Abstract

Gaussian decomposition is used as a theoretical infrastructure with which Z-scan experiments are analyzed. This procedure is extended here to the interesting, from a practical point of view, case in which the laser beam used is not perfectly Gaussian. We follow a perturbative approach to obtain the far-field pattern of the beam after the beam passes through a nonlinear sample. The procedure is based on the decomposition of the electric field at the exit plane of the sample to a linear combination of Hermite–Gaussian functions. To a first-order approximation, each mode of the incident beam is decomposed to a linear combination of different-order modes that do not exceed the order of the original mode. Finally, the effects of the simultaneous presence of first and higher-order refractive nonlinearities or first-order refractive nonlinearity and nonlinear absorption are studied.

© 2003 Optical Society of America

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References

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  1. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  2. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single beam Z-scan:  measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
    [CrossRef]
  3. W. Zhao and P. Palffy-Muhoray, “Z-scan using top-hat beams,” Appl. Phys. Lett. 63, 1613-1615 (1993).
    [CrossRef]
  4. R. E. Bridges, G. L. Fischer, and R. W. Boyd, “Z-scan measurement technique for non-Gaussian beams and arbitrary sample thickness,” Opt. Lett. 20, 1821-1823 (1995).
    [CrossRef]
  5. S. M. Mian, B. Taheri, and J. P. Wicksted, “Effects of beam ellipticity in Z-scan measurements,” J. Opt. Soc. Am. B 13, 856-863 (1996).
    [CrossRef]
  6. B. K. Rhee, J. S. Byun, and E. W. Van Stryland, “Z scan using circularly symmetric beams,” J. Opt. Soc. Am. B 13, 2720-2723 (1996).
    [CrossRef]
  7. P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional Z scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043-7050 (1999).
    [CrossRef]
  8. Y.-L. Huang and C.-K. Sun, “Z-scan measurement with an astigmatic Gaussian beam,” J. Opt. Soc. Am. B 17, 43-47 (2000).
    [CrossRef]
  9. D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, “Effect of low-power nonlinear refraction on laser-beam propagation in InSb,” Opt. Lett. 4, 331–333 (1979).
    [CrossRef] [PubMed]
  10. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  11. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).
  12. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).

2000 (1)

1999 (1)

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional Z scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043-7050 (1999).
[CrossRef]

1997 (1)

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single beam Z-scan:  measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

1996 (2)

1995 (1)

1993 (1)

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

1990 (1)

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

1979 (1)

Boyd, R. W.

Bridges, R. E.

Byun, J. S.

Chapple, P. B.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single beam Z-scan:  measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

Chen, P.

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional Z scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043-7050 (1999).
[CrossRef]

Fischer, G. L.

Hagan, D. J.

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

Hermann, J. A.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single beam Z-scan:  measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

Huang, Y.-L.

McDuff, R. G.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single beam Z-scan:  measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

McKay, T. J.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single beam Z-scan:  measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

Mian, S. M.

Miller, D. A. B.

Oulianov, D. A.

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional Z scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043-7050 (1999).
[CrossRef]

Palffy-Muhoray, P.

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

Rentzepis, P. M.

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional Z scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043-7050 (1999).
[CrossRef]

Rhee, B. K.

Said, A. A.

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

Smith, S. D.

Staromlynska, J.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single beam Z-scan:  measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

Sun, C.-K.

Taheri, B.

Tomov, I. V.

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional Z scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043-7050 (1999).
[CrossRef]

Van Stryland, E. W.

B. K. Rhee, J. S. Byun, and E. W. Van Stryland, “Z scan using circularly symmetric beams,” J. Opt. Soc. Am. B 13, 2720-2723 (1996).
[CrossRef]

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

Weaire, D.

Wei, T. H.

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

Wherrett, B. S.

Wicksted, J. P.

Zhao, W.

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

Appl. Phys. Lett. (1)

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

IEEE J. Quantum Electron. (1)

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

J. Appl. Phys. (1)

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional Z scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043-7050 (1999).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single beam Z-scan:  measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (2)

Other (3)

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

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

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).

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Figures (2)

Fig. 1
Fig. 1

Theoretical close-aperture Z-scan plots (S=0.4) for a circular near-Gaussian incident beam. The location of the beam waist is at z0=0. The curves correspond to the modes involved in the near-Gaussian beam, as shown. For all cases ε=0.1 and Δφ0(1) (z0, t)=1.

Fig. 2
Fig. 2

Theoretical close-aperture Z-scan plots (S=0.4) for elliptical near-Gaussian incident beams of increasing waist separation: (a) z0y-z0x=2zR and (b) z0y-z0x=3zR. The beam widths at the waists for the two principal dimensions have been set equal to each other (w0x=w0y=w0), leading to equal values for the Rayleigh lengths (zRx=zRy=zR). The beam waists are symmetrically located near z0=0. The curves correspond to the modes involved in the near-Gaussian beam, as shown. For all cases ε=0.1 and Δφ0(1) (z0, t)=1.

Equations (73)

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E(x, y, z, t)=c0E0(x, y, z, t)+εEh(x, y, z, t),
Eh(x, y, z, t)=l,m clmElm(x, y, z, t).
|c0|2+ε2l,m|clm|2=1.
Elm(x, y, z, t)
=A(z, t)(2l2ml!m!)-1/2Hl2xwx(z)Hm2ywy(z)
×exp-x2wx2(z)-y2wy2(z)×exp-ikx22Rx(z)-iky22Ry(z)×exp[-ikz+i(l+m+1)θ(z)].
wx,y(z)=w0x,y1+z-z0x,yzRx,y21/2,
Rx,y(z)=z-z0x,y+zRx,y2z-z0x,y,
θ(z)=12tan-1z-z0xzRx+12tan-1z-z0yzRy,
zRx,y=kw0x,y22.
cε0n02 |A(z, t)|2=2P(t)πwx(z)wy(z),
Ee(x, y, zs, t)
=E(x, y, zs, t)exp(-a0L/2)exp[iΔφ(x, y, zs, t)],
 
Δφ(x, y, zs, t)=-kγ(1)I(x, y, zs, t)Leff(1).
n(x, y, zs, t)=nlin+γ(1)I(x, y, zs, t),
Leff(1)=1-exp(-a0L)a0,
I(x, y, zs, t)=cε0n02 |E(x, y, zs, t)|2=cε0n02 |c0E0(x, y, zs, t)+εEh(x, y, zs, t)|2.
Ee(x, y, zs, t)=E(x, y, zs, t)exp(-a0L/2)×q=0iqq! [Δφ(x, y, zs, t)]q.
Δφ(x, y, zs, t)=-kγ(1)cε0n02 |c0E0(x, y, zs, t)+εEh(x, y, zs, t)|2Leff(1)=-kγ(1)cε0n02 |A(zs, t)|2|c0×E0(x, y, zs, t)+εEh(x, y, zs, t)|2Leff(1),
E0(x, y, zs, t)=E0(x, y, zs, t)A(zs, t),
Eh(x, y, zs, t)=Eh(x, y, zs, t)A(zs, t).
Δφ0(1)(zs, t)=-kγ(1)cε0n02 |A(zs, t)|2Leff(1),
Δφ(x, y, zs, t)=Δφ0(1)(zs, t)|c0E0(x, y, zs, t)+εEh(x, y, zs, t)|2.
Ee(x, y, zs, t)=A(zs, t)[c0E0(x, y, zs, t)+εEh(x, y, zs, t)]exp(-a0L/2)×q=0[iΔφ0(1)(zs, t)]qq!× |c0E0(x, y, zs, t)+εEh(x, y, zs, t)|2q.
Ee(x, y, zs, t)=A(zs, t)exp(-a0L/2)q=0[iΔφ0(1)(zs, t)]qq!× {|c0|2qc0|E0(x, y, zs, t)|2qE0(x, y, zs, t)+ε(q+1)|c0|2q|E0(x, y, zs, t)|2q×Eh(x, y, zs, t)+εq|c0|2(q-1)c02|×E0(x, y, zs, t)|2(q-1)×[E0(x, y, zs, t)]2[Eh(x, y, zs, t)]*}.
Ee(x, y, zs, t)=A(zs, t)exp(-a0L/2)q=0[iΔφ0(1)(zs, t)]qq!×exp-(2q+1)x2wx2(zs)-(2q+1)y2wy2(zs)×exp-ikx22Rx(zs)-iky22Ry(zs)×exp[-ikzs+iθ(zs)]×|c0|2qc0+ε(q+1)|c0|2q×l,m clm(2l2ml!m!)-1/2×Hl2xwx(zs)Hm2ywy(zs)×exp[i(l+m)θ(zs)]+εq|c0|2(q-1)c02×l,m clm(2l2ml!m!)-1/2Hl2xwx(zs)Hm2ywy(zs)×exp[-i(l+m)θ(zs)].
wx=wx(zs)2q+1,wy=wy(zs)2q+1.
Hl2xwx(zs)=r=0l pr2xwx(zs)r=r=0l pr(2q+1)-r/22xwxr,
wx,y(q)(zs)=w0x,y(q)1+zs-z0x,y(q)zRx,y(q)21/2=wx,y,
Rx,y(q)(z)=z-z0x,y(q)+[zRx,y(q)]2z-z0x,y(q)=Rx,y(zs).
z0x,y(q)=zs-[Bx,y(q)]2Rx,y(zs)[Bx,y(q)]2+Rx,y2(zs),
zRx,y(q)=Bx,y(q)Rx,y2(zs)[Bx,y(q)]2+Rx,y2(zs),
Bx,y(q)=k2wx,y2(zs)2q+1.
E(x, y, zs, t)=c0E0(x, y, zs, t)+ε[c11E11(x, y, zs, t)+c22E22(x, y, zs, t)+c33E33(x, y, zs, t)].
Ee(x, y, zs, t)=A(zs, t)exp(-a0L/2)q=0[iΔφ0(1)(zs, t)]qq!×exp-(2q+1)x2wx2(zs)-(2q+1)y2wy2(zs)×exp-ikx22Rx(zs)-iky22Ry(zs)exp[-ikzs+iθ(zs)]×|c0|2qc0+ε(q+1)|c0|2ql=13cll2ll!× Hl2xwx(zs)Hl2ywy(zs)exp[2ilθ(zs)]×εq|c0|2(q-1)c02l=13cll2ll! Hl2xwx(zs)Hl2ywy(zs)×exp[-2ilθ(zs)],
wx,y(q)(zs)=wx,y(zs)2q+1,
Rx,y(q)(z)=Rx,y(zs).
H12xwx(zs)=1(2q+1)1/2 H12xwx(q)(zs),
H22xwx(zs)=12q+1H22xwx(q)(zs)-4q,
H32xwx(zs)=1(2q+1)3/2H32xwx(q)(zs)-12qH12xwx(q)(zs).
Ee(x, y, zs, t)
=A(zs, t)exp(-a0L/2)exp[-ikzs+iθ(zs)]×q=0[iΔφ0(1)(zs, t)]qq!|c0|2qc0E0(q)(x, y, zs)+ε(q+1)|c0|2qc112(2q+1)exp[2iθ(zs)]×E11(q)(x, y, zs)+c228(2q+1)2exp[4iθ(zs)]×[E22(q)(x, y, zs)-4qE20(q)(x, y, zs)-4q×E02(q)(x, y, zs)+16q2E0(q)(x, y, zs)]+c3348(2q+1)3exp[6iθ(zs)][E33(q)(x, y, zs)-12qE31(q)(x, y, zs)-12qE13(q)(x, y, zs)+144q2E11(q)(x, y, zs)]+εq|c0|2(q-1)c02c112(2q+1)exp[-2iθ(zs)]×E11(q)(x, y, zs)+c228(2q+1)2exp[-4iθ(zs)]×[E22(q)(x, y, zs)-4qE20(q)(x, y, zs)-4qE02(q)(x, y, zs)+16q2E0(q)(x, y, zs)]+c3348(2q+1)3exp[-6iθ(zs)][E33(q)(x, y, zs)-12qE31(q)(x, y, zs)-12qE13(q)(x, y, zs)+144q2E11(q)(x, y, zs)],
Elm(q)(x, y, zs)=Hl2xwx(q)(zs)Hm2ywy(q)(zs)exp-xwx(q)(zs)2
-ywy(q)(zs)2exp-ikx22Rx(q)(zs)-iky22Ry(q)(zs).
Elm(q)(x, y, D)
=A(q)(D)Hl2xwx(q)(D)Hm2ywy(q)(D)×exp-xwx(q)(D)2-ywy(q)(D)2×exp-ikx22Rx(q)(D)-iky22Ry(q)(D)×exp{-ik(D-zs)+i(l+m+1)[×θ(q)(D)+δ(q)]}.
A(q)(D)=wx(q)(zs)wy(q)(zs)wx(q)(D)wy(q)(D)1/2.
w0x,y(q)=2zRx,y(q)k1/2,
wx,y(q)(D)=w0x,y(q)1+D-z0x,y(q)zRx,y(q)21/2,
Rx,y(q)(D)=[D-z0x,y(q)]1+zRx,y(q)D-z0x,y(q)2,
θ(q)(D)=12tan-1D-z0x(q)zRx(q)+12tan-1D-z0y(q)zRy(q),
δ(q)=-12tan-1zs-z0x(q)zRx(q)-12tan-1zs-z0y(q)zRy(q).
n(x, y, zs, t)=nlin+γ(1)I(x, y, zs, t)+γ(2)I2(x, y, zs, t),
Δφ(x, y, zs, t)=-kγ(1)I(x, y, zs, t)Leff(1)-kγ(2)I2(x, y, zs, t)Leff(2),
Leff(2)=1-exp(-2a0L)2a0.
Δφ(x, y, zs, t)=Δφ0(1)(zs, t)|c0E0(x, y, zs, t)+εEh(x, y, zs,t)|2+Δφ0(2)×(zs, t)|×c0E0(x, y, zs, t)+εEh(x, y, zs, t)|4,
Δφ0(2)(zs, t)=-kγ(2)cε0n02 |A(zs, t)|22Leff(2).
Ee(x, y, zs, t)
=A(zs, t)[c0E0(x, y, zs, t)+εEh(x, y, zs, t)]×exp(-a0L/2)q=01q! [iΔφ0(1)(zs, t)×|c0E0(x, y, zs, t)+εEh(x, y, zs, t)|2+iΔφ0(2)(zs, t)|c0E0(x, y, zs, t)+εEh(x, y, zs, t)|4]q.
Ee(x, y, zs, t)
=A(zs, t)[c0E0(x, y, zs, t)+εEh(x, y, zs, t)]×exp(-a0L/2)q=0[iΔφ0(1)(zs, t)]qq!×|c0E0(x, y, zs, t)+εEh(x, y, zs, t)|2q+[iΔφ0(1)(zs, t)]q-1[iΔφ0(2)(zs, t)](q-1)!×|c0E0(x, y, zs, t)+εEh(x, y, zs, t)|2(q+1).
lc(x, y, zs, t)=I(x, y, zs, t)exp(-a0L)1+Q(x, y, zs, t),
Δφ(x, y, zs, t)=kγ(1)β(1)ln[1+Q(x, y, zs, t)],
Q(x, y, zs, t)=β(1)I(x, y, zs, t)Leff(1).
Ee(x, y, zs, t)=A(zs, t)[c0E0(x, y, zs, t)+εEh(x, y, zs, t)]exp(-a0L/2)×[1+Q(x, y, ,zs, t)]ikγ(1)/β(1)-1/2.
[1+Q(x, y, zs, t)]ikγ(1)/β(1)-1/2
=q=0 F(q)[Q(x, y, zs, t)]qq!,
F(q)=r=1q[ikγ(1)/β(1)-1/2-r+1],q1.
[1+Q(x, y, zs, t)]ikγ(1)/β(1)-1/2
=q=0 F(q)[Q0(zs, t)]qq! |c0E0(x, y, zs, t)+εEh(x, y, zs, t)|2q,
Q0(zs, t)=β(1)cε0n02 |A(zs, t)|2Leff(1).
Ee(x, y, zs, t)=A(zs, t)[c0E0(x, y, zs, t)+εEh(x, y, zs, t)]exp(-a0L/2)×q=0 F(q)[Q0(zs, t)]qq! |c0×E0(x, y, zs, t)+εEh(x, y, zs, t)|2q.

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