## Radiation Pressure on Submerged Mirrors: Implications for the Momentum of Light in Dielectric Media

Optics Express, Vol. 15, Issue 5, pp. 2677-2682 (2007)

http://dx.doi.org/10.1364/OE.15.002677

Acrobat PDF (108 KB)

### Abstract

Radiation pressure measurements on mirrors submerged in dielectric liquids have consistently shown an effective Minkowski momentum for the photons within the liquid. Using an exact theoretical calculation based on Maxwell’s equations and the Lorentz law of force, we demonstrate that this result is a consequence of the fact that conventional mirrors impart, upon reflection, a 180° phase shift to the incident beam of light. If the mirror is designed to impart a different phase, then the effective momentum will turn out to be anywhere between the two extremes of the Minkowski and Abraham momenta. Since all values in the range between these two extremes are equally likely to be found in experiments, we argue that the photon momentum inside a dielectric host has the arithmetic mean value of the Abraham and Minkowski momenta.

© 2007 Optical Society of America

## 1. Introduction

1. R. V. Jones and J. C. S. Richards, Proc. Roy. Soc. A **221**, 480 (1954). [CrossRef]

2. R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. Roy. Soc. London, Series A **360**, 347–363 (1978). [CrossRef]

*n*

_{o}of the submerging liquid. Such findings, in turn, have been used to support the argument that the photons inside the liquid have the Minkowski momentum

*n*

_{o}

*hf*

_{o}/

*c*, where

*h*is Planck’s constant,

*f*

_{o}is the light’s frequency, and

*c*is the speed of light in vacuum [3

3. A. Ashkin and J. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. **30**, 139–142 (1973). [CrossRef]

4. J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A **8**, 14–21 (1973). [CrossRef]

*iϕ*), has a phase ϕ ≈ 180°. If, however,

*ϕ*is allowed to have other values, the radiation pressure on the submerged mirror will be reduced and, in particular, when

*ϕ*approaches zero, the effective photon momentum will be found to reach the Abraham value of

*hf*

_{o}/(

*n*

_{o}

*c*). Our analysis thus suggests that, depending on the chosen value of

*ϕ*for the mirror, the measured radiation pressure inside a dielectric medium would favor a photon momentum anywhere in the range between the Abraham and Minkowski values.

*ε*, to show that, when

*n*

_{o}≠ 1, the exactly calculated radiation pressure will have a strong dependence on

*ϕ*. Within the submerging liquid, an ideal flat mirror (i.e., one with 100% reflectance), sets up a perfect standing wave between the incident and reflected plane-waves. When

*ϕ*= 180°, the mirror’s surface will be at the null point of the standing

*E*-field; this is essentially the situation when conventional mirrors are used in the experiment, and the results of our calculations for the case of

*ϕ*= 180ϕ confirm the well-documented experimental findings [1

1. R. V. Jones and J. C. S. Richards, Proc. Roy. Soc. A **221**, 480 (1954). [CrossRef]

2. R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. Roy. Soc. London, Series A **360**, 347–363 (1978). [CrossRef]

*ϕ*deviates from 180°, the calculated radiation pressure drops; all else being the same, the ratio of the pressures on two mirrors, one with

*ϕ*= 180° and the other with

*ϕ*= 0°, is found to be equal to

*n*

_{o}

^{2}.

*not*limited to certain (idealized) types of mirror, but are a general property of standing waves in a dielectric host. It will be shown that the radiation pressure, if observed locally within a standing wave, would be a function of location within the interference fringe. A submerged pressure sensor would thus detect either the Abraham or the Minkowski momentum depending on whether the sensor is located at the peak or the valley of an interference fringe. Since all locations within a fringe are equally accessible, the average photon momentum associated with a standing wave in a dielectric will thus have the arithmetic mean value of the Abraham and Minkowski momenta.

## 2. Radiation pressure on an ideal submerged mirror

*n*

_{1}(i.e., the mirror’s dielectric constant,

*ε*= -

*n*

_{1}

^{2}, is a negative real number). The incidence medium is a transparent dielectric of refractive index

*n*

_{o}. The normally incident plane-wave has frequency

*f*

_{o}, free-space wavelength

*λ*

_{o}=

*c*/

*f*

_{o}, wave-number

*k*=

_{z}*n*

_{o}

*k*

_{o}= 2

*πn*

_{o}/

*λ*

_{o}, and electromagnetic field amplitudes (

*E*,

_{x}*H*) = (

_{y}*E*

_{o},

*n*

_{o}

*E*

_{o}/

*Z*

_{o}), where

*Z*

_{o}= (

*μ*

_{o}/

*ε*

_{o})

^{½}is the impedance of the free space [5, 6].

*ρ*= (

*n*

_{o}- i

*n*

_{1})/(

*n*

_{o}+ i

*n*

_{1}) of the submerged mirror has unit magnitude for all values of

*n*

_{1}, but its phase angle,

*ϕ*= -2 arctan (

*n*

_{1}/

*n*

_{o}), can be anywhere in the range from 0° to 180° depending on the value of

*n*

_{1}. Beneath the surface of the mirror, the transmitted beam is an inhomogeneous plane-wave with an imaginary propagation vector

**= i(2**

*k**πn*

_{1}/

*λ*

_{o})

**̂, which causes the beam amplitude to drop exponentially along the**

*z**z*-axis [6]. The transmitted

*E*- and

*H*-fields have a relative phase of 90°, yielding a time-averaged Poynting vector <

*S*> = 1/2 Re (

**×**

*E*

*H*^{*}) = 0 everywhere inside the mirror; this, of course, is consistent with the mirror surface’s 100% reflectance (i.e., ∣ρ∣

^{2}= 1).

**=**

*F**ρ*+

_{b}**E***×*

**J**_{b}**exerted on the bound charge density**

*B**ρ*= -

_{b}**∇**∙

**= -**

*P**ε*

_{o}(

*ε*- 1)

**∇**∙

**= 0, and also on the bound current density**

*E**=*

**J**_{b}*∂*/

**P***∂t*= -i

*ω*

*ε*

_{o}(

*ε*- 1)

**[7**

*E*7. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Optics Express **12**, 5375–5401 (2004). [CrossRef] [PubMed]

**, when integrated over the penetration depth of the light beam, yields**

*F**F*> is the time-averaged force per unit area of the mirror surface. For typical metallic mirrors,

_{z}*n*

_{1}≫

*n*

_{o}and the above formula reduces to <

*F*>≈

_{z}*ε*

_{o}

*n*

_{o}

^{2}

*E*

^{2}

_{o}. This is the expected result of assigning a Minkowski momentum density,

*p*_{M}= 1/2 Re[

**×**

*D*

*B*^{*}], to the incident and reflected beams [7–10

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

*n*

_{1}, Eq. (1) yields <

*F*> ≈ ε

_{z}_{o}

*E*

_{o}

^{2}, which is consistent with the presence of the Abraham momentum density,

*= ½ Re[*

**P**_{A}**×**

*E*

*H*^{*}]/

*c*

^{2}, in the dielectric medium. Although ordinary metals at visible wavelengths have a large value of

*n*

_{1}, at higher frequencies (i.e., just below the plasma resonance frequency

*ω*[5]) the metal’s dielectric constant

_{p}*ε*assumes small negative values, leading to small (imaginary) values for the refractive index.

*ϕ*of the Fresnel coefficient. In fact, Eq. (1

1. R. V. Jones and J. C. S. Richards, Proc. Roy. Soc. A **221**, 480 (1954). [CrossRef]

*ϕ*(with no explicit reference to the mirror’s refractive index i

*n*

_{1}) as follows:

*ϕ*can be readily designed, we expect the entire range of radiation pressures between ε

_{o}

*E*

_{o}

^{2}and

*ε*

_{o}

*n*

_{o}

*E*

_{o}

^{2}predicted by Eq. (2) to be amenable to experimental verification.

## 3. Electromagnetic momentum in standing waves

*z*) create a standing-wave inside a dielectric medium of refractive index

*n*

_{o}. For each wave the

*E*- and

*H*-field amplitudes are (

*E*,

_{x}*H*) = (

_{y}*E*

_{o},

*n*

_{o}

*E*

_{o}/

*Z*

_{o}). The standing waves may thus be expressed as follows:

*λ*

_{o}/2

*n*

_{o}depicts the intensity of the standing

*E*-field, with the origin of the coordinate system chosen to coincide with one of its nulls. The standing

*H*-field profile (not shown) is similar to that of the

*E*-field, but shifted along the

*z*-axis by

*λ*

_{o}/4

*n*

_{o}. The energy densities of the

*E*- and

*H*-fields, while stationary in space, oscillate in quadrature in time, so that the total optical energy swings back and forth between its electric and magnetic components.

*λ*

_{o}is opened at

*z*=

*z*

_{o}to pierce into the medium in an attempt to discern the nature of the local

*E*- and

*H*-fields at that particular location within the standing wave. While the gap is too narrow to affect in any significant way the standing wave profiles, the

*E*- and

*H*-fields inside the gap differ substantially from those within the dielectric host. We denote by

*E*the

_{g}*E*-field amplitude for each of the two counter-propagating plane-waves inside the gap; the corresponding

*H*-field amplitude is then

*H*=

_{y}*E*/

_{g}*Z*

_{o}. The standing fields inside the gap are thus given by

*ψ*is an as-yet-undetermined phase angle. The gap is sufficiently narrow that its upper and lower boundaries may be assumed to be effectively at the same location along the

*z*-axis, namely, at

*z*=

*z*

_{o}. Comparing Eqs. (3) and (4) shows that the continuity of (

*E*,

_{x}*H*) at the gap boundaries requires the following identities:

_{y}*ψ*and the amplitude ratio

*E*/

_{g}*E*

_{o}, yield

*z*

_{o}is varied from 0 to

*λ*

_{o}/4

*n*

_{o}, the value of

*E*would range from

_{g}*n*

_{o}

*E*

_{o}to

*E*

_{o}. The gap field, of course, is the superposition of two identical counter-propagating beams in free space, each with its own (well-defined) momentum density ±½(ε

_{o}

*E*

_{g}^{2}/

*c*)

*. At the null and peak positions of the*

**ẑ***E*-field intensity within the dielectric, the plane-waves of the gap carry, respectively, the Minkowski and Abraham momenta of the dielectric medium. When the

*E*

_{g}^{2}of Eq. (6b) is averaged over all values of

*z*

_{o}, each of the gap’s plane-waves is seen to have an average momentum density equal to the arithmetic mean of the Abraham and Minkowski values associated with each plane-wave of the dielectric host. This is essentially the same conclusion as reached in our earlier papers [7

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

11. M. Mansuripur, “Radiation pressure and the linear momentum of light in dispersive dielectric media,” Optics Express **13**, 2245–2250 (2005). [CrossRef] [PubMed]

*ϕ*of the reflection coefficient ρ plays a role similar to that of

*z*

_{o}in Eq. (6b). Thus when

*ϕ*varies from 180° to 0°, it is as though the position of the mirror within the standing wave (produced by interference between incident and reflected beams) has shifted from

*z*

_{o}= 0 toward

*z*

_{o}=

*λ*

_{o}4

*n*

_{o}. This is tantamount to substituting sin(

*ϕ*/2) for cos(

*n*

_{o}

*k*

_{o}

*z*

_{o}) in Eq. (6b), which would then reproduce the result in Eq. (2

2. R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. Roy. Soc. London, Series A **360**, 347–363 (1978). [CrossRef]

## 4. Momentum of a plane-wave

*z*-axis in a dielectric host of refractive index

*n*

_{o}; the field amplitudes inside the medium are (

*E*,

_{x}*H*) = (

_{y}*E*

_{o},

*H*

_{o}) = (

*E*

_{o},

*n*

_{o}

*E*

_{o}/

*Z*

_{o}). Let us now imagine a narrow gap of width

*δ*≪

*λ*

_{o}in the host medium, and examine the nature of the fields inside this gap. The plane of the gap is

*yz*in (a) and

*xz*in (b). The electromagnetic field inside the gap is the superposition of two evanescent plane-waves, each of which must satisfy the constraints

**∙**

*k***=**

*k**k*

^{2}

_{o},

**∙**

*k***= 0, and (**

*E**/*

**k***k*

_{o}) ×

*E*=

*Z*

_{o}

*imposed by Maxwell’s equations. The combined*

**H***E*- and

*H*-fields of these evanescent waves must also satisfy the boundary conditions on both walls of the gap. In the case depicted in Fig. 3(a), continuity is required of the perpendicular

*D*-field,

*D*=

_{x}*ε*

_{o}

*n*

_{o}

^{2}

*E*

_{o}, and the tangential

*H*-field,

*H*=

_{y}*H*

_{o}, whereas in the case of Fig. 3(b) it is

*E*and

_{x}*B*=

_{y}*μ*

_{o}

*H*that must be continuous. In Fig. 3(a) the two (co-propagating) gap fields have the following

_{y}*-vector and field amplitudes:*

**k***p*in the gap is derived from the Poynting vector component

*S*along the propagation direction, namely,

_{z}*xz*, yields

*λ*

_{o}, and cylinder axis aligned with the

*z*-axis (i.e., the direction of propagation of the beam). All possible momentum densities will then occur at different locations around the circumference of this cylindrical shell, and the overall momentum density in the shell will coincide with the aforementioned average of the Minkowski and Abraham momenta.

## Acknowledgements

## References

1. | R. V. Jones and J. C. S. Richards, Proc. Roy. Soc. A |

2. | R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. Roy. Soc. London, Series A |

3. | A. Ashkin and J. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. |

4. | J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A |

5. | J. D. Jackson, |

6. | L. Landau and E. Lifshitz, |

7. | M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Optics Express |

8. | R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Mod. Opt. |

9. | R. Loudon, “Radiation pressure and momentum in dielectrics,” Fortschr. Phys. |

10. | S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” J. Phys. B: At. Mol. Opt. Phys. |

11. | M. Mansuripur, “Radiation pressure and the linear momentum of light in dispersive dielectric media,” Optics Express |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 17, 2007

Manuscript Accepted: February 22, 2007

Published: March 5, 2007

**Virtual Issues**

Vol. 2, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Masud Mansuripur, "Radiation Pressure on Submerged Mirrors: Implications for the Momentum of Light in Dielectric Media," Opt. Express **15**, 2677-2682 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2677

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### References

- R. V. Jones and J. C. S. Richards, Proc. R. Soc. A 221, 480 (1954). [CrossRef]
- R. V. Jones and B. Leslie, "The measurement of optical radiation pressure in dispersive media," Proc. R. Soc. London, Series A, 360, 347-363 (1978). [CrossRef]
- A. Ashkin and J. Dziedzic, "Radiation pressure on a free liquid surface," Phys. Rev. Lett. 30, 139-142 (1973). [CrossRef]
- J. P. Gordon, "Radiation forces and momenta in dielectric media," Phys. Rev. A 8, 14-21 (1973). [CrossRef]
- J. D. Jackson, Classical Electrodynamics, 2nd edition, (Wiley, New York, 1975).
- L. Landau, E. Lifshitz, Electrodynamics of Continuous Media, (Pergamon, New York, 1960).
- M. Mansuripur, "Radiation pressure and the linear momentum of the electromagnetic field," Opt. Express 12, 5375-5401 (2004). [CrossRef] [PubMed]
- R. Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt. 49, 821-838 (2002). [CrossRef]
- R. Loudon, "Radiation pressure and momentum in dielectrics," Fortschr. Phys. 52, 1134-1140 (2004). [CrossRef]
- S. M. Barnett and R. Loudon, "On the electromagnetic force on a dielectric medium," J. Phys. B 39, S671-S684 (2006). [CrossRef]
- M. Mansuripur, "Radiation pressure and the linear momentum of light in dispersive dielectric media," Opt. Express 13, 2245-2250 (2005). [CrossRef] [PubMed]

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