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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 25 — Dec. 5, 2011
  • pp: 25263–25278
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Total longitudinal momentum in a dispersive optical waveguide

Jianhui Yu, Chunyan Chen, Yanfang Zhai, Zhe Chen, Jun Zhang, Lijun Wu, Furong Huang, and Yi Xiao  »View Author Affiliations


Optics Express, Vol. 19, Issue 25, pp. 25263-25278 (2011)
http://dx.doi.org/10.1364/OE.19.025263


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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

1. Introduction

The controversy over the correct momentum formula in dielectric media has been debated over the past century [1

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]

,2

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

]. Minkowksi proposed an asymmetry energy-momentum tensor (EMT) with a momentum density of pM = D × B [3

3. 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).

,4

4. J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986), Chap. VII.

], and expressed an optical pulse’s total momentum as PM = npU/c [5

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

,6

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

]. In order to satisfy the symmetry requirement for EMT in dispersive bulk media from the view point of angular momentum conservation, Abraham proposed another symmetrical EMT with a momentum density of pA = E × H/c2 [2

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

,6

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

,7

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

] and revised the optical pulse total momentum expression as PA = U/(ngc) [5

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

,6

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

], where U is the total energy for the pulse, and np and ng are respectively the phase and group refractive index of the dispersive bulk medium. Noether theorem [8

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

,9

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

] and Lorentz covariance [10

10. C. Wang, “ Wave four-vector in a moving medium and the Lorentz covariance of Minkowski's photon and electromagnetic momentums,” arXiv.org, arXiv:1106.1163v11 (2011).

] supported the argument that the Minkowksi momentum is the conserved (total) momentum in non-dispersive medium. However, the controversy is not over. Recently, Loudon, Barnett [11

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

,12

12. 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]

] and Mansuripur [13

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

] proposed using dipole and bound-charge Lorentz force to explain the kinetic interaction between an electromagnetic wave and a dielectric material and to obtain the mechanical momentum. In addition, these momentum formulas were shown to be equivalent to each other [14

14. 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]

], and can be derived from Maxwell’s stress tensor [15

15. B. A. Kemp, T. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express 13(23), 9280–9291 (2005). [CrossRef] [PubMed]

]. The bound-charge Lorentz force combined with the Abraham momentum was proved to be consistent with macroscopic Maxwell equations and the Poynting vector postulate [16

16. 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]

]. Recently, by applying dipole Lorentz force to the analysis of momentum transfer to atom or point-like dipole, Barnett [17

17. 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]

,18

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

] claimed the resolution of the Abraham-Minkowski dilemma and showed that the Minkowsk momentum is the canonical momentum and the Abraham momentum is the kinetic momentum. Saldanda proposed the division of mechanical and electromagnetic momentum densities in non-dispersive and magnetic bulk medium by using macroscopic Lorentz force law [19

19. 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]

]. Moreover, these Lorentz forces can be very conveniently computed using FDTD and applied to the analysis of optical force densities in various situations of interest [20

20. 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]

].

Despite numerous theoretical efforts in attempting to resolve the controversy [1

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]

,2

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

], only a few experiments have been carried out [1

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]

]. Recently, by observing the propagation of a pulsed (~0.25s) and a continuous-wave (CW) laser around a bend of a free-suspending subwavelength-diameter (SD) silica optical fiber, She et al. [21

21. 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]

] concluded that the end face of SD fiber should feel a push force as predicted by the Abraham momentum. However, using the Lorentz force and FDTD method, Mansuripur and Zakharian [22

22. 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]

], and H. Yu et al. [23

23. 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]

] theoretically analyzed the force exerted by an optical pulse (~10fs) and a CW laser emerging from an SD fiber, and they draw an opposite conclusion that an SD fiber should feel a pull force when the mechanical momentum carried by medium is considered. Brevik and Ellingsen [24

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

] argue that the transverse optical force is due to the asymmetric distribution of refractive index of the optical SD fiber. Although She et al.'s conclusion is questioned [25

25. 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]

27

27. 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]

], their experiment provided a simpler method to study optical momentum in a dielectric medium. Although the formula for a longitudinal total momentum in a dispersive bulk medium has been presented [5

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

,6

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

], the formulas for the longitudinal total momentum along a dispersive waveguide have not been well established. In this paper, by directly using the Lorentz force law, we derived the formulas for both the total momentum and the mechanical momentum along a dispersive optical waveguide having continuous translational symmetry. The low-order dispersion of the waveguide has been taken into consideration in our derivation. We believe that the presently derived formulas can be applied to the analysis of permanent momentum transfer associated with a finite waveguide and therefore, they may be useful in the design of opto-mechanical devices such as opto-mechanical all-optical switches [28

28. 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]

].

In section 2, the total Abraham longitudinal momentum and the total Minkowksi longitudinal momentum along a waveguide are derived. Longitudinal components of the total (conserved) and mechanical momentum are derived in section 3. In section 4, the momentum formulas without dispersive term are verified by the FDTD and Lorentz force method. In addition, elegant formulas for the force exerted by an outgoing optical pulse and the permanent transfer of the momentum to the waveguide are derived using the longitudinal total momentum and momentum conservation, and are verified by the FDTD method.

2. Abraham and Minkowski momentums in a dispersive waveguide

In this section, we derive Abraham and Minkowski momentum expressions along a waveguide. In our derivation, except for the continuous translational symmetry of the waveguide, no special waveguide structure is assumed. Thus the momentum formulas can be applied to a variety of waveguides as long as the waveguides has a continuous translational symmetry. Examples include SD optical fibers [29

29. 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]

,30

30. 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]

], photonic crystal fibers [31

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

], micro-structure fibers [32

32. B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9(13), 698–713 (2001). [CrossRef] [PubMed]

], and slot waveguides [33

33. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

,34

34. Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. 29(14), 1626–1628 (2004). [CrossRef] [PubMed]

]. When the dispersion of group velocity or the spectrum bandwidth of an optical pulse is small enough, the Abraham momentum is reduced to a simpler form, U/(ngc), and the Minkowski momentum is neffU/c, where U denotes the total energy of the optial pulse, neff and ng are the effective and group refractive index respectively, and c is the speed of light in vacuum. In the following, nonlinear effect of the waveguide medium is ignored, and Maxwell equations are therefore assumed linear.

The mode field of a waveguide with continuous translational symmetry at a given optical frequency ω can be written as (see detailed derivation in Appendix A)
[E˜(r,ω)H˜(r,ω)]=E˜(ω)[e(x,y,ω)h(x,y,ω)]exp[iβ(ω)z]
(1)
where β(ω) is the corresponding propagation constant, E˜(ω)is the amplitude for the monochromatic light wave component, e(x,y,ω) and h(x,y,ω) are respectively the normalized electric and magnetic eigenmode field supported by the waveguide.

Due to the linearity of the Maxwell’s equations, an arbitrary optical pulse propagating along a waveguide can rigorously be expressed as a superposition of eigenmode fields at different frequencies, that is,
E(r,t)=12πE˜(ω)e(x,y,ω)exp[iβziωt]dω
(2a)
H(r,t)=12πE˜(ω)h(x,y,ω)exp[iβziωt]dω
(2b)
where E˜(-ω)=E˜*(ω),β(ω)=β(ω), e(ω)=e*(ω), h(ω)=h*(ω).

Using Eqs. (2a) and (2b), we can obtain the total energy of an optical pulse propagating along the waveguide as

U=+dt+dx+dy[E×H]z=+dt+dx+dy+dω+dω'{E˜(ω)E˜(ω')[e(ω)×h(ω)]zexp[i(β+β')zi(ω+ω')t]}=+dx+dy+dω|E˜(ω)|2[e(ω)×h*(ω)]z
(3)

The total energy of the pulse can also be obtained in another way as shown below:
U=12+dz+dx+dy[DE+BH]=12πε0+dz+dx+dy+dω+dω{ε(ω)(E˜(ω)E˜(ω')[e(ω)e(ω')]exp[i(β+β')zi(ω+ω')t]}=ε0+dx+dy+dω+dβ'ω'β'{ε(ω)E˜(ω)E˜(ω')[e(ω)e*(ω')]exp[i(ω+ω')t]δ(β+β')}=ε0+dx+dy+dω[ε(ω)ωβ|E˜(ω)|2|e(ω)|2]
(4)
where the identities, DE=BH, 12π+dzexp[i(β+β)z]=δ(β+β), e(ω)=e*(ω) and h(ω)=h*(ω) are used in the derivation of Eq. (3) and Eq. (4).

Substituting Eq. (2a) and Eq. (2b) into the Abraham momentum density expression pA = E × H/c2 [1

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]

,2

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

] and integrating over the whole space, we can obtain the total Abraham momentum as

PA=dzdxdypA=dzdxdy[1c2E×H]z=1c2+dx+dy+dωωβ[|E˜(ω)|2[e(ω)×h*(ω)]z]z^=vgc2Uz^+PDA=1ngUcz^+PDA
(5)

Note that herez^is the unit vector along the longitudinal z axis, vg(ω)=dωdβ is the group velocity. In derivation of Eq. (5), Taylor’s expansion of vg(ω) around central optical frequency ω0 and the pulse energy expression of Eq. (3) have been used.

The momentum part induced by the dispersion of group velocity in the above expression (5) is
PDA=[m1αmA1vg(ω0)mvgωm|ω0]vg(ω0)c2Uz^
(6)
where

αmA=1m!+dx+dy+dω[(ωω0)m|E˜(ω)|2[e(ω)×h*(ω)]z]ε0+dx+dy+dω[ε(ω)ωβ|E˜(ω)|2|e(ω)|2],
(7)

If spectral bandwidth of a light pulse is small enough, the group velocity dispersion term PDA in Eq. (5) can be neglected. Thus the total Abraham momentum can be reduced to be the same as that in a bulk dispersive material, i.e.,

PA=vgc2Uz^=1ngUcz^
(8)

Similarly, by substituting Eq. (2a) and Eq. (2b) into the Minkowski momentum density pM=1c2D×B [1

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]

,2

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

] and integrating over the whole space, we can obtain the total Minkowski momentum as

PM=dzdxdypM=dzdxdy[ε(ω)c2E×H]zz^=1c2+dx+dy+dωωβ[|E˜(ω)|2[ε(ω)e(ω)×h*(ω)]z]z^=1cε0+dx+dy+dω[neff(ω)ε(ω)ωβ|E˜(ω)|2|e(ω)|2]z^
(9)

In the above expression, the effective refractive index is neff(ω)=β(ω)/c which gives the ratio of phase velocity in waveguides to light speed in vacuum, and can be expressed as [35

35. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983)

],

neff(ω)=+dx+dy[e(ω)×h*(ω)]zcε0+dx+dyε(ω)|e(ω)|2,
(10)

By combining Eq. (9) with Eq. (4) and using Taylor’s expansion of the neff(ω) with respect to ω, the total Minkowski momentum becomes

PM=neffUcz^+PDM
(11)

The dispersive contribution in Eq. (11), including all of neff dispersion, is
PDM=[m1αmM1neff(ω0)mneffωm|ω0]neff(ω0)Ucz^,
(12a)
where

αmM=1m!ε0+dx+dy+dω[(ωω0)mε(ω)ωβ|E˜(ω)|2|e(ω)|2]ε0+dx+dy+dω[ε(ω)ωβ|E˜(ω)|2|e(ω)|2]
(12b)

If the neff dispersion term PDMin Eq. (11) can be neglected, the Minkowski momentum associated with the waveguide mode can be reduced to the canonical momentum,

PM=neffUcz^
(13)

Based on the definition of effective refractive index given by β=neffωc, Eq. (13) indicates that when the dispersion of neff is neglected, the Minkowski momentum of a single photon equals to Ñβ. Here Ñβ is a canonical momentum of a single photon which roots in the continuous translational symmetry of the waveguide and the wave property of photons since β is corresponding to the eigenvalue of translational operator of the waveguide supported mode (see Appendix A). This result shows that the Minkowski momentum is not identical to canonical momentum except that the contribution PDMfrom neff dispersion in Eq. (11) can be neglected.

3. Mechanical and total momentum in a dispersive waveguide

In terms of a kinetic view [13

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

,17

17. 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]

,18

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

], the total optical momentum has two parts: the field momentum (Abraham momentum) carried by electromagnetic field alone, and the mechanical momentum (kinetic momentum) carried by the dielectric medium, which is associated with the Lorentz force [17

17. 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]

],

f=(P)E+Pt×B
(14)

Using one of Maxwell’s equations,
×E=tB
(15)
the Lorentz force can be written in another way as
f=(P)E+P×(×E)+t(P×B)=f1+f2
(16)
with
f1=12ε0ε|E|2,
(17)
f2=t(P×B),
(18)
where f1is the dipole force and f2 is the Abraham force [2

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

,5

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

].

Using Eq. (2a) and (2b), the part of mechanical momentum associated with the dipole force f1 can be expressed as
P1Dip=+dx+dy+dz+dtf1=12ε0+dx+dy+dω[χ(ω)|E˜(ω)|2|e(ω)|2]z^=UDievgz^P1DDip
(19)
with the dispersive term given by
P1DDip=[m1vgγmmβωm|ω0]UDievgz^
(20)
where

γm=1m!+dx+dy+dω[(ωω0)mχ(ω)ωβ|E˜(ω)|2|e(ω)|2]+dx+dy+dω[χ(ω)ωβ|E˜(ω)|2|e(ω)|2].
(21)

In Eqs. (19) and (20), UDieis the energy stored in the dielectric medium of the waveguide, and can be expressed as
UDie=12ε0+dx+dy+dω[χ(ω)ωβ|E˜(ω)|2|e(ω)|2],=12ε0+dx+dy+dω[(ε(ω)1)ωβ|E˜(ω)|2|e(ω)|2]
(22)
whereχ(ω)=ε(ω)1 is the electric susceptibility of the dielectric medium of the waveguide.

The other part of the mechanical momentum associated with the Abraham force f2 can be expressed as the difference between the Abraham momentum and the Minkowski momentum, i.e.,

P2AF=dzdxdy[ε(ω)1c2E×H]zz^=PMPA=(neff1ng)Ucz^+P2DAF
(23)

The second term of the last line of Eq. (23) is the dispersive contribution to the mechanical momentum associated with f2 and can be expressed as

P2DAF=[m1(αmMmneffωm|ω0αmA1vgmvgωm|ω0)]Ucz^
(24)

Therefore, the total mechanical momentum can be written as
PMech=P1Dip+P2AF=UDievgz^P1DDip+(neff1ng)Ucz^+P2DAF
(25)
and the total momentum can be written as

PTot=P1Dip+P2AF+PA=P1Dip+PM=UDievgz^+neffUcz^P1DDip+PDM
(26)

If the spectral bandwidth of the optical pulse propagating in the waveguide is small enough, the dispersive terms in Eq. (25) and Eq. (26) can be neglected. Therefore, the mechanical momentum and the total momentum can be further reduced, respectively, to
PMech=UDievgz^+(neff1ng)Ucz^,
(27)
and

PTot=UDievgz^+neffUcz^
(28)

The above expression of Eq. (28) can be compared to the total momentum in a bulk dispersive medium and it can be found that they are of the same form with the only exception that the effective index neff in Eq. (28) is replaced with the phase index np of a bulk medium. However, it should be noted that the dispersion relationship of a waveguide depends not only on the properties of the waveguide material, but also on the transverse geometrical structure of the waveguide [31

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

34

34. Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. 29(14), 1626–1628 (2004). [CrossRef] [PubMed]

]. Consequently, the group velocity of an optical pulse propagating along a waveguide can be tuned by engineering the transverse structure of the waveguide. In the case of a non-dispersive bulk medium, the group and effective refractive indexes are both reduced to the phase refractive index of the medium, and therefore, Eq. (28) becomes the average of Abraham and Minkowski momentum as proposed by Mansuripur in Ref [13

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

]. Therefore, we believe our formula, Eq. (28), is more general and rigorous than Mansuripur’s.

From Eq. (28), it can be found that neither the Minkowski nor the Abraham momentum presents a complete or total momentum that is conserved as required per the theoretic frame of classical electrodynamics with seven postulates proposed by Mansuripur in his publication [22

22. 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]

]. The total momentum for an optical pulse propagating in a waveguide has three contributions: the momentum from the dipole force, which can be expressed as a negative dielectric energy divided by the group velocity; the momentum from the Abraham force, which is the difference between the Minkowski momentum and the Abraham momentum; and the Abraham momentum, which is the ‘true’ electromagnetic field momentum, as has been well accepted by many authors of the literatures [1

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]

,5

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

,6

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

] and the references cited therein.

4. Numerical simulations

Finite difference time domain (FDTD) method [36

36. A. Talflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Boston, 2005)

] provides a powerful tool to verify if our derived formulas are valid. In our simulation, a ~10fs Gaussian pulse, carried by the basic TE waveguide mode at a wavelength of 800nm, is launched into a planar waveguide along the z direction. The planar waveguide has a 200nm slab core with a silica refractive index of 1.46 and an air cladding above and below the waveguide core. The instant profile of the y component electric field Ey(x,z,t) at 93.5fs is shown in 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
. The material dispersion is assumed to be zero and the waveguide dispersion is automatically accounted for in the FDTD simulation. Here, the grid size Δx = Δz = 0.02μm and the simulated region size is 40μm × 13μm with a perfect match layer of 1μm thickness.

4.1 Verification of the momentum formulas in an infinite waveguide

Figure 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.
shows the computed instant power of the optical pulse as it propagates to three positions (z = 10μm, 15μm, 20μm) by integrating the Poynting vector over x, Sz(x,z,t)dx. The FDTD simulation as shown in Figs. 1 and 2 produced a group refractive index ng of 1.3920 and a group velocity vg of 215.37μm/ps. The corresponding effective refractive index neff for TE fundamental mode at the central wavelength of 800nm is found to be 1.1938, which agreed well with a theoretical computation result of 1.1939 reported in Ref. [37

37. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 2.

]. Due to the waveguide dispersion, the pulse width increases as the pulse propagates along the waveguide, and the FWHM are 4.38fs, 4.80fs, 5.20fs respectively at the three positions [see Fig. 2]. A strong group-velocity dispersion of ~656.44fs2/mm is obtained according to the formula in Ref. [37

37. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 2.

]. Since the medium is assumed lossless, the energy of the pulse, obtained by integrating the instant power over time, is found to be U = 187.27fN⋅um/m and is unchanged along the waveguide.

To compute the optical momentum in an optical waveguide by FDTD, an integral region (x∈ [2, 11]μm and z∈ [12, 17]μm) as indicated by the black dashed line rectangle in Fig. 1 is selected to contain an optical pulse with ~3μm length in our simulations.

By directly integrating Abraham and Minkowski momentum densities over the selected region, the Abraham momentum is found to be 0.4468fN⋅ps/m and the Minkowski momentum is found to be 0.7418fN⋅ps/m, If Eqs. (8) and (13) are used, the Abraham momentum would be 0.4462fN⋅ps/m and the Minkowski momentum would be 0.7458fN⋅ps/m. This shows that Eq. (8) and Eq. (13) can produce almost the same Abraham and Minkowski momentum even if the momentum contribution from strong dispersion of group velocity is neglected.

Using FDTD and the formula,
PTot=dtdxdz[f(x,z,t)]+PA
(29)
where the first term to the right is the mechanical momentum carried by the waveguide dielectric media, the total momentum is found to be 0.5712fN⋅ps/m and the mechanical momentum is found to be 0.1244fN⋅ps/m. Integrating dielectric energy density by FDTD, we obtain 37.12fN⋅μm/m for the dielectric energy of the pulse, and −0.1733fN⋅ps/m for the part of mechanical momentum from the dipole force.

On the other hand, by inserting the total and dielectric energy, the group and effective index into Eq. (27) and Eq. (28), we found that the mechanical momentum is 0.1255fN⋅ps/m and the total momentum is 0.5725fN⋅ps/m.

Compared with the above computations from FDTD, we obtain a relative error of ~0.22% for the total momentum and ~0.88% for the mechanical momentum.

Other pulses at different central wavelength have also been studied to investigate validity of the formulas without group velocity dispersion in sections 2 and 3. Table 1

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

table-icon
View This Table
shows the FDTD-computed effective and group indexes, the total and dielectric energies for pulses of the central wavelength. It should be noted that the energy of pulses increases linearly with the central wavelength because in our simulations the width of pulse is set to always contain only five optical oscillations. Figure 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.
shows the comparisons of the four FDTD results with those obtained using the formulas derived in sections 2 and 3 for the Abarham, Minkowski, mechanical and total momentum. From Fig. 3, it can be seen that the relative errors for the Abraham, Minkowski and total momentum are all less than 1.1%. The maximum relative error is 2.1% for the mechanical momentum at the central wavelength of 860nm. This relatively large error is attributed to the fact that the integral region in FDTD simulation cannot entirely contain the pulse anymore. Thus we can see that, Eqs. (8), (11), (28) and (27) can all provide enough accuracy and can be conveniently used to analyze the transfer of momentum to a finite waveguide with high precision when an optical pulse leaves a waveguide, as will be shown in subsection 4.2.

4.2 Conservation and transfer of optical momentum in a finite waveguide

When an optical pulse propagates through a finite waveguide, if the pulse and the waveguide are considered as a closed system, the system momentum must be conserved. Figure 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.
shows the FDTD simulation results of an optical pulse travelling through a lossless rigid waveguide with a length of 22μm. The pulse has a central wavelength of 800nm and the same parameters as discussed in subsection 4.1. The simulation shows that it is the total momentum rather than the Abraham or the Minkowski momentum that is conserved.

To compute the total momentum, we selected a larger integral region (x∈ [2, 11]μm and z∈ [15, 38]μm, i.e. the black dashed rectangle in Fig. 4) which is divided into a left region (z∈ [15, 22]μm, red dashed line) and a right region (z∈[22.02,38]μm, blue dash line) with the border at the end of the waveguide. A smaller integral region (x∈ [2, 11]μm and z∈ [12, 17] μm, the pink dash-dotted rectangle in Fig. 4) is selected for computing the total waveguide momentum reflected by the waveguide end surface. The profile of the pulse propagating through the finite waveguide is also shown in Fig. 4. Figures 4(a)-(d) respectively shows the pulse profile at different times of t = 110.01fs, 125.99fs, 137.5fs, and 160.01fs. The pulse reaches the waveguide end at ~120fs and leaves the end at ~130fs.

Figures 5(a)-(c)
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.
respectively show the FDTD computed Abraham momentum, Minkowski momentum, and total momentum. In Fig. 5, the momentums are integrated over four regions as plotted by the cyan dotted line (pink dash region), the black solid line (black dash region), the red dotted line (left part of the largest region), and the blue dash-dotted line (the largest right part). The cyan dotted curve shows that the Abraham and the Minkowski momentums are conserved until the pulse leaves the region (z∈ [12, 17]μm) and no momentum is transferred to this part of waveguide after the pulse leaves the region (z∈ [12, 17]μm). This indicates that in the case of an infinite waveguide one cannot tell what is the ‘true’ conserved momentum. However, in the integral region as indicated by the black solid line in Fig. 4, which contains the waveguide end face, the Abraham and the Minkowski momentum are not conserved as the pulse propagates through the region and there is a permanent transfer of momentum from the pulse to the waveguide. As shown by the red, blue and black lines in Figs. 5(a)-(b), the Abraham momentum increases and the Minkowski momentum decreases, when the pulse leaves the waveguide end face. If we look at the kinetic picture, we will find that the Abraham momentum is the same as in that in vacuum and the mechanical momentum carried by the polarization of the waveguide medium should be taken into consideration as well. An intuitive explanation is that, when the pulse leaves the waveguide end face, the polarization of waveguide medium at the end face will radiate energy mostly into the vacuum and little is bounced back into the waveguide. Thus part of mechanical momentum is turned back into the electromagnetic wave. As a result, the Abraham momentum increases and the waveguide end should feel a recoil force, i.e. a push force. This interaction process is mediated by Lorentz force as shown in Fig. 6(a)
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.
. Figure 6 shows that the total Lorentz and dipole forces (open circles, red) integrated over the region (z∈ [15, 38]μm) are exactly equal to the rate of decrease of the Abraham momentum. Furthermore, Figs. 6 implies that the total momentum is kept conserved at any time when the light pulse emerges out from the waveguide.

From Fig. 5(c), we can see that when the pulse leaves the waveguide end in the time period of 120fs-135fs, the input total momentum PFDTDi with a value of 0.5712fN⋅ps/m as computed by Eq. (29) using FDTD, is conserved. After leaving the waveguide at ~130fs, the pulse spreads out, with only a little amount of the pulse emerging out at the top and bottom boundaries of the black integral region at 145fs. After 145fs, the total momentum declines slowly as shown by the black solid line in Fig. 5(c). The decline becomes quicker after 180fs due to fact that the pulse is now leaving the right boundary of the simulated region. After the pulse completely leaves the integral regions (z∈ [15, 38]μm) at ~188fs, the total momentum does not vanish as there is a small amount of momentum PFDTDw with a value of 0.0497 fN⋅ps/m, that is transferred to the waveguide. This can be seen by the overlap of the red dotted line with the black solid line after 188fs. The reflected pulse reaches the region (z∈ [12, 17]μm) at ~145fs and leaves it at ~175fs. Accompanied with the pulse is a reflected total momentum PFDTDr of −0.0151fN⋅ps/m, as shown by the cyan dotted line in Fig. 5(c). The transmitted total momentum PFDTDt can be found to be 0.5366fN⋅ps/m from the blue dash-dot line in Fig. 5(c). Thus, the identity, PFDTDi = PFDTDr + PFDTDt + PFDTDw, is confirmed. This shows that the total momentum is the true conserved momentum when considering an optical pulse propagating along a waveguide. Using Eq. (28), we can conveniently determine that the permanent transfer of momentum to the waveguide can be given by
Pw=[(neffρng)(1+R)T]Ucz^
(30)
where ρ = 19.22% is the ratio of the dielectric energy over the total energy of the pulse, and R and T are the reflectance and transmittance of the energy of the pulse at the waveguide end face, respectively. In our simulation, R = 6.87% and T = 91.00% and they are computed based on the instant optical powers obtained at z = 20μm and z = 22.02μm in 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.
. Note that 2.13% of incident energy is scattered into other directions. By inserting these quantities into Eq. (30), the permanently transferred momentum is found to be PEqw = 0.0490fN⋅ps/m. Compared with FDTD-computed transfer of the momentum PFDTDw, Eq. (30) has a small relative error of 1.4%. We therefore believe that Eq. (30) provide a relatively precise expression for the momentum transferred to a finite waveguide.

By taking the derivative of Eq. (30) with respect to time, we find that the average force exerted by a pulse emerging out from a finite waveguide is,
fw=[(neffρng)(1+R)T]Wcz^,
(31)
where W is the average power of input light wave which can be directly measured. It is worth noting that Eq. (31) can be more easily verified in the experiment that She et al. carried out [21

21. 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]

].

5. Conclusions

Optical momentum formulas for an optical pulse as a waveguide mode of a dispersive waveguide with continuous translational symmetry are derived for the first time, to the best of our knowledge. It is found that Abraham momentum is the same as the momentum in vacuum, and Minkowski momentum is equal to neff times the momentum in vacuum. However, neither Abraham nor Minkowski momentum represents the correct total (conserved) momentum in the waveguide. Only the total momentum is ‘truely’ conserved along a waveguide. The total momentum is expressed as PTot = -UDie/vg + neff U/c, which has three contributions: (1) the Abraham momentum in waveguide, PA = (1/ng)U/c; (2) the momentum from the Abraham force, i.e. P2AF = (neff −1/ng)U/c, which is the difference of the Minkowski and Abraham momentum; and (3) the momentum due to the dipole force, Pcanmech = -UDie/vg, which equals to the negative dielectric energy divided by the group velocity. In Abraham (kinetic) terms, the mechanical momentum carried by the dielectric medium of a waveguide equals to the sum of the last two contributions, i.e. Pkinmech = -UDie/vg + (neff −1/ng)U/c. By using FDTD, we verified that the momentum formulas without dispersive term can exactly give the total (conserved) momentum in a waveguide. A simpler formula for the force exerted by an outgoing optical pulse from a waveguide is also derived. This shows the convenience of total momentum formula for the analysis of optical forces and the transfer of optical momentum in a finite waveguide.

Appendix A

It is shown below how Eq. (1) in section 2 can be deduced by using the continuous translational symmetry of a waveguide. From Maxwell equations, Helmholtz equations for the electric and magnetic fields at a given optical frequency ω can be deduced piecewise in a uniform dielectric medium with boundary conditions at the interfaces between different mediums,

[2+n2(x,y,ω)k02][E˜(x,y,z,ω)H˜(x,y,z,ω)]=0
(A1)

where [E˜(x,y,z,ω),H˜(x,y,z,ω)]T are one of the possible solutions. Translational operator T^ can be defined as

T^(δz)[E˜(x,y,z,ω)H˜(x,y,z,ω)]=[E˜(x,y,z+δz,ω)H˜(x,y,z+δz,ω)]
(A2)

where T^(δz) denotes the translation of mode fields by infinitesimal change δz along a waveguide. Using Taylor’s expansion of Eq. (A2), we have

T^(δz)[E˜(x,y,z,ω)H˜(x,y,z,ω)]=m=0+1m!(δzz)m[E˜(x,y,z,ω)H˜(x,y,z,ω)],
(A3)

where the operator T^(δz)can be written as,

T^(δz)=exp(δzz),
(A4)

with zbeing the derivative with respective to z. In terms of commutation relations of [x,z]=0, [y,z]=0, [z,z]=0, the following commutation relation can be obtained,

H^T^T^H^=0,
(A5)

where the operator is H^=2+n2(x,y)k02. It is worth noting that continuous translational symmetry of the waveguide [i.e., zn(x,y)=0] should be used in deriving Eq. (A5). Applying the operator T^(δz) to both sides of Eq. (A1) and using the commutation relation of Eq. (A5), we obtain

[2+n2(x,y)k02]{T^(δz)[E˜(x,y,z,ω)H˜(x,y,z,ω)]}=0
(A6)

Equation (A6) indicates that the translated fields T^(δz)[E˜(x,y,z,ω),H˜(x,y,z,ω)]T are the solutions of Eq. (A1) with the same eigenvalue as the mode fields [E˜(x,y,z,ω),H˜(x,y,z,ω)]T. According to Maxwell equations, it is necessary to ensure that the electric and magnetic fields can be excited by each other mutually and thus the electromagnetic wave can propagate along the waveguide steadily, the field mode must be a periodic function along the waveguide. Assuming that the spatial period of field mode along the waveguide is λeff for a given optical frequency ω, we have

T^(λeff)[E˜(x,y,z,ω)H˜(x,y,z,ω)]=[E˜(x,y,z,ω)H˜(x,y,z,ω)]
(A7)

where the condition

[E˜(x,y,z,ω)H˜(x,y,z,ω)]=[E˜(x,y,z+λeff,ω)H˜(x,y,z+λeff,ω)]
(A8)

has been used.

From Eq. (A7), we can infer that the eigenvalue of the operator T^(λeff)=exp[λeffz]is exp[il2π]. Thus the eigenvalue of the operator zis iβl=i(l2π/λeff), (l=±1, ±2, ±3…, ±nmax). βl corresponds to the propagation constant for lth mode at a given optical frequency ω, where the sign of l indicates the direction of light wave propagation, and nmax is determined by eigenequations of βl and ω. By considering only the positive direction of light wave propagation in the waveguide and by inserting the eigenvalue iβlof operator z into Eq. (A4), the field mode translated by a distance z becomes

[E˜(x,y,z+z,ω)H˜(x,y,z+z,ω)]=exp[iβlz][E˜(x,y,z,ω)H˜(x,y,z,ω)]
(A9)

The arbitrary nature of the translational distance value z in Eq. (A9) indicates that the field mode should be written as,

[E˜(x,y,z,ω)H˜(x,y,z,ω)]=E˜(ω)[e(x,y,ω)h(x,y,ω)]exp[iβl(ω)z]
(A10)

where E˜(ω)is the amplitude of the monochromatic light wave component at an optical frequency of ω, and e(x,y,ω) and h(x,y,ω) are respectively the normalized electric and magnetic eigenmode field guided by the waveguide. It should be noted that only continuous translational symmetry of a waveguide and Helmholtz equations from Maxwell equations are used in the above derivation.

Acknowledgments

This work is supported by National Nature Science Foundation of China (NSFC) under Grant Nos. 11004086, 61027010, 10776099, 61177075, by the National High Technology Research and Development Program of China under Grant No. 2009AA04Z315, by Foundation for Distinguished Young Talents in Higher Education of Guangdong of China under Grant No. LYM10024, by the Fundamental Research Funds for the Central Universities of China under Grant Nos. 21609508, 21609421, 21611602 and 21611516, and by Project of High-level Professionals in the Universities of Guangdong Province. The authors thank Prof. Yan Zhou for his language assistance.

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OCIS Codes
(000.2690) General : General physics
(230.7370) Optical devices : Waveguides
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Physical Optics

History
Original Manuscript: September 2, 2011
Revised Manuscript: October 3, 2011
Manuscript Accepted: November 13, 2011
Published: November 23, 2011

Citation
Jianhui Yu, Chunyan Chen, Yanfang Zhai, Zhe Chen, Jun Zhang, Lijun Wu, Furong Huang, and Yi Xiao, "Total longitudinal momentum in a dispersive optical waveguide," Opt. Express 19, 25263-25278 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25263


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References

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