December 15, 2014 / Vol. 39, No. 24 / OPTICS LETTERS
6919
Optical systems for controlled specular depolarization Myriam Zerrad,1,* Clément Luitot,2 Jacques Berthon,2 and Claude Amra1 1
Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel, UMR 7249, 13013 Marseille, France 2
CNES, 18 avenue Edouard Belin, 31 401 Toulouse CEDEX 9, France *Corresponding author: [email protected]
Received September 1, 2014; revised November 3, 2014; accepted November 5, 2014; posted November 6, 2014 (Doc. ID 220743); published December 11, 2014 Optical coatings are known to be highly polarizing at oblique incidence. On the other hand, they cannot reduce the degree of polarization of light. We show how to overcome this difficulty and reach spatial depolarization with significant efficiency. Applications concern space optical systems. © 2014 Optical Society of America OCIS codes: (310.5448) Polarization, other optical properties; (260.2130) Ellipsometry and polarimetry; (030.6600) Statistical optics. http://dx.doi.org/10.1364/OL.39.006919
Optical coatings are well known to be highly polarizing, in the sense that they may produce an output polarization (reflected or transmitted) strongly different from the input (incident) polarization. Such property was extensively used to produce polarizing devices [1–5] these last decades; oblique incidence is generally required since thin films are commonly isotropic, though anisotropic films produced at oblique deposition were also considered [6]. For all these devices, polarization is modified or selected, but light remains totally polarized, which means that the temporal and local degree of polarization [7–9] remains unity (DOP 1). To go further, optical coatings do not reduce the DOP of fully polarized incident light [10–15]. This result constitutes a limitation for specific systems (space applications) that are polarization sensitive and hence cannot be calibrated with enough accuracy. Several authors have proposed solutions to solve this point [16], but a demand for significant progress is still expected. In this Letter, we propose an alternative solution to overcome this difficulty. We considered another type of depolarization, the spatial depolarization [12,17], which consists in mixing a high number of polarization states within the receiver aperture or bandwidth. To reach an efficient depolarizing device, variation of polarization must be very fast versus position, wavelength, or incidence, which led us to involve specific multilayers. All these new results and procedures are presented in this Letter. A quasi-monochromatic illumination beam in the far field is here described with an electromagnetic field E0 x; y; z; t propagating on its two analytic transverse polarization modes s and p: ⃗E 0 x; y; z; t E 0S x; y; z; t . (1) E 0P x; y; z; t In relation (1), the propagation direction is z, x; y are the transverse coordinates, and t is for the time. The temporal degree of polarization (DOP) of this field is defined as s detJ DOP0t x; y 1 − 4 2 0t tr J 0t
(2)
0146-9592/14/246919-04$15.00/0
with detJ 0t & trJ 0t , respectively the determinant and trace of the mutual coherence matrix defined in Eq. (3) below: J 0t x; y hjE 0S x; yj2 it hE 0S x; y E 0P x; yit
hE 0S x; yE 0P x; y it hjE 0P x; yj2 it
. (3)
In relation (3), the brackets hit are for temporal averages, and A is the complex conjugate of A. Notice here that we used the subscribe t for most quantities in order to recall that the averages are temporal, in opposition to spatial averages that will appear further in the report. Relation (2) can also be written versus the polarization ratio β0t and the complex mutual coherence degree μ0t as defined below: 8 > hjE 0P x; yj2 it > > < β0t x; y hjE 0S x; yj2 it ; hE 0S x; yE 0P x; y it > > p μ x; y > 0t : hjE 0S x; yj2 it hjE 0P x; yj2 it
4
s β0t x; y 1 − jμ0t x; yj2 . DOP0t x; y 1 − 4 1 β0t x; y2 (5) At this step, all average processes are temporal, so that the degree of polarization is a local quantity within the interval [0,1]. Notice in relation (5) that the DOP is written in a plane of altitude z. Temporal depolarization is a property of light resulting from the average process of optical detectors that cannot follow the high optical frequencies. Hence such property is difficult to control. For this reason, we will here consider another depolarization process resulting from a spatial average over the receiver aperture. This phenomenon has been highlighted in recent papers [12,17] because it spontaneously occurs in the case of a fully polarized incident light scattered by inhomogeneous volumes or rough surfaces. In this specific configuration, the local scattered field still remains fully polarized but presents a high dispersion of polarization states in © 2014 Optical Society of America
6920
OPTICS LETTERS / Vol. 39, No. 24 / December 15, 2014
only a few speckle areas. This leads to a global depolarization resulting from the integration of many speckle areas. The theoretical study of this spatial depolarization is similar to the analysis of temporal depolarization, where the temporal averages are replaced by spatial averages. So, the degree of polarization of the illumination beam is now defined by: s detJ x; y DOP0ΔΩ x; y 1 − 4 2 0ΔΩ ; tr J 0ΔΩ x; y
(6)
where ΔΩ is the domain of spatial integration from the departure point x; y and J 0;ΔΩ x; y the mutual coherence matrix defined Eq. (7) with a¯ the spatial average of variable a over ΔΩ: J 0ΔΩ x; y
hjE 0S x; yj2 it
hE 0S x; yE 0P x; y it
hE 0S x; y E 0P x; yit
hjE 0P x; yj2 it
Fig. 1. Specular depolarizer with a linear filter, configuration of use.
! . (7)
Notice here that J 0;ΔΩ involves quantities that are double averages on time (hit ) and space − . In the same way, the polarization ratio and mutual coherence degree are extended to spatial depolarization as
spatial repartition of the analytic reflected field is given Eq. (11): p r S RS x; yejδS x;y r p ; r P RP x; yejδP x;y
8 < β0ΔΩ x; y hjE0P x; yj2 it ∕hjE 0S x; yj2 it q ; : μ0ΔΩ x; y hE 0S x; yE 0P x; y it ∕ hjE 0S x; yj2 it hjE 0P x; yj2 it
r x; yE 0S . E r x; y; t S r P x; yE 0P
and the resulting degree of polarization is: s β0ΔΩ x; y 1 − jμ0ΔΩ x; yj2 . DOP0ΔΩ x; y 1 − 4 1 β0ΔΩ x; y2 (9) Formula (9) gives a general definition of the degree of polarization taking into account both spatial and temporal averages. Such quantity is a function of the illumination beam, the illuminated medium, and the receiver aperture. Contrary to the previous works devoted to spatial depolarization and which involved scattering patterns from disordered media, we here plan to propose optical components that provide a fully depolarized but specular beam when illuminated with fully or partially polarized collimated specular light. For that we use, an optical filter whose complex reflection coefficient varies with location x; y in the filter plane, as shown in Fig. 1. Such spatially variable filter can be obtained with a low slope linear filter [18], but other techniques can also be used. As for the complex reflection coefficient of the filter, it is defined on the polarization basis Eq. (10) and the
(10)
8
(11)
Notice in these equations that reflection is assumed not to be wavelength dependent, which results from the assumption of a quasi-monochromatic incident beam. The local polarization parameters in the reflected plane (P r ) are expressed versus the incident light properties and the spatial parameters of the filter. They allow to express the local degree of polarization in the reflected beam as
x;y β0 βR x; y RRPS x;y ; jδ x;y−δ S P x;y μ μR x; y e 0
12
s RS x; yRP x; yβ0 DOPR x; y 1 − 4 1 − jμ0 j2 . RS x; y RP x; yβ0 2 (13) We can notice that in the specific case of a reflection without energy losses RS x; y RP x; y∀ x; y, that the reflected DOP equals the incident DOP at any location in the reflected plane.
December 15, 2014 / Vol. 39, No. 24 / OPTICS LETTERS
Now using a single sensor collecting the whole reflected beam, a spatial average is performed and results in a DOP which can be expressed as s αR;ΔΩ β0 1 − jγ R;ΔΩ μ0 j2 14 DOPR;ΔΩ 1 − 4 1 αR;ΔΩ β0 2 with RR ΔΩ r S x; yr P x; ydxdy ; (15) γ R;ΔΩ q RR RR 2 dxdy 2 dxdy jr x; yj jr x; yj S P ΔΩ ΔΩ RR jr x; y:j2 dxdy αR;ΔΩ RRΔΩ P . 2 ΔΩ jr S x; y:j dxdy
(16)
In this particular configuration, ΔΩ must be large enough to integrate the whole reflected specular beam. We notice that the global DOP of the reflected beam is a function of both the incident polarization parameters β0; μ0 of the illumination beam and the spatial repartition of the complex reflection coefficient in terms of phase and amplitude γ R;ΔΩ and αR;ΔΩ . Equations (15) and (16) also allow to write the polarimetric characteristics of the integrated reflected beam in a simplified form: μR;ΔΩ μ0 γ R;ΔΩ
and
βR;ΔΩ β0 αR;ΔΩ .
(17)
An example of application is given Fig. 2. The incident polarization beam is first chosen with equal energy on each polarization axis (x and y) and fully polarized with an arbitrary elliptic polarization state. The light is said to be “balanced” in the chosen axis, which means that the beam energy is equally divided on the two polarization modes, that is, β0 : 1 in the specific referential. In order to simplify the problem, we consider a device that reflects the same energy for each incident polarization (this may happen at total internal reflection, or with specific multilayers). On the other hand, the polarimetric phase shift at reflection is assumed to be linear on xdirection and varying from 0 to 2π. We can notice at this
Fig. 2. Dispersion of polarization states (red points) taken in the reflected beam after reflection of a fully polarized and balanced incident light (blue point) on the filter shown on the left. The resulting polarization states measured by a sensor are reduced to a single central point at the center of the sphere (black point).
6921
step that the linear function can be replaced by any kind of polarimetric phase shift reparation, even random, while γ R;ΔΩ factor equals zero. Under these assumptions the reflection coefficient of the depolarizer is written as Rs x; y Rp x; y ∀ x; y ∈ 0; L2 . δp x; y π Lx −δs x; y
(18)
As a consequence, this depolarizer fulfills γ R;ΔΩ 0 and αR;ΔΩ 1. With such a configuration, the resulting measured reflected beam follows μR;ΔΩ 0 and βR;ΔΩ β0 1 so that is uncorrelated (μR;ΔΩ 0), and fully depolarized DOPR;ΔΩ 0. At this step we can notice that if the reflection coefficients are chosen with a unitary modulus, the full depolarization can be reached without energy losses. A synthesis of this application is given Fig. 2. The spatial repartition of relative parameters (Rp ∕Rs and δp − δs ) of the designed filter given Eq. (18) are plotted on the left side. The incident light is located by the blue point on the Poincaré sphere, and the circle defined by red points is the locus of the polarizations taken in the reflected beam before integration by the sensor. This representation confirms that the local degree of polarization of the reflected beam equals the illumination degree of polarization because all the red points are equidistant to the centre of the sphere. At last, the resulting light after integration by the sensor is plotted in black (central point). Because RS x; y RP x; y∀ x; y, the result is full depolarization (DOPR;ΔΩ 0). We can notice here that this result remains valid for any incoming light checking the property β0 1 whatever its other polarimetric characteristics. Until now the incident light was fully polarized and balanced (DOP0 1 & β0 1) in the referential, and hence very specific. It is now necessary to quantify the efficiency of the procedure with an arbitrary incident light. To this aim, 256 incident polarization states were randomly generated with a DOP in the range (0,1), and are plotted in blue Fig. 3(a) (data points are in the volume and at the surface of the sphere). The resulting light measured by the sensor is located [Fig. 3(b)] in the Poincaré sphere. We can see that whatever the incident DOP, the light measured by the sensor after reflection by the designed component is on the diameter of the sphere passing through s (transverse electric) and p (transverse magnetic) points. Therefore, in the case of an arbitrary incident polarization, the specular depolarizer projects
Fig. 3. (a) Localization of the 256 random incident fields in the Poincaré sphere. (b) Resulting polarization states after reflection by the linear filter and integration by the sensor.
6922
OPTICS LETTERS / Vol. 39, No. 24 / December 15, 2014
Fig. 4. Degree of polarization of the reflected beam after integration by the sensor (DOPR;ΔΩ ) versus the incident polarization ratio for a component designed for γ R;ΔΩ 0 and αR;ΔΩ 1.
the polarization states on a straight line within the sphere. We notice that the performances of the procedure depend on the incident field. More specifically, because γ R;ΔΩ 0, the incident temporal correlation μ0 has no influence on the resulting DOP, which is only a function of the incident polarization ratio β0 . Its evolution is plotted Fig. 4. Role of filter parameters: At this step, a last point is to quantify the efficiency of the depolarizing device versus the spatial properties of its reflection. γ R;ΔΩ is managed by the polarimetric phase shift at reflection, and αR;ΔΩ is controlled by the ratio of the reflected amplitudes. We have to notice here that to conserve the energy of illumination after reflection, we have to impose relationship (19) and so αR;ΔΩ 1: Rs x; y Rp x; y ∀ x; y ∈ 0; L2 .
(19)
Such a condition can be approximated reached with specifically designed multilayers. The variations of polarimetric phase shift induced by the reflection are a key point to control the dispersion of possible polarization states after integration by the sensor. To extend progressively these variations, we will consider a series of reflectors whose phase delay at reflection is defined as δp x; y kπ
x −δs x; y∀ x; y ∈ 0; L2 . L
(20)
The corresponding polarization states at the output of the system are calculated for the 256 random incident illuminations defined Fig. 3(a) and are given [Fig. 5] for k f0; 1∕5; 2∕5; 3∕5; 4∕5; 1g. The locus of the resulting electromagnetic fields is an ellipsoid whose revolution axis is the segment linking s and p polarization states and whose transverse plane is aρ radius circle. ρ is the degree of polarization obtained when the incident light is a balanced and fully polarized beam: sinkπ . (21) kπ The capability to create or control depolarization with specular devices has been demonstrated. Optical component designed to generate a strong dispersion of polarization states within the reflected beam is required. The measurement of the light reflected by such a filter ρ
Fig. 5. Repartition of the resulting polarization states calculated for 256 random polarization states defined Fig. 3(a) with k f0; 1∕5; 2∕5; 3∕5; 4∕5; 1g:
has an apparent degree of polarization which is a function of both the incident light and spatial characteristics of the depolarizing device. It is possible to conserve all the incident energy after reflection. In this specific case, if the phase shift at the reflection takes a multiplicity of values in [0, 2π], the specular depolarizer will give totally unpolarized output light whatever the incident polarization provided that the illumination beam is balanced (β0 1). At last, the performances of the specular depolarizers have been quantified as a function of the filters parameters and the polarimetric characteristics of incoming light. References 1. H. A. MacLeod and A. MacLeod, eds., Thin Film Optical Filters (Taylor & Francis, 2001), p. 3. 2. J. A. Dobrowolski, in Handbook of Optics, BASS (McGrawHill, 1995). 3. P. W. Baumeister, Optical Coating Technology (SPIE Book, 2004). 4. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989). 5. S. A. Furman and A. V. Tikhonrarov, Basics of Optics of Multilayer Systems (Éditions Frontières, 1992). 6. D. Liang, R. W. Smith, and D. J. Srolovitz, J. Appl. Phys. 80, 5682 (1996). 7. E. Wolf, ed., Theory of Coherence and Polarization of Light (Cambridge University, (2007). 8. C. Brosseau, Fundamentals of Polarized Light, A Statistical Approach (Wiley, 1998). 9. J. W. Goodman, Statistical Optics (Wiley, 2000). 10. R. Barakat, Opt. Commun. 4–6, 123 (1996). 11. P. Réfrégier and A. Luis, J. Opt. Soc. Am. A 25, 2749 (2008). 12. M. Zerrad, J. Sorrentini, G. Soriano, and C. Amra, Opt. Express 18, 15832 (2010). 13. J. Sorrentini, M. Zerrad, G. Soriano, and C. Amra, Opt. Express 19, 21313 (2011). 14. P. Réfrégier, M. Zerrad, and C. Amra, Opt. Lett. 37, 2055 (2012). 15. M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, Opt. Express 21, 2787 (2013). 16. R. M. Illing, in Polarization Science and Remote Sensing IV, J. A. Shaw and J. Scott Tyo, eds. (SPIE, 2009), Vol. 7461, p. 746104. 17. J. Broky and A. Dogariu, Opt. Express 18, 20105 (2010). 18. L. Abel-Tibérini, F. Lemarquis, and M. Lequime, Appl. Opt. 47, 5706 (2008).
6919
Optical systems for controlled specular depolarization Myriam Zerrad,1,* Clément Luitot,2 Jacques Berthon,2 and Claude Amra1 1
Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel, UMR 7249, 13013 Marseille, France 2
CNES, 18 avenue Edouard Belin, 31 401 Toulouse CEDEX 9, France *Corresponding author: [email protected]
Received September 1, 2014; revised November 3, 2014; accepted November 5, 2014; posted November 6, 2014 (Doc. ID 220743); published December 11, 2014 Optical coatings are known to be highly polarizing at oblique incidence. On the other hand, they cannot reduce the degree of polarization of light. We show how to overcome this difficulty and reach spatial depolarization with significant efficiency. Applications concern space optical systems. © 2014 Optical Society of America OCIS codes: (310.5448) Polarization, other optical properties; (260.2130) Ellipsometry and polarimetry; (030.6600) Statistical optics. http://dx.doi.org/10.1364/OL.39.006919
Optical coatings are well known to be highly polarizing, in the sense that they may produce an output polarization (reflected or transmitted) strongly different from the input (incident) polarization. Such property was extensively used to produce polarizing devices [1–5] these last decades; oblique incidence is generally required since thin films are commonly isotropic, though anisotropic films produced at oblique deposition were also considered [6]. For all these devices, polarization is modified or selected, but light remains totally polarized, which means that the temporal and local degree of polarization [7–9] remains unity (DOP 1). To go further, optical coatings do not reduce the DOP of fully polarized incident light [10–15]. This result constitutes a limitation for specific systems (space applications) that are polarization sensitive and hence cannot be calibrated with enough accuracy. Several authors have proposed solutions to solve this point [16], but a demand for significant progress is still expected. In this Letter, we propose an alternative solution to overcome this difficulty. We considered another type of depolarization, the spatial depolarization [12,17], which consists in mixing a high number of polarization states within the receiver aperture or bandwidth. To reach an efficient depolarizing device, variation of polarization must be very fast versus position, wavelength, or incidence, which led us to involve specific multilayers. All these new results and procedures are presented in this Letter. A quasi-monochromatic illumination beam in the far field is here described with an electromagnetic field E0 x; y; z; t propagating on its two analytic transverse polarization modes s and p: ⃗E 0 x; y; z; t E 0S x; y; z; t . (1) E 0P x; y; z; t In relation (1), the propagation direction is z, x; y are the transverse coordinates, and t is for the time. The temporal degree of polarization (DOP) of this field is defined as s detJ DOP0t x; y 1 − 4 2 0t tr J 0t
(2)
0146-9592/14/246919-04$15.00/0
with detJ 0t & trJ 0t , respectively the determinant and trace of the mutual coherence matrix defined in Eq. (3) below: J 0t x; y hjE 0S x; yj2 it hE 0S x; y E 0P x; yit
hE 0S x; yE 0P x; y it hjE 0P x; yj2 it
. (3)
In relation (3), the brackets hit are for temporal averages, and A is the complex conjugate of A. Notice here that we used the subscribe t for most quantities in order to recall that the averages are temporal, in opposition to spatial averages that will appear further in the report. Relation (2) can also be written versus the polarization ratio β0t and the complex mutual coherence degree μ0t as defined below: 8 > hjE 0P x; yj2 it > > < β0t x; y hjE 0S x; yj2 it ; hE 0S x; yE 0P x; y it > > p μ x; y > 0t : hjE 0S x; yj2 it hjE 0P x; yj2 it
4
s β0t x; y 1 − jμ0t x; yj2 . DOP0t x; y 1 − 4 1 β0t x; y2 (5) At this step, all average processes are temporal, so that the degree of polarization is a local quantity within the interval [0,1]. Notice in relation (5) that the DOP is written in a plane of altitude z. Temporal depolarization is a property of light resulting from the average process of optical detectors that cannot follow the high optical frequencies. Hence such property is difficult to control. For this reason, we will here consider another depolarization process resulting from a spatial average over the receiver aperture. This phenomenon has been highlighted in recent papers [12,17] because it spontaneously occurs in the case of a fully polarized incident light scattered by inhomogeneous volumes or rough surfaces. In this specific configuration, the local scattered field still remains fully polarized but presents a high dispersion of polarization states in © 2014 Optical Society of America
6920
OPTICS LETTERS / Vol. 39, No. 24 / December 15, 2014
only a few speckle areas. This leads to a global depolarization resulting from the integration of many speckle areas. The theoretical study of this spatial depolarization is similar to the analysis of temporal depolarization, where the temporal averages are replaced by spatial averages. So, the degree of polarization of the illumination beam is now defined by: s detJ x; y DOP0ΔΩ x; y 1 − 4 2 0ΔΩ ; tr J 0ΔΩ x; y
(6)
where ΔΩ is the domain of spatial integration from the departure point x; y and J 0;ΔΩ x; y the mutual coherence matrix defined Eq. (7) with a¯ the spatial average of variable a over ΔΩ: J 0ΔΩ x; y
hjE 0S x; yj2 it
hE 0S x; yE 0P x; y it
hE 0S x; y E 0P x; yit
hjE 0P x; yj2 it
Fig. 1. Specular depolarizer with a linear filter, configuration of use.
! . (7)
Notice here that J 0;ΔΩ involves quantities that are double averages on time (hit ) and space − . In the same way, the polarization ratio and mutual coherence degree are extended to spatial depolarization as
spatial repartition of the analytic reflected field is given Eq. (11): p r S RS x; yejδS x;y r p ; r P RP x; yejδP x;y
8 < β0ΔΩ x; y hjE0P x; yj2 it ∕hjE 0S x; yj2 it q ; : μ0ΔΩ x; y hE 0S x; yE 0P x; y it ∕ hjE 0S x; yj2 it hjE 0P x; yj2 it
r x; yE 0S . E r x; y; t S r P x; yE 0P
and the resulting degree of polarization is: s β0ΔΩ x; y 1 − jμ0ΔΩ x; yj2 . DOP0ΔΩ x; y 1 − 4 1 β0ΔΩ x; y2 (9) Formula (9) gives a general definition of the degree of polarization taking into account both spatial and temporal averages. Such quantity is a function of the illumination beam, the illuminated medium, and the receiver aperture. Contrary to the previous works devoted to spatial depolarization and which involved scattering patterns from disordered media, we here plan to propose optical components that provide a fully depolarized but specular beam when illuminated with fully or partially polarized collimated specular light. For that we use, an optical filter whose complex reflection coefficient varies with location x; y in the filter plane, as shown in Fig. 1. Such spatially variable filter can be obtained with a low slope linear filter [18], but other techniques can also be used. As for the complex reflection coefficient of the filter, it is defined on the polarization basis Eq. (10) and the
(10)
8
(11)
Notice in these equations that reflection is assumed not to be wavelength dependent, which results from the assumption of a quasi-monochromatic incident beam. The local polarization parameters in the reflected plane (P r ) are expressed versus the incident light properties and the spatial parameters of the filter. They allow to express the local degree of polarization in the reflected beam as
x;y β0 βR x; y RRPS x;y ; jδ x;y−δ S P x;y μ μR x; y e 0
12
s RS x; yRP x; yβ0 DOPR x; y 1 − 4 1 − jμ0 j2 . RS x; y RP x; yβ0 2 (13) We can notice that in the specific case of a reflection without energy losses RS x; y RP x; y∀ x; y, that the reflected DOP equals the incident DOP at any location in the reflected plane.
December 15, 2014 / Vol. 39, No. 24 / OPTICS LETTERS
Now using a single sensor collecting the whole reflected beam, a spatial average is performed and results in a DOP which can be expressed as s αR;ΔΩ β0 1 − jγ R;ΔΩ μ0 j2 14 DOPR;ΔΩ 1 − 4 1 αR;ΔΩ β0 2 with RR ΔΩ r S x; yr P x; ydxdy ; (15) γ R;ΔΩ q RR RR 2 dxdy 2 dxdy jr x; yj jr x; yj S P ΔΩ ΔΩ RR jr x; y:j2 dxdy αR;ΔΩ RRΔΩ P . 2 ΔΩ jr S x; y:j dxdy
(16)
In this particular configuration, ΔΩ must be large enough to integrate the whole reflected specular beam. We notice that the global DOP of the reflected beam is a function of both the incident polarization parameters β0; μ0 of the illumination beam and the spatial repartition of the complex reflection coefficient in terms of phase and amplitude γ R;ΔΩ and αR;ΔΩ . Equations (15) and (16) also allow to write the polarimetric characteristics of the integrated reflected beam in a simplified form: μR;ΔΩ μ0 γ R;ΔΩ
and
βR;ΔΩ β0 αR;ΔΩ .
(17)
An example of application is given Fig. 2. The incident polarization beam is first chosen with equal energy on each polarization axis (x and y) and fully polarized with an arbitrary elliptic polarization state. The light is said to be “balanced” in the chosen axis, which means that the beam energy is equally divided on the two polarization modes, that is, β0 : 1 in the specific referential. In order to simplify the problem, we consider a device that reflects the same energy for each incident polarization (this may happen at total internal reflection, or with specific multilayers). On the other hand, the polarimetric phase shift at reflection is assumed to be linear on xdirection and varying from 0 to 2π. We can notice at this
Fig. 2. Dispersion of polarization states (red points) taken in the reflected beam after reflection of a fully polarized and balanced incident light (blue point) on the filter shown on the left. The resulting polarization states measured by a sensor are reduced to a single central point at the center of the sphere (black point).
6921
step that the linear function can be replaced by any kind of polarimetric phase shift reparation, even random, while γ R;ΔΩ factor equals zero. Under these assumptions the reflection coefficient of the depolarizer is written as Rs x; y Rp x; y ∀ x; y ∈ 0; L2 . δp x; y π Lx −δs x; y
(18)
As a consequence, this depolarizer fulfills γ R;ΔΩ 0 and αR;ΔΩ 1. With such a configuration, the resulting measured reflected beam follows μR;ΔΩ 0 and βR;ΔΩ β0 1 so that is uncorrelated (μR;ΔΩ 0), and fully depolarized DOPR;ΔΩ 0. At this step we can notice that if the reflection coefficients are chosen with a unitary modulus, the full depolarization can be reached without energy losses. A synthesis of this application is given Fig. 2. The spatial repartition of relative parameters (Rp ∕Rs and δp − δs ) of the designed filter given Eq. (18) are plotted on the left side. The incident light is located by the blue point on the Poincaré sphere, and the circle defined by red points is the locus of the polarizations taken in the reflected beam before integration by the sensor. This representation confirms that the local degree of polarization of the reflected beam equals the illumination degree of polarization because all the red points are equidistant to the centre of the sphere. At last, the resulting light after integration by the sensor is plotted in black (central point). Because RS x; y RP x; y∀ x; y, the result is full depolarization (DOPR;ΔΩ 0). We can notice here that this result remains valid for any incoming light checking the property β0 1 whatever its other polarimetric characteristics. Until now the incident light was fully polarized and balanced (DOP0 1 & β0 1) in the referential, and hence very specific. It is now necessary to quantify the efficiency of the procedure with an arbitrary incident light. To this aim, 256 incident polarization states were randomly generated with a DOP in the range (0,1), and are plotted in blue Fig. 3(a) (data points are in the volume and at the surface of the sphere). The resulting light measured by the sensor is located [Fig. 3(b)] in the Poincaré sphere. We can see that whatever the incident DOP, the light measured by the sensor after reflection by the designed component is on the diameter of the sphere passing through s (transverse electric) and p (transverse magnetic) points. Therefore, in the case of an arbitrary incident polarization, the specular depolarizer projects
Fig. 3. (a) Localization of the 256 random incident fields in the Poincaré sphere. (b) Resulting polarization states after reflection by the linear filter and integration by the sensor.
6922
OPTICS LETTERS / Vol. 39, No. 24 / December 15, 2014
Fig. 4. Degree of polarization of the reflected beam after integration by the sensor (DOPR;ΔΩ ) versus the incident polarization ratio for a component designed for γ R;ΔΩ 0 and αR;ΔΩ 1.
the polarization states on a straight line within the sphere. We notice that the performances of the procedure depend on the incident field. More specifically, because γ R;ΔΩ 0, the incident temporal correlation μ0 has no influence on the resulting DOP, which is only a function of the incident polarization ratio β0 . Its evolution is plotted Fig. 4. Role of filter parameters: At this step, a last point is to quantify the efficiency of the depolarizing device versus the spatial properties of its reflection. γ R;ΔΩ is managed by the polarimetric phase shift at reflection, and αR;ΔΩ is controlled by the ratio of the reflected amplitudes. We have to notice here that to conserve the energy of illumination after reflection, we have to impose relationship (19) and so αR;ΔΩ 1: Rs x; y Rp x; y ∀ x; y ∈ 0; L2 .
(19)
Such a condition can be approximated reached with specifically designed multilayers. The variations of polarimetric phase shift induced by the reflection are a key point to control the dispersion of possible polarization states after integration by the sensor. To extend progressively these variations, we will consider a series of reflectors whose phase delay at reflection is defined as δp x; y kπ
x −δs x; y∀ x; y ∈ 0; L2 . L
(20)
The corresponding polarization states at the output of the system are calculated for the 256 random incident illuminations defined Fig. 3(a) and are given [Fig. 5] for k f0; 1∕5; 2∕5; 3∕5; 4∕5; 1g. The locus of the resulting electromagnetic fields is an ellipsoid whose revolution axis is the segment linking s and p polarization states and whose transverse plane is aρ radius circle. ρ is the degree of polarization obtained when the incident light is a balanced and fully polarized beam: sinkπ . (21) kπ The capability to create or control depolarization with specular devices has been demonstrated. Optical component designed to generate a strong dispersion of polarization states within the reflected beam is required. The measurement of the light reflected by such a filter ρ
Fig. 5. Repartition of the resulting polarization states calculated for 256 random polarization states defined Fig. 3(a) with k f0; 1∕5; 2∕5; 3∕5; 4∕5; 1g:
has an apparent degree of polarization which is a function of both the incident light and spatial characteristics of the depolarizing device. It is possible to conserve all the incident energy after reflection. In this specific case, if the phase shift at the reflection takes a multiplicity of values in [0, 2π], the specular depolarizer will give totally unpolarized output light whatever the incident polarization provided that the illumination beam is balanced (β0 1). At last, the performances of the specular depolarizers have been quantified as a function of the filters parameters and the polarimetric characteristics of incoming light. References 1. H. A. MacLeod and A. MacLeod, eds., Thin Film Optical Filters (Taylor & Francis, 2001), p. 3. 2. J. A. Dobrowolski, in Handbook of Optics, BASS (McGrawHill, 1995). 3. P. W. Baumeister, Optical Coating Technology (SPIE Book, 2004). 4. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989). 5. S. A. Furman and A. V. Tikhonrarov, Basics of Optics of Multilayer Systems (Éditions Frontières, 1992). 6. D. Liang, R. W. Smith, and D. J. Srolovitz, J. Appl. Phys. 80, 5682 (1996). 7. E. Wolf, ed., Theory of Coherence and Polarization of Light (Cambridge University, (2007). 8. C. Brosseau, Fundamentals of Polarized Light, A Statistical Approach (Wiley, 1998). 9. J. W. Goodman, Statistical Optics (Wiley, 2000). 10. R. Barakat, Opt. Commun. 4–6, 123 (1996). 11. P. Réfrégier and A. Luis, J. Opt. Soc. Am. A 25, 2749 (2008). 12. M. Zerrad, J. Sorrentini, G. Soriano, and C. Amra, Opt. Express 18, 15832 (2010). 13. J. Sorrentini, M. Zerrad, G. Soriano, and C. Amra, Opt. Express 19, 21313 (2011). 14. P. Réfrégier, M. Zerrad, and C. Amra, Opt. Lett. 37, 2055 (2012). 15. M. Zerrad, G. Soriano, A. Ghabbach, and C. Amra, Opt. Express 21, 2787 (2013). 16. R. M. Illing, in Polarization Science and Remote Sensing IV, J. A. Shaw and J. Scott Tyo, eds. (SPIE, 2009), Vol. 7461, p. 746104. 17. J. Broky and A. Dogariu, Opt. Express 18, 20105 (2010). 18. L. Abel-Tibérini, F. Lemarquis, and M. Lequime, Appl. Opt. 47, 5706 (2008).
