Abstract

By variational approach, we analyze the characteristics of beam propagation through a cubic optical nonlinear medium using a laser beam that has a transverse elliptic Gaussian profile. The analytic solution to the normalized transmittance at the center of the far field as a function of medium position and the beam characteristics is obtained and compared with the numerical simulation, which is realized by a combination of algorithms. We also analyze the peak-valley transmittance difference as a function of medium length, ellipticity, and astigmatism. The relationship between peak-valley normalized transmittance difference of the z-scan trace and aperture size or the slit width are obtained. Meanwhile, the comparison of z-scan characteristics with an elliptic Gaussian beam with those using a circular symmetric Gaussian beam is made.

© 2003 Optical Society of America

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References

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  1. M. Sheik-Bahae, A. A. Said, E. W. Vanstryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [CrossRef]
  2. S. M. Mian, B. Taheri, J. P. Wicksted, “Effects of beam ellipticity on Z-scan measurements,” J. Opt. Soc. Am. B 13, 856–862 (1996).
    [CrossRef]
  3. Y.-L. Huang, C.-K. Sun, “Z-scan measurement with an astigmatic Gaussian beam,” J. Opt. Soc. Am. B 17, 43–47 (2000).
    [CrossRef]
  4. J. H. Marburger, “Self-focusing theory.” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  5. F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
    [CrossRef]
  6. S. M. Mian, B. Taheri, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
    [CrossRef]
  7. J.-G. Tian, W.-P. Zang, C.-Z. Zhang, “Analysis of beam propagation through thick nonlinear medium by variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).
  8. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1284 (1990).
    [CrossRef] [PubMed]
  9. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.
  10. L. Lapidus, G. F. Pinder, Numerical Solution of Paritial Differential Equationals in Science and Engineering (Wiley, New York, 1982), Chap. 6.
  11. S. Hughes, J. M. Burzer, G. Spruce, B. S. Wherrett, “Fast Fourier transform techniques for efficient simulation of Z-scan measurements,” J. Opt. Soc. Am. B 12, 1888–1893 (1995).
    [CrossRef]

2000

1996

1995

S. M. Mian, B. Taheri, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

S. Hughes, J. M. Burzer, G. Spruce, B. S. Wherrett, “Fast Fourier transform techniques for efficient simulation of Z-scan measurements,” J. Opt. Soc. Am. B 12, 1888–1893 (1995).
[CrossRef]

1994

J.-G. Tian, W.-P. Zang, C.-Z. Zhang, “Analysis of beam propagation through thick nonlinear medium by variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).

1990

Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1284 (1990).
[CrossRef] [PubMed]

F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, E. W. Vanstryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

1975

J. H. Marburger, “Self-focusing theory.” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Burzer, J. M.

Cornolti, F.

F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Huang, Y.-L.

Hughes, S.

Lapidus, L.

L. Lapidus, G. F. Pinder, Numerical Solution of Paritial Differential Equationals in Science and Engineering (Wiley, New York, 1982), Chap. 6.

Lucchesi, M.

F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Marburger, J. H.

J. H. Marburger, “Self-focusing theory.” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Mian, S. M.

S. M. Mian, B. Taheri, J. P. Wicksted, “Effects of beam ellipticity on Z-scan measurements,” J. Opt. Soc. Am. B 13, 856–862 (1996).
[CrossRef]

S. M. Mian, B. Taheri, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

Pinder, G. F.

L. Lapidus, G. F. Pinder, Numerical Solution of Paritial Differential Equationals in Science and Engineering (Wiley, New York, 1982), Chap. 6.

Said, A. A.

M. Sheik-Bahae, A. A. Said, E. W. Vanstryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, E. W. Vanstryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Silberberg, Y.

Spruce, G.

Sun, C.-K.

Taheri, B.

S. M. Mian, B. Taheri, J. P. Wicksted, “Effects of beam ellipticity on Z-scan measurements,” J. Opt. Soc. Am. B 13, 856–862 (1996).
[CrossRef]

S. M. Mian, B. Taheri, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

Tian, J.-G.

J.-G. Tian, W.-P. Zang, C.-Z. Zhang, “Analysis of beam propagation through thick nonlinear medium by variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).

Vanstryland, E. W.

M. Sheik-Bahae, A. A. Said, E. W. Vanstryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Wherrett, B. S.

Wicksted, J. P.

S. M. Mian, B. Taheri, J. P. Wicksted, “Effects of beam ellipticity on Z-scan measurements,” J. Opt. Soc. Am. B 13, 856–862 (1996).
[CrossRef]

S. M. Mian, B. Taheri, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.

Zambon, B.

F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Zang, W.-P.

J.-G. Tian, W.-P. Zang, C.-Z. Zhang, “Analysis of beam propagation through thick nonlinear medium by variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).

Zhang, C.-Z.

J.-G. Tian, W.-P. Zang, C.-Z. Zhang, “Analysis of beam propagation through thick nonlinear medium by variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).

Acta Phys. Sin.

J.-G. Tian, W.-P. Zang, C.-Z. Zhang, “Analysis of beam propagation through thick nonlinear medium by variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).

IEEE J. Quantum Electron.

M. Sheik-Bahae, A. A. Said, E. W. Vanstryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

J. Appl. Phys.

S. M. Mian, B. Taheri, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Opt. Lett.

Prog. Quantum Electron.

J. H. Marburger, “Self-focusing theory.” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Other

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.

L. Lapidus, G. F. Pinder, Numerical Solution of Paritial Differential Equationals in Science and Engineering (Wiley, New York, 1982), Chap. 6.

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Figures (11)

Fig. 1
Fig. 1

z-scan experimental setup. D2/D1 is measured as a function of the medium position z. D2, D1, detectors.

Fig. 2
Fig. 2

z-scan traces with different length of medium and c = 0 mm.

Fig. 3
Fig. 3

The peak-valley transmittance difference ΔT p-v versus the medium length L for different ellipticity e and c = 0 mm, where e = 1, 2, 3, and 5, respectively.

Fig. 4
Fig. 4

The z-scan traces for different beam-waist separation c and e = 1, where c = 0, 2, 4, and 6 mm, respectively.

Fig. 5
Fig. 5

The peak-valley transmittance difference ΔT p-v versus the medium length L for different beam-waist separation c and e = 1, where c = 0, 2, 4, and 6 mm, respectively.

Fig. 6
Fig. 6

The z-scan traces for ellipticity e = 2 and c = 0. Solid curve variational solution; dotted curve, numerical solution.

Fig. 7
Fig. 7

The z-scan traces for beam-waist separation c = 2 mm and e = 1. Solid curves, variational solution; dotted curve, numerical solution.

Fig. 8
Fig. 8

The normalized peak-valley transmittance difference ΔTp-v¯ versus the aperture’s radius r a for different ellipticity e and c = 0 mm, where e = 1, 2, 3, and 5, respectively.

Fig. 9
Fig. 9

The normalized peak-valley transmittance difference ΔTp-v¯ versus the aperture’s radius r a for different beam-waist separation c and e = 1, where c = 0, 1, 2, and 3 mm.

Fig. 10
Fig. 10

The normalized peak-valley transmittance difference ΔTp-v¯ versus the slit’s width d for different ellipticity e and c = 0 mm, where e = 1, 2, 3, and 5.

Fig. 11
Fig. 11

The normalized peak-valley transmittance difference ΔTp-v¯ versus the slit’s width d for different beam-waist separation c and e = 1, where c = 0, 1, 2, and 3 mm.

Equations (26)

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2Ex2+ 2Ey2-2ik Ez+ 2n2k2n0 |E|2E=0,
L=Ex2+Ey2-ikE E*z-E* Ez- n2k2n0 |E|4.
Er, z=Azexp- x22ax2z- y22ay2z+ibxzx2+ibyzy2,
δ --- Ldxdydz=0.
ddzaxay|A|2=0,  axay|A|2=ax0ay0A02,
bx,y= -k2ax,ydax,ydz,
d2ax,ydz2- 1k2ax,y31-P=0,
dax,ydz=±Aax,y2+Bx,y1/2=±Πax,y,
A= 1k2P-1, Bx,y=y0x,0y2-P-1Aa0x,0y2,
alx,ly=a0x,0y2+2LA+Bx,ya0x,0y21/2+Bx,yL21/2,
blx,ly=-kBx,yL+A+Bx,ya0x,0y21/22a0x,0y2+2LA+Bx,ya0x,0y21/2+Bx,yL2.
alx,ly=a0x,0y2-2LA+Bx,ya0x,0y21/2+Bx,yL21/2,
blx,ly=-kBx,yL-A+Bx,ya0x,0y21/22a0x,0y2-2LA+Bx,ya0x,0y21/2+Bx,yL2.
alx,ly=a0x,0y2+2LA+Bx,ya0x,0y21/2+Bx,yL21/2,
blx,ly=-kBx,yL+A+Bx,ya0x,0y22a0x,0y2+2LA+Bx,ya0x,0y21/2+Bx,yL2.
1qlx,ly= 1Rlx,ly- 2ikwlx,ly2.
wax,ay2=1+ n0dRlx,ly2+ 4n02d2k2wlx,ly4wlx,ly2n0,
1Rax,ay= n0Rlx,ly1+ n0dRlx,ly+ 4dn02k2wlx,ly41+n0dRlx,ly2+ 4dn02k2wlx,ly4.
Tx= waxp=0wayp=0waxpwayp,
Iap, r, z, t=C 1waxwayexp-2r2fθ, z,
fθ, z= cos2 θwax2z+ sin2 θway2z.
T= waxp=0wayp=002π1fθ, z, p1-exp-2ra2fθ, z, pdθwaxpwayp02π1fθ, z, p=01-exp-2ra2fθ, z, p=0dθ.
T= waxp=0-d/2d/2exp-x2/wax2pdxwaxp-d/2d/2exp-x2/wax2p=0dx.
Ex, y, z, t=E0tw0xwxzw0ywyz1/2 exp-ikz-ηzexp-x21wx2z+ ik2Rxz-y21wy2z+ ik2Ryz,
a0x,0y=12 wx,y2z1/2,
dax,ydzax,y=a0x,0y= a0x,0yR0x,0y.

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