Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Current Optics and Photonics 2017; 1(2): 85-89
Published online April 25, 2017 https://doi.org/10.3807/COPP.2017.1.2.085
Copyright © Optical Society of Korea.
Xinyu Peng, Dong Ye, Guo Zheng, Qi Zhao^{*}, and Minmin Song
Corresponding author: zhaoqi@njust.edu.cn
It is shown that polarization singularities of a new type, namely
Keywords: Polarization, Coherence, Singular optics, Spatial light modulator
Since 1974 [1], when Nye and Berry first found dislocations in optical fields, singular optics have widely attracted attention. While phase singularities (wave dislocations, or optical vortices) are frequently encountered in the interference of scalar waves [2, 3], they evolve into polarization singularities when the vector nature of light is retained. The main subject of coherent singular optics often refers to the singular optics of vector fields [4-10]. Within the framework of vector singular optics, one considers the set of
Since both
In this paper, two mutually incoherent and orthogonally linearly polarized beams with different distributions of intensity are coaxially mixed. Then, the condition for the occurrence of
Let us consider vector singularities in partially coherent optical beams by mixing two mutually incoherent and orthogonally linearly polarized beams with different distributions of intensity.
To analyze the conditions for the occurrence of optical singularities, we need to proceed from the Jones vectors of two orthogonally linear polarized beams, an
where
The coherence matrix of the beam is defined as [17]
The angular brackets denote an ensemble average, and an asterisk denotes a complex conjugate.
For our purpose, we first consider the limiting case when the two beams are completely mutually coherent. With the disappearance of any optical-path difference between two beams, we believe that two components are completely mutually coherent. The coherence matrix can then be written as
Combining the elements of the coherence matrix, one can find the full Stokes parameters:
where Δ =
where
The azimuthal angle
Here we only consider the polarization ellipse with respect to the distribution of intensities of two components. To simplify, we set .
When the intensities of the two components are equal,
Analogously, when one of the intensities of the mixed components is zero,
Before considering the most general case of partial mutual coherence of the mixed orthogonally polarized beams, let us consider another limiting case,
It is clear that when two components become equal in intensity, the normalized Stokes parameters of the combined beam become {1,0,0,0}. The point for such elements is completely unpolarized, and a
If one of the intensities of the mixed components is equal to zero, then the degree of polarization equals unity, which is referred to as a P (completely polarized) point. Its location is determined by the vanishing component of the combined beam. The set of
We emphasize that the conditions under which
According to the analysis above, we know that the degree of polarization is just related to the intensities of the mixed, completely incoherent beams. The degree of polarization can be represented in terms of the coherence matrix:
where det and tr respectively denote the determinant and trace of the coherence matrix. Substituting Eq. (9) into Eq. (10), we obtain the following relationship:
where is the intensity of the mixed beam. From Eq. (11), we can also come to the conclusion following Eq. (9). We can use this relationship to generate
The laser emitted from polarizer
Figures 5 and 6 show the one-dimensional intensity distributions for the two mixed beams, and the degree of polarization of the combined beam. We can see that when the intensities of the components are equal, the degree of polarization is zero-a
Also, comparing Figs. 5 and 6, the distribution of degree of polarization in experiment is in quite satisfactory qualitative agreement with the simulation result, except for the edge of the light caused by the fluctuating intensity, which is caused by the rotating ground-disk. The intensity on the edge is weak. According to Eq. (11), when small fluctuations occur, the degree of polarization may experience large fluctuations. We can see, though, that the U and P singularities are generated very well. So, we can generate any desired two-dimensional distribution of degree of polarization by combining two mutually incoherent beams.
At last, we can obtain the vector skeleton of the combined beam, as shown in Fig. 7. The contour of the
In this paper, we first introduced new polarization singularities
Current Optics and Photonics 2017; 1(2): 85-89
Published online April 25, 2017 https://doi.org/10.3807/COPP.2017.1.2.085
Copyright © Optical Society of Korea.
Xinyu Peng, Dong Ye, Guo Zheng, Qi Zhao^{*}, and Minmin Song
Correspondence to:zhaoqi@njust.edu.cn
It is shown that polarization singularities of a new type, namely
Keywords: Polarization, Coherence, Singular optics, Spatial light modulator
Since 1974 [1], when Nye and Berry first found dislocations in optical fields, singular optics have widely attracted attention. While phase singularities (wave dislocations, or optical vortices) are frequently encountered in the interference of scalar waves [2, 3], they evolve into polarization singularities when the vector nature of light is retained. The main subject of coherent singular optics often refers to the singular optics of vector fields [4-10]. Within the framework of vector singular optics, one considers the set of
Since both
In this paper, two mutually incoherent and orthogonally linearly polarized beams with different distributions of intensity are coaxially mixed. Then, the condition for the occurrence of
Let us consider vector singularities in partially coherent optical beams by mixing two mutually incoherent and orthogonally linearly polarized beams with different distributions of intensity.
To analyze the conditions for the occurrence of optical singularities, we need to proceed from the Jones vectors of two orthogonally linear polarized beams, an
where
The coherence matrix of the beam is defined as [17]
The angular brackets denote an ensemble average, and an asterisk denotes a complex conjugate.
For our purpose, we first consider the limiting case when the two beams are completely mutually coherent. With the disappearance of any optical-path difference between two beams, we believe that two components are completely mutually coherent. The coherence matrix can then be written as
Combining the elements of the coherence matrix, one can find the full Stokes parameters:
where Δ =
where
The azimuthal angle
Here we only consider the polarization ellipse with respect to the distribution of intensities of two components. To simplify, we set .
When the intensities of the two components are equal,
Analogously, when one of the intensities of the mixed components is zero,
Before considering the most general case of partial mutual coherence of the mixed orthogonally polarized beams, let us consider another limiting case,
It is clear that when two components become equal in intensity, the normalized Stokes parameters of the combined beam become {1,0,0,0}. The point for such elements is completely unpolarized, and a
If one of the intensities of the mixed components is equal to zero, then the degree of polarization equals unity, which is referred to as a P (completely polarized) point. Its location is determined by the vanishing component of the combined beam. The set of
We emphasize that the conditions under which
According to the analysis above, we know that the degree of polarization is just related to the intensities of the mixed, completely incoherent beams. The degree of polarization can be represented in terms of the coherence matrix:
where det and tr respectively denote the determinant and trace of the coherence matrix. Substituting Eq. (9) into Eq. (10), we obtain the following relationship:
where is the intensity of the mixed beam. From Eq. (11), we can also come to the conclusion following Eq. (9). We can use this relationship to generate
The laser emitted from polarizer
Figures 5 and 6 show the one-dimensional intensity distributions for the two mixed beams, and the degree of polarization of the combined beam. We can see that when the intensities of the components are equal, the degree of polarization is zero-a
Also, comparing Figs. 5 and 6, the distribution of degree of polarization in experiment is in quite satisfactory qualitative agreement with the simulation result, except for the edge of the light caused by the fluctuating intensity, which is caused by the rotating ground-disk. The intensity on the edge is weak. According to Eq. (11), when small fluctuations occur, the degree of polarization may experience large fluctuations. We can see, though, that the U and P singularities are generated very well. So, we can generate any desired two-dimensional distribution of degree of polarization by combining two mutually incoherent beams.
At last, we can obtain the vector skeleton of the combined beam, as shown in Fig. 7. The contour of the
In this paper, we first introduced new polarization singularities