Abstract

Nonlinear optical absorption within thick optical media is analyzed within the paraxial approximation, starting with the nonlinear wave equation for the optical field. Both irradiance-dependent and fluence-dependent absorbers are considered. An approximate solution for the transmitted optical power (and pulse energy) is shown to be useful within the strongly nonlinear regime.

© 1997 Optical Society of America

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References

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  1. J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
    [CrossRef]
  2. P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin- and thick-sample limits,” J. Opt. Soc. Am. B 11, 975–982 (1994).
    [CrossRef]
  3. J. A. Hermann, P. B. Chapple, J. Staromlynska, and P. J. Wilson, “Design criteria for optical power limiters,” Proc. SPIE 2229, 167–178 (1994).
    [CrossRef]
  4. J. A. Hermann and P. J. Wilson, “Factors affecting optical-limiting and scanning with thin nonlinear samples,” Int. J. Nonlinear Opt. Phys. 2, 613–629 (1993).
    [CrossRef]
  5. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” submitted to J. Nonlin. Opt. Phys. Mater.
  6. 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]
  7. R. G. McDuff, “Nonlinear self-focusing and beam propagation using Gaussian-Laguerre modal decomposition,” Ph.D. dissertation (University of Queensland, Queensland, Australia, 1994).
  8. J. A. Hermann, “Beam propagation and optical power limiting with nonlinear media,” J. Opt. Soc. Am. B 1, 729–736 (1984).
    [CrossRef]
  9. P. B. Chapple and J. A. Hermann, “Power saturation effects in thick, single-element, optical limiters,” submitted to Opt. Commun.
  10. D. J. Hagan, E. W. Van Stryland, M. J. Soileau, and Y. Y. Wu, “Self-protecting semiconductor optical limiters,” Opt. Lett. 13, 315–318 (1988).
    [CrossRef] [PubMed]
  11. E. W. Van Stryland, Y. Y. Wu, D. J. Hagan, M. J. Soileau, and K. Mansour, “Optical limiting with semiconductors,” J. Opt. Soc. Am. B 5, 1980–1996 (1988).
    [CrossRef]
  12. R. G. McDuff, A. E. Smith, N. R. Heckenberg, and J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” Int. J. Nonlinear Opt. Phys. 1, 265–286 (1992).
    [CrossRef]
  13. P. A. Miles, “Bottleneck optical limiters: the optimal use of excited-state absorbers,” Appl. Opt. 33, 6965–6979 (1994).
    [CrossRef] [PubMed]
  14. P. A. Miles, “Material figures of merit for saturated excited state absorptive limiters,” Proc. SPIE 2143, 251–262 (1994).
    [CrossRef]

1994 (4)

P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin- and thick-sample limits,” J. Opt. Soc. Am. B 11, 975–982 (1994).
[CrossRef]

J. A. Hermann, P. B. Chapple, J. Staromlynska, and P. J. Wilson, “Design criteria for optical power limiters,” Proc. SPIE 2229, 167–178 (1994).
[CrossRef]

P. A. Miles, “Bottleneck optical limiters: the optimal use of excited-state absorbers,” Appl. Opt. 33, 6965–6979 (1994).
[CrossRef] [PubMed]

P. A. Miles, “Material figures of merit for saturated excited state absorptive limiters,” Proc. SPIE 2143, 251–262 (1994).
[CrossRef]

1993 (2)

J. A. Hermann and P. J. Wilson, “Factors affecting optical-limiting and scanning with thin nonlinear samples,” Int. J. Nonlinear Opt. Phys. 2, 613–629 (1993).
[CrossRef]

J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
[CrossRef]

1992 (1)

R. G. McDuff, A. E. Smith, N. R. Heckenberg, and J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” Int. J. Nonlinear Opt. Phys. 1, 265–286 (1992).
[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]

1988 (2)

1984 (1)

Chapple, P. B.

J. A. Hermann, P. B. Chapple, J. Staromlynska, and P. J. Wilson, “Design criteria for optical power limiters,” Proc. SPIE 2229, 167–178 (1994).
[CrossRef]

P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin- and thick-sample limits,” J. Opt. Soc. Am. B 11, 975–982 (1994).
[CrossRef]

P. B. Chapple and J. A. Hermann, “Power saturation effects in thick, single-element, optical limiters,” submitted to Opt. Commun.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” submitted to J. Nonlin. Opt. Phys. Mater.

Hagan, D. J.

Heckenberg, N. R.

R. G. McDuff, A. E. Smith, N. R. Heckenberg, and J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” Int. J. Nonlinear Opt. Phys. 1, 265–286 (1992).
[CrossRef]

Hermann, J. A.

J. A. Hermann, P. B. Chapple, J. Staromlynska, and P. J. Wilson, “Design criteria for optical power limiters,” Proc. SPIE 2229, 167–178 (1994).
[CrossRef]

J. A. Hermann and P. J. Wilson, “Factors affecting optical-limiting and scanning with thin nonlinear samples,” Int. J. Nonlinear Opt. Phys. 2, 613–629 (1993).
[CrossRef]

J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
[CrossRef]

R. G. McDuff, A. E. Smith, N. R. Heckenberg, and J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” Int. J. Nonlinear Opt. Phys. 1, 265–286 (1992).
[CrossRef]

J. A. Hermann, “Beam propagation and optical power limiting with nonlinear media,” J. Opt. Soc. Am. B 1, 729–736 (1984).
[CrossRef]

P. B. Chapple and J. A. Hermann, “Power saturation effects in thick, single-element, optical limiters,” submitted to Opt. Commun.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” submitted to J. Nonlin. Opt. Phys. Mater.

Mansour, K.

McDuff, R. G.

P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin- and thick-sample limits,” J. Opt. Soc. Am. B 11, 975–982 (1994).
[CrossRef]

J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
[CrossRef]

R. G. McDuff, A. E. Smith, N. R. Heckenberg, and J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” Int. J. Nonlinear Opt. Phys. 1, 265–286 (1992).
[CrossRef]

R. G. McDuff, “Nonlinear self-focusing and beam propagation using Gaussian-Laguerre modal decomposition,” Ph.D. dissertation (University of Queensland, Queensland, Australia, 1994).

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” submitted to J. Nonlin. Opt. Phys. Mater.

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,” submitted to J. Nonlin. Opt. Phys. Mater.

Miles, P. A.

P. A. Miles, “Material figures of merit for saturated excited state absorptive limiters,” Proc. SPIE 2143, 251–262 (1994).
[CrossRef]

P. A. Miles, “Bottleneck optical limiters: the optimal use of excited-state absorbers,” Appl. Opt. 33, 6965–6979 (1994).
[CrossRef] [PubMed]

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, A. E.

R. G. McDuff, A. E. Smith, N. R. Heckenberg, and J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” Int. J. Nonlinear Opt. Phys. 1, 265–286 (1992).
[CrossRef]

Soileau, M. J.

Staromlynska, J.

P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin- and thick-sample limits,” J. Opt. Soc. Am. B 11, 975–982 (1994).
[CrossRef]

J. A. Hermann, P. B. Chapple, J. Staromlynska, and P. J. Wilson, “Design criteria for optical power limiters,” Proc. SPIE 2229, 167–178 (1994).
[CrossRef]

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” submitted to J. Nonlin. Opt. Phys. Mater.

Van Stryland, E. W.

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]

Wilson, P. J.

J. A. Hermann, P. B. Chapple, J. Staromlynska, and P. J. Wilson, “Design criteria for optical power limiters,” Proc. SPIE 2229, 167–178 (1994).
[CrossRef]

J. A. Hermann and P. J. Wilson, “Factors affecting optical-limiting and scanning with thin nonlinear samples,” Int. J. Nonlinear Opt. Phys. 2, 613–629 (1993).
[CrossRef]

Wu, Y. Y.

Appl. Opt. (1)

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]

Int. J. Nonlinear Opt. Phys. (2)

J. A. Hermann and P. J. Wilson, “Factors affecting optical-limiting and scanning with thin nonlinear samples,” Int. J. Nonlinear Opt. Phys. 2, 613–629 (1993).
[CrossRef]

R. G. McDuff, A. E. Smith, N. R. Heckenberg, and J. A. Hermann, “Generalised description of the effects of a thin nonlinear medium upon the propagation of an optical beam,” Int. J. Nonlinear Opt. Phys. 1, 265–286 (1992).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Lett. (1)

Proc. SPIE (2)

P. A. Miles, “Material figures of merit for saturated excited state absorptive limiters,” Proc. SPIE 2143, 251–262 (1994).
[CrossRef]

J. A. Hermann, P. B. Chapple, J. Staromlynska, and P. J. Wilson, “Design criteria for optical power limiters,” Proc. SPIE 2229, 167–178 (1994).
[CrossRef]

Other (3)

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” submitted to J. Nonlin. Opt. Phys. Mater.

P. B. Chapple and J. A. Hermann, “Power saturation effects in thick, single-element, optical limiters,” submitted to Opt. Commun.

R. G. McDuff, “Nonlinear self-focusing and beam propagation using Gaussian-Laguerre modal decomposition,” Ph.D. dissertation (University of Queensland, Queensland, Australia, 1994).

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Figures (6)

Fig. 1
Fig. 1

Arrangement of the optical beam, the active medium, and the detector (D).

Fig. 2
Fig. 2

Typical apertureless Z-scan profiles T(ζ0) for the nonlinear parameter α=0.25 and for negligible linear absorption α0, with increasing values of the sample thickness (scaled in Rayleigh lengths) l=L/zr: (i) l=1, (ii) l=5, (iii) l=10, (iv) l=20.

Fig. 3
Fig. 3

Behavior of the absorption dip, ΔT, as a function of the sample thickness. The three nonlinear values used are (i) α =0.0252, (ii) α=0.252, (iii) α=2.52.

Fig. 4
Fig. 4

Relationship between the sample thickness ls, corresponding to 90% of the saturation value of ΔT, and the nonlinear parameter α.

Fig. 5
Fig. 5

Demonstration of the saturation effect in the absorption dip ΔT with increasing nonlinearity (for α0=0.0, l=10): (i) α=5, (ii) α=10, (iii) α=20, (iv) α=40.

Fig. 6
Fig. 6

Normalized transmittance for a sample of thickness of 10 Rayleigh lengths, as a function of the distance from the waist ζ0, for different values of the linear absorption: (i) α0=0.0, (ii) α0=0.1.

Tables (2)

Tables Icon

Table 1 Comparison of the Normalized Transmittance Values at the Dip Position, as Predicted by the Low-Power and Saturation Approximations, with the Values Obtained from an Exact Numerical Solution of the Nonlinear Wave Equation for the Absorption Parameter α=0.252a

Tables Icon

Table 2 Comparison of the Normalized Transmittance Values at the Dip Position, as Predicted by the Saturation Approximation, with the Values Obtained from an Exact Numerical Solution of the Nonlinear Wave Equation for the Absorption Parameter α=2.520a

Equations (82)

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2E+k2E=0,
E=E·exp(-ik0rz),
γ(Eζζ-μ2E)+i(Eζ+μE)+σEσσ+Eσ+12BIE=0,
γ=(w0/2zr)2,
μ=k0izr,
BI=w02k0k1,I=|E/E0|2.
Incident:Iin=exp[-2σ/(1+ζ12)]/(1+ζ12),
Internal:Iint=exp{-2σ/[1+(ζ-ζ0)2]}/[1+(ζ-ζ0)2],
Transmitted:Itr=exp{-2σ/[1+(ζ2-Δζ)2]/[1+(ζ2-Δζ)2],
E/E0=exp(-k0iz)exp(-wσ)×w[1+2γw(1-12w2σ2)],
w=[1-i(ζ0-ζ)]-1.
P/P0=exp(-2k0iz)[1+3γ+O(γ2)],
P0=12πw02|E0|2.
E=ELIN[1+12iBg(v, y)+O(B2)],
v=2σ/[1+(ζ-ζ0)2],
y=tan-1 ζ0+tan-1(ζ-ζ0),
gy=exp(-v-α0ζ)+2i[vgvv+(1-v)gv].
g0(v, y)=14i[E1(v)-E1(uv)],
u(y)=[2 exp(2iy)-1]-1.
g(v, y)=0y exp[-α0(ζ-ζ)]g0(v, y)ydy.
I/ILIN=1-[αgr(σ, ζ0, ζ)+βgi(σ, ζ0, ζ)],
ILIN=exp{-2σ/[1+(ζ-ζ0)2]}[1+(ζ-ζ0)2],
Nonlinearfocalshift:ζ0-ζw=12α(1+3ζ02)-βζ0(1+9ζ02).
g0(v, y)=14i1uexp(-vu)udu
E=E/ELIN=1-18Bλbλaexp(xλ)λdλ,
x=2σ/(1+ζ22),
λa,b=1-iν(θa,b)3+iν(θa,b),
ν(θ)=θ-(1+θ2)/(ζ2+θ)
θa=ζ0,θb=ζ0-ζm
ζ2=ζ2-Δζ
E=1-18B ln(λa/λb).
E=1-18B[Ei(λax)-Ei(λbx)].
|E(x)|2=1-12πα.
|E(x)|2=1-14πα-14β[Ei(13x)+E1(x)].
|E(x)|2=1-14πα+14β[Ei(13x)+E1(x)].
ΔT(1)=1-P/(PinS)=12πα+O(α2),
ΔΦ(1)NL=14πβ+O(β2),
U(ζ, ζ0)=0exp(-v)gdv
P(ζ, ζ0)=P/Pin=exp(-α0ζ)[1-αU(ζ, ζ0)]+O(α2),
U(ζ, ζ0)=120ζexp(-α0ζ)dζ1+(ζ-ζ0)2=120y exp-α0(1+ζ02)tan y1+ζ0 tan ydy.
Pζ=-12α exp(-2α0ζ)1+(ζ-ζ0)2-α0P+O(α2).
Pζ0=0,
ΔT=1-Pmin=α tan-1(12ζm).
ΔT=α tan-1(12ζm)1+α tan-1(12ζm).
Pζ=-12αP21+(ζ-ζ0)2.
Pζ=-12αP21+(ζ-ζ0)2-α0P,
P(ζ, ζ0)=P/Pin=exp(-α0ζ)[1+αU(ζ, ζ0)]-1.
U(ζ, ζ0)=Uthin=12ζeff1+ζ02,ζeff=1-exp(-α0ζ)α0.
P(ζ, ζ0)=exp(-α0ζ)ln(1+2αUthin)/2αUthin,
E/ζ+12γenE+12NγgE-iσT2E=0,
n/t-γgω(N-n)|E|2+n/τ=0,
n=α0ω-t exp[(t-t)/τ]×exp[(F-F)γg/ω]|E(t)|2dt,
F=-t|E(t)|2dt.
n(σ, ζ, t)=N{1-exp[-α0F(σ, ζ, t)/Nω]}.
FL(t)=exp(-α0ζ)W(t)|E0|2 exp(-v)1+(ζ-ζ0)2,
W(t)=-t|f(t)|2dt.
gy=exp(-α0ζ)W(t)|E0|2n=0(-η)n exp[-(n+1)v](n+1)!+2i[vgvv+(1-v)gv],
η=exp(-α0ζ)W(t)|E0|2α0/Nω1+(ζ-ζ0)2.
P=P/Pin=exp(-α0ζ)1-γeα0ωW(t)|E0|2×0ζexp(-α0ζ)Φ(ζ, ζ0, t)1+(ζ-ζ0)2dζ,
Φ(ζ, ζ0, t)=exp(-η)-1+ηη2=12-16η+124η2-.
Pζ=-γeα0ωW(t)|E0|2 exp(-2α0ζ)Φ(ζ, ζ0, t)1+(ζ-ζ0)2-α0P.
P=exp(-α0ζ)1+γeα0ωW(t)|E0|2U(ζ, ζ0).
(i)Irradiancedependent:T(S=1)=(1+αU)-1,
(ii)Fluencedependent:T(S=1)=ln[1+R(tp)]/R(tp),
R(tp)=γeα0|E0|2tpU(ζm, ζ0)ω.
f(t)=exp(-12t2/tg2),
(i)Irradiancedependent:T=2π1/20exp(-x2)dx1+αU exp(-x2)=n=0(-αU)n(1+n)1/2,
(ii)Fluencedependent:T=ln[1+π1/2R(tg)]/R(tg)=π1/2n=0[-π1/2R(tg)]nn+1.
Tthin(S=1)=1-q022+q0233-q038+.
Tthick(S=1)=1-q122+q1243-q1316+ ,
gy=exp(-v)F(y)+2i[vgvv+(1-v)gv]
f(s, p)=00 exp[-(sv+py)]g(v, y)dvdy
f(s, p)=-12iG(p)1+sk=01-s1+skBk+1, -12ip,
G(p)=0 exp(-py)F(y)dy
g(v, y)=0y exp[-α0(ζ-ζ)]g0(v, y)ydy.
g(0, y)=0ζexp[-α0(ζ-ζ)](1-iζ)(1+3iζ)dζ.
g(v, y)=1-exp(-α0ζ)α0ζ×g0(v, y)-16iα0 exp(-v)y3+O(y4).
g(v, y)=14i1uexp[-α0(ζ-ζ)]exp(-vu)udu,
hy=exp(-v)H1(y)+2i[vgvv+(1-v)gv]-3vgv2,
fy=exp(-v)H2(y)+2i[vhvv+(1-v)hv]-8vgvhv,
H1(y)=-3 Im g(0, y),
H2(y)=-4 Im h(0, y)+23H1(y)2.

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