Abstract

A circularly polarized plane-wave is known to have no angular momentum when examined through Maxwell’s equations. This, however, contradicts the experimentally observed facts, where finite segments of plane waves are known to be capable of imparting angular momentum to birefringent platelets. Using a superposition of four plane-waves propagating at slightly different angles to a common direction, we derive an expression for the angular momentum density of a single plane-wave in the limit when the propagation directions of the four beams come into alignment. We proceed to use this four-beam technique to analyze the conservation of angular momentum when a plane-wave enters a dielectric slab from the free space. The angular momentum of the beam is shown to decrease upon entering the dielectric medium, by virtue of the fact that the incident beam exerts a torque on the slab surface at the point of entry. When the beam leaves the slab, it imparts an equal but opposite torque to the exit facet, thus recovering its initial angular momentum upon re-emerging into the free-space. Along the way, we derive an expression for the outward-directed force of a normally incident, finite-diameter beam on a dielectric surface; the possible relationship between this force and the experimentally observed bulging of a liquid surface under intense illumination is explored.

© 2005 Optical Society of America

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References

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  1. R. A. Beth, Phys. Rev. 50, 115 (1936).
    [CrossRef]
  2. M. Friese, T. Niemenen, N. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
    [CrossRef]
  3. L. Allen, S. Barnett, and M. Padgett, Optical Angular Momentum (Institute of Physics Publishing, 2003).
    [CrossRef]
  4. A. Stewart, �??Angular momentum of the electromagnetic field: the plane wave paradox resolved,�?? Eur. J. Phys. 26, 635-641 (2005).
    [CrossRef]
  5. L. Allen, M. Padgett, and M. Babiker, �??The orbital angular momentum of light,�?? Progress in Optics 39, 291-372 (1999).
    [CrossRef]
  6. A. O'Neil, I. MacVicar, L. Allen, and M. Padgett, �??Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,�?? Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef] [PubMed]
  7. R. Loudon, �??Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,�?? Phys. Rev. A. 68, 013806 (2003).
    [CrossRef]
  8. J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, 1975.
  9. M. Mansuripur, �??Radiation pressure and the linear momentum of the electromagnetic field,�?? Opt. Express 12, 5375-5401 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5375</a>.
    [CrossRef] [PubMed]
  10. M. Mansuripur, A. R. Zakharian, and J. V. Moloney, �??Radiation pressure on a dielectric wedge,�?? Opt. Express 13, 2064-2074 (2005), <a href "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2064">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2064</a>.
    [CrossRef] [PubMed]
  11. M. Mansuripur, "Radiation pressure and the linear momentum of light in dispersive dielectric media," Opt. Express 13, 2245-2250 (2005), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2245">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2245</a>.
    [CrossRef] [PubMed]
  12. A. R. Zakharian, M. Mansuripur, and J. V. Moloney, "Radiation pressure and the distribution of electromagnetic force in dielectric media," Opt. Express 13, 2321-2336 (2005), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2321">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2321</a>.
    [CrossRef] [PubMed]
  13. A. Ashkin and J. Dziedzic, �??Radiation pressure on a free liquid surface,�?? Phys. Rev. Lett. 30, 139-142 (1973).
    [CrossRef]
  14. M. Padgett, S. Barnett, and R. Loudon, �??The angular momentum of light inside a dielectric,�?? J. Mod. Opt. 50, 1555-1562 (2003).

Eur. J. Phys. (1)

A. Stewart, �??Angular momentum of the electromagnetic field: the plane wave paradox resolved,�?? Eur. J. Phys. 26, 635-641 (2005).
[CrossRef]

J. Mod. Opt. (1)

M. Padgett, S. Barnett, and R. Loudon, �??The angular momentum of light inside a dielectric,�?? J. Mod. Opt. 50, 1555-1562 (2003).

Nature (1)

M. Friese, T. Niemenen, N. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

Opt. Express (4)

Phys. Rev. (1)

R. A. Beth, Phys. Rev. 50, 115 (1936).
[CrossRef]

Phys. Rev. A. (1)

R. Loudon, �??Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,�?? Phys. Rev. A. 68, 013806 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

A. O'Neil, I. MacVicar, L. Allen, and M. Padgett, �??Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,�?? Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

A. Ashkin and J. Dziedzic, �??Radiation pressure on a free liquid surface,�?? Phys. Rev. Lett. 30, 139-142 (1973).
[CrossRef]

Progress in Optics (1)

L. Allen, M. Padgett, and M. Babiker, �??The orbital angular momentum of light,�?? Progress in Optics 39, 291-372 (1999).
[CrossRef]

Other (2)

L. Allen, S. Barnett, and M. Padgett, Optical Angular Momentum (Institute of Physics Publishing, 2003).
[CrossRef]

J. D. Jackson, Classical Electrodynamics, 2nd edition, Wiley, New York, 1975.

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Figures (3)

Fig. 1.
Fig. 1.

Plots of E-field magnitude (top row) and phase (bottom row) in the xy cross-sectional plane for a superposition of four circularly-polarized plane waves. (Left to right: Ex , Ey , Ez ). The assumed wavelength is λ o=1.0 µm, and the four plane-waves are tilted by θ=1° relative to the z-axis; the central lobe’s corners are thus located at x, yπ/(k osinθ)=28.65 µm. A white square is overlaid on the Ez magnitude and phase plots to delineate the central lobe’s boundaries. The color coded amplitude plots (top row) are on a logarithmic scale, with the maximum amplitude in each frame coded as red, minimum as blue, and the values in between assigned the rainbow color spectrum. (In each frame, the amplitude plots are truncated at the point where the E-field intensity drops below 10-3×Peak_Intensity.). In the Ex and Ey phase plots (bottom row), the central lobe is 180° phase-shifted relative to its four adjacent neighbors. At any given point in the xy-plane, Ey is 90° ahead of Ex . As for the Ez phase plot (bottom, right), the rainbow colors are assigned such that Blue=0°, Green=180°, Red=360°. In the region of the central lobe (as in all other lobes) Ez shows a vortex-like behavior.

Fig. 2.
Fig. 2.

Left to right: plots of the Poynting vector components Sx , Sy , Sz in the xy cross-sectional plane of the beam depicted in Fig. 1. (The overlaid white square in each frame delineates the central lobe’s boundaries.) In each frame, the maximum value of the depicted function is coded as red, the minimum value as blue, and the values in between are assigned the rainbow color spectrum. The Sx and Sy plots thus show that, in each lobe, the projection of the Poynting vector in the xy-plane circulates around the z-axis.

Fig. 3.
Fig. 3.

Four circularly polarized plane-waves enter a medium of dielectric constant ε. The k-vectors of the plane-waves are k 1 , 3 = k o ( ± σ , 0 , ε σ 2 ) and k 2 , 4 = k o ( 0 , ± σ , ε σ 2 ) . Here σ=sinθ, where θ is the angle of incidence in the free space.

Equations (34)

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E 1 ( x , y , z ) = E o ( cos θ x ̂ + i y ̂ sin θ z ̂ ) exp [ i k o ( x sin θ + z cos θ ) ] ,
H 1 ( x , y , z ) = ( E o Z o ) ( i cos θ x ̂ + y ̂ + i sin θ z ̂ ) exp [ i k o ( x sin θ + z cos θ ) ] .
E ( x , y , z ) = 2 E o { [ cos θ cos ( k o sin θ x ) + cos ( k o sin θ y ) ] x ̂ + i [ cos ( k o sin θ x ) + cos θ cos ( k o sin θ y ) ] y ̂
sin θ [ i sin ( k o sin θ x ) sin ( k o sin θ y ) ] z ̂ } exp ( i k o cos θ z ) ,
H ( x , y , z ) = 2 E o Z o { i [ cos θ cos ( k o sin θ x ) + cos ( k o sin θ y ) ] x ̂ + [ cos ( k o sin θ x ) + cos θ cos ( k o sin θ y ) ] y ̂
sin θ [ sin ( k o sin θ x ) + i sin ( k o sin θ y ) ] z ̂ } exp ( i k o cos θ z ) ,
S = 4 ( E o 2 Z o ) { sin θ sin ( k o sin θ y ) [ cos ( k o sin θ x ) + cos θ cos ( k o sin θ y ) ] x ̂
+ sin θ sin ( k o sin θ x ) [ cos θ cos ( k o sin θ x ) + cos ( k o sin θ y ) ] y ̂
+ { cos θ [ cos 2 ( k o sin θ x ) + cos 2 ( k o sin θ y ) ] + ( 1 + cos 2 θ ) cos ( k o sin θ x ) cos ( k o sin θ y ) } } z ̂
< P o > = [ 2 π 2 ( k o 2 sin 2 θ ) ] 1 S z d x d y = 4 ( E o 2 Z o ) Central lobe cos θ .
< L z > = 4 ( E o 2 Z o c 2 ) [ 2 π 2 ( k o 2 sin 2 θ ) ] 1 { 1 2 sin θ cos θ [ x sin ( 2 k o sin θ x ) + y sin ( 2 k o sin θ y ) ] Central lobe
+ sin θ [ x sin ( k o sin θ x ) cos ( k o sin θ y ) + y cos ( k o sin θ x ) sin ( k o sin θ y ) ] } d x d y
= 4 ε o E o 2 ( 2 π f ) .
E ( x , y , z ) = 2 E o { ( 1 + r ) [ cos θ cos ( k o sin θ x ) + cos ( k o sin θ y ) ] x ̂
+ i ( 1 + r ) [ cos ( k o sin θ x ) + cos θ cos ( k o sin θ y ) ] y ̂
( 1 r ) ε 1 sin θ [ i sin ( k o sin θ x ) sin ( k o sin θ y ) ] z ̂ } exp ( i k o ε sin 2 θ z ) ,
E ( x , y , z ) = 2 ( E o Z o ) { i ( 1 r ) [ cos θ cos ( k o sin θ x ) + cos ( k o sin θ y ) ] x ̂
+ ( 1 r ) [ cos ( k o sin θ x ) + cos θ cos ( k o sin θ y ) ] y ̂
( 1 + r ) sin θ [ sin ( k o sin θ x ) i sin ( k o sin θ y ) ] z ̂ } exp ( i k o ε sin 2 θ z ) ,
σ b ( x , y , z = 0 ) = 2 ε o E o ( 1 r ) ( 1 ε 1 ) sin θ [ i sin ( k o sin θ x ) sin ( k o sin θ y ) ] .
F z ( x , y , z = 0 ) = 4 ( n 1 ) ( n 2 + 1 ) ε o E o 2 n 2 ( n + 1 ) sin 2 θ [ sin 2 ( k o sin θ x ) + sin 2 ( k o sin θ y ) ] .
Central lobe F z ( x , y , z = 0 ) d x d y = 2 ( n 1 ) ( n 2 + 1 ) n 2 ( n + 1 ) ε o E o 2 λ o 2 = ( n 1 ) ( n 2 + 1 ) < P o > λ o 2 2 n 2 ( n + 1 ) c ,
F x x ̂ + F y y ̂ = 8 ( n 1 ) ε o E o 2 n ( n + 1 ) sin θ [ cos ( k o sin θ x ) + cos ( k o sin θ y ) ] [ sin ( k o sin θ y ) x ̂ + sin ( k o sin θ x ) y ̂ ]
< T z ( surface ) > = 8 ( n 1 ) ε o E o 2 n ( n + 1 ) [ 2 π 2 ( k o 2 sin 2 θ ) ] 1 × sin θ Central lobe [ 1 2 x sin ( 2 k o sin θ x ) + 1 2 y sin ( 2 k o sin θ y )
+ x sin ( k o sin θ x ) cos ( k o sin θ y ) + y cos ( k o sin θ x ) sin ( k o sin θ y ) ] d x d y
= 8 ( n 1 ) ε o E o 2 n ( n + 1 ) k o .
F ( x , y , z ) = 1 2 Real ( J b × B * ) = 1 2 Real [ i ω ε o ( ε 1 ) E × μ o H * ]
16 k o ε o E o 2 ( n + 1 ) 2 { ( n + n 1 ) sin θ [ cos ( k o sin θ x ) + cos ( k o sin θ y ) ] [ sin ( k o sin θ y ) x ̂ + sin ( k o sin θ x ) y ̂ ]
+ ( n 2 + 1 ) [ cos ( k o sin θ x ) + cos ( k o sin θ y ) ] 2 z ̂ } κ exp ( 2 k o κ z ) .
< F z ( bulk ) > = [ 2 π 2 ( k o 2 sin 2 θ ) ] 1 Central lobe d x d y 0 F z ( x , y , z ) d z = 8 ( n 2 + 1 ) ( n + 1 ) 2 ε o E o 2
< F z ( bulk ) > = [ 2 π 2 ( k o 2 sin 2 θ ) ] 1 Central lobe { x 0 F y ( x , y , z ) d z y 0 F x ( x , y , z ) d z } d x d y
= 8 ( n + n 1 ) ε o E o 2 k o ( n + 1 ) 2
< T z ( surface ) + T z ( bulk ) > = 16 n ε o E o 2 k o ( n + 1 ) 2 = [ 4 ε o E o 2 c ( 2 π f ) ] ( 1 r 2 ) .
L z = 1 2 ( 1 + n 2 ) [ 4 ε o ε ( τ E o ) 2 ( 2 π f ) ] .

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