Abstract

We have critically reconsidered the standard representation of optical signals in the form of quasimonochromatic electromagnetic waves. We propose an alternative formulation in terms of ortho-normalized series of elementary wave packages (EWP). We show that this allows the synthesis and analysis of superwideband optical signals, for example, femtosecond optical pulses (fs pulse). In addition, it includes the general properties of electromagnetic waves. We also propose a technique for describing the polarization and angular momentum of the fs pulse. As examples of the application of the proposed EWP representation, we investigated nonstationary diffraction of fs pulse of any duration by a slit and interaction of the fs pulse with a two-level system. We have established the resonant character of these processes over the duration of the fs pulse.

© 2012 Optical Society of America

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References

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  1. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, The Theory of Waves, 2nd ed. (Nauka, 1990).
  2. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, 1992).
  3. P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley, 2007).
  4. L. U. Astanin and A. A. Kostyliov, The Base of the Ultra-wideband Radiolocation Measurements (Radio i Sviaz, 1989).
  5. IEEE International Conference on Ultra-Wideband,Nanjing, China, September 20–23, 2010.
  6. B. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).
  7. D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optics media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
    [CrossRef]
  8. M. Cherqui, D. Jonas, E. Riedle, R. Schoenlein, and A. Taylor in Ultrafast Phenomena XVII: Proceedings of the 17th International Conference (Oxford University, 2010).
  9. Z. Chang and P. Corcum, “Attosecond photon sources: the first decade and beyond,” J. Opt. Soc. Am. B 27, B9–B17 (2010).
    [CrossRef]
  10. E. G. Bessonov, “Conditionally—strange electromagnetic waves,” Quantum Electron. 19, 35–39 (1992).
  11. G. Arfken, Mathematical Methods for Physicists (Academic, 1967).
  12. V. S. Ovechko, “Optics of the coherent ultrawideband pulses,” in Proceedings of the Kyiv National Taras Schevchenko University (Pedagogyka, 2004), 190–198.
  13. V. S. Ovechko, “Orthogonal normalized base for radio signals,” Bulletin of University of Kyiv Series: Physics & Mathematics N1, 196–202 (2009).
  14. A. Papoulis, Signal Analysis (McGraw-Hill, 1977).
  15. W. A. Shurkliff, Polarized Light (Harvard University, 1962).
  16. V. S. Ovechko, “Diffraction of the femtosecond optical impulses,” Bulletin of University of Kyiv Series: Physics & Mathematics N4, 175–178 (2009).
  17. E. Fermi, Notes on Quantum Mechanics (University of Chicago, 1960).
  18. V. S. Ovechko, “Wavelet-analysis of the time structure of the electromagnetic signal through their expansion in terms of elementary waves packets,” Bulletin of University of Kyiv Series: Physics & Mathematics N1, 184–186 (2010).
  19. V. S. Ovechko, “Expansion of the Fourier-spectrum of the radio-signal into a series of elementary waves packets,” Bulletin of University of Kyiv Series: Physics & Mathematics N2, 216–218(2010).

2010

Z. Chang and P. Corcum, “Attosecond photon sources: the first decade and beyond,” J. Opt. Soc. Am. B 27, B9–B17 (2010).
[CrossRef]

V. S. Ovechko, “Wavelet-analysis of the time structure of the electromagnetic signal through their expansion in terms of elementary waves packets,” Bulletin of University of Kyiv Series: Physics & Mathematics N1, 184–186 (2010).

V. S. Ovechko, “Expansion of the Fourier-spectrum of the radio-signal into a series of elementary waves packets,” Bulletin of University of Kyiv Series: Physics & Mathematics N2, 216–218(2010).

2009

V. S. Ovechko, “Orthogonal normalized base for radio signals,” Bulletin of University of Kyiv Series: Physics & Mathematics N1, 196–202 (2009).

V. S. Ovechko, “Diffraction of the femtosecond optical impulses,” Bulletin of University of Kyiv Series: Physics & Mathematics N4, 175–178 (2009).

1992

E. G. Bessonov, “Conditionally—strange electromagnetic waves,” Quantum Electron. 19, 35–39 (1992).

1984

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optics media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, 1992).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, 1967).

Astanin, L. U.

L. U. Astanin and A. A. Kostyliov, The Base of the Ultra-wideband Radiolocation Measurements (Radio i Sviaz, 1989).

Auston, D. H.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optics media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Bessonov, E. G.

E. G. Bessonov, “Conditionally—strange electromagnetic waves,” Quantum Electron. 19, 35–39 (1992).

Born, B. M.

B. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

Chang, Z.

Cherqui, M.

M. Cherqui, D. Jonas, E. Riedle, R. Schoenlein, and A. Taylor in Ultrafast Phenomena XVII: Proceedings of the 17th International Conference (Oxford University, 2010).

Cheung, K. P.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optics media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, 1992).

Corcum, P.

Fermi, E.

E. Fermi, Notes on Quantum Mechanics (University of Chicago, 1960).

Jonas, D.

M. Cherqui, D. Jonas, E. Riedle, R. Schoenlein, and A. Taylor in Ultrafast Phenomena XVII: Proceedings of the 17th International Conference (Oxford University, 2010).

Kleinman, D. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optics media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Kostyliov, A. A.

L. U. Astanin and A. A. Kostyliov, The Base of the Ultra-wideband Radiolocation Measurements (Radio i Sviaz, 1989).

Ovechko, V. S.

V. S. Ovechko, “Wavelet-analysis of the time structure of the electromagnetic signal through their expansion in terms of elementary waves packets,” Bulletin of University of Kyiv Series: Physics & Mathematics N1, 184–186 (2010).

V. S. Ovechko, “Expansion of the Fourier-spectrum of the radio-signal into a series of elementary waves packets,” Bulletin of University of Kyiv Series: Physics & Mathematics N2, 216–218(2010).

V. S. Ovechko, “Diffraction of the femtosecond optical impulses,” Bulletin of University of Kyiv Series: Physics & Mathematics N4, 175–178 (2009).

V. S. Ovechko, “Orthogonal normalized base for radio signals,” Bulletin of University of Kyiv Series: Physics & Mathematics N1, 196–202 (2009).

V. S. Ovechko, “Optics of the coherent ultrawideband pulses,” in Proceedings of the Kyiv National Taras Schevchenko University (Pedagogyka, 2004), 190–198.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, 1977).

Riedle, E.

M. Cherqui, D. Jonas, E. Riedle, R. Schoenlein, and A. Taylor in Ultrafast Phenomena XVII: Proceedings of the 17th International Conference (Oxford University, 2010).

Rudenko, O. V.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, The Theory of Waves, 2nd ed. (Nauka, 1990).

Schoenlein, R.

M. Cherqui, D. Jonas, E. Riedle, R. Schoenlein, and A. Taylor in Ultrafast Phenomena XVII: Proceedings of the 17th International Conference (Oxford University, 2010).

Shurkliff, W. A.

W. A. Shurkliff, Polarized Light (Harvard University, 1962).

Sukhorukov, A. P.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, The Theory of Waves, 2nd ed. (Nauka, 1990).

Taylor, A.

M. Cherqui, D. Jonas, E. Riedle, R. Schoenlein, and A. Taylor in Ultrafast Phenomena XVII: Proceedings of the 17th International Conference (Oxford University, 2010).

Ufimtsev, P. Ya.

P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley, 2007).

Valdmanis, J. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optics media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Vinogradova, M. B.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, The Theory of Waves, 2nd ed. (Nauka, 1990).

Vysloukh, V. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, 1992).

Wolf, E.

B. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

Bulletin of University of Kyiv Series: Physics & Mathematics

V. S. Ovechko, “Orthogonal normalized base for radio signals,” Bulletin of University of Kyiv Series: Physics & Mathematics N1, 196–202 (2009).

V. S. Ovechko, “Diffraction of the femtosecond optical impulses,” Bulletin of University of Kyiv Series: Physics & Mathematics N4, 175–178 (2009).

V. S. Ovechko, “Wavelet-analysis of the time structure of the electromagnetic signal through their expansion in terms of elementary waves packets,” Bulletin of University of Kyiv Series: Physics & Mathematics N1, 184–186 (2010).

V. S. Ovechko, “Expansion of the Fourier-spectrum of the radio-signal into a series of elementary waves packets,” Bulletin of University of Kyiv Series: Physics & Mathematics N2, 216–218(2010).

J. Opt. Soc. Am. B

Phys. Rev. Lett.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optics media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Quantum Electron.

E. G. Bessonov, “Conditionally—strange electromagnetic waves,” Quantum Electron. 19, 35–39 (1992).

Other

G. Arfken, Mathematical Methods for Physicists (Academic, 1967).

V. S. Ovechko, “Optics of the coherent ultrawideband pulses,” in Proceedings of the Kyiv National Taras Schevchenko University (Pedagogyka, 2004), 190–198.

M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov, The Theory of Waves, 2nd ed. (Nauka, 1990).

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, 1992).

P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley, 2007).

L. U. Astanin and A. A. Kostyliov, The Base of the Ultra-wideband Radiolocation Measurements (Radio i Sviaz, 1989).

IEEE International Conference on Ultra-Wideband,Nanjing, China, September 20–23, 2010.

B. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1986).

M. Cherqui, D. Jonas, E. Riedle, R. Schoenlein, and A. Taylor in Ultrafast Phenomena XVII: Proceedings of the 17th International Conference (Oxford University, 2010).

A. Papoulis, Signal Analysis (McGraw-Hill, 1977).

W. A. Shurkliff, Polarized Light (Harvard University, 1962).

E. Fermi, Notes on Quantum Mechanics (University of Chicago, 1960).

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Figures (8)

Fig. 1.
Fig. 1.

Elementary waves functions: (a) odd and (b) even functions: n=2, 1, 0 (from left to right).

Fig. 2.
Fig. 2.

(a) Sample of the femtosecond optical pulse (fs pulse) with amplitude modulation (E(t)=E0[t(1t/τ)/τ]sin[6πt/τ]; (b) EWP spectrum A2n+1.

Fig. 3.
Fig. 3.

Hodograph of the electric component of fs pulse U2(x): F=0.

Fig. 4.
Fig. 4.

Angular momentum |N¯(τ)| of the even EWP pulse of the “0”-th order relationship to ω0τ:Γ·τ/2=0.01.

Fig. 5.
Fig. 5.

EWP spectrum of the pulse U2(x), which is diffracted on the slit aperture.

Fig. 6.
Fig. 6.

Space distribution of the diffracted pulse Ed(x):a/τc=1/2, (tr0/c)/τ=1/2, r0Const. The spatial distribution (Figure 6) is a function of parameter 2a/cτ. The temporal pulse shape Ed(t) depends upon the same parameter as the space one spatial distribution and is also dependent on the direction of the diffraction wave x/r0. The characteristic case is shown in Fig. 7(b), where the maximum time delay due to the slit width is equal to the length of the fs pulse: 2ax/τcr0=1. The leading edge of the diffracting pulse is similar to the original pulse. The trailing edge is distorted by the interference effect.

Fig. 7.
Fig. 7.

Time distribution of the diffracted pulse Ed(t): (a) x=0, (b) x/r0=1; a/τ·c=1/2, A=(tr0/c)/τ.

Fig. 8.
Fig. 8.

Probability of the absorption Pif plotted as a function of parameter ωfiτ.

Tables (1)

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Table 1. Elementary Waves Packets

Equations (37)

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I=E00f(t)dt=0,
I=E00τf(t)dt=0,
f(t<0)=f(t>τ)=0,f(0tτ)=f(t).
f(t)t|t=0=f(t)t|t=τ=0.
f(x)=0(A2n+1U2n+1(x)+B2n+2U2n+2(x)),
U2n+1(x)=[2(n+1)π(n+2)]1/2·[sin[x(2n+2)]2(n+1)sin[x]cos[x(2n+3)]],
U2n+2(x)=[2(n+1)π(n+2)]1/2·[12(n+1)(sin[x(2n+3)]sin[x]1)cos[x(2n+4)]]0xπ(0tτ).
A2n+1=0πf(x)U2n+1(x)dx,
B2n+2=0πf(x)U2n+2(x)dx.
0π|U2n+1(x)|2dx=0π|U2n+2(x)|2dx=1.
f(x)=xπ(1xπ)sin[mx]0xπ,m=2,3
0π|f(x)|2dx=n=0(A2n+12+B2n+22).
Ex(x)=E0f0(x)cos(ωmxF),
Ey(x)=E0f0(x)sin(ωmx+F),
I1x=0πf01(x)cos(ωmxF)dx=8ωm(sin[F]+sin[Fωmπ])910ωm2+ωm4=0,
I1y=0πf01(x)sin(ωmx+F)dx=8ωm(cos[F]+cos[F+ωmπ])910ωm2+ωm4.
I2x=0πf02(x)cos(ωmxF)dx=12ωm(sin[F]+sin[Fωmπ])6420ωm2+ωm4=0,
I2y=0πf02(x)sin(ωmx+F)dx=12ωm(cos[F]+cos[F+ωmπ])6420ωm2+ωm4.
W(x)=14π(Ex2+Ey2)==E024πf02(x)(1+sin[2nx]sin[2F]).
0πW(x)dx=E024π.
E(φ)=exEx(φ)+eyEy(φ),
r¨+Γr˙+ω02r=emE,
dNdt=e[r×E]=eze(xEyyEx).
(x(t)y(t))=2eω0motexp[Γ2(tt)]×sin[(4ω02Γ2)1/2(tt)/2]E0f0(t)×(cos[mπt/τ]sin[mπt/τ])dt.
E(x,y,z,t)=1+z/r4πcr0S0E0t|t|r¯r¯0|cdx0dy0.
n=0(AnU+2n+1BnU2n+2)|t=1+z/r4πcr0S0n=0An0U2n+10t+Bn0U2n+20t|t|r¯r¯0|cdx0dy0,
n=0AnU2n+1(t,r¯)=1+z/r4πcr0S0n=0Bn0U2n+20t|t|r¯r¯0|cdx0dy0,
n=0BnU2n+2(t,r¯)=1+z/r4πcr0S0n=0An0U2n+10t|t|r¯r¯0|cdx0dy0.
U2n+2(t,x0,y0)=E0f02(t)=E0π1/2(cos[2πt/τ]cos[4πt/τ]).
Ed(x,y)=2E0π1/2ab1+z/r04πcr0×(2sin[2πA]sinc[2πBx]sinc[2πBy]+4sin[4πA]sinc[4πBx]sinc[4πBy]),
Ed(x,y)=2E0π1/2a1+z/r04πcr0×(2sin[2πA]sinc[2πBx]+4sin[4πA]sinc[4πBx]).
A=[Δt/τ,1+Δt/τ].
τ(x)=τ(1+2ax/r0cτ).
Ed(x,y,t)=1+z/r04πcr0E0aabb(sign[Q(t,x)]sign[Q(t,x)2(x0x+yy0)cτ0τ1])×(sin[2πQ(t,x)]+2sin[4πQ(t,x)])dx0dy0,
Pif(t)=|Cf(t)|2,
Pif(t)=|dE0|fi242×{[(C+(ω1)C(ω1)C+(ω2)+C(ω2))]2+[(S+(ω1)+S(ω1)S+(ω2)S(ω2))]2},
Pif(φ)=576π4Pif0φ4sin2[φ2](4π2φ2)2(16π2φ2)2,

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