## Partial polarization of light induced by random defects at surfaces or bulks

Optics Express, Vol. 16, Issue 14, pp. 10372-10383 (2008)

http://dx.doi.org/10.1364/OE.16.010372

Acrobat PDF (288 KB)

### Abstract

Partial polarization may be the result of a scattering process from a fully polarized incident beam. It is shown how the “loss of polarization” is connected with the nature of scatterers (surface roughness, bulk heterogeneity) and on the receiver solid angle. These effects are theoretically predicted and confirmed via multiscale polarization measurements in the speckle pattern of rough surfaces. “Full” polarization can be recovered when reducing the receiver aperture.

© 2008 Optical Society of America

## 1. Introduction

1. J. Li, G. Yao, and L. V. Wang, “Degree of polarization in laser speckles from turbid media: implications in tissue optics”, J. Biomed.Opt. **7**, 307 (2002). [CrossRef] [PubMed]

2. D. J. L. Graham, A. S. Parkins, and L. R. Watkins, “Ellipsometry with polarisation-entangled photons,” Opt. Express **14**, 7037–7045 (2006). [CrossRef] [PubMed]

3. M. J. R. Previte and C. D. Geddes, “Fluorescence microscopy in a microwave cavity,” Opt. Express **15**, 11640–11649 (2007). [CrossRef] [PubMed]

4. J. Moreau, V. Loriette, and A. -C. Boccara, “Full-Field Birefringence Imaging by Thermal-Light Polarization-Sensitive Optical Coherence Tomography. I. Theory,” Appl. Opt. **42**, 3800–3810 (2003). [CrossRef] [PubMed]

5. C. Amra, C. Deumié, and O. Gilbert, “Elimination of polarized light scattered by surface roughness or bulk heterogeneity,” Opt. Express **13**, 10854–10864 (2005). [CrossRef] [PubMed]

6. G. Georges, C. Deumié, and C. Amra, “Selective probing and imaging in random media based on the elimination of polarized scattering,” Opt. Express **15**, 9804–9816 (2007). [CrossRef] [PubMed]

## 2. Basic principles

### 2-1 Polarization of a plane wave

**E**(

**ρ**) the complex electric field associated with a plane wave (Fig. 1):

**A**the vector complex amplitude,

**ρ**=(

**r**,z) =(x,y,z) the space coordinates,

**k**the wave vector and

**σ**the spatial pulsation:

**A**lies in a plane perpendicular to the wave vector (Fig. 2), it can be splitted into two polarization vector components:

**u**,

**v**) plane as:

**ρ**the field direction is constant when δ

_{S}=δ

_{P}, in which case polarization is said to be linear. In other situations (δ=δ

_{S}-δ

_{P}≠0), rotation of the field occurs in the plane perpendicular to the wave vector, and polarization is said to be elliptic. The case of non polarized (natural) light is traditionally addressed with δ as a random variable within the interval (0,2π).

### 2-2 Measurements procedure

_{P}/A

_{S}|. In the case |A

_{S}|=|A

_{P}|, the contrast of the curve is given by 2cosδ, which constitutes a signature of the polarized interference (Fig. 3). Other techniques allow accurate analysis of polarization, in particular those involving specific devices to modulate the incident polarization. Most often with these techniques Eq. (9) is unchanged but a time function η(t) is added to the polarimetric phase term δ, which creates several harmonics in the resulting field.

### 2-3 Procedure limits

_{S}|=|A

_{P}|). Mueller matrices [7

7. T. Novikova, A. De Martino, P. Bulkin, Q. Nguyen, B. Drévillon, V. Popov, and A. Chumakov, “Metrology of replicated diffractive optics with Mueller polarimetry in conical diffraction,” Opt. Express **15**, 2033–2046 (2007). [CrossRef] [PubMed]

_{S}-η

_{P}the polarization phase delay of the plate. With this additional parameter, a maximum contrast can be reached since an ellipsometric zero of I” is obtained for tgψ= |A

_{S}/A

_{P}| provided that the retardation term η is matched such as δ+η=(2n+1)π. From the point of view of experiment, the zero value will be replaced by a minimum connected with the relative efficiency of crossed analyzers and polarizers. These points will be helpful at the end of the next section.

## 3. Polarization of a wave packet

**r**=>

**σ**) Fourier Transform

**Ê**(

**σ**,z) of the field, which is valid for waves of finite spatial extension. Therefore the field given in Eq. (12) can be the far field result of a diffraction or scattering process. Since Maxwell equations offer rigorous solutions to these optical interactions, each elementary component

**Ê**(

**σ**,z) under the integral (12) is a “fully” polarized plane wave provided that the original (incident) field is fully polarized. Therefore all polarization parameters δ(

**σ**) and r(

**σ**)=|A

_{P}/A

_{S}|(

**σ**) can be theoretically derived from the knowledge of the sample under illumination. Strictly speaking, the phase difference δ(

**σ**) will be characteristic of the elliptic or linear polarization of the Fourier component

**Ê**(

**σ**,z) at infinity in the far field.

### 3.1 Measurements procedure

**E**. Measurements at a finite distance involve a solid angle ΔΩ in which the polarization parameters δ(

**σ**) and r(

**σ**) should not vary in order to be able to define a polarization state in a classical way. If it is not the case, the angular frequency or angular variations create an averaging process within the receiver solid angle ΔΩ, resulting in an apparent “loss” of polarization. To investigate and quantify this effect, we first write the wave packet after passing through the analyzer, as an algebric sum on the analyzer axis:

**E*** which is carried within the receiver. For this purpose we limit the frequency support Δσ of integral (13) to spatial frequencies that give rise to scattering angles collected in the far field within the receiver solid angle. After direct analytical calculation, this flux can be expressed as:

_{S}/F

_{P}of the wave packet in a way similar to that of plane waves.)

### 3-2 Equivalent or apparent polarization

5. C. Amra, C. Deumié, and O. Gilbert, “Elimination of polarized light scattered by surface roughness or bulk heterogeneity,” Opt. Express **13**, 10854–10864 (2005). [CrossRef] [PubMed]

6. G. Georges, C. Deumié, and C. Amra, “Selective probing and imaging in random media based on the elimination of polarized scattering,” Opt. Express **15**, 9804–9816 (2007). [CrossRef] [PubMed]

**σ**) has been replaced by δ(

**σ**)+η, with η the tunable phase delay. When the angular variations can be neglected in the integral (21), we obtain a relationship similar to that of a plane wave (see section 4–1), and zero values of intensity can again be obtained provided that the condition δ+η=(2n+1)π is fulfilled. In all other situations, extinction of the field cannot be reached, due to the constant η value in opposition to the variable phase term δ(

**σ**). In other words, the extreme values ±1 cannot be reached by parameter β in the general case of a wave packet with high angular polarization variations. The result is that the ellipsometric zeros of the intensity curve F(ψ,η) will be replaced by minima. Therefore equivalent polarization cannot here be introduced, and the equivalent phase δ* will be said to be characteristic of partial polarization.

5. C. Amra, C. Deumié, and O. Gilbert, “Elimination of polarized light scattered by surface roughness or bulk heterogeneity,” Opt. Express **13**, 10854–10864 (2005). [CrossRef] [PubMed]

_{0}that allow to neglect the variations of parameters δ(

**σ**) and r(

**σ**) within the integrals, so that “full” polarization can be recovered. The correlation length of scatterers will be responsible for the minimum value ΔΩ

_{0}.

## 4- Defect-induced partial polarization

### 4-1 The case of low-scattering levels

8. J. M. Elson, J. P. Rahn, and J. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties,” Appl. Opt. **22**, 3207–19 (1983). [CrossRef] [PubMed]

^{d}to be proportional to the Fourier Transform ĝ(σ) of defects, with g(

**r**) the surface topography (case of surface scattering) or the relative variations in the bulk permittivity (case of volume scattering). These theories have shown successful results when the ratio of roughness to wavelength, or the root mean square of bulk heterogeneity, is much less than unity [9

9. C. Amra, C. Grezes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. **32**, 5492–503 (1993). [CrossRef] [PubMed]

**d**~ĝ(σ)), it is immediate to show that the polarimetric phase term δ(

**σ**) does not depend on the microstructure of the scattering sample:

6. G. Georges, C. Deumié, and C. Amra, “Selective probing and imaging in random media based on the elimination of polarized scattering,” Opt. Express **15**, 9804–9816 (2007). [CrossRef] [PubMed]

_{SorP}optical coefficients are connected with polarization, refractive index, wavelength and incidence…, but not on microstructure. Moreover, these coefficients exhibit low variations versus scattering angle in the whole range (0°, 90°) [5

**13**, 10854–10864 (2005). [CrossRef] [PubMed]

**σ**)=(4π

^{2}/S)|ĝ(σ)|

^{2}the roughness or permittivity spectrum of defects, and S the illuminated area on the sample.

### 4.2 Arbitrary scattering levels

11. L. Arnaud, G. Georges, C. Deumié, and C. Amra, “Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis”, Optics Communications , Volume **281**, Issue 6, 15 March 2008, Pages1739–1744. [CrossRef]

12. O. Gilbert, C. Deumié, and C. Amra, “Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks,” Opt. Express **13**, 2403–2418 (2005). [CrossRef] [PubMed]

**σ**) of these samples may be uniformly distributed within (0°, 360°) and emphasize rapid variations versus scattering angle when the correlation length is decreased. Such variations can no more be neglected in the previous integrals (17–19). In this case measurements may detect partial polarization as an average process over the receiver aperture, unless we reduce the receiver solid angle to recover full polarization. We address this point in the section 5.

### 4.3 Comparison with Stokes formalism

1. J. Li, G. Yao, and L. V. Wang, “Degree of polarization in laser speckles from turbid media: implications in tissue optics”, J. Biomed.Opt. **7**, 307 (2002). [CrossRef] [PubMed]

_{3}. In these condtiions the dynamic d=F

_{max}/F

_{min}that we measure in the next sections and which is obtained when ψ is varied, can be expressed as the ratio:

## 5- Experimental data on rough surfaces

_{ij}(ψ,τ) delivered by one or several pixels (i,j), which allows to emphasize polarization averages versus solid angle. Each pixel is 5µm in length and the integration time is 2s. The illumination incidence is 56°, whereas the measurement angle is close to the specular direction (120° from the incident beam). Notice that the speckle size was approximately given by λ/a=0.011°, with a=2mm the radius of the spot size on the sample.

_{max}and minima F

_{min}for each pixel or collection of pixels of the sensor. For the sake of simplicity, we will be focused on the dynamic d=F

_{max}/F

_{min}of the data rather that on the phase term δ that can be derived from the whole F(ψ,τ) curve.

### 5-1 Case of a polished surface

^{-4}) is specific of first-order scattering. The scattering speckle is given in figure 5, for a 2080×2600 µm

^{2}field view. Since the measurement distance is 50cm from the sample, the angular aperture Δθ is 0.30° for the whole sensor field.

**15**, 9804–9816 (2007). [CrossRef] [PubMed]

### 5-2 Case of an arbitrary rough surface: multiscale polarization data

12. O. Gilbert, C. Deumié, and C. Amra, “Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks,” Opt. Express **13**, 2403–2418 (2005). [CrossRef] [PubMed]

^{2}.

_{min}, τ

_{min}) for minimization are different for each pixel, in opposition to the previous case of low-level scattering. In other words, from one pixel to another the variations are independent when the 2 plates (analyzer and quarterwave) are rotated. As discussed in the previous section, this is due to the fact that the phase term δ(

**σ**) is now microstructural dependent and exhibits high variations versus scattering angle (here versus pixel). For this reason we studied different zones of the CCD area that are fitted into each other:

- The second zone (2) is included in zone (1), and appears at the middle of Fig. 7. The field view is 165×300 µm
^{2}, which corresponds to an angular aperture (Δθ) of 0.034°. - The third zone (3) is included in zone (2), and appears at the right of Fig. 7. It is related to one pixel (5×5µm
^{2}), which corresponds to an angular aperture (Δθ) of 5.7 10^{-4}°.

_{max}/I

_{min}) of the I(ψ,τ) curve. The results are given in table 1, together with the corresponding value of solid angle. As predicted in the previous sections, we observe a noticeable increase of the dynamic (from 3.7 to 26.6) when the angular aperture is reduced, which indicates a progressive recovering of polarization. These dynamic data could be more accurate with a specific zooming around the minima positions, but the procedure is time consuming. The maximum dynamic is reached for the single pixel (zone 3) for which the angular aperture is lower than the speckle size. Notice that in the absence of the procedure here discussed, an apparent phase term δ* or partial polarization would be measured for each angular aperture.

## 6- Conclusion

**13**, 10854–10864 (2005). [CrossRef] [PubMed]

**15**, 9804–9816 (2007). [CrossRef] [PubMed]

## References and links

1. | J. Li, G. Yao, and L. V. Wang, “Degree of polarization in laser speckles from turbid media: implications in tissue optics”, J. Biomed.Opt. |

2. | D. J. L. Graham, A. S. Parkins, and L. R. Watkins, “Ellipsometry with polarisation-entangled photons,” Opt. Express |

3. | M. J. R. Previte and C. D. Geddes, “Fluorescence microscopy in a microwave cavity,” Opt. Express |

4. | J. Moreau, V. Loriette, and A. -C. Boccara, “Full-Field Birefringence Imaging by Thermal-Light Polarization-Sensitive Optical Coherence Tomography. I. Theory,” Appl. Opt. |

5. | C. Amra, C. Deumié, and O. Gilbert, “Elimination of polarized light scattered by surface roughness or bulk heterogeneity,” Opt. Express |

6. | G. Georges, C. Deumié, and C. Amra, “Selective probing and imaging in random media based on the elimination of polarized scattering,” Opt. Express |

7. | T. Novikova, A. De Martino, P. Bulkin, Q. Nguyen, B. Drévillon, V. Popov, and A. Chumakov, “Metrology of replicated diffractive optics with Mueller polarimetry in conical diffraction,” Opt. Express |

8. | J. M. Elson, J. P. Rahn, and J. M. Bennett, “Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties,” Appl. Opt. |

9. | C. Amra, C. Grezes-Besset, and L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. |

10. | S. Kassam, A. Duparre, K. Hehl, P. Bussemer, and J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. |

11. | L. Arnaud, G. Georges, C. Deumié, and C. Amra, “Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis”, Optics Communications , Volume |

12. | O. Gilbert, C. Deumié, and C. Amra, “Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks,” Opt. Express |

**OCIS Codes**

(120.6150) Instrumentation, measurement, and metrology : Speckle imaging

(240.5770) Optics at surfaces : Roughness

(260.2110) Physical optics : Electromagnetic optics

(260.2130) Physical optics : Ellipsometry and polarimetry

(290.5820) Scattering : Scattering measurements

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: January 2, 2008

Revised Manuscript: March 7, 2008

Manuscript Accepted: March 21, 2008

Published: June 27, 2008

**Citation**

Claude Amra, Myriam Zerrad, Laure Siozade, Gaelle Georges, and Carole Deumié, "Partial polarization of light induced by random defects at surfaces or bulks," Opt. Express **16**, 10372-10383 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-14-10372

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### References

- J. Li, G. Yao, and L. V. Wang, "Degree of polarization in laser speckles from turbid media: implications in tissue optics," J. Biomed. Opt. 7, 307 (2002). [CrossRef] [PubMed]
- D. J. L. Graham, A. S. Parkins, and L. R. Watkins, "Ellipsometry with polarisation-entangled photons," Opt. Express 14, 7037-7045 (2006). [CrossRef] [PubMed]
- M. J. R. Previte and C. D. Geddes, "Fluorescence microscopy in a microwave cavity," Opt. Express 15, 11640-11649 (2007). [CrossRef] [PubMed]
- J. Moreau, V. Loriette, and A. -C. Boccara, "Full-Field Birefringence Imaging by Thermal-Light Polarization-Sensitive Optical Coherence Tomography. I. Theory," Appl. Opt. 42, 3800-3810 (2003). [CrossRef] [PubMed]
- C. Amra, C. Deumié, and O. Gilbert, "Elimination of polarized light scattered by surface roughness or bulk heterogeneity," Opt. Express 13,10854-10864 (2005). [CrossRef] [PubMed]
- G. Georges, C. Deumié, and C. Amra, "Selective probing and imaging in random media based on the elimination of polarized scattering," Opt. Express 15, 9804-9816 (2007). [CrossRef] [PubMed]
- T. Novikova, A. De Martino, P. Bulkin, Q. Nguyen, B. Drévillon, V. Popov, and A. Chumakov, "Metrology of replicated diffractive optics with Mueller polarimetry in conical diffraction," Opt. Express 15, 2033-2046 (2007). [CrossRef] [PubMed]
- J. M. Elson, J. P. Rahn, and J. M. Bennett, "Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties," Appl. Opt. 22, 3207-19 (1983). [CrossRef] [PubMed]
- C. Amra, C. Grezes-Besset, and L. Bruel, "Comparison of surface and bulk scattering in optical multilayers," Appl. Opt. 32, 5492-503 (1993). [CrossRef] [PubMed]
- S. Kassam, A. Duparre, K. Hehl, P. Bussemer, and J. Neubert, "Light scattering from the volume of optical thin films: theory and experiment," Appl. Opt. 31, 1304-13 (1992). [CrossRef] [PubMed]
- L. Arnaud, G. Georges, C. Deumié, and C. Amra, "Discrimination of surface and bulk scattering of arbitrary level based on angle-resolved ellipsometry: Theoretical analysis," Opt. Commun. 281, 1739-1744 (2008). [CrossRef]
- O. Gilbert, C. Deumié, and C. Amra, "Angle-resolved ellipsometry of scattering patterns from arbitrary surfaces and bulks," Opt. Express 13, 2403-2418 (2005). [CrossRef] [PubMed]

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