; = 0, 73;. Angle position 30°: Nus = 4, 82; Grs = 7, 28×104; = 29, 7 K [Photo according to Hauf (1966)]. = 0, 73;. Angle position 30°: Nus = 4, 82; Grs = 7, 28×104; Interferogram of a horizontal annulus with infinite fringe field. di = 40 mm; da = 98 mm; s = 29 mm; = 0, 73;. Angle position 30°: Nus = 4, 82; Grs = 7, 28×104; = 29, 7 K [Photo according to Hauf (1966)]. = 29, 7 K [Photo according to Hauf (1966)].

REFERENCES

Mach, L. (1892) Über einen Interferenzrefraktor. Z. Instrumentenk. 12:89-93.

Zehnder, L. (1891) Ein neuer Interferenzrefraktor. Z. Instrumentenk. 11:275-285.

Grigull, U. and Hauf, W. (1966) Natural convection in horizontal cylindrical annuli. Proc. Inter. Heat Transfer Conf. 3rd. 11:182-195. Chicago.

Bennett, F. D. and Kahl, G. D. (1953) A generalized vector theory for the Mach-Zehnder-Interferometer. J. Opt. Soc. Am. 43:71-78.

Price, E. W. (1952) Initial adjustment of the Mach-Zehnder-Interferometer. Rev. Sci. Instr. 23:162.

References

  1. Mach, L. (1892) Über einen Interferenzrefraktor. Z. Instrumentenk. 12:89-93.
  2. Zehnder, L. (1891) Ein neuer Interferenzrefraktor. Z. Instrumentenk. 11:275-285.
  3. Grigull, U. and Hauf, W. (1966) Natural convection in horizontal cylindrical annuli. Proc. Inter. Heat Transfer Conf. 3rd. 11:182-195. Chicago.
  4. Bennett, F. D. and Kahl, G. D. (1953) A generalized vector theory for the Mach-Zehnder-Interferometer. J. Opt. Soc. Am. 43:71-78.
  5. Price, E. W. (1952) Initial adjustment of the Mach-Zehnder-Interferometer. Rev. Sci. Instr. 23:162. DOI: 10.1063/1.1746214

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