## INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS (2000)

### BibTeX

@MISC{00instituteof,

author = {},

title = {INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS},

year = {2000}

}

### OpenURL

### Abstract

Optical multilayers as a tool for visualizing special relativity

### Citations

107 |
Electrodynamics and Classical Theory of Fields and Particles
- Barut
- 1980
(Show Context)
Citation Context ...the three-dimensional rotation group SO(3) and the two-dimensional unimodular-unitary group SU(2) [50]. The homomorphism between SL(2,C) and SO(3, 1) is usually shown explicitly in the following form =-=[51]-=- we recall for clarity: Let σµ be the set of four Hermitian matrices σ0 = I (the identity) and σk (the Pauli matrices). With each four-vector x µ we associate the (2 × 2) Hermitian matrix X = x µ � 0 ... |

60 |
Classical Electrodynamics, 3rd edn
- Jackson
- 1998
(Show Context)
Citation Context ... another boost, but renders an additional spatial rotation [25, 26]. The angle of this rotation is generally called the Wigner angle [27], and is the kinematic effect underlying the Thomas precession =-=[28]-=-. Physically, the best known manifestation of this precession is the appearance of a factor 1/2 in the spin–orbit coupling in atomic physics, which is caused by the extra rotation an electron in a cir... |

20 |
A 1992 Spacetime Physics: Introduction to Special Relativity
- Taylor, Wheeler
- 1982
(Show Context)
Citation Context ... the question of the Wigner rotation, showing that it can be easily observed in multilayers. The Lorentz group plays a fundamental role not only in special relativity, but in many branches of physics =-=[1]-=-. In modern optics (both classical and quantum) it is difficult to overestimate its importance [2]. By way of example, it is worth recalling that the symplectic group Sp(2,R), which is locally isomorp... |

16 |
Quantum-mechanical noise in an interferometer,” Phys
- Caves
- 1981
(Show Context)
Citation Context ...gue computer for special relativity, just as the LCR circuit is an analogue of the forced harmonic oscillator [17]. However, most of the proposed experimental tests involve the use of squeezed states =-=[18]-=-–[20] (which belong to the realm of quantum optics) and require a relatively new and complicated technology that is not available in every laboratory. In this respect it is worth emphasizing that it i... |

10 |
A O and R¸aczka R 1977 Theory
- Barut
(Show Context)
Citation Context ...ce we are going to discuss is a straightforward generalization of the well known homomorphism between the three-dimensional rotation group SO(3) and the two-dimensional unimodular-unitary group SU(2) =-=[50]-=-. The homomorphism between SL(2,C) and SO(3, 1) is usually shown explicitly in the following form [51] we recall for clarity: Let σµ be the set of four Hermitian matrices σ0 = I (the identity) and σk ... |

10 |
Principles of Optics 7th edn
- Born, Wolf
- 1999
(Show Context)
Citation Context ...hat is, �1�2 ←→ M(�1)M(�2) = M(�1�2). On the other hand, from our basic field variables � (+) � E e = (16) E (−) for both ambient and substrate media, we can construct the associated coherency matrix =-=[52]-=- defined as E = e ⊗ e † � (+) 2 |E | = E (+) E (−)∗ E (+)∗E (−) |E (−) | 2 � (17)s44 J J MonzónandLLSánchez-Soto which transforms as Ea = MasEsM † as . (18) That is just the same law as the space–time... |

6 |
On the matrix formulation of the theory of partial polarization in terms of observables. Nuovo Cimento
- Parent, Roman
- 1960
(Show Context)
Citation Context ... great many uses in geometrical optics [4]–[6]. There have been also many interesting attempts to construct a consistent mathematical representation for polarization optics based on the Lorentz group =-=[7]-=-–[13]. Likewise, the importance in quantum optics of the group SU(1, 1), also locally isomorphic to SO(2, 1) (especially in connection with squeezed states of light), has been emphasized [14]–[16]. In... |

6 |
On the Winger angle and its relation with the defect of a triangle in hyperbolic geometry, J.Geom.Phys
- Chen, Ge
(Show Context)
Citation Context ...his Wigner angle can be viewed as the geometric phase or anholonomy (for a complete account of the achievements in this field see [32]) associated with a triangular circuit in the rapidity space [34]–=-=[37]-=-. It is worth mentioning that the geometric phases associated with the group SU(1, 1) (or SO(2, 1)) have been previously identified, being always proportional to the area of a closed circuit on the un... |

5 | Lie algebraic theory of geometrical optics and optical aberrations - Dragt - 1982 |

4 | What is common to both polarization optics and relativistic kinematics? Optik (Stuttgart) 90 - Pellat-Finet, Buasset - 1992 |

4 | Physical holonomy: Thomas precession, and Clifford algebra - Urbantke - 1990 |

3 | Theory of the coherency matrix for light of arbitrary bandwidth - Barakat - 1963 |

3 |
3 rd Edn
- Hecht, Optics
- 1998
(Show Context)
Citation Context ...at allows us to trace a triangle on the hyperboloid and to explicitly compute the geometric phase as the area enclosed by the loop. Multilayer optics is a common topic in undergraduate optics courses =-=[45, 46]-=-, but it is usually approached from a practical point of view, in relation to antireflection coatings and the like. We wish to stress that the benefit of our formulation does not lie in any inherent a... |

3 |
A “bicycle wheel” proof of the Gauss-Bonnet theorem, Exposition
- Levi
- 1994
(Show Context)
Citation Context ...true as proved by Aravind and others, see [34]–[37]. We stress that the interpretation of the Wigner angle as an area is a keystone for its identification as a geometric phase. In addition, we recall =-=[54, 55]-=- that for a hyperbolic (spherical) triangle the sum of angles is less (greater) than π, with the angular defect (excess) being equal to the area of the triangle on the unit hyperboloid (sphere). In ou... |

2 |
The relativistic velocity composition paradox and the Thomas rotation
- Ungar
- 1989
(Show Context)
Citation Context ...tion. In this paper we wish to concentrate on the well known, yet striking, fact that the composition of two non-collinear pure boosts is not another boost, but renders an additional spatial rotation =-=[25, 26]-=-. The angle of this rotation is generally called the Wigner angle [27], and is the kinematic effect underlying the Thomas precession [28]. Physically, the best known manifestation of this precession i... |

2 |
Factors of 2 in magnetic moments, spin-orbit coupling, and Thomas precession
- Corben
- 1993
(Show Context)
Citation Context ...on of this precession is the appearance of a factor 1/2 in the spin–orbit coupling in atomic physics, which is caused by the extra rotation an electron in a circular orbit feels in its own rest frame =-=[29, 30]-=-. In a recent paper, Vigoureux and Labeke [31] pointed out that both the Thomas precession in special relativity and the Pancharatnam phase in polarization optics [32] have a structure quite reminisce... |

2 |
Lorentz-group Berry phases in squeezed light Phys
- Chiao, Jordan
- 1988
(Show Context)
Citation Context ...es, polarization optics [38, 39] and the Gouy effect [40] for Gaussian beams have been exploited to highlight this phenomenon. Experimental schemes to measure these phases by parametric amplification =-=[41]-=-–[43] and squeezed states of light [20] have been also analysed. For lossless multilayers the rapidity space is also the unit hyperboloid associated with SU(1, 1) [44] and in this work we show a simpl... |

2 |
Lorentz-group Berry phases via two-mode squeezing
- Leonhardt
- 1993
(Show Context)
Citation Context ...olarization optics [38, 39] and the Gouy effect [40] for Gaussian beams have been exploited to highlight this phenomenon. Experimental schemes to measure these phases by parametric amplification [41]–=-=[43]-=- and squeezed states of light [20] have been also analysed. For lossless multilayers the rapidity space is also the unit hyperboloid associated with SU(1, 1) [44] and in this work we show a simple opt... |

2 |
The Classical Theory of Fields (New
- Landau, Lifshitz
- 1975
(Show Context)
Citation Context ...the velocity β is replaced by the hyperbolic parameter ζ = tanh −1 (β), which constitutes the usual rapidity space and whose line element has a Lobachevskian metric, as has been known for a long time =-=[53]-=-. Stated in global terms, this implies that a triangle in this rapidity space obeys a non-Euclidean geometry and, in our context, this results in the fact that the parameter γ in (32) for the compound... |

1 |
Symmetry in Lie optics Ann
- Wolf
- 1986
(Show Context)
Citation Context ...hic to the (2 + 1)-dimensional Lorentz group SO(2, 1), is the natural mathematical parlance for linear canonical transformations [3] and has recently found a great many uses in geometrical optics [4]–=-=[6]-=-. There have been also many interesting attempts to construct a consistent mathematical representation for polarization optics based on the Lorentz group [7]–[13]. Likewise, the importance in quantum ... |

1 | Geometrical approach to polarization optics: I-Geometrical structure of polarized light Optik 87 - Pellat-Finet - 1991 |

1 | Non-image forming polarization optical devices and Lorentz transformations—an analogy Phys - Opatrny, Peˇrina - 1993 |

1 |
Y S and Noz M E 1997 Stokes parameters as a Minkowskian four-vector Phys
- Han, Kim
(Show Context)
Citation Context .... The interval remains invariant (e 0 s )2 − (e 1 s )2 − (e 2 s )2 − (e 3 s )2 = (e 0 a )2 − (e 1 a )2 − (e 2 a )2 − (e 3 a )2 = 0 (20) and it is light like. To finish, it is worth mentioning that in =-=[12, 13]-=- it has been shown that both the Jonesmatrix formalism and the Stokes parameters in polarization optics can be reformulated as two different representations of the same Lorentz group. In this context,... |

1 |
Y S and Noz M E 1999 Wigner rotations and Iwasawa decompositions in polarization optics Phys
- Han, Kim
(Show Context)
Citation Context ...at many uses in geometrical optics [4]–[6]. There have been also many interesting attempts to construct a consistent mathematical representation for polarization optics based on the Lorentz group [7]–=-=[13]-=-. Likewise, the importance in quantum optics of the group SU(1, 1), also locally isomorphic to SO(2, 1) (especially in connection with squeezed states of light), has been emphasized [14]–[16]. In cons... |

1 |
Y S and Noz M E 1988 Linear canonical transformations of coherent and squeezed states in the Wigner phase space Phys
- Han, Kim
(Show Context)
Citation Context ...ntz group [7]–[13]. Likewise, the importance in quantum optics of the group SU(1, 1), also locally isomorphic to SO(2, 1) (especially in connection with squeezed states of light), has been emphasized =-=[14]-=-–[16]. In consequence, it has been suggested that modern optics can serve as an analogue computer for special relativity, just as the LCR circuit is an analogue of the forced harmonic oscillator [17].... |

1 |
Special relativity and interferometers Phys
- Han, Kim
- 1998
(Show Context)
Citation Context ... [14]–[16]. In consequence, it has been suggested that modern optics can serve as an analogue computer for special relativity, just as the LCR circuit is an analogue of the forced harmonic oscillator =-=[17]-=-. However, most of the proposed experimental tests involve the use of squeezed states [18]–[20] (which belong to the realm of quantum optics) and require a relatively new and complicated technology th... |

1 | Generation of squeezed states in degenerate four-wave mixing Phys - Reid, Walls - 1985 |

1 |
Thomas precession and squeezed states of light Phys
- Han, Hardekopf, et al.
- 1989
(Show Context)
Citation Context ...omputer for special relativity, just as the LCR circuit is an analogue of the forced harmonic oscillator [17]. However, most of the proposed experimental tests involve the use of squeezed states [18]–=-=[20]-=- (which belong to the realm of quantum optics) and require a relatively new and complicated technology that is not available in every laboratory. In this respect it is worth emphasizing that it is als... |

1 |
Grossel Ph 1993 A relativistic-like presentation of optics in stratified planar media Am
- Vigoureux
(Show Context)
Citation Context ...e in every laboratory. In this respect it is worth emphasizing that it is also possible to reformulate multilayer optics in a fully relativistic-like manner. The pioneer work of Vigoureux and Grossel =-=[21]-=- has drawn attention to the formal analogy between (complex) field amplitudes at the multilayer and (real) coordinates in a (1 + 1)-dimensional space–time. In this way, they arrived at a similarity be... |

1 |
Sánchez-Soto L L 1999 Fully relativisticlike formulation of multilayers optics
- Monzón
(Show Context)
Citation Context ...039+13$30.00 © 2001 IOP Publishing Ltd Printed in the UK 39s40 J J MonzónandLLSánchez-Soto We have devoted some effort to straighten out the ultimate basis of this analogy. To this end, we have shown =-=[22]-=- that the matrix describing a lossy multilayer is an element of the group SL(2,C) of unimodular (2 × 2) complex matrices, which is locally isomorphic to the (3 + 1)-dimensional Lorentz group SO(3, 1).... |

1 |
Sánchez-Soto L L 1999 Lossless multilayers and Lorentz transformations: more than an analogy Opt
- Monzón
(Show Context)
Citation Context ...ar (2 × 2) complex matrices, which is locally isomorphic to the (3 + 1)-dimensional Lorentz group SO(3, 1). For the outstanding case of lossless multilayers, this matrix belongs to the group SU(1, 1) =-=[23]-=-. Therefore, a natural identification of the multilayer reflection and transmission coefficients with the parameters of a Lorentz transformation emerges. This is more than a formal equivalence, since ... |

1 |
Sánchez-Soto L L 1999 Multilayer optics as an analog computer for testing special relativity Phys
- Monzón
(Show Context)
Citation Context ...valence, since one can be now confident of the fact that experiments using multilayers have to be in full agreement with the kinematics of special relativity. In particular, within this framework, in =-=[24]-=- we have discussed the optical implementations of two of the more popular predictions of the latter; namely, length contraction and time dilation. In this paper we wish to concentrate on the well know... |

1 |
WMP1986 Special relativity completed: the source of some 2s in the magnitude of physical phenomena
- Strandberg
(Show Context)
Citation Context ...on of this precession is the appearance of a factor 1/2 in the spin–orbit coupling in atomic physics, which is caused by the extra rotation an electron in a circular orbit feels in its own rest frame =-=[29, 30]-=-. In a recent paper, Vigoureux and Labeke [31] pointed out that both the Thomas precession in special relativity and the Pancharatnam phase in polarization optics [32] have a structure quite reminisce... |

1 |
Labeke D V 1998 A geometric phase in optical multilayers
- Vigoureux
(Show Context)
Citation Context ...r 1/2 in the spin–orbit coupling in atomic physics, which is caused by the extra rotation an electron in a circular orbit feels in its own rest frame [29, 30]. In a recent paper, Vigoureux and Labeke =-=[31]-=- pointed out that both the Thomas precession in special relativity and the Pancharatnam phase in polarization optics [32] have a structure quite reminiscent of the phase appearing in the composition o... |

1 |
Sánchez-Soto L L 1999 Origin of the Thomas rotation that arises in lossless multilayers
- Monzón
(Show Context)
Citation Context ...elativity and the Pancharatnam phase in polarization optics [32] have a structure quite reminiscent of the phase appearing in the composition of the reflected amplitudes from multilayers. In fact, in =-=[33]-=- we identified the Wigner angle as the phase of the transmission coefficient of the compound multilayer that results from putting together two lossless multilayers, which allows for a simple experimen... |

1 |
The Wigner angle as an anholonomy in rapidity space Am
- Aravind
- 1997
(Show Context)
Citation Context ...er, this Wigner angle can be viewed as the geometric phase or anholonomy (for a complete account of the achievements in this field see [32]) associated with a triangular circuit in the rapidity space =-=[34]-=-–[37]. It is worth mentioning that the geometric phases associated with the group SU(1, 1) (or SO(2, 1)) have been previously identified, being always proportional to the area of a closed circuit on t... |

1 | Berry phases and unitary transformations - Jordan - 1988 |

1 |
Yabuzaki T 1989 Observation of Lorentz-group Berry phases in polarization optics Phys
- Kitano
(Show Context)
Citation Context ...ted with the group SU(1, 1) (or SO(2, 1)) have been previously identified, being always proportional to the area of a closed circuit on the unit hyperboloid. Among other examples, polarization optics =-=[38, 39]-=- and the Gouy effect [40] for Gaussian beams have been exploited to highlight this phenomenon. Experimental schemes to measure these phases by parametric amplification [41]–[43] and squeezed states of... |

1 |
Quantum theory of partially polarizing devices and SU(1, 1) Berry phases in polarization optics Opt
- Agarwal
- 1991
(Show Context)
Citation Context ...ted with the group SU(1, 1) (or SO(2, 1)) have been previously identified, being always proportional to the area of a closed circuit on the unit hyperboloid. Among other examples, polarization optics =-=[38, 39]-=- and the Gouy effect [40] for Gaussian beams have been exploited to highlight this phenomenon. Experimental schemes to measure these phases by parametric amplification [41]–[43] and squeezed states of... |

1 |
Bargmann invariant and the geometry of the Güoy effect Phys
- Simon, Mukunda
- 1993
(Show Context)
Citation Context ...or SO(2, 1)) have been previously identified, being always proportional to the area of a closed circuit on the unit hyperboloid. Among other examples, polarization optics [38, 39] and the Gouy effect =-=[40]-=- for Gaussian beams have been exploited to highlight this phenomenon. Experimental schemes to measure these phases by parametric amplification [41]–[43] and squeezed states of light [20] have been als... |

1 | Berry’s phases in the degenerate parametric amplifier Phys - C - 1989 |

1 |
Sánchez-Soto L L 2001 A simple optical demonstration of geometric phases from multilayer stacks: the Wigner angle as an anholonomy
- Monzón
(Show Context)
Citation Context ...ases by parametric amplification [41]–[43] and squeezed states of light [20] have been also analysed. For lossless multilayers the rapidity space is also the unit hyperboloid associated with SU(1, 1) =-=[44]-=- and in this work we show a simple optical experiment that allows us to trace a triangle on the hyperboloid and to explicitly compute the geometric phase as the area enclosed by the loop. Multilayer o... |

1 |
Introduction to Optics (Englewood Cliffs, NJ: Prentice-Hall) ch 22
- Pedrotti, Pedrotti
- 1987
(Show Context)
Citation Context ...at allows us to trace a triangle on the hyperboloid and to explicitly compute the geometric phase as the area enclosed by the loop. Multilayer optics is a common topic in undergraduate optics courses =-=[45, 46]-=-, but it is usually approached from a practical point of view, in relation to antireflection coatings and the like. We wish to stress that the benefit of our formulation does not lie in any inherent a... |

1 |
Ellipsometry and Polarized Light (Amsterdam: North-Holland) section 4.6
- AzzamRMAandBashara
- 1987
(Show Context)
Citation Context ... plane wave in the substrate, denoted by E (+) s . Since the equations that govern the propagation of light are linear, the amplitudes of the output fields are related to those of the input fields by =-=[47, 48]-=- E (−) a (+) (−) = RasE a + TsaE s E (+) s = TasE (+) a + RsaE (−) s where Ras and Tas are, respectively, the overall reflection and transmission coefficients for a wave incident from the ambient and ... |

1 |
Sánchez-Soto L L 1996 On the concept of absorption for a Fabry–Perot interferometer Am
- Monzón
(Show Context)
Citation Context ... plane wave in the substrate, denoted by E (+) s . Since the equations that govern the propagation of light are linear, the amplitudes of the output fields are related to those of the input fields by =-=[47, 48]-=- E (−) a (+) (−) = RasE a + TsaE s E (+) s = TasE (+) a + RsaE (−) s where Ras and Tas are, respectively, the overall reflection and transmission coefficients for a wave incident from the ambient and ... |

1 |
Theory of Reflection (Dordrecht
- Lekner
- 1987
(Show Context)
Citation Context ..., since there is a one-to-one correspondence between the propagation in planar stratified media of electromagnetic waves and of the non-relativistic particle waves satisfying the Schrödinger equation =-=[49]-=-. (1)s42 J J MonzónandLLSánchez-Soto The field amplitudes at each side of the multilayer are related by a (2 × 2) complex matrix Mas, we shall call a multilayer matrix, in the form � (+) E a E (−) � �... |