External forces displace atoms in the glass matrix, giving rise to a stress tensor that determines the local birefringence. In a matrix representation, stress and birefringence are related in the following fashion where σ11 and σ22 are the values of principal stress and R is the 2×2 rotation matrix identifying the direction of principal stress:
The retardance in optical path Γ = ηt is the product of the birefringence, η = |Δn
1 - Δn
2| with material thickness, t. Δn
1 and Δn
2 have the usual dependence on the stress: Δn
1 = k
11σ11 + k
12σ22 and Δn
2 = k
11σ22 + k
12σ11 are assumed and kij
are the stress optical coefficients.
In a manner similar to that of Ref. [6
Y. Unno, ”Distorted wave front produced by a high-resolution projection optical system having rotationally symmetric birefringence,” Appl. Opt.
37, 7241–7247 (1998). [CrossRef]
], we assume an optical model in which the optical waves traveling through the window are confined to a paraxial region about the direction perpendicular to the window (z
axis). While the general theory of wave propagation in such a medium has a spatially varying set of principal axes in three
dimensions, we assume that every point has one principal axis in the direction of propagation, with the remaining two principal axes aligned with the directions of transverse
principal stress. This allows us to define a space-variant Jones matrix assuming a principal stress direction θ
and a magnitude of retardance δ
which are assumed to be functions of the normalized polar coordinates of the window:
In general, the input polarization will be uniform and specified by a single Jones vector Ein
. We are specifically interested in circularly polarized illumination; without loss of generality, we may assume right hand circular input, in which e-iωt
time dependence is assumed and the polarization rotation follows a right hand rule along the direction of propagation. The output polarization is space-variant, and given by
For right hand circular input, the multiplication yields
A right hand circular analyzer may be modeled by left-multiplying by , yielding
To gather the left hand circular component of the output field, a left hand circular analyzer may be modeled by left-multiplying by , yielding
As in Ref. [1
], we will assume that the principal stress direction rotates in proportion to the polar angle (θ = βϕ).β is determined from the symmetry of the stress according to the rule β = (2 - m
)/2, in which m
denotes the symmetry of the applied stress. Finite element modeling verifies that, for m
= 3, the retardance is proportional to radius near the center of the piece; for the remainder of the paper we will take δ = c
ρ in which c is a coefficient depending on wavelength, thickness and rate of change of birefringence. c is proportional to the external applied force and represents the rate of change of the phase retardance near the center of the piece. For trigonal (m
= 3) stress, θ = -ϕ/2 and the right hand circular and left hand circular components of the output field can then be expressed as follows:
and the left hand circular component becomes
The left hand circular component exhibits a phase vortex while the right hand circular component has no phase vortex. The charge of the phase vortex in the left hand circular component is determined by the symmetry of the stress. In this case, m = 3 stress yields a vortex of +1 topological charge. The right hand circular component shows a symmetric apodization that varies as cos(cρ/2).
The point spread function of the waveplate may be modeled using the tools of polarization modeling combined with Fourier optics. A space-variant Jones matrix is defined as described in Eq. (2
) and is multiplied by a Jones vector describing circularly polarized light. Each Cartesian component of the field amplitude is then Fourier transformed, and combined in vector fashion to deduce the electric field distribution near focus. The fields in the pupil are related to the fields in the image through the following relation:
where (x,y) represent the coordinates of the image and (fx
) represent the spatial frequencies of the image with . An imaging system can then be modeled by defining the incident pupil field, computing the transmitted field using the space-variant Jones matrix, transforming the pupil coordinates into dimensions of spatial frequency and assembling the vector point spread function. To model defocus, a quadratic phase error is applied to the pupil function as is customary in Fourier optics and aberration theory.
A thermal compression technique may be used to create trigonal (m = 3) stress distribution over the interior of an optical window. The procedure utilizes two materials, a window and metal housing, having different thermal expansion coefficients. For this investigation, windows
of BK7 (for visible applications) and fused silica (for UV applications) are studied. While BK7 has a larger thermal expansion coefficient than fused silica, the stress optical coefficient of BK7 is smaller than that of fused silica. The metal housing, which applies trigonal symmetric force to the window, is chosen so as to have a larger thermal expansion coefficient than the thermal expansion coefficient of the glass.
Before force is applied, a machine quality finish is applied to the edges of the window to provide uniform contact with the metal housing and to ensure the edges are perpendicular to the faces. In our case, the nominal outer diameter of the window is 1.26 cm; to utilize thermal compression, a hole, about 25 μm smaller than the diameter of the window, is cut in the metal housing. Material is then removed from the inside of the metal housing in three regions, each 120° apart, to provide trigonal symmetry and produce at least one wave of retardation at the edge of the pupil of the system.
Both the glass and the metal are heated to a temperature of greater than 300°C to allow sufficient differential expansion. The window is then placed in the metal housing and both materials are cooled. The result is a fixed and stable stress distribution which is nearly uniform in the axial direction. The magnitude of stress is proportional to the difference in outer diameter between the window and metal housing at room temperature, and can therefore be controlled by the choice of materials and the fabrication precision.
shows a general schematic of the system used for point spread function (PSF) measurement and imaging. A microscope objective and Bertrand lens are configured to relay the pupil image to a plane containing the stressed window. A third lens, L
, functions as a relay lens to transfer the image to a CCD.
To explore the effect of a space-variant pupil polarization on the through-focus PSF, a star test is constructed using a spatially filtered/collimated beam from a HeNe laser. A 40x objective (NA=0.65) mounted on a computer controlled translation stage produces a point source. Through-focus images are then acquired, and analyzed, to extract the axial irradiance. A similar set of through-focus measurements are also taken, with the point source replaced by an Air Force target illuminated by an LED, holographic diffuser, and condenser lens.
Fig. 1. Experimental design. The Air Force target is placed at plane P
1, the front focal plane of the objective L
1, and the trigonally stressed window is placed at plane P
2 is a Bertrand lens arranged to create an image of the aperture stop (P
2) in the plane of the stressed window and L
3 is a relay lens. An intermediate image is formed between L
2 and P
3. For the optical system, NA=0.06.
Two types of calibration are necessary for comparison of theory and experiment. Our analysis shows that knowledge of the wavelength, numerical aperture (NA), and dimensionless stress parameter c
are sufficient to completely describe the PSF. The NA is measured both by generating a diffraction limited PSF from an unstressed window and by estimating the cutoff frequency in the image of the Air Force target. The value of c
) is set by examining the right hand circular pupil image and adjusting an iris diaphram to align with an integer number of waves of retardation. Figure 3c
shows an example of the pupil image.
Both experiments employ circularly polarized light. For the star test, a linearly polarized
HeNe is followed by a quarter waveplate. For the imaging experiment, the unpolarized LED was passed through a linear polarizer and quarter waveplate. In order to analyze the polarization content of the image, an analyzer is placed after the stressed window. To analyze the content of circular polarization, for example, a combination quarter waveplate and linear polarizer (such as is typical in photography) is employed.
4. Results and discussion
It is interesting to examine the pupil irradiance and the PSF for a pupil window having trigonal stress. When the stress is sufficient to produce two or more waves of retardation at the edge of the pupil (c
), circularly polarized illumination produces a transmitted beam having alternating dark and bright fringes. These alternating dark and bright fringes are observed in the photographs in Fig. 2
, where the right photograph is an enlargement of the central portion of the left photograph.
Fig. 2. Rings of equal retardance in a window held under trigonal symmetric stress. (a) Photograph of stressed window held between circular polarizers. (b) An expanded view of the center region of the photograph.
Contours of equal birefringence may be viewed using circular polarizers. The alternating dark and bright fringes correspond to alternating left and right circularly polarized rings. In Fig. 2
, circularly polarized light is passed through the window and then through a circular analyzer, which passes right hand circular (RHC) light but blocks left hand circular (LHC) light. In such a view, the dark regions are contours of half-integer retardance (eg., half-wave, one and a half-wave, etc.), while the bright regions are regions of retardance equal to a multiple of one wavelength. Rings therefore have a wavelength-dependent radius; this property will be important in understanding the structure of the PSF.
A full waveplate will, under rotation, leave the phase of the transmitted wave unchanged. However, a half-waveplate (which transforms RHC to LHC light) will impart a phase proportional to twice the angle of the retarder. Thus, the half-wave regions produce the LHC vortex predicted in Eq. (8
). The total PSF at the paraxial focus is the superposition of two nearly identical irradiance distributions: A set of RHC rings without a vortex and a set of LHC rings with integer vortices. The result is a dominant ring with a half-integer
Because a full wave of retardation leaves the phase of a wave unchanged, the rings of right circular polarization exhibit no phase vortex. This was shown theoretically in Eq. (7
), a result which also shows that successive rings have a π
phase difference. In contrast, a half-wave of retardation does modulate the phase of circular polarization: Right circular is converted to left circular with a phase that changes with twice the angle of the fast axis, as shown in Eq. (8
). Figure 3a
shows the computed irradiance of the first half-wave ring color coded with the phase, while 3b shows a photograph of a stressed window taken through a left circular analyzer.
shows a pupil image (LED illumination) of right circular light, and Fig. 3d
shows the equivalent monochromatic pupil image of left circular light, showing the vortex rings.
Fig. 3. (a) Plot of the irradiance of the LHC light color coded with phase. (b) Photograph of the LHC component of the trigonally stressed window between circular polarizers. (c) Pupil image of RHC light. (d) Pupil image of LHC light.
To study the through-focus PSF, a slice of the PSF is computed for values of the defocus varying from -2λ
). The accompanying movie shows fascinating through-focus behavior. As the defocus changes, the PSF collapses to a nearly diffraction limited spot at about ωD
and, after passing through the aberrated paraxial focus, collapses again
, producing a second diffraction limited spot at ωD
shows a plot of the through-focus axial irradiance, and clearly shows an axial splitting of the focus induced by the trigonal stress; for a stress induced retardance of two waves at the edge of the aperture (c
), the separation is about 300 μm
. For the through-focus axial irradiance plot in Fig. 5
, the circles represent the experimental data and the solid line represents the theoretical curve. The numerical aperture (0.06) determined from the experiment was subsequently used in generating the theoretical curve. Figure 6
shows a side by side comparison of experimental and theoretical PSFs, for an NA=0.06 system.
The computed axial irradiance peaks at about 25% of that for a perfect system, although the width of the focal spot is very close to a perfect Airy disk (Fig. 4
). The magnitude of the irradiance can be explained as follows: The transmitted light shown in Fig. 5
has both right and left circular components. The left circular component, which is roughly half of the total transmitted light, does not contribute to the axial irradiance. The remaining half of the total transmitted light, the right circular component, is split equally between the two foci, yielding about 25% of the energy in each focus.
One way to understand this double focus system created by our stressed window is to picture the alternating transmitted RHC rings using the zone plate style amplitude modulation shown in Fig. 7
. The zone plate has alternating states of polarization with right hand circularly polarized rings and neighboring π
out of phase rings with a vortex that are left hand circularly polarized. By analyzing the RHC rings, the zone plate appears as a long period diffraction grating with a period just short enough to produce two distinct foci.
Simulation of the PSF slice of the trigonally stressed window for varying values of defocus (file size: 2MB). [Media 1
Fig. 5. Axial irradiance as a function of focal shift, measured in microns. The circles represent the experiment and the solid line represents the theoretical curve.
We consider the pupil as a small section of a zone plate consisting of a radial grating period Λ, and apply small angle diffraction theory as follows: If the phase has the radial dependence Δϕ = k
0κρt, in which k
0 = 2π/λ is the vacuum wave vector, κ is the rate of change of birefringence near the center of the window, the grating period (measured in units normalized to the pupil radius) is given as Λ = λ/κt. For small diffraction angles, an axial ray will be deflected by θ = ±κt. The first order optics of Fig. 8 illustrates that, if the zone plate is placed in the front focal plane, the longitudinal separation of the focal spots, |2εL
|, will be
We now relate this to the normalized stress parameter c (c = 2πκt/λ) and express the focal
separation as follows:
(a) Experimental through-focus PSF of the trigonally stressed window in the pupil plane of an NA
= 0.06 imaging system. At each stage the PSF is normalized to its peak irradiance (file size: 388 KB). [Media 2
] (b) Theoretical simulation of the through-focus PSF of the trigonally stressed window in the pupil plane plotted in normalized units of λ
(file size: 1.6 MB). [Media 3
Fig. 7. Zone plate with alternating rings of RHC and LHC light. The period of the rings is given as Λ.
It is important, and instructive, to relate the pupil defocus to the physical shift. The longitudinal shift εL
of an axial ray is proportional to the gradient of the wavefront. For defocus,
Equating the expressions for longitudinal separation, |2εL
|, gives defocus in terms of the normalized stress parameter and wavelength:
By varying the amount of stress, c, the separation of the foci changes. Figure 9
illustrates axial irradiance plots for various values of applied stress, c. As the applied stress increases, the focal separation increases as well. The bar above each irradiance plot shows the focal splitting is proportional to c, which is consistent with the geometrical theory. For the case where c
Fig. 8. Schematic illustrating the origin of the stress parameter, c. |2εL
| is the separation between the two foci.
Fig. 9. Relative axial irradiance for various values of the stress parameter, c.
) summarize the dependence of the focal splitting on NA and wavelength, both in terms of the defocus parameter and the absolute focal shift. It is important to realize that the wavelength dependence of the retarder balances the usual dispersion of a zone plate to yield a diffraction angle that is approximately achromatic, depending on λ
only through the material dispersion. A lens could, in principle, be designed to balance the dispersion of the birefringence and produce a truly achromatic focal splitting.
If the two focal planes and the paraxial focus are isolated, the theoretical and experimental PSFs can be compared with the real effect of defocus on an Air Force target. Figure 10
illustrates the first focus, paraxial focus and second focus for the star test theory and experiment as well as for the Air Force target. The Fig. shows the two foci are approximately one wave of defocus in each direction from the paraxial focus.
Fig. 10. (a) Experimental PSFs. (b) Predicted PSFs. (c) Air Force target positioned at each focus (left and right) and midway between (middle image).
For wide field imaging, an extended source illuminates an entire area of a specimen and collects both in focus and out of focus image planes simultaneously. Confocal microscopy offers increased effective resolution and improved image contrast by mapping a specimen point by point and rejecting out of plane scattering light. We believe stress induced focal splitting could have important applications in both confocal and wide field imaging.
This double focus imaging mode, with suitable image processing, could apply to applications ranging from digital photography to wide field microscopy. Wide field imaging, as can be seen from the Air Force target images, produces a ’ghost’ background due to the out of focus light from the second focal plane. This ’ghost’ background could likely be removed by image processing.
Our axial irradiance measurements of the trigonally stressed window show that rejecting the out of focus light in a confocal point scanning or confocal slit scanning arrangement would give simultaneous sharp images at two separate focal planes. Specific applications of confocal imaging include reticle and wafer inspection in lithography and a variety of different modes of biomedical imaging. This could include ordinary confocal imaging, confocal fluorescence imaging, multiphoton imaging, or optical coherence tomography.
Masks and reticles for optical lithography, for example, often have several planar regions that each require inspection, but are difficult to simultaneously capture with very high NA microscopes. The focal splitting described here would offer the possibility of simultaneously imaging top and bottom layers of photoresist on wafers.
The trigonally stressed window could also be used for extended depth of field imaging to get the highest possible resolution. A conventional system has a depth of field of approximately 1.3λ/NA
2, so our focal splitting can be thought of as appearing just beyond the normal depth of focus. With the stressed window, the longitudinal separation of the focal spots can exceed 2.6λ/NA
2 and, as the axial irradiance plot shows, the stressed window could be used to elongate the axial region to obtain twice the depth of field of traditional imaging.