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Abstract
Interferometric image processing systems based on image inversion normally use multiple paths with inversion mirrors. Since such systems must meet strict requirements of alignment and stability, a common-path implementation using polarization channels and six anisotropic optical elements was recently introduced. We demonstrate here the operation of a common-path polarization-based image-inversion interferometeric system using only two anisotropic lenses. Applications such as spatial parity analysis and image centroid measurements are examined theoretically and demonstrated experimentally.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Image inversion interferometery has found applications in areas including spatial parity analysis [1,2], particle or beam tracking [3], two-point imaging [4,5], aberration-free microscopy [6–11] and spatial-mode demultiplexing [12], to name a few. Whatever the purpose — be it for information processing or parameter estimation — the illumination collected by the interferometer is split into two modes: $h_M$, which undergoes either one-or two-dimensional inversion, and $h_I$, which is imaged or propagated through the system without inversion. Many such systems have been implemented by use of interferometers with spatially separated paths using either imaging systems or free-space propagation and mirror or prism reflections to implement $h_I$ and $h_M$.
More recently, it was demonstrated [13] that it is possible to use a single path with two orthogonal polarization channels creating — in the same plane — an inverted image in one polarization and an upright image in the other. The two polarizations are then combined to achieve the required interferometric operation. It was shown that six anisotropic lenses are necessary for this operation. Since alignment of six optical elements can also be difficult and less robust than systems with smaller blueprint, we investigate in this paper the possibility of using fewer anisotropic elements to implement an operation approximating image inversion, and reap benefits similar to those afforded in applications of the perfect image-inversion interferometer.
While it is apparent that there is little that can be done with a single anisotropic lens towards reproducing the behavior of the ideal six-element imaging system, we show here that, surprisingly, only two lenses are required to approximately meet the performance of the six-lens system. Starting with the original six-lens design, which inverts the images with one polarization and reproduces the image of the other polarization, we keep the two lenses responsible for inversion, and remove the lenses that reproduce the image of the orthogonal polarization, letting it propagate through the system instead.
It turns out that for appropriately selected focal lengths and propagation distances in relation to the beam diameter, our goal of approximating the perfect system is achieved and is adequate for certain applications. Moreover, it was found that the system is tolerant to slight deviations of its parameters from their optimal values.
In this paper, we consider two applications of image-inversion interferometry: spatial-parity analysis, i.e., separation of even and odd modes of an optical beam, and measurement of lateral displacement of an optical beam. In both cases, based on measures such as interferogram visibility and parameter estimation Fisher information, we assess the performance of the approximate system in comparison with the perfect system and also with the best possible performance dictated by the corresponding quantum Fisher information. This analysis allows us to both compare the performance of our system against a perfect image inversion interferometer and optimize the system parameters used in our design.
The proposed system was implemented experimentally by use of anisotropic lens doublets constructed by placing a diffractive waveplate with polarization-dependent focal lengths $\pm f$ in contact with a conventional refractive lens with focal length $f$. The resulting doublet is a lens that has a focal length of $f/2$ for right-circular polarization and no optical power for left-circular polarization. It also changes the handedness of the polarizations upon transmission, but this is an effect that is readily canceled out by introducing a waveplate or reversing the incidence side of subsequent doublets in the beam path [14]. Such combined refractive-diffractive elements prove to be one of the more attractive methods of anisotropic wavefront modulation: when compared to pure birefringent modulation from say, purely liquid crystal devices, the doublets are capable of offering greater modulation for fixed crystal poling resolution, owing to the power provided by the refractive lens in the doublet.
The paper begins with a summary of the basic theory and applications of the image inversion interferometer, and proceeds to present the two-lens interferometer and its optimization for spatial parity analysis and displacement measurement. Experimental confirmation and conclusions follow.
2. Image inversion interferometer
The image-inversion interferometer combines a copy of an optical field $E(x)$ with an inverted version therefrom to generate two superpositions
which are proportional to the even and odd components of the field distribution. A natural application is spatial parity analysis [15,16]. The powers in the even and odd components are given by the integrals $P_{\pm }(x)=\int dx |E_{\pm }(x)|^2$.A second application is coherent projection on even and odd functions. This is implemented by transmitting $E(x)$ and $E(-x)$ in the two branches of the interferometer through identical systems with impulse response function $h(x)$ so that the interferometer outputs are $[E(x)\pm E(-x)]\otimes h(x)=E(x)\otimes h_\pm (x)$, where $h_\pm (x)=h(x)\pm h(-x)$ are even and odd functions. At $x=0$, we obtain the projections $\int dx E(x)h_\pm (x)$, which are useful for applications such as superresolution [5,17].
A third, related application is the measurement of lateral displacement of an optical beam from a reference optical axis. Such displacement alters the even and odd components of the field distributions, and may be assessed by projection on even and odd functions [3].
A fourth application is optical computation of the sine and cosine transforms, concurrently. The even and odd components of the spatial Fourier transform of the field $\int dx e^{ik_xx}E(x)$, which are simply the sine and cosine transforms, will be generated concurrently at the two interferometer output ports.
A fifth application is the optical computation of the Wigner distribution function [18]. This is accomplished by displacing the field and tilting its wavefront and using the function $E(x+s)e^{ikx}$ as an input to the interferometer. The difference between the output intensities will yield the Wigner function $W(k,s)=\int dx E^*(s-x)E(s+x)e^{i2kx}$.
Image-inversion interferometry also has a number of applications in incoherent optical processing and imaging. Although the light is initially incoherent, the interferometer lines up the inverted image with the original image, whereupon they interfere [19]. If an incoherent field $E(t)$ with intensity $I(x)=\langle |E(x)|^2 \rangle$ is subjected to a filter with impulse response function $h(x)$ before transmission through the interferometer, then the measured intensities at the output, will be $I_\pm (x)=\int dx h_\pm (x,x')I(x')$, where $h_\pm (x,x')=|h(x-x') \pm h(x+x')|^2$. The system, which is shift-variant with a narrower width, has been used for fluorescent microscopy [9,20]. The system can also implement the sine and cosine Fourier transforms on incoherent images [21].
3. Two-lens interferometer
As illustrated in Fig. 1, the optical system uses two identical anisotropic doublet lenses separated by a distance $z_2$, with the first lens located at a distance $z_1$ from the object plane. Each lens is composed of a diffractive waveplate in contact with a refractive lens. For right-circularly polarized (RCP) light, the first lens, which has a focal length $f$ forms an image at the output plane and the second lens modulates its phase. The lenses have no effect on left-circularly polarized (LCP) light, which undergoes free-space propagation by a distance $z_T=z_1+z_2$.
Fig. 1. Schematic for an optical system using anisotropic lenses with focal lengths $f$ for RCP light, but no power for LCP light. Images with even or odd symmetry maintain their symmetry, but an image with odd symmetry is inverted in only the RCP channel.
The effect of the system on the transverse distributions of the fields is modeled by defining two linear shift-invariant systems with impulse response functions $h_r(x)$ and $h_l(x)$ for RCP and LCP light, respectively. If the optical field in the input plane is $E(x)$, then the optical fields of the RCP and LCP in the output plane are:
where $\otimes$ represents the operation of convolution. The minus sign in the RCP case takes account of the fact the lens introduces inversion (which renders the system actually shift-variant). The LCP function $h_l(x)$ represents propagation in free space a distance $z_T$, while the RCP function $h_r(x)$ represents propagation through a single lens of focal length $f$ with distances $z_1$ and $z_2$ before and after, followed by another identical lens. The impulse response function corresponding to a distance $z$ in free space is $h_z(x)=\tfrac {1}{\sqrt {\lambda z}} \ e^{ikx^2/2z}$ and the amplitude transmittance of a lens of focal length $f$ is $t_f(x)=e^{-ikx^2/2f}$, where $k=2\pi /\lambda$ and $\lambda$ is the wavelength, so thatSince $h_z(x)$ and $t_f(x)$ are complex Gaussian functions, the impulse response functions $h_r(x)$ and $h_l(x)$ are also complex Gaussian. If the input field $E(x)$ is also Gaussian, then both $E_r(x)$ and $E_l(x)$ will also be Gaussian with center of symmetry at the origin. Any other function with even symmetry will maintain its symmetry, although its shape would be modified. Any input function with odd symmetry will also maintain its odd symmetry about the origin, but will be inverted and its shape will be modified. The field distributions in the output plane can be evaluated by performing the convolution with the complex Gaussian functions $h_r(x)$ and $h_l(x)$ numerically.
Our aim is to select the system parameters $z_1$, $z_2$, and $f$ such that the magnitude and phase changes of the upright and inverted images have minimal effect on the image processing task dictated by the application.
If the distances satisfy the imaging condition ($1/z_1 + 1/z_2 = 1/f$), an inverted, magnified copy of the intensity distribution of the RCP light will be formed, along with an excess quadratic phase [22]. The LCP light, which travels in free space, produces a diffracted, magnified, uninverted version of the intensity distribution, along with the phase associated with wavefront curvature.
Fortunately, the phase difference between the two fields can be compensated by use of the second lens, which adds to the RCP component a phase that approximately matches the phase acquired by the LCP wave upon propagation through the distance $z_2$ between the lenses. As we will show, this compensation of the phase difference is what leads to high-visibility polarization interference, which enables high-fidelity image-inversion interferometry.
Note that if the phase compensation is perfect, then the detection plane may be moved to the right of the second lens, without affecting the performance of the system. After all, both waves would be traveling the same distance in free space. Hence, the parameters $z_1$, $z_2$, and $f$ uniquely determine the performance of the system for a given input and given task.
Since the eigenmodes of spatial-parity give binary results in Eq. (1), it is elucidating to see how the imaging system performs with either a purely even or purely odd input, which we plot in Fig. 2.
Fig. 2. Intensity (top) and phase (bottom) profiles of the fields of the even (left) and odd (right) modes at the input (red) and output (blue) of the two-lens image inversion system. The intensities are matched reasonably well, and although the odd modes are inverted and their distributions have the similar profiles at the input and output, they do not match well.
4. System optimization
4.1 Optimization for parity analysis
To test the efficacy of the two-lens anisotropic imaging system as a spatial parity analyzer, we consider an input field that is a linear superposition
of an even mode $E_e(x)=E_e(-x)$ and an odd mode $E_o(x)=-E_o(-x)$, normalized such that $\int dx |E_e(x)|^2 = \int dx |E_o(x)|^2 =1$, so that the powers in the even and odd components in the tested field are $P_e(\theta ) = \cos ^2 \theta$ and $P_o(\theta ) = \sin ^2 \theta$, respectively, and $P_e(\theta )+P_o(\theta )= 1$.To estimate the relative power of the even and odd components in the superposition, and hence the degree of spatial parity of the field distribution, we need to estimate the value of $\theta$. This is accomplished by our image-inversion interferometer, which measures the power in the sum and difference of the received fields at the output polarization channels. These are the powers in the horizontal and vertical polarization components composed from the sum and difference of the RCP and LCP polarization components,
and where $E_r(x;\theta ) = h_r(x) \otimes E(x;\theta )$ and $E_l(x;\theta ) = h_l(x) \otimes E(x;\theta )$. Ideally, the optical system conserves power, so that $P_V(\theta )+P_H(\theta )=1$.To quantify the efficacy of the system in the task of estimating $\theta$, we regard $P_V(\theta )$ and $P_H(\theta )=1$ as probabilities and calculate the Fisher information
The Fisher information is a function of $\theta$ and the system parameters $z_1/f$ and $z_2/f$. We have conducted a parametric study of the effect of these parameters on $F(\theta )$ assuming that $E_e(x)=\sqrt {2/\pi \sigma ^2} e^{-x^2/\sigma ^2}$ is a Gaussian function of width $\sigma$ and $E_o(x) = [2H(x)-1]E_e(x)$, where $H(x)$ is the Heaviside function. As shown subsequently, it turns out that $F(\theta )$ is maximized at $z_1/f \approx 2$, but is rather insensitive to $z_2$. The results are shown in Figs. 3 and 4. Figure 3 shows the dependence of the powers $P_H(\theta )$ and $P_V(\theta )$ of the projections and the corresponding Fisher information $F(\theta )$ on $\theta$, for $z_2/f=1.99$, $z_1/f=30$, and $\sigma /f = 7.75$ cm.
Fig. 3. (a) Projected powers $P_H$ and $P_V$ used to estimate the spatial parity parameter $\theta$. (b) Fisher information $F(\theta )$ normalized to the maximum value governed by the quantum Fisher information. In these plots $z_2=1.99 f$, $z_1=30 f$ and $f=7.75$ cm.
The maximum Fisher information, which is achieved for all $z_1,z_2$ and $f$ at $\theta = \pi /4$, is plotted in Fig. 4 as a function of $z_1$ and $z_2$ for several values of $f/z_0$, where $z_0 = \pi \sigma ^2/\lambda$ is the beam depth of focus. Relatively high performance is achieved for a large range of values of $z_1/f$ near $z_2/f=2$. Although a larger value of $f/z_0$ causes the system to have good performance for a wider range of values of $z_2/f$ and a narrower range of values of $z_1/f$, we find that high performance is still achieved for values of $z_2$ near $2f$.
Fig. 4. Fisher information $F(\theta )$ for estimation of the spatial parity parameter $\theta$ calculated for $\theta =\pi /4$ using image-inversion interferometry with anisotropic lenses of focal length $f$ equal to (a) $z_0/2$, (b) $z_0$, and (c) $2z_0$, where $z_0=\pi \sigma ^2/\lambda$ is the beam depth of focus. In each case, optimal performance is found near $z_2=2f$.
While there are some values of $\theta$ for which the Fisher information drops significantly, it is possible to use — as an example — a spatial light modulator in the object plane to introduce a ’bias’ degree of asymmetry [1]. Whether a spatial light modulator or some other method is used, this leads to an input field given by
where $\theta _0$ is the known degree of asymmetry introduced by the SLM in the image plane. By updating the controlled value of the bias degree of symmetry, it is possible to lock the operation of the system to the point where measurements provide the greatest value of Fisher information.In each case, we compare our measurement sensitivity to the quantum Fisher information, which dictates the best possible sensitivity that could be achieved by any measurement apparatus that probes the optical field to estimate an underlying parameter [23]. For the case of discrimination between two pure states, the quantum Fisher information for spatial parity analysis is equal to 4 per detected photon. Hence, in Figs. 3(b) and 4, we plot the Fisher information of our system as a ratio to the maximum achievable value dictated by the quantum Fisher information [25]. We find that it is possible for an ideal image inversion interferometer to achieve a sensitivity that is $90\%$ of the maximum achievable.
4.2 Displacement measurement
Measurement of beam displacement is of practical importance in metrological applications ranging from adaptive imaging to particle tracking. As reported in [3], the image inversion interferometer is able to measure the lateral displacement of a symmetric optical distribution with sensitivity that is nearly perfect when measuring small displacements. To test the efficacy of the common-path polarization-based two-lens system in performing this task, we have calculated the Fisher information for estimating the displacement $s$ for an optical beam with even symmetry,
Following the same procedure that led to Eqs. (5) and (6) leaves us with probabilities $P_H(s)$ and $P_V(s)$ that are now parametrized by $s$ rather than $\theta$, and the Fisher information
As shown in Fig. 5, the maximum achievable Fisher information for estimation of $s$ is found in the region around $z_2/f=2$, as is the case with spatial parity measurement. As shown in Fig. 6(b), the Fisher information is maximal at a finite displacement $s=s_m$ away from $s=0$, where the Fisher information is maximized when using an ideal image inverting interferometer. This outcome, which may appear surprising or unexpected, is actually not new or unexpected in interferometric imaging systems that deviate from their ideal operation scenario [26]. In the case of displacement measurement, however, a lateral scanning system can be designed to adaptively shift the axis of the system such that the measured displaced beam is near the optimal displacement $s_m$.
Fig. 5. Maximal Fisher information $F(s_m)$ for estimation of beam displacement $s$ using image-inversion interferometry with anisotropic lenses of focal length equal to a) $z_0/2$, b) $z_0$, and c) $2z_0$, where $z_0=\pi \sigma ^2/\lambda$ is the beam depth of focus. In each case, optimal performance is found near $z_2=2f$.
Fig. 6. (a) Projected powers $P_H$ and $P_V$ used to estimate the displacement parameter s (normalized to the beam width $\sigma$). (b) Associated Fisher information $F(\theta )$ normalized to the quantum Fisher information (blue). Also shown is the Fisher information achieved for an ideal image-inversion interferometer (red). In these plots, $z_2=1.99 f$, $z_1=30 f$ and $f=7.75$ cm.
5. Experimental confirmation
To confirm the utility of the two-lens anisotropic imaging system as a spatial parity analyzer, we generated a beam with a variable mix of even and odd functions, as in Eq. (4), and used it as the input to the two-lens system. As the mixing parameter $\theta$ is varied, the system outputs $P_H$ and $P_V$ follow an interference pattern with visibility that can be directly correlated with the achievable Fisher information.
The input field is generated by use of a spatial light modulator (SLM) with a binary phase step across its reflective face,
where $H(x)$ is the Heaviside step function centered at $x=0$. By beginning with a well-collimated beam in an even mode $E_e(x)$, reflection off the SLM face creates a superposition with form identical to Eq. (4) with $E_o(x) = [2H(x)-1]E_e(x)$.As illustrated schematically in Fig. 7, the setup uses a fiber-coupled laser source (Thorlabs S1FC808 ) to interrogate the face of the SLM (Hammamatsu LCOS-SLM X10468-02). The modulated field is then sent through the two-lens anisotropic imaging system. After transmission through the second anisotropic doublet, the light passes through a polarization analyzer that projects it onto the horizontal and vertical polarization bases. The illumination is then collected by a fiber-coupling lens and total power is measured as a function of $\theta$. To align the system, a CCD camera is used to center the optical components along the optic axes, maximizing for polarization interference at the output of the system.
Fig. 7. Experimental Setup. SLM:Spatial light modulator; POL:Polarizer; BS:Beam splitter; HWP: Half-wave plate;
We use the calculated optimal distance $z_2\approx 2f$ as a guide for finding the actual optimal distance. Practically, the distance ends up being longer than the calculated distance. This comes, in large part, from the fact that the lenses and waveplates are not perfectly in contact, reducing the effective power of each doublet. Nonetheless, it is still straightforward to use the CCD to optimize the distance between the lenses by monitoring changes in the interference visibility as the doublets are translated relative to one another.
The measured power in each output polarization mode is plotted in Fig. 8 as a function of the phase difference between sides of the SLM face. Due to the limited cone of collection of the fiber coupling setup the total power $P_H+P_V$ varies slightly as a function of $\theta$. To compensate for this effect, we divide each of the polarization projections by the total measured power of both polarization components. The final interferogram shows an interference visibility $V=0.86$.
Fig. 8. Measured normalized powers $P_H$ and $P_V$ as functions of the phase $\theta$ introduced by the SLM.
To compare the visibility with the predicted Fisher information, we consider a sinusoidal signal with imperfect visibility. In such a case, the power in either output mode as a function of $\theta$ is given by
The Fisher information for such a case is given by
6. Conclusion
We have demonstrated the successful operation of a two-lens anisotropic imaging system that effectively achieves the operation of an image-inversion interferometer. In the context of both spatial parity analysis and estimation of beam lateral displacement, relaxing the requirement on production of perfect upright and inverted images in the same plane does not significantly diminish the utility of the interferometer. This allows for a device that is much simpler than the previously proposed common-path image-inverting interferometer.
In the context of spatial-parity analysis, we showed that the two-lens anisotropic imaging system achieves a sensitivity of $86\%$ of the maximum achievable as dictated by the quantum Fisher information. To confirm this, we used the system in an experiment with optical fields of known spatial parity, measuring an interference visibility that indicates a sensitivity corresponding to a Fisher information of $76\%$ of the maximum achievable, standing in good confirmation with our calculations.
Funding
Air Force Research Laboratory (FA8651-19-2-0001).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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