Abstract

We analyze the influence of different objects on the sensitivity of using a 4f imaging system for nonlinear optical measurements. By defining the diffraction efficiency as the physical quantity to measure, it is possible to perform different measuring methods (Z-scan, eclipsing Z-scan, I-scan, degenerate four-wave mixing) by matching the field stop in the image plane with the object at the entry. One, two, three, and four waves mixing are considered in order to compare their related sensitivities. We provide simple quadratic relations for each object that allow the characterization of the cubic optical nonlinearity. A systematic comparison is done showing that one circular aperture object gives the highest sensitivity. The experimental measurements are performed in order to validate our simulation.

© 2009 Optical Society of America

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  1. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  2. G. Boudebs, K. Fedus, C. Cassagne, and H. Leblond, “Degenerate multi-wave mixing using Z-scan technique,” Appl. Phys. Lett. 93, 021118 (2008).
    [CrossRef]
  3. R. W. Boyd, Nonlinear Optics (Academic, 1992), chap. 6.
  4. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hill, 1996).
  5. G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
    [CrossRef]
  6. Y. R. Shen, Nonlinear Optics (Wiley, 1984), chap. 3.
  7. J. A. Hermann, “Beam propagation and optical power limiting with nonlinear media,” J. Opt. Soc. Am. B 1, 729-736 (1984).
    [CrossRef]
  8. S. Cherukulappurath, G. Boudebs, and A. Monteil, “4-f coherent system imager and application to nonlinear optical measurements,” J. Opt. Soc. Am. B 21, 273-279 (2004).
    [CrossRef]
  9. T. Xia, D. I. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317-319 (1994).
    [CrossRef] [PubMed]
  10. W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613-1615 (1993).
    [CrossRef]
  11. B. Taheri, H. Liu, B. Jassemnejad, D. Appling, R. C. Powell, and J. J. Song, “Intensity scan and two photon absorption and nonlinear refraction of C60 in toluene,” Appl. Phys. Lett. 68, 1317-1319 (1996).
    [CrossRef]
  12. Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
    [CrossRef]
  13. G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
    [CrossRef]
  14. G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using 4f coherent imaging system with phase objects,” Phys. Rev. A 69, 053813 (2004).
    [CrossRef]

2008

G. Boudebs, K. Fedus, C. Cassagne, and H. Leblond, “Degenerate multi-wave mixing using Z-scan technique,” Appl. Phys. Lett. 93, 021118 (2008).
[CrossRef]

2004

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using 4f coherent imaging system with phase objects,” Phys. Rev. A 69, 053813 (2004).
[CrossRef]

S. Cherukulappurath, G. Boudebs, and A. Monteil, “4-f coherent system imager and application to nonlinear optical measurements,” J. Opt. Soc. Am. B 21, 273-279 (2004).
[CrossRef]

2003

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

1998

G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
[CrossRef]

G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
[CrossRef]

1996

B. Taheri, H. Liu, B. Jassemnejad, D. Appling, R. C. Powell, and J. J. Song, “Intensity scan and two photon absorption and nonlinear refraction of C60 in toluene,” Appl. Phys. Lett. 68, 1317-1319 (1996).
[CrossRef]

1994

1993

W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

1990

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

1984

Appling, D.

B. Taheri, H. Liu, B. Jassemnejad, D. Appling, R. C. Powell, and J. J. Song, “Intensity scan and two photon absorption and nonlinear refraction of C60 in toluene,” Appl. Phys. Lett. 68, 1317-1319 (1996).
[CrossRef]

Boudebs, G.

G. Boudebs, K. Fedus, C. Cassagne, and H. Leblond, “Degenerate multi-wave mixing using Z-scan technique,” Appl. Phys. Lett. 93, 021118 (2008).
[CrossRef]

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using 4f coherent imaging system with phase objects,” Phys. Rev. A 69, 053813 (2004).
[CrossRef]

S. Cherukulappurath, G. Boudebs, and A. Monteil, “4-f coherent system imager and application to nonlinear optical measurements,” J. Opt. Soc. Am. B 21, 273-279 (2004).
[CrossRef]

G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
[CrossRef]

G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992), chap. 6.

Cassagne, C.

G. Boudebs, K. Fedus, C. Cassagne, and H. Leblond, “Degenerate multi-wave mixing using Z-scan technique,” Appl. Phys. Lett. 93, 021118 (2008).
[CrossRef]

Cherukulappurath, S.

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using 4f coherent imaging system with phase objects,” Phys. Rev. A 69, 053813 (2004).
[CrossRef]

S. Cherukulappurath, G. Boudebs, and A. Monteil, “4-f coherent system imager and application to nonlinear optical measurements,” J. Opt. Soc. Am. B 21, 273-279 (2004).
[CrossRef]

Chis, M.

G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
[CrossRef]

G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
[CrossRef]

Creekmore, S.

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

Fedus, K.

G. Boudebs, K. Fedus, C. Cassagne, and H. Leblond, “Degenerate multi-wave mixing using Z-scan technique,” Appl. Phys. Lett. 93, 021118 (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hill, 1996).

Hagan, D.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Hagan, D. I.

Hermann, J. A.

Jassemnejad, B.

B. Taheri, H. Liu, B. Jassemnejad, D. Appling, R. C. Powell, and J. J. Song, “Intensity scan and two photon absorption and nonlinear refraction of C60 in toluene,” Appl. Phys. Lett. 68, 1317-1319 (1996).
[CrossRef]

Jung, S.

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

Kim, S. Y.

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

Leblond, H.

G. Boudebs, K. Fedus, C. Cassagne, and H. Leblond, “Degenerate multi-wave mixing using Z-scan technique,” Appl. Phys. Lett. 93, 021118 (2008).
[CrossRef]

Liu, H.

B. Taheri, H. Liu, B. Jassemnejad, D. Appling, R. C. Powell, and J. J. Song, “Intensity scan and two photon absorption and nonlinear refraction of C60 in toluene,” Appl. Phys. Lett. 68, 1317-1319 (1996).
[CrossRef]

Min, N.

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

Monteil, A.

S. Cherukulappurath, G. Boudebs, and A. Monteil, “4-f coherent system imager and application to nonlinear optical measurements,” J. Opt. Soc. Am. B 21, 273-279 (2004).
[CrossRef]

G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
[CrossRef]

G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
[CrossRef]

Mott, A.

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

Palffy-Muhoray, P.

W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

Powell, R. C.

B. Taheri, H. Liu, B. Jassemnejad, D. Appling, R. C. Powell, and J. J. Song, “Intensity scan and two photon absorption and nonlinear refraction of C60 in toluene,” Appl. Phys. Lett. 68, 1317-1319 (1996).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Seo, J. T.

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

Sheik-Bahae, M.

T. Xia, D. I. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317-319 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Shen, Y. R.

Y. R. Shen, Nonlinear Optics (Wiley, 1984), chap. 3.

Song, J. J.

B. Taheri, H. Liu, B. Jassemnejad, D. Appling, R. C. Powell, and J. J. Song, “Intensity scan and two photon absorption and nonlinear refraction of C60 in toluene,” Appl. Phys. Lett. 68, 1317-1319 (1996).
[CrossRef]

Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Taheri, B.

B. Taheri, H. Liu, B. Jassemnejad, D. Appling, R. C. Powell, and J. J. Song, “Intensity scan and two photon absorption and nonlinear refraction of C60 in toluene,” Appl. Phys. Lett. 68, 1317-1319 (1996).
[CrossRef]

Temple, D.

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

Van Stryland, E. W.

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Xia, T.

Yang, Q.

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

Yoo, K. P.

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

Zhao, W.

W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

Appl. Phys. Lett.

G. Boudebs, K. Fedus, C. Cassagne, and H. Leblond, “Degenerate multi-wave mixing using Z-scan technique,” Appl. Phys. Lett. 93, 021118 (2008).
[CrossRef]

W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

B. Taheri, H. Liu, B. Jassemnejad, D. Appling, R. C. Powell, and J. J. Song, “Intensity scan and two photon absorption and nonlinear refraction of C60 in toluene,” Appl. Phys. Lett. 68, 1317-1319 (1996).
[CrossRef]

Q. Yang, J. T. Seo, S. Creekmore, D. Temple, A. Mott, N. Min, K. P. Yoo, S. Y. Kim, and S. Jung, “Distortions in Z-scan spectroscopy,” Appl. Phys. Lett. 82, 19-21 (2003).
[CrossRef]

IEEE J. Quantum Electron.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
[CrossRef]

G. Boudebs, M. Chis, and A. Monteil, “Contrast increasing by third order nonlinear image processing: a numerical study for microscopic rectangular objects,” Opt. Commun. 150, 287-296 (1998).
[CrossRef]

Opt. Lett.

Phys. Rev. A

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using 4f coherent imaging system with phase objects,” Phys. Rev. A 69, 053813 (2004).
[CrossRef]

Other

Y. R. Shen, Nonlinear Optics (Wiley, 1984), chap. 3.

R. W. Boyd, Nonlinear Optics (Academic, 1992), chap. 6.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hill, 1996).

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Figures (5)

Fig. 1
Fig. 1

Schematic of the 4 f coherent system imager. The nonlinear material is placed in the Fourier plane ( z = 0 ) . The labels refer to the object O ( x , y ) , lenses ( L 1 L 2 ) , field stops (sf.), focal length ( f ) , and field distribution in image plane U ( x , y ) .

Fig. 2
Fig. 2

(a) Gaussian beam image profiles for a different nonlinear phase shift. The solid curve (1) is obtained for φ NL 0 = 0 ; (2), (3), and (4) are obtained for φ NL 0 = 1 , 2, and 2.7 rad , respectively. The arrows indicate the points of intersection corresponding to all these profiles. (b) Numerical simulation showing the positive part of the image subtraction corresponding to (4)–(1). The coordinates ( x , y ) are in pixels.

Fig. 3
Fig. 3

The case of a Gaussian beam at the entry and eclipsing disk in the image plane. (a) Diffraction efficiency versus S, the fraction of light blocked by the obscuration disk ( φ NL 0 = 0.1 ) . (b) Diffraction efficiency versus φ NL 0 for S = 0.89 .

Fig. 4
Fig. 4

(a) Numerical simulation showing the images of the diffracted beams with objects composed of one, two, three, or four circular apertures. The adapted field stops appear in black in the image plane. (b) Diffraction efficiency versus the normalized opaque disk radius; filled circles, stars, empty circles, and the solid curve are for one, two, three, and four apertures, respectively.

Fig. 5
Fig. 5

Diffraction efficiency ( η ) versus the nonlinear phase shift ( φ NL 0 ) ; (1), (2), and (3) for one, two, and three apertures, respectively. (a) simulation with opaque disks of radii a = R . (b) Simulation with opaque disks of radii a = 2 R . (c) Experimental data obtained with the object composed of three circular apertures and adapted field stops with radii a = 2 R . The dotted curve is the numerical simulation (3) in Fig. 5b.

Equations (4)

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

S ( u , v ) = F ̃ [ O ( x , y ) ] = + + O ( x , y ) exp [ j 2 π ( u x + v y ) ] d x d y ,
I im ( x , y ) = U ( x , y ) 2 = F ̃ 1 [ S ( u , v ) × T ( u , v ) ] 2 ,
T ( u , v ) = exp [ j 2 π n 2 L I ( u , v ) λ ] ,
η = ( E DNL E DL ) E IN ,

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