Abstract

A novel ultrastable widefield interferometer is presented. This uses a modulated light camera (MLC) to capture and stabilise the interferogram in the widefield heterodyne interferometer. This system eliminates the contribution of piston phase to the interferogram without the need for common path optics and results in a highly stable widefield interferometer. The MLC uses quadrature demodulation circuitry built into each pixel to demodulate the light signal and extract phase information using an electronic reference signal. In contrast to the work previously presented [Opt. Express 19, 24546 (2011)], the reference signal is derived from one of the pixels on board the MLC rather than an external source. This local reference signal tracks the instantaneous modulation frequency detected by the other pixels and eliminates the contribution of piston phase to the interferogram, substantially removing the contributions of unwanted vibrations and microphonics to the interferogram. Interferograms taken using the ultrastable system are presented with one of the interferometer mirrors moving at up to 85 mm s−1 over a variety of frequencies from 18 Hz to 20 kHz (giving a variation in optical path length of 220 μm, or 350 wavelengths at 62 Hz). This limit was the result of complex motion in the mirror mount rather than the stability limit of the system. The system is shown to be insensitive to pure piston phase variations equivalent to an object velocity of over 3 m s−1.

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References

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  1. H. Osterberg, “An interferometer method of studying the vibrations of an oscillating quartz plate,” J. Opt. Soc. Am.22, 19–34 (1932).
    [CrossRef]
  2. R. Patel, S. Achamfuo-Yeboah, R. Light, and M. Clark, “Widefield heterodyne interferometry using a custom CMOS modulated light camera,” Opt. Express19, 24546–24556 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
  4. K. Birch and M. Okaji, “Stable interferometer supporting system,” J. Phys. E: Sci. Instrum.19, 361–363 (1986).
    [CrossRef]
  5. W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, “Inverse electromechanical effect in mechanically oriented sc*-elastomers examined by means of an ultra-stable Michelson interferometer,” Ferroelectr.208, 373–383 (1998).
    [CrossRef]
  6. H. Kikuta, S. Asai, H. Yasukochi, and K. Iwata, “Force microscopy using common-path optical-heterodyne interferometer,” Jpn. J. Appl. Phys. Part 130, 587–590 (1991).
    [CrossRef]
  7. Y. Park and K. Cho, “Heterodyne interferometer scheme using a double pass in an acousto-optic modulator,” Opt. Lett.36, 331–333 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. R. Smythe and R. Moore, “Common path scanning heterodyne optical profilometer for absolute phase measurement,” Opt. Eng.23, 361–364 (1984).
  14. B. K. A. Ngoi, K. Venkatakrishnan, and N. R. Sivakumar, “Phase-shifting interferometry immune to vibration,” Appl. Opt.40, 3211–3214 (2001).
    [CrossRef]
  15. P. Dmochowski, B. Hayes-Gill, M. Clark, J. Crowe, M. Somekh, and S. Morgan, “Camera pixel for coherent detection of modulated light,” Electron. Lett.40, 1403–1404 (2004).
    [CrossRef]
  16. M. J. Griffin, “An introduction to whole body vibration,” in Handbook of Human Vibration, (Academic Press, 1996), 27–42.
  17. A. Ismail, M. Nuawi, N. Kamaruddin, and R. Bakar, “Comparative assessment of the whole-body vibration exposure under different car speed based on Malaysian road profile,” J. Appl. Sci.10, 1428–1434 (2010).
    [CrossRef]
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  19. W. W. Hays (U.S.), Facing Geologic and Hydrologic Hazards: Earth-Science Considerations (U.S. Dept. of the Interior, Geological Survey, 1981).
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    [CrossRef]

2011 (2)

2010 (1)

A. Ismail, M. Nuawi, N. Kamaruddin, and R. Bakar, “Comparative assessment of the whole-body vibration exposure under different car speed based on Malaysian road profile,” J. Appl. Sci.10, 1428–1434 (2010).
[CrossRef]

2009 (1)

J. Chieh, S. Yang, H.-E. Horng, C.-Y. Hong, and H. Yang, “Measurements of the complex transmission/reflection coefficient of a material using mixed-type common-path heterodyne interferometery,” IEEE Trans. Instrum. Meas.58, 1878–1885 (2009).
[CrossRef]

2008 (1)

P. Stafford, F. Strasser, and J. Bommer, “An evaluation of the applicability of the NGA models to ground-motion prediction in the Euro-Mediterranean region,” Bull. Earthquake Eng.6, 149–1772008.
[CrossRef]

2006 (1)

N. A. Riza, M. A. Arain, and F. N. Ghauri, “Self-calibrating hybrid wavelength, polarization, and time-multiplexed heterodyne interferometers for angstrom precision measurements,” Opt. Eng.45, 125603 (2006).
[CrossRef]

2004 (1)

P. Dmochowski, B. Hayes-Gill, M. Clark, J. Crowe, M. Somekh, and S. Morgan, “Camera pixel for coherent detection of modulated light,” Electron. Lett.40, 1403–1404 (2004).
[CrossRef]

2001 (1)

1999 (1)

1998 (2)

W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, “Inverse electromechanical effect in mechanically oriented sc*-elastomers examined by means of an ultra-stable Michelson interferometer,” Ferroelectr.208, 373–383 (1998).
[CrossRef]

N. Sawyer, C. See, M. Clark, M. Somekh, and J. Goh, “Ultrastable absolute-phase common-path optical profiler based on computer-generated holography,” Appl. Opt.37, 6716–6720 (1998).
[CrossRef]

1992 (1)

C. Gordon, “Generic criteria for vibration-sensitive equipment,” Proc. SPIE Int. Soc. Opt. Eng.1619, 71–85 (1992).

1991 (1)

H. Kikuta, S. Asai, H. Yasukochi, and K. Iwata, “Force microscopy using common-path optical-heterodyne interferometer,” Jpn. J. Appl. Phys. Part 130, 587–590 (1991).
[CrossRef]

1989 (1)

M. J. Offside, M. Somekh, and C. See, “Common path scanning heterodyne optical profilometer for absolute phase measurement,” Appl. Phys. Lett.55, 2051–2053 (1989).
[CrossRef]

1986 (1)

K. Birch and M. Okaji, “Stable interferometer supporting system,” J. Phys. E: Sci. Instrum.19, 361–363 (1986).
[CrossRef]

1984 (1)

R. Smythe and R. Moore, “Common path scanning heterodyne optical profilometer for absolute phase measurement,” Opt. Eng.23, 361–364 (1984).

1983 (1)

1932 (1)

Achamfuo-Yeboah, S.

Arain, M. A.

N. A. Riza, M. A. Arain, and F. N. Ghauri, “Self-calibrating hybrid wavelength, polarization, and time-multiplexed heterodyne interferometers for angstrom precision measurements,” Opt. Eng.45, 125603 (2006).
[CrossRef]

Asai, S.

H. Kikuta, S. Asai, H. Yasukochi, and K. Iwata, “Force microscopy using common-path optical-heterodyne interferometer,” Jpn. J. Appl. Phys. Part 130, 587–590 (1991).
[CrossRef]

Bakar, R.

A. Ismail, M. Nuawi, N. Kamaruddin, and R. Bakar, “Comparative assessment of the whole-body vibration exposure under different car speed based on Malaysian road profile,” J. Appl. Sci.10, 1428–1434 (2010).
[CrossRef]

Birch, K.

K. Birch and M. Okaji, “Stable interferometer supporting system,” J. Phys. E: Sci. Instrum.19, 361–363 (1986).
[CrossRef]

Bommer, J.

P. Stafford, F. Strasser, and J. Bommer, “An evaluation of the applicability of the NGA models to ground-motion prediction in the Euro-Mediterranean region,” Bull. Earthquake Eng.6, 149–1772008.
[CrossRef]

Chieh, J.

J. Chieh, S. Yang, H.-E. Horng, C.-Y. Hong, and H. Yang, “Measurements of the complex transmission/reflection coefficient of a material using mixed-type common-path heterodyne interferometery,” IEEE Trans. Instrum. Meas.58, 1878–1885 (2009).
[CrossRef]

Cho, K.

Clark, M.

Crowe, J.

P. Dmochowski, B. Hayes-Gill, M. Clark, J. Crowe, M. Somekh, and S. Morgan, “Camera pixel for coherent detection of modulated light,” Electron. Lett.40, 1403–1404 (2004).
[CrossRef]

Dmochowski, P.

P. Dmochowski, B. Hayes-Gill, M. Clark, J. Crowe, M. Somekh, and S. Morgan, “Camera pixel for coherent detection of modulated light,” Electron. Lett.40, 1403–1404 (2004).
[CrossRef]

Eckert, T.

W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, “Inverse electromechanical effect in mechanically oriented sc*-elastomers examined by means of an ultra-stable Michelson interferometer,” Ferroelectr.208, 373–383 (1998).
[CrossRef]

Finkelmann, H.

W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, “Inverse electromechanical effect in mechanically oriented sc*-elastomers examined by means of an ultra-stable Michelson interferometer,” Ferroelectr.208, 373–383 (1998).
[CrossRef]

Gattinger, P.

W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, “Inverse electromechanical effect in mechanically oriented sc*-elastomers examined by means of an ultra-stable Michelson interferometer,” Ferroelectr.208, 373–383 (1998).
[CrossRef]

Ghauri, F. N.

N. A. Riza, M. A. Arain, and F. N. Ghauri, “Self-calibrating hybrid wavelength, polarization, and time-multiplexed heterodyne interferometers for angstrom precision measurements,” Opt. Eng.45, 125603 (2006).
[CrossRef]

Ghebremichael, F.

Goh, J.

Gordon, C.

C. Gordon, “Generic criteria for vibration-sensitive equipment,” Proc. SPIE Int. Soc. Opt. Eng.1619, 71–85 (1992).

Griffin, M. J.

M. J. Griffin, “An introduction to whole body vibration,” in Handbook of Human Vibration, (Academic Press, 1996), 27–42.

Hayes-Gill, B.

P. Dmochowski, B. Hayes-Gill, M. Clark, J. Crowe, M. Somekh, and S. Morgan, “Camera pixel for coherent detection of modulated light,” Electron. Lett.40, 1403–1404 (2004).
[CrossRef]

Hays, W. W.

W. W. Hays (U.S.), Facing Geologic and Hydrologic Hazards: Earth-Science Considerations (U.S. Dept. of the Interior, Geological Survey, 1981).

Hong, C.-Y.

J. Chieh, S. Yang, H.-E. Horng, C.-Y. Hong, and H. Yang, “Measurements of the complex transmission/reflection coefficient of a material using mixed-type common-path heterodyne interferometery,” IEEE Trans. Instrum. Meas.58, 1878–1885 (2009).
[CrossRef]

Horng, H.-E.

J. Chieh, S. Yang, H.-E. Horng, C.-Y. Hong, and H. Yang, “Measurements of the complex transmission/reflection coefficient of a material using mixed-type common-path heterodyne interferometery,” IEEE Trans. Instrum. Meas.58, 1878–1885 (2009).
[CrossRef]

Ismail, A.

A. Ismail, M. Nuawi, N. Kamaruddin, and R. Bakar, “Comparative assessment of the whole-body vibration exposure under different car speed based on Malaysian road profile,” J. Appl. Sci.10, 1428–1434 (2010).
[CrossRef]

Iwata, K.

H. Kikuta, S. Asai, H. Yasukochi, and K. Iwata, “Force microscopy using common-path optical-heterodyne interferometer,” Jpn. J. Appl. Phys. Part 130, 587–590 (1991).
[CrossRef]

Kamaruddin, N.

A. Ismail, M. Nuawi, N. Kamaruddin, and R. Bakar, “Comparative assessment of the whole-body vibration exposure under different car speed based on Malaysian road profile,” J. Appl. Sci.10, 1428–1434 (2010).
[CrossRef]

Keck, M.

W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, “Inverse electromechanical effect in mechanically oriented sc*-elastomers examined by means of an ultra-stable Michelson interferometer,” Ferroelectr.208, 373–383 (1998).
[CrossRef]

Kikuta, H.

H. Kikuta, S. Asai, H. Yasukochi, and K. Iwata, “Force microscopy using common-path optical-heterodyne interferometer,” Jpn. J. Appl. Phys. Part 130, 587–590 (1991).
[CrossRef]

Knize, R.

Kremer, F.

W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, “Inverse electromechanical effect in mechanically oriented sc*-elastomers examined by means of an ultra-stable Michelson interferometer,” Ferroelectr.208, 373–383 (1998).
[CrossRef]

Lehmann, W.

W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, “Inverse electromechanical effect in mechanically oriented sc*-elastomers examined by means of an ultra-stable Michelson interferometer,” Ferroelectr.208, 373–383 (1998).
[CrossRef]

Light, R.

Moore, R.

R. Smythe and R. Moore, “Common path scanning heterodyne optical profilometer for absolute phase measurement,” Opt. Eng.23, 361–364 (1984).

Morgan, S.

P. Dmochowski, B. Hayes-Gill, M. Clark, J. Crowe, M. Somekh, and S. Morgan, “Camera pixel for coherent detection of modulated light,” Electron. Lett.40, 1403–1404 (2004).
[CrossRef]

Mutoh, K.

Ngoi, B. K. A.

Nuawi, M.

A. Ismail, M. Nuawi, N. Kamaruddin, and R. Bakar, “Comparative assessment of the whole-body vibration exposure under different car speed based on Malaysian road profile,” J. Appl. Sci.10, 1428–1434 (2010).
[CrossRef]

Offside, M. J.

M. J. Offside, M. Somekh, and C. See, “Common path scanning heterodyne optical profilometer for absolute phase measurement,” Appl. Phys. Lett.55, 2051–2053 (1989).
[CrossRef]

Okaji, M.

K. Birch and M. Okaji, “Stable interferometer supporting system,” J. Phys. E: Sci. Instrum.19, 361–363 (1986).
[CrossRef]

Osterberg, H.

Park, Y.

Patel, R.

Riza, N. A.

N. A. Riza, M. A. Arain, and F. N. Ghauri, “Self-calibrating hybrid wavelength, polarization, and time-multiplexed heterodyne interferometers for angstrom precision measurements,” Opt. Eng.45, 125603 (2006).
[CrossRef]

Sawyer, N.

See, C.

N. Sawyer, C. See, M. Clark, M. Somekh, and J. Goh, “Ultrastable absolute-phase common-path optical profiler based on computer-generated holography,” Appl. Opt.37, 6716–6720 (1998).
[CrossRef]

M. J. Offside, M. Somekh, and C. See, “Common path scanning heterodyne optical profilometer for absolute phase measurement,” Appl. Phys. Lett.55, 2051–2053 (1989).
[CrossRef]

Sivakumar, N. R.

Smythe, R.

R. Smythe and R. Moore, “Common path scanning heterodyne optical profilometer for absolute phase measurement,” Opt. Eng.23, 361–364 (1984).

Somekh, M.

P. Dmochowski, B. Hayes-Gill, M. Clark, J. Crowe, M. Somekh, and S. Morgan, “Camera pixel for coherent detection of modulated light,” Electron. Lett.40, 1403–1404 (2004).
[CrossRef]

N. Sawyer, C. See, M. Clark, M. Somekh, and J. Goh, “Ultrastable absolute-phase common-path optical profiler based on computer-generated holography,” Appl. Opt.37, 6716–6720 (1998).
[CrossRef]

M. J. Offside, M. Somekh, and C. See, “Common path scanning heterodyne optical profilometer for absolute phase measurement,” Appl. Phys. Lett.55, 2051–2053 (1989).
[CrossRef]

Stafford, P.

P. Stafford, F. Strasser, and J. Bommer, “An evaluation of the applicability of the NGA models to ground-motion prediction in the Euro-Mediterranean region,” Bull. Earthquake Eng.6, 149–1772008.
[CrossRef]

Stein, P.

W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, “Inverse electromechanical effect in mechanically oriented sc*-elastomers examined by means of an ultra-stable Michelson interferometer,” Ferroelectr.208, 373–383 (1998).
[CrossRef]

Strasser, F.

P. Stafford, F. Strasser, and J. Bommer, “An evaluation of the applicability of the NGA models to ground-motion prediction in the Euro-Mediterranean region,” Bull. Earthquake Eng.6, 149–1772008.
[CrossRef]

Takeda, M.

Venkatakrishnan, K.

Yang, H.

J. Chieh, S. Yang, H.-E. Horng, C.-Y. Hong, and H. Yang, “Measurements of the complex transmission/reflection coefficient of a material using mixed-type common-path heterodyne interferometery,” IEEE Trans. Instrum. Meas.58, 1878–1885 (2009).
[CrossRef]

Yang, S.

J. Chieh, S. Yang, H.-E. Horng, C.-Y. Hong, and H. Yang, “Measurements of the complex transmission/reflection coefficient of a material using mixed-type common-path heterodyne interferometery,” IEEE Trans. Instrum. Meas.58, 1878–1885 (2009).
[CrossRef]

Yasukochi, H.

H. Kikuta, S. Asai, H. Yasukochi, and K. Iwata, “Force microscopy using common-path optical-heterodyne interferometer,” Jpn. J. Appl. Phys. Part 130, 587–590 (1991).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

M. J. Offside, M. Somekh, and C. See, “Common path scanning heterodyne optical profilometer for absolute phase measurement,” Appl. Phys. Lett.55, 2051–2053 (1989).
[CrossRef]

Bull. Earthquake Eng. (1)

P. Stafford, F. Strasser, and J. Bommer, “An evaluation of the applicability of the NGA models to ground-motion prediction in the Euro-Mediterranean region,” Bull. Earthquake Eng.6, 149–1772008.
[CrossRef]

Electron. Lett. (1)

P. Dmochowski, B. Hayes-Gill, M. Clark, J. Crowe, M. Somekh, and S. Morgan, “Camera pixel for coherent detection of modulated light,” Electron. Lett.40, 1403–1404 (2004).
[CrossRef]

Ferroelectr. (1)

W. Lehmann, P. Gattinger, M. Keck, F. Kremer, P. Stein, T. Eckert, and H. Finkelmann, “Inverse electromechanical effect in mechanically oriented sc*-elastomers examined by means of an ultra-stable Michelson interferometer,” Ferroelectr.208, 373–383 (1998).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

J. Chieh, S. Yang, H.-E. Horng, C.-Y. Hong, and H. Yang, “Measurements of the complex transmission/reflection coefficient of a material using mixed-type common-path heterodyne interferometery,” IEEE Trans. Instrum. Meas.58, 1878–1885 (2009).
[CrossRef]

J. Appl. Sci. (1)

A. Ismail, M. Nuawi, N. Kamaruddin, and R. Bakar, “Comparative assessment of the whole-body vibration exposure under different car speed based on Malaysian road profile,” J. Appl. Sci.10, 1428–1434 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. E: Sci. Instrum. (1)

K. Birch and M. Okaji, “Stable interferometer supporting system,” J. Phys. E: Sci. Instrum.19, 361–363 (1986).
[CrossRef]

Jpn. J. Appl. Phys. Part 1 (1)

H. Kikuta, S. Asai, H. Yasukochi, and K. Iwata, “Force microscopy using common-path optical-heterodyne interferometer,” Jpn. J. Appl. Phys. Part 130, 587–590 (1991).
[CrossRef]

Opt. Eng. (2)

N. A. Riza, M. A. Arain, and F. N. Ghauri, “Self-calibrating hybrid wavelength, polarization, and time-multiplexed heterodyne interferometers for angstrom precision measurements,” Opt. Eng.45, 125603 (2006).
[CrossRef]

R. Smythe and R. Moore, “Common path scanning heterodyne optical profilometer for absolute phase measurement,” Opt. Eng.23, 361–364 (1984).

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE Int. Soc. Opt. Eng. (1)

C. Gordon, “Generic criteria for vibration-sensitive equipment,” Proc. SPIE Int. Soc. Opt. Eng.1619, 71–85 (1992).

Other (2)

W. W. Hays (U.S.), Facing Geologic and Hydrologic Hazards: Earth-Science Considerations (U.S. Dept. of the Interior, Geological Survey, 1981).

M. J. Griffin, “An introduction to whole body vibration,” in Handbook of Human Vibration, (Academic Press, 1996), 27–42.

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Figures (6)

Fig. 1.
Fig. 1.

A schematic of the MLC camera array and of an RFout pixel. Each pixel in the array contains a transimpedance amplifier to convert photocurrent into a voltage, an amplifier, in phase and quadrature mixers (Gilbert cells) and low phase filters (around 2 kHz) for the I, Q and DC outputs. Four pixels in the array contain additional circuitry that allows them to output the raw RF signal. In the normal mode of operation, the I and Q inputs are driven by a signal generator. In ultrastable mode, they are driven by signals derived from the RFout signal from one of the four special pixels.

Fig. 2.
Fig. 2.

Modified Mach-Zehnder interferometer. The interferometer uses a fixed pinhole and a movable lens in the same arm as a Bragg cell which permits the system to be change from heterodyne to homodyne operation whilst retaining the same interferogram. One mirror in this arm was mounted on a loud speaker, when this was driven it introduced large amplitude vibrations into the OPL. A Polytech vibrometer was used to measure the amplitude of vibration of this mirror.

Fig. 3.
Fig. 3.

(top) The reference signal from a signal generator used to drive the Bragg cell in Fig. 2. (bottom) The signal captured from one of the RFout pixels (left) without driving the mirror and (right) with driving the mirror.

Fig. 4.
Fig. 4.

(top) Traces of measured vibration velocity on the mirror mount using the vibrometer. (middle) Standard heterodyne interferogram images captured using an external signal as the LO, stability mode off. (bottom) Ultrastable heterodyne interferogram images captured using the RFout pixel feedback for the LO, stability mode on. Images taken with (a)(d)(g) no vibration induce, (b)(e)(h) 1 Hz square wave induced, (c)(f)(i) 62 Hz sine wave induce on the mirror mount. All fringe pattern images were taken with 20 averages.

Fig. 5.
Fig. 5.

Modified Michelson interferometer. The Bragg cell splits the beam into two orders with different frequencies that take different paths after going through a polarising beamsplitter. One beam is reflected off the object, and both beams are interfered when they pass through the beam splitter the second time.

Fig. 6.
Fig. 6.

Interferograms captured by the ultrastable Michelson interferometer as the modulation frequency was swept from 11MHz to 16MHz. The fringe pattern remains unchanged by the change in frequency because of the feedback from the RFout signal of one of the pixels to the LO input, meaning the MLC tracks the modulation frequency. At both ends of the frequency range, the noise is significantly increased because the signal is now at the edge of the system bandwidth. This is determined by the filters in the feedback path. The ability to handle this bandwidth is equivalent to immunity from piston phase change for an object moving at up to 3.1 m s−1. The inset shows an AFM image (100μm×100μm) of the grating indicating the grating height (140±20nm) which is in agreement with the measure phase in the interferograms.

Tables (1)

Tables Icon

Table 1. Table summarising the immunity to vibration for the ultrastable interferometer. The right hand columns indicate the amplitude of sinusoidal vibration at various frequencies.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

E R ( x , y , t ) = a r cos ( ω r t + ϕ r ( x , y ) + ψ r ( t ) ) E I ( x , y , t ) = a i cos ( ω i t + ϕ i ( x , y ) + ψ i ( t ) )
I = I dc + I RF = I dc + A cos ( ω d t + ϕ d ( x , y ) + ψ d ( t ) )
i lo = B cos ( ω d t ) q lo = B sin ( ω d t )
i d ( x , y ) = B A cos [ ϕ d ( x , y ) + ψ d ( t ) ] q d ( x , y ) = B A sin [ ϕ d ( x , y ) + ψ d ( t ) ]
ϕ d ( x , y ) + ψ d ( t ) = arctan ( q d i d )
I pixel 1 = I dc 1 + I rfout = I dc 1 + A 1 cos [ ω d t + ϕ d 1 + ψ d 1 ( t ) ]
ϕ d ( x , y ) + ψ d ( t ) ϕ d 1 ψ d 1 ( t ) = arctan ( q d i d )
ϕ d ( x , y ) = arctan ( q d i d )

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