1. Introduction

Over the past decade, several research groups have described polarization vortices that are a class of space-variant polarization states [1–10]. An optical vortex is a point which exhibits a phase singularity with a null in the field at that point. The phase of the field evolves from 0 to 2π (or some multiple thereof) in any circular path traced about that point. A polarization vortex is an optical vortex that has a space-variant polarization orientation that evolves about the singularity. Radial and azimuthal beams are examples of the lowest-order polarization vortex beams, and they are single mode solutions to the vector wave equations [1, 2].

The present work examines the union of polarization vortices and optical coherence theory. For much of its history, optical coherence theory dealt primarily with scalar fields of homogeneous polarization. Hopkins and others recognized the important role of coherence in image formation [11–13]. Only recently has the study of coherence included vector fields [14–21]. Extending on the ideas of Zernike [22], the degree of coherence, degree of correlation, and degree of polarization were adopted to describe the coherence properties of an inhomogeneous vector field. It is only recently that the first combined measurements of polarization and coherence properties of an optical source have been carried out [23, 24].

The rigorous formulation of optical coherence theory is usually framed as a propagation problem. It is in this context that Gbur, Visser, and Wolf [25] and Swartzlander and coworkers [26–29] have explored the behavior of partially coherent scalar vortex fields. An alternative way to formulate coherence theory was outlined by Hopkins [11], in which he showed how the coherent impulse response of an imaging system could be used to construct the coherence function of light at the image plane.

In this paper, we describe an experiment in which a critical illumination system forms a superposition of uncorrelated polarization vortices at the object plane of an imaging system. We show the theoretical description and experimental determination of the correlation matrix at an object plane for a partially correlated source created by placing an azimuthal polarization vortex mode converter in the pupil plane of a critical illumination system. Finally, we show an experimental situation in which the polarization-dependent correlations have a profound impact on image contrast.

2. Theory

Following Zernike [22] and Hopkins [11], we model a critical illumination system in which a condenser system images a spatially incoherent source of uniform irradiance to an object plane conjugate to the source (Fig. 1(a)). The condenser is equipped with optics that transform each point on the source into a polarization vortex field distribution over the plane of the object under illumination. The illumination field at the object plane may, quite generally, be written as follows:

in which A _j(ρ) is the jth electric field component of the source (a paraxial approximation with only transverse components considered) and P _ij(r-ρ) is the shift-invariant impulse response of the condenser system. The subscripts (i, j) indicate that the distribution of the ith component of the electric field in the object plane is produced from the jth component of the electric field in the source plane. As depicted in Fig. 1(a), the spatial coordinates of the source and object planes are ρ = ux̂+vŷ and r=xx̂+yŷ, respectively.

The unification of the subjects of coherence and polarization has mainly centered around the electric cross-spectral density matrix that is the temporal Fourier transform of the electric mutual coherence matrix [21]. If E _i(r) and E _j(r) are components of the electric field and are members of suitably constructed statistical ensembles, the cross-spectral density may be expressed as a correlation matrix [21]. We are interested in predicting and measuring the correlation matrix for the illumination field at the object plane with the form:

where we have substituted Eq. (1). The frequency dependence of the correlation matrix is implied. Since our experiments employ a nearly monochromatic source, the frequency dependence is unimportant in this work.

We consider a uniform, spatially incoherent source described by a diagonal polarization matrix such that:

in which S _k represents the spectral density of the k ^th polarization component of the source at the point ρ ₁. For a source with this polarization matrix, Eq. (2) then reduces to:

We use a model impulse response tensor similar to that formulated by Borghi and Santarsiero [30]:

This form for P⃡ has the important property that an x-polarized source (S _x = 1, S _y = 0) will produce the azimuthal impulse response [-(y-v)x+(x-u)ŷ]exp[-β((x-u)²+(y-v)²)] and a y-polarized source (S _x = 0, S _y = 1) will produce the radial response [(x-u)x̂+(y-v)ŷ]exp[-β (((x-u)²+(y-v)²)]. The quantity β is dependent on the wavelength of light used, the details of the polarization vortex mode converter, and the choice of condenser optical components. In the remaining discussion we will consider a uniform, x-polarized source of infinite extent that produces an azimuthal impulse response.

Using a substitution of variables where:

${\Delta }_x=x_1-x_2,{\Delta }_y=y_1-y_2,x_0=\frac{x_1+x_2}{2},\mathrm{and}{y}_0=\frac{y_1+y_2}{2},$

we can write the correlation matrix for the illumination field at the object plane as a function of the horizontal (Δ_x) and vertical (Δ_y) separation between the two spatial coordinates r ₁ and r ₂ and of the mean horizontal (x ₀) and vertical (y ₀) position between r ₁ and r ₂. Using a trace normalization at Δ_x = Δ_y = 0, the correlation matrix for the illumination field at the object plane of this partially correlated azimuthal vortex (PCAV) illumination system is simply:

This important result has a very simple form that is only dependent on Δ_x, Δ_y, and β. We point out that the W _xx component of the correlation matrix becomes negative when ${\Delta }_y>\sqrt{1⁄\beta }$, and the W _yy component becomes negative when ${\Delta }_x>\sqrt{1⁄\beta }$.

Consequentially, this illumination system produces vertically polarized fields (W _yy) that are anti-correlated (180 degrees out of phase) for positions in the object plane that are separated horizontally by a distance larger than$\sqrt{1⁄\beta }$. At the same time, horizontally polarized fields (W _xx) are anti-correlated for positions in the object plane that are separated vertically by a distance larger than $\sqrt{1⁄\beta }$. On the contrary, W _xx is correlated (in phase) for horizontally separated features and W _yy is correlated for vertically separated features. It can be easily shown that W⃡ is invariant under rotation, consistent with the azimuthal symmetry of the vortex and the symmetry of the partially coherent critical illumination system.

3. Experimental details

The experimental construct of our PCAV illumination system is shown in Fig. 1(b). A spatially filtered, quasi-monochromatic (λ = 532nm) laser beam is first focused onto a rotating ground glass diffuser. Light from this nearly uniform, spatially incoherent source is then propagated through an azimuthal polarization vortex mode converter placed in the pupil plane of a critical illumination system. This mode converter creates an azimuthal vortex state of polarization using circular polarizers, an azimuthal polarization analyzer (TSI, Inc.), and a space-variant stress-birefringent glass window originally described by Spilman and Brown [10].

To experimentally determine the correlation matrix for the illumination field at the object plane of this PCAV illumination system, we utilized a reversed-wavefront Young interferometer (RWYI) originally described by Borghi and Santarsiero [31] and recently modified by Brown et al. [23]. As shown in Fig. 2, the slowly converging light from the illumination system is sent through a (nominally) non-polarizing 50/50 beamsplitter cube that passes the original illumination wavefront (denoted by R in the figure) and creates a laterally offset, horizontally reversed-wavefront replica (denoted by the reversed R in the figure). In brief, the modified RWYI used in our experiments utilizes polarization analyzers, optical shutters, and a Young’s double pinhole mask to determine the magnitude and phase of the four polarization-and space-dependent correlation matrix components.

 

Fig. 1. (a) Illustration of the conjugate planes and coordinate systems used in this paper. (b) Partially correlated azimuthal vortex (PCAV) illumination system. A spatially-filtered, quasi-monochromatic laser beam is focused with a lens (focal length f ₁) onto a ground glass diffuser creating an array of independent point sources. The light is then relayed to an image plane with lenses of focal length f ₂ and f ₃. A left-hand circular (LHC) polarizer, space-variant stress-birefringent glass window, right-hand circular (RHC) analyzer, and azimuthal analyzer create an azimuthal polarization vortex within the entire pupil. The condenser lens creates a superposition of azimuthal polarization vortices at the object plane.

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Due to the non-standard orientation of the beamsplitter, both polarization-dependent losses (e.g. from Fresnel reflections) and amplitude/phase errors in the immersed dielectric layers are present, requiring careful calibration of the optical system [23]. This calibration was carried out with a uniformly coherent plane-wave input source having equal components of vertical and horizontal polarizations.

Because of the reversed-wavefront geometry, the shutters, polarization analyzers, and pinhole mask can move in a simple horizontal translation to determine the correlation matrix components. This geometry also results in a constant fringe period. The horizontal translation is equivalent to a horizontal pinhole separation (Δ_x) on the original wavefront that can vary from zero to the full extent of the beam with a fixed vertical pinhole separation of zero (Δ_y = 0). Light passing through the two pinholes is interfered and imaged onto a CCD camera. By changing the orientations of the two polarization analyzers, four sets of interference fringes are analyzed for visibility and phase at each step of Δ_x. The measured visibility and phase correspond to the magnitude and phase, respectively, of the four correlation matrix components. See Ref. [23] for more information on the analysis.

 

Fig. 2. Reversed-wavefront Young interferometer (RWYI) with converging illumination source. A non-polarizing 50/50 beamsplitter cube creates reversed-wavefront replica. Polarization analyzers can be changed to select the different polarization components of each beam. Optical shutters can block each beam to capture the irradiance from each pinhole separately. A Young’s double pinhole mask translates horizontally and samples each beam. Condensing and imaging optics then interfere the light from each pinhole and optically magnify the interference fringes onto a CCD camera.

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4. Discussion

Experimentally implementing a PCAV illumination system and sending it into the RWYI with the Young’s double pinhole mask aligned at the object plane, we were able to experimentally determine the correlation matrix for a PCAV illumination system. Figure 3 compares our experimental results for the diagonal components of the correlation matrix with the theoretical description discussed above in Eq. (6). Both components have been normalized to unity at Δ_x = 0. The plot on the left is the magnitude of the diagonal components, and the plot on the right is the phase. One can see excellent agreement between the experimental results and the theoretical predictions.

The most striking feature of the diagonal components of the correlation matrix for a PCAV illuminator is an anti-correlation that exists for some values of Δ_x for one polarization but not the other. In this experiment, for values of Δ_x that are larger than about 100 microns, the W _yy component becomes anti-correlated while the W _xx component remains correlated. This means that if two points in the object plane are horizontally separated by more than about 100 microns, the vertical polarization component of light passing through these two points will be 180 degrees out of phase. At the same time, the horizontal polarization component of light passing through these two points will be in phase.

It is interesting to note that the strong polarization dependence of W⃡ exists despite the fact that the degree of polarization (DoP) is zero. Using the usual definition [18]:

and Eq. (6), it can be easily shown that the DoP is zero over the entire object plane. This result can be understood as follows: since every independent point on the source creates an azimuthal impulse response, the illumination system creates a superposition of azimuthal polarization vortices at the object plane. This means every point in the object plane contains an incoherent superposition of all possible linear polarization states. This superposition results in an unpolarized field in which the correlation properties of points in the object plane are polarization dependent.

 

Fig. 3. Diagonal components of the correlation matrix for the illumination field at the object plane of a PCAV illumination system. The magnitude is shown on the left, and the phase is shown on the right. The symbols correspond to the experimental data, and the lines correspond to the theoretical predictions. The vertical polarization component, W _yy, (red squares and magenta line) exhibits an anti-correlation property for values of Δ_x larger than 100 microns, but the horizontal polarization component, W _xx, (blue circles and cyan line) is partially coherent for all non-zero values of Δ_x. Both components are fully coherent at Δ_x = 0.

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The polarization-dependent phase of the field correlations has a profound impact on image contrast. To study this, we incorporated PCAV illumination into a simple video microscope and observed the effects on image performance due to the measured properties of the correlation matrix for this illumination at the object plane. Figure 4 shows the effects of placing a horizontal or vertical polarization analyzer in a low numerical aperture (NA = 0.004) imaging system with a 1951 USAF resolution target as the object. The center-to-center distance for the lines of interest is 140.3 microns. As expected from the RWYI experimental results, the contrast of horizontally separated features that are separated by more than 100 microns is higher when using a vertical polarization analyzer then when using a horizontal polarization analyzer. This is due to the anti-correlation property in theW _yy component of the correlation matrix.

This important result can be scaled to imaging systems with a higherNA by scaling the NA of the PCAV illumination system proportionally. Figure 5 demonstrates this scalability. For these images, the NA of the illumination and imaging systems is 0.25, which is 62.5 times higher than the NA used in the case shown in Fig. 4. This means the anti-correlation property in the W _yy component of the correlation matrix will improve image contrast for horizontally separated features separated by more than about 1.6 microns. In Fig. 5, the middle set of bars have a center-to-center separation of 1.74±0.09 microns, and the contrast is again higher when using a vertical polarization analyzer compared to using a horizontal polarization analyzer.

This discussion can also be extended to include vertically separated features in the object plane. From the correlation matrix at the object plane of a PCAV illumination system (Eq. (6)), one can see that the W _xx component is now anti-correlated for points in an object that are sufficiently separated by a vertical distance, Δ_y, but have Δ_x = 0. At the same time, the W _yy component is now correlated. One can observe in Figs. 4 and 5 that vertically separated features exhibit contrast improvements similar to horizontally separated features except the improvement is seen with the opposite polarization analyzer.

 

Fig. 4. Partially correlated azimuthal illumination of 1951 USAF resolution target. NA=0.004. Top image is vertically analyzed; bottom image is horizontally analyzed. Plot on the right shows a single slice through horizontally separated features on each image as shown.

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Fig. 5. Partially correlated azimuthal illumination of 1951 USAF resolution target. NA=0.25. Top image is vertically analyzed; bottom image is horizontally analyzed. Plot on the right shows a single slice through horizontally separated features on each image as shown.

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In general, because the correlation matrix for a PCAV illuminator is invariant under rotation, there will exist a combination of Δ_x and Δ_y that gives an anti-correlation property for any arbitrary orientation of the polarization analyzer. This can be seen in Fig. 6. Here, the NA of the illumination and imaging systems is again 0.25, and the object is a pitted and scratched silver coated glass slide. By rotating the polarization analyzer appropriately, one can see a contrast improvement in lines separated by approximately 1.74 microns in directions that are not horizontal or vertical but something arbitrary. As can be seen in the line profile plot of the two images, three lines (one very weak on the left-hand side of the plot) and one strong line are visible with the anti-correlated component of the PCAV illumination due to destructive interference. With the correlated component, however, only the strong line is clearly visible because constructive interference between the light passing through the four lines greatly decreases the contrast.

5. Conclusions

In conclusion, we have developed a theoretical description for the correlation matrix of a PCAV illumination system that is created by placing an azimuthal polarization vortex mode converter in the pupil plane of a critical illumination system. Using a RWYI, we verified the accuracy of the model. We have shown that strong polarization-dependent anti-correlations exist for certain separations of points in the object plane of an imaging system. We demonstrated that when using PCAV illumination with a simple video microscope an increase in the contrast occurred for horizontally separated features when using the vertical polarization component and for vertically separated features when using the vertical polarization component. This contrast improvement was shown to be scalable to an imaging system with a larger NA by increasing the NA of the PCAV illumination system. Finally, we demonstrated that this polarization-selective anti-correlation property of a PCAV illumination system may be used for features separated at an arbitrary direction if the polarization analyzer in the imaging system is oriented correctly. This result is significant for metrology, microscopy, and lithography.

 

Fig. 6. Example of use in metrology. The polarization analyzer is at the orientation shown for each image (-10° and 80° clockwise from a vertical orientation). The blue (solid) curve corresponds to a slice through the left image as shown, and the red (dashed) curve corresponds to a slice through the right image as shown.

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