Abstract

We present, for the first time to our knowledge, an optomechanical integration method for finite functions. This technique allows for the integration of any finite function by combining optical and mechanical principals. The integrated function can then be determined using curve fitting methods. Furthermore, the original function can be reproduced through numerical or analytical integration.

© 2007 Optical Society of America

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References

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  1. H. Eves, An Introduction to the History of Mathematics, 4th ed. (Holt, Rinehart and Winston, 1976).
  2. I. Newton, Principia, 1st ed. (University of California Press, 1966).
  3. A. Augustus, Fundamentals of Numerical Analysis, 1st ed. (Ronald Press, 1963).
  4. J. Geremia and H. Rabitz, "Quadratically convergent algorithm for fractional occupation numbers in density functional theory," J. Chem. Phys. 118, 5369-5381 (2002).
    [CrossRef]
  5. T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65, 779-782 (1997).
    [CrossRef]
  6. E. Miller, C. Shadbolt, and L. Holm, "Use of an optical planimeter for measuring leaf area," Plant Physiol. 31, 484-486 (1956).
    [CrossRef] [PubMed]
  7. T. Kawaguchi, Y. Akasaka, and M. Maeda, "Size measurements of droplets and bubbles by advanced interferometric laser imaging technique," Meas. Sci. Technol. 13, 308-316 (2002).
    [CrossRef]
  8. L. Zeng, H. Matsumoto, and K. Kawachi, "Two-directional scanning method for reducing the shadow effects in laser triangulation," Meas. Sci. Technol. 8, 262-266 (1997).
    [CrossRef]
  9. D. Burt, J. Magnes, G. Schwarz, and J. Hartke, "Discovering integration through a physical phenomenon," Primus (to be published).
  10. H. Anton, Calculus with Analytic Geometry, 2nd ed. (Wiley, 1984).
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    [CrossRef] [PubMed]
  12. W. Plass, R. Maestle, K. Wittig, A. Voss, and A. Giesen, "High-resolution knife-edge laser beam profiling," Opt. Commun. 134, 21-24 (1997).
    [CrossRef]
  13. J. M. Khosrofian and B. A. Garetz, "Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data," Appl. Opt. 22, 3406-3410 (1983).
    [CrossRef] [PubMed]
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  15. R. Dorn, S. Quabis, and G. Leuchs, "The focus of light--linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).
  16. B. J. Pernick, "Two-dimensional light-distribution measurement with a 90° cornered knife edge," Appl. Opt. 32, 3610-3613 (1993).
    [CrossRef] [PubMed]
  17. D. Karabacak, T. Kouh, C. C. Huangand, and K. L. Ekinci, "Optical knife-edge technique for nanomechanical displacement detection," Appl. Phys. Lett. 88, 193122 (2006).
    [CrossRef]

2006 (1)

D. Karabacak, T. Kouh, C. C. Huangand, and K. L. Ekinci, "Optical knife-edge technique for nanomechanical displacement detection," Appl. Phys. Lett. 88, 193122 (2006).
[CrossRef]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, "The focus of light--linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).

2002 (2)

J. Geremia and H. Rabitz, "Quadratically convergent algorithm for fractional occupation numbers in density functional theory," J. Chem. Phys. 118, 5369-5381 (2002).
[CrossRef]

T. Kawaguchi, Y. Akasaka, and M. Maeda, "Size measurements of droplets and bubbles by advanced interferometric laser imaging technique," Meas. Sci. Technol. 13, 308-316 (2002).
[CrossRef]

1997 (3)

L. Zeng, H. Matsumoto, and K. Kawachi, "Two-directional scanning method for reducing the shadow effects in laser triangulation," Meas. Sci. Technol. 8, 262-266 (1997).
[CrossRef]

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65, 779-782 (1997).
[CrossRef]

W. Plass, R. Maestle, K. Wittig, A. Voss, and A. Giesen, "High-resolution knife-edge laser beam profiling," Opt. Commun. 134, 21-24 (1997).
[CrossRef]

1993 (1)

1992 (1)

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).

1984 (2)

1983 (1)

1976 (1)

H. Eves, An Introduction to the History of Mathematics, 4th ed. (Holt, Rinehart and Winston, 1976).

1966 (1)

I. Newton, Principia, 1st ed. (University of California Press, 1966).

1963 (1)

A. Augustus, Fundamentals of Numerical Analysis, 1st ed. (Ronald Press, 1963).

1956 (1)

E. Miller, C. Shadbolt, and L. Holm, "Use of an optical planimeter for measuring leaf area," Plant Physiol. 31, 484-486 (1956).
[CrossRef] [PubMed]

Akasaka, Y.

T. Kawaguchi, Y. Akasaka, and M. Maeda, "Size measurements of droplets and bubbles by advanced interferometric laser imaging technique," Meas. Sci. Technol. 13, 308-316 (2002).
[CrossRef]

Anton, H.

H. Anton, Calculus with Analytic Geometry, 2nd ed. (Wiley, 1984).

Augustus, A.

A. Augustus, Fundamentals of Numerical Analysis, 1st ed. (Ronald Press, 1963).

Baumert, T.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65, 779-782 (1997).
[CrossRef]

Brixner, T.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65, 779-782 (1997).
[CrossRef]

Burt, D.

D. Burt, J. Magnes, G. Schwarz, and J. Hartke, "Discovering integration through a physical phenomenon," Primus (to be published).

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, "The focus of light--linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).

Ekinci, K. L.

D. Karabacak, T. Kouh, C. C. Huangand, and K. L. Ekinci, "Optical knife-edge technique for nanomechanical displacement detection," Appl. Phys. Lett. 88, 193122 (2006).
[CrossRef]

Eves, H.

H. Eves, An Introduction to the History of Mathematics, 4th ed. (Holt, Rinehart and Winston, 1976).

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).

Garetz, B. A.

Gerber, G.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65, 779-782 (1997).
[CrossRef]

Geremia, J.

J. Geremia and H. Rabitz, "Quadratically convergent algorithm for fractional occupation numbers in density functional theory," J. Chem. Phys. 118, 5369-5381 (2002).
[CrossRef]

Giesen, A.

W. Plass, R. Maestle, K. Wittig, A. Voss, and A. Giesen, "High-resolution knife-edge laser beam profiling," Opt. Commun. 134, 21-24 (1997).
[CrossRef]

Hartke, J.

D. Burt, J. Magnes, G. Schwarz, and J. Hartke, "Discovering integration through a physical phenomenon," Primus (to be published).

Holm, L.

E. Miller, C. Shadbolt, and L. Holm, "Use of an optical planimeter for measuring leaf area," Plant Physiol. 31, 484-486 (1956).
[CrossRef] [PubMed]

Huangand, C. C.

D. Karabacak, T. Kouh, C. C. Huangand, and K. L. Ekinci, "Optical knife-edge technique for nanomechanical displacement detection," Appl. Phys. Lett. 88, 193122 (2006).
[CrossRef]

Karabacak, D.

D. Karabacak, T. Kouh, C. C. Huangand, and K. L. Ekinci, "Optical knife-edge technique for nanomechanical displacement detection," Appl. Phys. Lett. 88, 193122 (2006).
[CrossRef]

Kawachi, K.

L. Zeng, H. Matsumoto, and K. Kawachi, "Two-directional scanning method for reducing the shadow effects in laser triangulation," Meas. Sci. Technol. 8, 262-266 (1997).
[CrossRef]

Kawaguchi, T.

T. Kawaguchi, Y. Akasaka, and M. Maeda, "Size measurements of droplets and bubbles by advanced interferometric laser imaging technique," Meas. Sci. Technol. 13, 308-316 (2002).
[CrossRef]

Khosrofian, J. M.

Kouh, T.

D. Karabacak, T. Kouh, C. C. Huangand, and K. L. Ekinci, "Optical knife-edge technique for nanomechanical displacement detection," Appl. Phys. Lett. 88, 193122 (2006).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, "The focus of light--linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).

Maeda, M.

T. Kawaguchi, Y. Akasaka, and M. Maeda, "Size measurements of droplets and bubbles by advanced interferometric laser imaging technique," Meas. Sci. Technol. 13, 308-316 (2002).
[CrossRef]

Maestle, R.

W. Plass, R. Maestle, K. Wittig, A. Voss, and A. Giesen, "High-resolution knife-edge laser beam profiling," Opt. Commun. 134, 21-24 (1997).
[CrossRef]

Magnes, J.

D. Burt, J. Magnes, G. Schwarz, and J. Hartke, "Discovering integration through a physical phenomenon," Primus (to be published).

Matsumoto, H.

L. Zeng, H. Matsumoto, and K. Kawachi, "Two-directional scanning method for reducing the shadow effects in laser triangulation," Meas. Sci. Technol. 8, 262-266 (1997).
[CrossRef]

McCally, R. L.

Miller, E.

E. Miller, C. Shadbolt, and L. Holm, "Use of an optical planimeter for measuring leaf area," Plant Physiol. 31, 484-486 (1956).
[CrossRef] [PubMed]

Newton, I.

I. Newton, Principia, 1st ed. (University of California Press, 1966).

Pernick, B. J.

Plass, W.

W. Plass, R. Maestle, K. Wittig, A. Voss, and A. Giesen, "High-resolution knife-edge laser beam profiling," Opt. Commun. 134, 21-24 (1997).
[CrossRef]

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, "The focus of light--linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).

Rabitz, H.

J. Geremia and H. Rabitz, "Quadratically convergent algorithm for fractional occupation numbers in density functional theory," J. Chem. Phys. 118, 5369-5381 (2002).
[CrossRef]

Schwarz, G.

D. Burt, J. Magnes, G. Schwarz, and J. Hartke, "Discovering integration through a physical phenomenon," Primus (to be published).

Seyfried, V.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65, 779-782 (1997).
[CrossRef]

Shadbolt, C.

E. Miller, C. Shadbolt, and L. Holm, "Use of an optical planimeter for measuring leaf area," Plant Physiol. 31, 484-486 (1956).
[CrossRef] [PubMed]

Strehle, M.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65, 779-782 (1997).
[CrossRef]

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).

Voss, A.

W. Plass, R. Maestle, K. Wittig, A. Voss, and A. Giesen, "High-resolution knife-edge laser beam profiling," Opt. Commun. 134, 21-24 (1997).
[CrossRef]

Wittig, K.

W. Plass, R. Maestle, K. Wittig, A. Voss, and A. Giesen, "High-resolution knife-edge laser beam profiling," Opt. Commun. 134, 21-24 (1997).
[CrossRef]

Zeng, L.

L. Zeng, H. Matsumoto, and K. Kawachi, "Two-directional scanning method for reducing the shadow effects in laser triangulation," Meas. Sci. Technol. 8, 262-266 (1997).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65, 779-782 (1997).
[CrossRef]

Appl. Phys. Lett. (1)

D. Karabacak, T. Kouh, C. C. Huangand, and K. L. Ekinci, "Optical knife-edge technique for nanomechanical displacement detection," Appl. Phys. Lett. 88, 193122 (2006).
[CrossRef]

J. Chem. Phys. (1)

J. Geremia and H. Rabitz, "Quadratically convergent algorithm for fractional occupation numbers in density functional theory," J. Chem. Phys. 118, 5369-5381 (2002).
[CrossRef]

J. Mod. Opt. (1)

R. Dorn, S. Quabis, and G. Leuchs, "The focus of light--linear polarization breaks the rotational symmetry of the focal spot," J. Mod. Opt. 50, 1917-1926 (2003).

Meas. Sci. Technol. (2)

T. Kawaguchi, Y. Akasaka, and M. Maeda, "Size measurements of droplets and bubbles by advanced interferometric laser imaging technique," Meas. Sci. Technol. 13, 308-316 (2002).
[CrossRef]

L. Zeng, H. Matsumoto, and K. Kawachi, "Two-directional scanning method for reducing the shadow effects in laser triangulation," Meas. Sci. Technol. 8, 262-266 (1997).
[CrossRef]

Opt. Commun. (1)

W. Plass, R. Maestle, K. Wittig, A. Voss, and A. Giesen, "High-resolution knife-edge laser beam profiling," Opt. Commun. 134, 21-24 (1997).
[CrossRef]

Plant Physiol. (1)

E. Miller, C. Shadbolt, and L. Holm, "Use of an optical planimeter for measuring leaf area," Plant Physiol. 31, 484-486 (1956).
[CrossRef] [PubMed]

Other (6)

H. Eves, An Introduction to the History of Mathematics, 4th ed. (Holt, Rinehart and Winston, 1976).

I. Newton, Principia, 1st ed. (University of California Press, 1966).

A. Augustus, Fundamentals of Numerical Analysis, 1st ed. (Ronald Press, 1963).

D. Burt, J. Magnes, G. Schwarz, and J. Hartke, "Discovering integration through a physical phenomenon," Primus (to be published).

H. Anton, Calculus with Analytic Geometry, 2nd ed. (Wiley, 1984).

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, 1992).

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Figures (6)

Fig. 1
Fig. 1

(a) Inverted parabola and (b) integrated parabola on the interval [ a , b ] .

Fig. 2
Fig. 2

Top view of principle experimental setup. The light travels from the source past the razor blade through a collection lens into a detector.

Fig. 3
Fig. 3

(a), (b) Masks are used to demonstrate the integration of known functions. In this case a straight line and an inverted parabola. (c) A mask in the shape of a house is used to demonstrate the integration of an unknown function.

Fig. 4
Fig. 4

(a)–(c) Optomechanically integrated representations of a straight line, a parabola, and a house, respectively. The data points in (a) and (b) allow for a second and third order polynomial fit obtaining the integral of the function in the mask. Error bars are too small to be shown on this scale. (d)–(f) The differentiated data corresponding to (a)–(c) and Fig. 3, verifying the validity of our method. The solid lines in (d) and (e) represent the fit to the numerical differentiated points, whereas the dashed lines are the analytically differentiated function to the fit in (a) and (b).

Fig. 5
Fig. 5

Function as defined in Eq. (3) is used to demonstrate the agreement with the original mask.

Fig. 6
Fig. 6

Razor blade is rotated 10° with respect to the mask. A lateral shift in the peak is apparent. The relative height of apexes change as well.

Equations (3)

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lim Δ x i 0 i = 1 n f ( x i ) Δ x i = a b f ( x ) d x ,
d P T d x = 1 2 ( y i + 1 y i x i + 1 x i + y i y i 1 x i x i 1 )
y ( x ) = a sin 2 ( b x + d ) x + c ,

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