Abstract

Using the Lorentz force law, we derived simpler expressions for the total longitudinal (conserved) momentum and the mechanical momentums associated with an optical pulse propagating along a dispersive optical waveguide. These expressions can be applied to an arbitrary non-absorptive optical waveguide having continuous translational symmetry. Our simulation using finite difference time domain (FDTD) method verified that the total momentum formula is valid in a two-dimensional infinite waveguide. We studied the conservation of the total momentum and the transfer of the momentum to the waveguide for the case when an optical pulse travels from a finite waveguide to vacuum. We found that neither the Abraham nor the Minkowski momentum expression for an electromagnetic wave in a waveguide represents the complete total (conserved) momentum. Only the total momentum as we derived for a mode propagating in a dispersive optical waveguides is the ‘true’ conserved momentum. This total momentum can be expressed as PTot = –UDie/vg + neff U/c. It has three contributions: (1) the Abraham momentum; (2) the momentum from the Abraham force, which equals to the difference between the Abraham momentum and the Minkowski momentum; and (3) the momentum from the dipole force which can be expressed as –UDie/vg. The last two contributions constitute the mechanical momentum. Compared with FDTD-Lorentz-force method, the presently derived total momentum formula provides a better method in terms of analyzing the permanent transfer of optical momentum to a waveguide.

© 2011 OSA

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    [CrossRef] [PubMed]
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2011 (1)

H. Yu, W. Fang, F. Gu, M. Qiu, Z. Yang, and L. Tong, “Longitudinal Lorentz force on a subwavelength-diameter optical fiber,” Phys. Rev. A 83(5), 053830 (2011).
[CrossRef]

2010 (5)

I. Brevik and S. A. Ellingsen, “Transverse radiation force in a tailored optical fiber,” Phys. Rev. A 81(1), 011806 (2010).
[CrossRef]

S. M. Barnett, “Resolution of the abraham-minkowski dilemma,” Phys. Rev. Lett. 104(7), 070401 (2010).
[CrossRef] [PubMed]

P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” Adv. Opt. Photonics 2(4), 519–553 (2010).
[CrossRef]

C. Baxter and R. Loudon, “Radiation pressure and the photon momentum in dielectrics,” J. Mod. Opt. 57(10), 830–842 (2010).
[CrossRef]

P. L. Saldanha, “Division of the momentum of electromagnetic waves in linear media into electromagnetic and material parts,” Opt. Express 18(3), 2258–2268 (2010).
[CrossRef] [PubMed]

2009 (7)

J. Yu, R. Feng, and W. She, “Low-power all-optical switch based on the bend effect of a nm fiber taper driven by outgoing light,” Opt. Express 17(6), 4640–4645 (2009).
[CrossRef] [PubMed]

M. Mansuripur and A. R. Zakharian, “Theoretical analysis of the force on the end face of a nanofilament exerted by an outing light pulse,” Phys. Rev. A 80(2), 023823 (2009).
[CrossRef]

I. Brevik, “Comment on “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light”,” Phys. Rev. Lett. 103(21), 219301, author reply 219302 (2009).
[CrossRef] [PubMed]

W. She, J. Yu, and R. Feng, “She et al. Reply,” Phys. Rev. Lett. 103(21), 219302 (2009).
[CrossRef]

M. Mansuripur, “Comment on “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light”,” Phys. Rev. Lett. 103(1), 019301 (2009).
[CrossRef] [PubMed]

M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 026608 (2009).
[CrossRef] [PubMed]

E. A. Hinds and S. M. Barnett, “Momentum exchange between light and a single atom: Abraham or Minkowski?” Phys. Rev. Lett. 102(5), 050403 (2009).
[CrossRef] [PubMed]

2008 (1)

W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. 101(24), 243601 (2008).
[CrossRef] [PubMed]

2007 (1)

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[CrossRef]

2006 (1)

S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. At. Mol. Opt. Phys. 39(15), S671–S684 (2006).
[CrossRef]

2005 (3)

2004 (6)

2003 (2)

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[CrossRef] [PubMed]

2001 (1)

1991 (1)

D. F. Nelson, “Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy,” Phys. Rev. A 44(6), 3985–3996 (1991).
[CrossRef] [PubMed]

1979 (1)

I. Brevik, “Experiment in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979).
[CrossRef]

1909 (1)

V. M. Abraham, “Zur Elektrodynamik bewegter Körper,” Rend. Circ. Matem. Palermo 28(1), 1–28 (1909).
[CrossRef]

1908 (1)

H. Minkowski, ““Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Ges. Wiss. Gottn Math.-Phys. Kl. 1908, 53–111 (1908); reprinted in Math Ann 68, 472–525 (1910).

Abraham, V. M.

V. M. Abraham, “Zur Elektrodynamik bewegter Körper,” Rend. Circ. Matem. Palermo 28(1), 1–28 (1909).
[CrossRef]

Almeida, V. R.

Ashcom, J. B.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Barnett, S. M.

S. M. Barnett, “Resolution of the abraham-minkowski dilemma,” Phys. Rev. Lett. 104(7), 070401 (2010).
[CrossRef] [PubMed]

E. A. Hinds and S. M. Barnett, “Momentum exchange between light and a single atom: Abraham or Minkowski?” Phys. Rev. Lett. 102(5), 050403 (2009).
[CrossRef] [PubMed]

S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. At. Mol. Opt. Phys. 39(15), S671–S684 (2006).
[CrossRef]

R. Loudon, S. M. Barnett, and C. Baxter, “Radation pressure and momentum transfer in dielectrics: The photon drag effect,” Phys. Rev. A 71(6), 063802 (2005).
[CrossRef]

Barrios, C. A.

Baxter, C.

C. Baxter and R. Loudon, “Radiation pressure and the photon momentum in dielectrics,” J. Mod. Opt. 57(10), 830–842 (2010).
[CrossRef]

R. Loudon, S. M. Barnett, and C. Baxter, “Radation pressure and momentum transfer in dielectrics: The photon drag effect,” Phys. Rev. A 71(6), 063802 (2005).
[CrossRef]

Boyd, R. W.

P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” Adv. Opt. Photonics 2(4), 519–553 (2010).
[CrossRef]

Brevik, I.

I. Brevik and S. A. Ellingsen, “Transverse radiation force in a tailored optical fiber,” Phys. Rev. A 81(1), 011806 (2010).
[CrossRef]

I. Brevik, “Comment on “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light”,” Phys. Rev. Lett. 103(21), 219301, author reply 219302 (2009).
[CrossRef] [PubMed]

I. Brevik, “Experiment in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979).
[CrossRef]

Eggleton, B. J.

Ellingsen, S. A.

I. Brevik and S. A. Ellingsen, “Transverse radiation force in a tailored optical fiber,” Phys. Rev. A 81(1), 011806 (2010).
[CrossRef]

Fang, W.

H. Yu, W. Fang, F. Gu, M. Qiu, Z. Yang, and L. Tong, “Longitudinal Lorentz force on a subwavelength-diameter optical fiber,” Phys. Rev. A 83(5), 053830 (2011).
[CrossRef]

Feigel, A.

A. Feigel, “Quantum vacuum contribution to the momentum of dielectric media,” Phys. Rev. Lett. 92(2), 020404 (2004).
[CrossRef] [PubMed]

Feng, R.

J. Yu, R. Feng, and W. She, “Low-power all-optical switch based on the bend effect of a nm fiber taper driven by outgoing light,” Opt. Express 17(6), 4640–4645 (2009).
[CrossRef] [PubMed]

W. She, J. Yu, and R. Feng, “She et al. Reply,” Phys. Rev. Lett. 103(21), 219302 (2009).
[CrossRef]

W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. 101(24), 243601 (2008).
[CrossRef] [PubMed]

Gattass, R. R.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Grzegorczyk, T.

Gu, F.

H. Yu, W. Fang, F. Gu, M. Qiu, Z. Yang, and L. Tong, “Longitudinal Lorentz force on a subwavelength-diameter optical fiber,” Phys. Rev. A 83(5), 053830 (2011).
[CrossRef]

Hale, A.

He, S.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Heckenberg, N. R.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[CrossRef]

Hinds, E. A.

E. A. Hinds and S. M. Barnett, “Momentum exchange between light and a single atom: Abraham or Minkowski?” Phys. Rev. Lett. 102(5), 050403 (2009).
[CrossRef] [PubMed]

Kemp, B. A.

Kerbage, C.

Kong, J. A.

Lipson, M.

Lou, J.

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12(6), 1025–1035 (2004).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Loudon, R.

C. Baxter and R. Loudon, “Radiation pressure and the photon momentum in dielectrics,” J. Mod. Opt. 57(10), 830–842 (2010).
[CrossRef]

S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. At. Mol. Opt. Phys. 39(15), S671–S684 (2006).
[CrossRef]

R. Loudon, S. M. Barnett, and C. Baxter, “Radation pressure and momentum transfer in dielectrics: The photon drag effect,” Phys. Rev. A 71(6), 063802 (2005).
[CrossRef]

R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. 52(11-12), 1134–1140 (2004).
[CrossRef]

Mansuripur, M.

M. Mansuripur and A. R. Zakharian, “Theoretical analysis of the force on the end face of a nanofilament exerted by an outing light pulse,” Phys. Rev. A 80(2), 023823 (2009).
[CrossRef]

M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 026608 (2009).
[CrossRef] [PubMed]

M. Mansuripur, “Comment on “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light”,” Phys. Rev. Lett. 103(1), 019301 (2009).
[CrossRef] [PubMed]

A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Radiation pressure and the distribution of electromagnetic force in dielectric media,” Opt. Express 13(7), 2321–2336 (2005).
[CrossRef] [PubMed]

M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12(22), 5375–5401 (2004).
[CrossRef] [PubMed]

Maxwell, I.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Mazur, E.

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12(6), 1025–1035 (2004).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Milonni, P. W.

P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” Adv. Opt. Photonics 2(4), 519–553 (2010).
[CrossRef]

Minkowski, H.

H. Minkowski, ““Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Ges. Wiss. Gottn Math.-Phys. Kl. 1908, 53–111 (1908); reprinted in Math Ann 68, 472–525 (1910).

Moloney, J. V.

Nelson, D. F.

D. F. Nelson, “Momentum, pseudomomentum, and wave momentum: Toward resolving the Minkowski-Abraham controversy,” Phys. Rev. A 44(6), 3985–3996 (1991).
[CrossRef] [PubMed]

Nieminen, T. A.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[CrossRef]

Panepucci, R. R.

Pfeifer, R. N. C.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[CrossRef]

Qiu, M.

H. Yu, W. Fang, F. Gu, M. Qiu, Z. Yang, and L. Tong, “Longitudinal Lorentz force on a subwavelength-diameter optical fiber,” Phys. Rev. A 83(5), 053830 (2011).
[CrossRef]

Rubinsztein-Dunlop, H.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[CrossRef]

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[CrossRef] [PubMed]

Saldanha, P. L.

She, W.

W. She, J. Yu, and R. Feng, “She et al. Reply,” Phys. Rev. Lett. 103(21), 219302 (2009).
[CrossRef]

J. Yu, R. Feng, and W. She, “Low-power all-optical switch based on the bend effect of a nm fiber taper driven by outgoing light,” Opt. Express 17(6), 4640–4645 (2009).
[CrossRef] [PubMed]

W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. 101(24), 243601 (2008).
[CrossRef] [PubMed]

Shen, M.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Tong, L.

H. Yu, W. Fang, F. Gu, M. Qiu, Z. Yang, and L. Tong, “Longitudinal Lorentz force on a subwavelength-diameter optical fiber,” Phys. Rev. A 83(5), 053830 (2011).
[CrossRef]

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12(6), 1025–1035 (2004).
[CrossRef] [PubMed]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Westbrook, P. S.

Windeler, R. S.

Xu, Q.

Yang, Z.

H. Yu, W. Fang, F. Gu, M. Qiu, Z. Yang, and L. Tong, “Longitudinal Lorentz force on a subwavelength-diameter optical fiber,” Phys. Rev. A 83(5), 053830 (2011).
[CrossRef]

Yu, H.

H. Yu, W. Fang, F. Gu, M. Qiu, Z. Yang, and L. Tong, “Longitudinal Lorentz force on a subwavelength-diameter optical fiber,” Phys. Rev. A 83(5), 053830 (2011).
[CrossRef]

Yu, J.

W. She, J. Yu, and R. Feng, “She et al. Reply,” Phys. Rev. Lett. 103(21), 219302 (2009).
[CrossRef]

J. Yu, R. Feng, and W. She, “Low-power all-optical switch based on the bend effect of a nm fiber taper driven by outgoing light,” Opt. Express 17(6), 4640–4645 (2009).
[CrossRef] [PubMed]

W. She, J. Yu, and R. Feng, “Observation of a push force on the end face of a nanometer silica filament exerted by outgoing light,” Phys. Rev. Lett. 101(24), 243601 (2008).
[CrossRef] [PubMed]

Zakharian, A. R.

M. Mansuripur and A. R. Zakharian, “Theoretical analysis of the force on the end face of a nanofilament exerted by an outing light pulse,” Phys. Rev. A 80(2), 023823 (2009).
[CrossRef]

M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 026608 (2009).
[CrossRef] [PubMed]

A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Radiation pressure and the distribution of electromagnetic force in dielectric media,” Opt. Express 13(7), 2321–2336 (2005).
[CrossRef] [PubMed]

Adv. Opt. Photonics (1)

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M. Mansuripur and A. R. Zakharian, “Theoretical analysis of the force on the end face of a nanofilament exerted by an outing light pulse,” Phys. Rev. A 80(2), 023823 (2009).
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H. Yu, W. Fang, F. Gu, M. Qiu, Z. Yang, and L. Tong, “Longitudinal Lorentz force on a subwavelength-diameter optical fiber,” Phys. Rev. A 83(5), 053830 (2011).
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Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 026608 (2009).
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Figures (7)

Fig. 1
Fig. 1

Instantaneous profile of Gaussian optical field Ey of a ~10fs temporal width pulse carried by a basic TE waveguide mode at 800nm wavelength at t = 93.5fs. The dashed rectangle shows the integral region (x∈ [2, 11]μm, z∈ [12, 17]μm) that is selected for the momentum computations using the FDTD method

Fig. 2
Fig. 2

Instantaneous power of the pulse propagating along a waveguide as computed at three positions. The power is obtained by integration of Poynting vector Sz(x,z,t) over x at positions, z = 10μm,15μm,20μm, and thus is a function of time. Due to waveguide dispersion, the widths of the pulse increases with time, and the FWHM of the pulse are respectively 4.38fs, 4.80fs, 5.20fs at the three positions.

Fig. 3
Fig. 3

Respective comparisons of the four FDTD results with the four momentum formulas at different central wavelengths of 560nm,640nm,700nm,760nm,800nm,860nm. (a)-(d) are the verifications for the Abraham momentum Eq. (8), the Minkowski momentum Eq. (13), the Mechanical momentum Eq. (27) and the total momentum Eq. (28), respectively. The right y axis denotes the relative error.

Fig. 4
Fig. 4

Profiles of a pulse with central wavelength of 800nm propagating along a finite waveguide at different instant of time, t = 110fs, 125.99fs, 137.5fs, 160.01fs. Four integral regions are selected for our momentum computation, i.e. z∈ [12, 17]μm, [15, 38]μm [15, 22], μm, [22.02,38]μm with x∈ [1, 12]μm. They are denoted by the pink, black, red and blue lined rectangles, respectively.

Fig. 5
Fig. 5

Abraham(a), Minkowski(b) and total(c) momentum at any instant of time as calculated through integration over the four integral regions with x∈ [1, 12]μm and z∈ [12, 17]μm, [15, 38]μm [15, 22], μm, and [22.02,38]μm, respectively. Red dotted line for the left region (z∈ [15, 22]μm) contains the end face and some part of the waveguide; Blue dashed-dotted line for the right vacuum region (z∈[22.02,38]μm) is outside the waveguide; Cyan dotted line for region (z∈ [12, 17]μm) contains part of waveguide without the waveguide tip; Black solid line for the region (z∈ [15, 38]μm) contains both part of the vacuum and part of the waveguide with the end face.

Fig. 6
Fig. 6

The plots of the total Lorentz force which is obtained by integrating the corresponding force density over the region (z∈ [15, 38]μm). Overlapped with Lorentz force FLz (open circle, red) is the decrease rate of the total Abraham (-dtPAz), which indicates the total longitudinal momentum is conserved at any time when the light pulse emerges out from the waveguide.

Fig. 7
Fig. 7

Instant power at z = 20um in the waveguide and at z = 22.02um when pulse emerges from the end of the waveguide.

Tables (1)

Tables Icon

Table 1 FDTD-computed effective and group indexes, total and dielectric energies of a pulse at different central wavelengths of 500nm, 640nm, 800nm, 850nm

Equations (43)

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[ E ˜ (r,ω) H ˜ (r,ω) ]= E ˜ (ω)[ e(x,y,ω) h(x,y,ω) ]exp[iβ(ω)z]
E(r,t)= 1 2π E ˜ (ω)e(x,y,ω)exp[iβziωt]dω
H(r,t)= 1 2π E ˜ (ω)h(x,y,ω)exp[iβziωt]dω
U= + dt + dx + dy [ E×H ] z = + dt + dx + dy + dω + dω '{ E ˜ (ω) E ˜ (ω') [ e(ω)×h(ω) ] z exp[ i(β+β')zi(ω+ω')t ] } = + dx + dy + dω | E ˜ (ω) | 2 [ e(ω)× h * (ω) ] z
U= 1 2 + dz + dx + dy [ DE+BH ] = 1 2π ε 0 + dz + dx + dy + dω + d ω { ε(ω)( E ˜ (ω) E ˜ (ω')[ e(ω)e(ω') ] exp[i(β+β')zi(ω+ω')t] } = ε 0 + dx + dy + dω + dβ' ω' β' { ε(ω) E ˜ (ω) E ˜ (ω')[ e(ω) e * (ω') ] exp[i(ω+ω')t]δ(β+β') } = ε 0 + dx + dy + dω [ ε(ω) ω β | E ˜ (ω) | 2 | e(ω) | 2 ]
P A = dz dx dy p A = dz dx dy [ 1 c 2 E×H ] z = 1 c 2 + dx + dy + dω ω β [ | E ˜ (ω) | 2 [ e(ω)×h*(ω) ] z ] z ^ = v g c 2 U z ^ + P D A = 1 n g U c z ^ + P D A
P D A =[ m1 α m A 1 v g ( ω 0 ) m v g ω m | ω 0 ] v g ( ω 0 ) c 2 U z ^
α m A = 1 m! + dx + dy + dω [ (ω ω 0 ) m | E ˜ (ω) | 2 [ e(ω)× h * (ω) ] z ] ε 0 + dx + dy + dω [ ε(ω) ω β | E ˜ (ω) | 2 | e(ω) | 2 ] ,
P A = v g c 2 U z ^ = 1 n g U c z ^
P M = dz dx dy p M = dz dx dy [ ε(ω) c 2 E×H ] z z ^ = 1 c 2 + dx + dy + dω ω β [ | E ˜ (ω) | 2 [ ε(ω)e(ω)×h*(ω) ] z ] z ^ = 1 c ε 0 + dx + dy + dω [ n eff (ω)ε(ω) ω β | E ˜ (ω) | 2 | e(ω) | 2 ] z ^
n eff (ω)= + dx + dy [ e(ω)×h*(ω) ] z c ε 0 + dx + dy ε(ω) | e(ω) | 2 ,
P M = n eff U c z ^ + P D M
P D M =[ m1 α m M 1 n eff ( ω 0 ) m n eff ω m | ω 0 ] n eff ( ω 0 )U c z ^ ,
α m M = 1 m! ε 0 + dx + dy + dω [ (ω ω 0 ) m ε(ω) ω β | E ˜ (ω) | 2 | e(ω) | 2 ] ε 0 + dx + dy + dω [ ε(ω) ω β | E ˜ (ω) | 2 | e(ω) | 2 ]
P M = n eff U c z ^
f=( P )E+ P t ×B
×E= t B
f=( P )E+P×( ×E )+ t ( P×B ) = f 1 + f 2
f 1 = 1 2 ε 0 ε | E | 2 ,
f 2 = t ( P×B ),
P 1 Dip = + dx + dy + dz + dt f 1 = 1 2 ε 0 + dx + dy + dω [ χ(ω) | E ˜ (ω) | 2 | e(ω) | 2 ] z ^ = U Die v g z ^ P 1D Dip
P 1D Dip =[ m1 v g γ m m β ω m | ω 0 ] U Die v g z ^
γ m = 1 m! + dx + dy + dω [ (ω ω 0 ) m χ(ω) ω β | E ˜ (ω) | 2 | e(ω) | 2 ] + dx + dy + dω [ χ(ω) ω β | E ˜ (ω) | 2 | e(ω) | 2 ] .
U Die = 1 2 ε 0 + dx + dy + dω [ χ(ω) ω β | E ˜ (ω) | 2 | e(ω) | 2 ], = 1 2 ε 0 + dx + dy + dω [ ( ε(ω)1 ) ω β | E ˜ (ω) | 2 | e(ω) | 2 ]
P 2 AF = dz dx dy [ ε(ω)1 c 2 E×H ] z z ^ = P M P A =( n eff 1 n g ) U c z ^ + P 2D AF
P 2D AF =[ m1 ( α m M m n eff ω m | ω 0 α m A 1 v g m v g ω m | ω 0 ) ] U c z ^
P Mech = P 1 Dip + P 2 AF = U Die v g z ^ P 1D Dip +( n eff 1 n g ) U c z ^ + P 2D AF
P Tot = P 1 Dip + P 2 AF + P A = P 1 Dip + P M = U Die v g z ^ + n eff U c z ^ P 1D Dip + P D M
P Mech = U Die v g z ^ +( n eff 1 n g ) U c z ^ ,
P Tot = U Die v g z ^ + n eff U c z ^
P Tot = dt dx dz [ f(x,z,t) ]+ P A
P w =[ ( n eff ρ n g )(1+R)T ] U c z ^
f w =[ ( n eff ρ n g )(1+R)T ] W c z ^ ,
[ 2 + n 2 (x,y,ω) k 0 2 ][ E ˜ (x,y,z,ω) H ˜ (x,y,z,ω) ]=0
T ^ (δz)[ E ˜ (x,y,z,ω) H ˜ (x,y,z,ω) ]=[ E ˜ (x,y,z+δz,ω) H ˜ (x,y,z+δz,ω) ]
T ^ (δz)[ E ˜ (x,y,z,ω) H ˜ (x,y,z,ω) ]= m=0 + 1 m! ( δz z ) m [ E ˜ (x,y,z,ω) H ˜ (x,y,z,ω) ],
T ^ (δz)=exp( δz z ),
H ^ T ^ T ^ H ^ =0,
[ 2 + n 2 (x,y) k 0 2 ]{ T ^ (δz)[ E ˜ (x,y,z,ω) H ˜ (x,y,z,ω) ] }=0
T ^ ( λ eff )[ E ˜ (x,y,z,ω) H ˜ (x,y,z,ω) ]=[ E ˜ (x,y,z,ω) H ˜ (x,y,z,ω) ]
[ E ˜ (x,y,z,ω) H ˜ (x,y,z,ω) ]=[ E ˜ (x,y,z+ λ eff ,ω) H ˜ (x,y,z+ λ eff ,ω) ]
[ E ˜ (x,y,z+ z ,ω) H ˜ (x,y,z+ z ,ω) ]=exp[ i β l z ][ E ˜ (x,y,z,ω) H ˜ (x,y,z,ω) ]
[ E ˜ (x,y,z,ω) H ˜ (x,y,z,ω) ]= E ˜ (ω)[ e(x,y,ω) h(x,y,ω) ]exp[i β l (ω)z]

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