Abstract

A two-modulator generalized ellipsometer is described that is capable of measuring all 16 elements of a sample Mueller matrix with four measurements made at different azimuthal orientations of the polarization state generator and polarization state detector. If the sample can be described with a Mueller–Jones matrix, only a single measurement is needed. Only two calibration steps are needed to determine the fundamental operating parameters of the instrument. A reflection measurement from silicon is presented as an example, which illustrates that the elements of the Mueller–Jones matrix can be measured to an accuracy of ∼0.1–0.2%.

© 1997 Optical Society of America

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References

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  1. G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
    [CrossRef]
  2. J. C. Kemp, “Piezo-optical birefringence modulators: new use for a long-known effect,” J. Opt. Soc. Am. 59, 950–954 (1969).
  3. G. E. Jellison, F. A. Modine, “Accurate calibration of a photoelastic modulator in a polarization modulation ellipsometry experiment,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).
  4. G. E. Jellison, F. A. Modine, “A simple implementation of a power supply for constant phototube current in light modulation spectroscopy,” Rev. Sci. Instrum. 60, 3345–3346 (1989).
    [CrossRef]
  5. F. A. Modine, “Circuit for maintaining constant phototube current in polarization modulation spectroscopy,” Rev. Sci. Instrum. 50, 386–387 (1979).
    [CrossRef] [PubMed]
  6. G. E. Jellison, “Optical functions of silicon determined by two-channel polarization modulation ellipsometry,” Opt. Mat. 1, 41–47 (1992).
    [CrossRef]
  7. G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
    [CrossRef] [PubMed]

1997 (1)

1992 (1)

G. E. Jellison, “Optical functions of silicon determined by two-channel polarization modulation ellipsometry,” Opt. Mat. 1, 41–47 (1992).
[CrossRef]

1990 (1)

1989 (1)

G. E. Jellison, F. A. Modine, “A simple implementation of a power supply for constant phototube current in light modulation spectroscopy,” Rev. Sci. Instrum. 60, 3345–3346 (1989).
[CrossRef]

1979 (1)

F. A. Modine, “Circuit for maintaining constant phototube current in polarization modulation spectroscopy,” Rev. Sci. Instrum. 50, 386–387 (1979).
[CrossRef] [PubMed]

1969 (1)

Jellison, G. E.

G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
[CrossRef]

G. E. Jellison, “Optical functions of silicon determined by two-channel polarization modulation ellipsometry,” Opt. Mat. 1, 41–47 (1992).
[CrossRef]

G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
[CrossRef] [PubMed]

G. E. Jellison, F. A. Modine, “A simple implementation of a power supply for constant phototube current in light modulation spectroscopy,” Rev. Sci. Instrum. 60, 3345–3346 (1989).
[CrossRef]

G. E. Jellison, F. A. Modine, “Accurate calibration of a photoelastic modulator in a polarization modulation ellipsometry experiment,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).

Kemp, J. C.

Modine, F. A.

G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
[CrossRef]

G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
[CrossRef] [PubMed]

G. E. Jellison, F. A. Modine, “A simple implementation of a power supply for constant phototube current in light modulation spectroscopy,” Rev. Sci. Instrum. 60, 3345–3346 (1989).
[CrossRef]

F. A. Modine, “Circuit for maintaining constant phototube current in polarization modulation spectroscopy,” Rev. Sci. Instrum. 50, 386–387 (1979).
[CrossRef] [PubMed]

G. E. Jellison, F. A. Modine, “Accurate calibration of a photoelastic modulator in a polarization modulation ellipsometry experiment,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Mat. (1)

G. E. Jellison, “Optical functions of silicon determined by two-channel polarization modulation ellipsometry,” Opt. Mat. 1, 41–47 (1992).
[CrossRef]

Rev. Sci. Instrum. (2)

G. E. Jellison, F. A. Modine, “A simple implementation of a power supply for constant phototube current in light modulation spectroscopy,” Rev. Sci. Instrum. 60, 3345–3346 (1989).
[CrossRef]

F. A. Modine, “Circuit for maintaining constant phototube current in polarization modulation spectroscopy,” Rev. Sci. Instrum. 50, 386–387 (1979).
[CrossRef] [PubMed]

Other (1)

G. E. Jellison, F. A. Modine, “Accurate calibration of a photoelastic modulator in a polarization modulation ellipsometry experiment,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the 2-MGE showing the details of the electrical connections.

Fig. 2
Fig. 2

Details of the PSG, for light going left to right, and of the PSD, for light going right to left.

Fig. 3
Fig. 3

Calibration data for the modulators used in the 2-MGE. Top, static strain of each modulator as a function of wavelength, where the solid curves indicate the fits to the data by using Eq. (8). Bottom, the modulator control voltage required to give A0 = A1 = 2.4048 rad, where the solid lines show the fits to the data by using Eq. (7). The data for modulator 1 are offset by 1 V to show all the data.

Fig. 4
Fig. 4

Five parameters expected to be zero for a measurement of silicon at azimuthal angles of θm0 = 0° and θm1 = 45° taken using the 2-MGE at an angle of incidence of 65.12°.

Fig. 5
Fig. 5

Results of a four-zone average of silicon taken by using the 2-MGE at an angle of incidence of 65.12°.

Equations (21)

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It=Idc+IX0X0+IY0Y0+IX1X1+IY1Y1+IX0X1X0X1+IX0Y1X0Y1+IY0X1Y0X1+IY0Y1Y0Y0,
X0=sinA0 sinω0t,
Y0=cosA0 sinω0t,
X1=sinA1 sinω1t,
Y1=cosA1 sinω1t.
Xi=sinAi sinωit=2k=1J2k-1Aisin2k-1ωit,
Yi=cosAi sinωit=J0Ai+2k=1J2kAicos2kωit,
It=Idc+2k=1Rkαk cosΩkt+βk sinΩkt.
IX0=0,
IY0=P01J0A1cos2θm-2 sin2θmεb1,
IX1=0,
IY1=P01J0A0cos2θm+2 sin2θmεb0,
IX0X1=-P01,
IX0Y1=-P01δ1+δ0 cos2θm,
IY0X1=-P01δ0+δ1 cos2θm,
IY0Y1=P01 cos2θm,
J0Ai=-0.51962.4048-Ai,
VmiE=KvAiEk=0γkiE2k,
δiE=kδiEk=0γkiE2k.
IY0=-sin2θb0N sin2θm0-sin2θb1CJ0A1,
IY1=-sin2θb1N sin2θm1-sin2θb0CJ0A0.

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