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Abstract
We present a collinear, common-path image-inversion interferometer using the polarization channels of a single optical beam. Each of the channels is an imaging system of unit magnification, one positive and the other negative (inverted). Image formation is realized by means of a set of anisotropic lenses, each offering refractive power in one polarization and none in the other. The operation of the interferometer as a spatial-parity analyzer is demonstrated experimentally by separating even- and odd-order orbital angular momentum modes of an optical beam. The common-path configuration overcomes the stability issues present in conventional two-path interferometers.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
From optical communication and information processing to imaging and metrology, there has been a growing need for analysis and synthesis of the spatial structure of the optical field. Information transmission via spatial modes is based on operations such as modulation for multiplexing, projections for demultiplexing and compression, and filtering and correlation for recognition and classification [1–5]. Interferometry plays a major role in such operations.
Optical Interferometry is based on mixing a reference optical wave with a delayed, displaced, or rotated version of itself; a comparison of the two enables differential measurement with subwavelength resolution. The interferometer is typically configured to generate both the sum and the difference of the reference and modified waves, whose intensities form two complementary interferograms. A common background in these interferograms may be removed by simple subtraction. This principle is utilized in the balanced homodyne detector, which is widely used for classical and quantum measurement [6]. In the context of spatial structures, the integrated intensities of the sum and difference fields provide binary measures, compressing a simple image into two projections.
A more recent entry into the interferometry toolbox is the image-inversion interferometer. This is based on mixing a reference image with an inverted copy of itself, so that the interferometer is sensitive to inversion symmetry. By producing spatial fields equal to the sum and difference between an original distribution and its inverted copy, the interferometer effectively decomposes the field into its even and odd components, thus serving as a spatial parity analyzer [7]. For example, an image-inversion interferometer decomposes an optical beam containing a superposition of orbital-angular-momentum (OAM) carrying modes into the even-order and the odd-order OAM modes [8,9]. As shown in previous contexts [10], this binary classification can be cascaded to provide finer classification of the OAM modes.
If each arm of the interferometer has a spatial filter with a given amplitude point spread function (APSF), then in the presence of image inversion in one arm, one output of the interferometer is filtered by a system with even APSF, while the other is filtered with odd APSF. The powers in the output beams provide projections on even and odd functions, which form a binary compression of the original field distribution. Such compression offers a useful platform for communication and information processing.
The image-inversion interferometer has also found applications in imaging and microscopy. If applied to an incoherent light field with each arm including an imaging system of narrow shift-invariant APSF, then the difference between the output intensities is itself an image of the original intensity distribution with shift-variant response function centered at the origin and with narrower width, so that the system can be used as a scanner with improved spatial resolution [11–15].
Spatial parity analysis is also useful in quantum optical information processing. It has been shown that a single photon with spatial distribution in either an even or an odd one-dimensional spatial mode, or a superposition thereof, forms a qubit [16], which can be used for quantum logic. One such mode can be converted into the other by use of either a phase plate or spatial light modulator (SLM), and the image-inversion interferometer may be used as a modal analyzer [7,17,18].
Additionally, it has been shown that estimation of parameters of a spatial distribution, such as the separation of the diffraction-limited image of a two-point source, by means of projections onto a complete set of spatial modes can provide accuracies unobtainable through standard imaging [19–21]. Moreover, the projection onto a complete set of spatial modes is unnecessary, since projection onto just two spatial-modes (one even, one odd) offers significant enhancement over traditional measurements [22–24].
There is no question that the image-inversion interferometer is an important asset for both classical and quantum information processing.
In previous implementations of the image-inversion interferometer, conventional two-path configurations have been adopted with either an extra reflector in one of the arms, or with imaging systems offering upright imaging in one arm and inverted imaging in the other [7,24–26]. In this configuration, the pathlengths must be kept stable to within a small fraction of a wavelength, and this is not trivial. Stabilization has been addressed by use of post-selection of data based on long scans from which projection fidelity is inferred. Clearly, these methods require a surplus of auxiliary photonic resources, either through longer integration time or stronger illumination. In either case, especially in the context of quantum systems, the overhead required to implement the device may prove to outweigh the benefits gained by its application in the first place. Additionally, the projections necessary for modal analysis have relied on holographic methods that are often inefficient, especially in the context of quantum applications of OAM [9,27–30].
This paper introduces an alternative: a common-path image-inversion interferometer utilizing two polarization channels. The beam splitters in the conventional Michelson interferometer are replaced with polarization analyzers and combiners. Image inversion in one of the channels is implemented by use of a set of anisotropic lenses providing upright imaging for one polarization and inverted imaging for the other. This collinear configuration obviates the stability limitations inherent in the conventional two-path interferometer.
The paper begins with an overview of the theory underlying the various applications of the image-inversion interferometer and proceeds to describe the polarization-based implementation and its experimental verification.
2. Theory
A conventional optical interferometer is a four-port system that mixes an optical field with a phase-shifted, time-delayed, or spatially-translated version of itself and generates the sum and difference fields at its output ports. Likewise, an image-inversion interferometer mixes a spatial field f(x) with an inverted version of itself f (−x), and generates at its output ports the fields and . Here, x = (x, y) are coordinates in the transverse plane. The inversion operation f(x) → f (−x) is easily implemented by reflection from a simple mirror, but other configurations involving production of an inverted image by use of a lens may be used. It is also possible to include in the two interferometer arms other linear transformations ℒ1 and ℒ2 and a (−π/2) phase shift, as illustrated in Fig. 1, so that the outputs are the fields
As summarized below, a number of useful applications may be implemented by means of various choices of ℒ1 and ℒ2. For simplicity, we will present this summary for one-dimensional images f(x), but the results are readily applicable to the two-dimensional case.
Fig. 1 Schematic of a conventional image inversion interferometer. An image inverter (INV) is placed in one branch and ℒ1 and ℒ2 are linear systems used for various applications.
2.1. Parity analysis
In the simplest image-inversion interferometer, ℒ1 and ℒ2 are identity operators so that the system generates from the input field f(x) two output fields with spatial distributions
that are proportional to the even and odd components of f(x). The two arms of the interferometer will have to be perfect imagers, implemented, e.g., by 4-f optical systems. The interferometer serves as a parity analyzer that separates even and odd spatial distributions. It can be used as a classifier, or demultiplexer, separating a superposition of spatial modes into its constituent even and odd modes, as will be demonstrated experimentally in the next section. This type of parity-based modal analysis can be crucial in the context of the measurement of quantum-optical states that employ even and odd spatial modes in lieu of spin states [7,16,18,31]. Also, by virtue of reciprocity, the interferometer can be used in reverse as a multiplexer, combining even and odd modes into a single spatial pattern [15].2.2. Projections onto even and odd functions
If the operators ℒ1 and ℒ2 represent multiplication by a prescribed function h(x) created, e.g., by an SLM, then the areas under the outputs g±(x) provide the projections
where are proportional to the even and odd components of h(x). Thus, α± represent the projections of the original function f(x) onto the even and odd functions h±(x). For example, if h(x) is a function with support in the interval [0, d], then h±(x) are even and odd functions with support in the [−d, d] interval.Projections on even and odd spatial modes have been used in measurements optimized to estimate the separation between two incoherent point sources by measurement on the optical field they emit [23,24].
2.3. Fourier cosine and sine transforms
If the operators ℒ1 and ℒ2 represent the spatial Fourier transform, as can be implemented by a single-lens in the 2-f configuration [32], then the interferometer outputs provide the Fourier cosine and sine transforms of the input:
If f(x) is a real function, then g±(x) provide separately the real and imaginary parts of the Fourier transform, from which the phase can be calculated. Also, the Hartley transform ∫ dx f(x)[cos(kx) + sin(kx)] may be determined by adding up these transforms.2.4. Incoherent image-inversion interferometry
If the operators ℒ1 and ℒ2 represent linear systems of APSF h1(x; x′) and h2(x; x′), then the interferometer produces the fields
where If f(x) is a random function representing a spatially incoherent optical field for which 〈f*(x)f(x′)〉 = Ii(x)δ(x − x′), where Ii(x) is the optical intensity, then the average intensities at the outputs of the two branches of the interferometer are Here, |h±(x; x′)|2 represent point spread functions (PSF) of the incoherent imaging systems in the two branches. In view of Eq. (7), the two images I±(x) exhibit interference, and when subtracted, a new image Io(x) = I+(x) − I−(x) is created for which the background terms are canceled out, and the cross terms remain, where The PSF in Eq. (10) is a product of a conjugated version of the impulse response function of the first system and an inverted version of that of the second system. If these impulse response functions are shift-invariant, i.e., functions of their coordinate differences, and assuming that they are identical functions h(x − x′), then the overall system is shift variant with PSF If h(x) is a narrow 1D function centered about x = 0, then hi(x; x′) will be a narrow 2D function centered near the origin, x = x′ = 0, as illustrated in Fig. 2. To demonstrate that this PSF offers enhanced resolution in a scanning configuration, consider as an example the Gaussian APSF h(x) = e(−x2/σ2) for which the overall PSF in Eq. (11) is For a point at x′ = 0 in the input plane, the response in the image plane is a Gaussian function e−2x2/σ2 with width smaller than that of h(x) by a factor of . Additionally, if the point in the object plane is offset from the center of inversion by a distance Δ, then the measured image is again a Gaussian function e−2x2/σ2 but with amplitude reduced by a factor e−2Δ2/σ2.Fig. 2 (a) Shift-invariant amplitude impulse response function of the systems ℒ1 and ℒ2. (b) Shift-variant point spread function of the interferometric system incoherent illumination.
It has been shown that integrating over the interferometric transfer function’s image plane coordinate, which is accomplished simply by means of an integrating detector, offers a factor of 2 resolution improvement over direct imaging, and cancels radially symmetric aberrations [11–13,25].
2.5. Fourier transform with incoherent light
A special case of the image-inversion interferometer described by Eq. (10) is that for which h(x, x′) = eiπ(x−x′)2/λd, which is the kernel for the Fresnel transform (propagation of light of wavelength λ through a distance d in free space in the paraxial approximation). In this case, Eq. (10) gives
which is the kernel for the Fourier cosine transform. The image-inversion interferometer then produces an image Io(x) that is the cosine transform of the intensity of the original image Ii(x). This is remarkable since the original image is incoherent. This lensless system may of course be implemented by use of a Fourier transform lens. This type of configuration has been previously used in two-path interferometers using corner cube mirrors to implement the image inversion [33–36].2.6. Fourier transform with partially coherent light
If the optical field f(x) is partially coherent with coherence function Γ(x′, x″) = 〈f*(x′)f(x″)〉, then
For a system implementing the Fourier transform h1(x, x′) = h2(x, x′) = ei2πxx′/λd using a lens of focal length d, so that Io(x) is proportional to a one-dimensional (1D) Fourier transform of the two-dimensional function Γ(x′, x″) calculated along the 135° direction in the (x′, x″) plane. If Γ(x′, x″) is sufficiently narrow so that , then the output of the interferometer Io(x) is a cosine Fourier transform of the input intensity Ii(x), as in Eq. (13), regardless of the actual shape of the complex degree of coherence γ(x). By contrast, the output of a conventional interferometer is proportional to the 1D Fourier transform of Γ(x′, x″) along the 45° direction, and Io(x) is a cosine Fourier transform of the input degree of coherence γ(x).3. The common-path interferometer
A single-path image-inversion interferometer may be implemented by using the two polarization components of the propagating field to carry the two interfering images, with one polarization producing an un-inverted image and the other an inverted image. The design requires use of anisotropic imaging components—refractive and/or diffractive. In this work we use an anisotropic optical element made of a combination of a polarization-sensitive diffractive waveplate (DW) and a conventional refractive lens.
The DW is designed to act on incident light in two ways. First, upon transmission, left-circularly polarized light experiences wavefront curvature identical to the effect of a positive lens of focal length fdw, while right-circularly polarized light experiences the effect of a negative lens of focal length − fdw. Second, right-circularly polarized light becomes left-circularly polarized, and vice-versa. The DW is placed in contact with a conventional lens with focal length fcl. If fcl = fdw, then the effective focal length of the combined anisotropic lens pair (doublet) is for left-circular polarization, and ∞ for right-circular polarization. The anisotropic doublet thus has the effect of a lens acting on a single polarization, leaving the other unchanged.
The interferometer uses a cascade of 6 anisotropic doublets, 4 with focal length fdb = 2″ and 2 with focal length fdb = 4″, in the configuration shown in Fig. 3. For this system, a left-circularly polarized image will see a cascade of two Fourier-transform imaging systems. Hence, in the output plane, the result will be an inverted image with unit magnification of the spatial distribution in the input plane. Conversely, a right-circularly polarized image will see a cascade of four Fourier-transform imaging systems, resulting in an image with unit magnification and no inversion. Since the handedness of the polarization switches upon transmission through each doublet, a half wave-plate is placed in the middle of the system.
Fig. 3 A set of six anisotropic doublets used as a common-path polarization-based interferometric spatial parity analyzer. Right-circular polarization sees a cascade of four Fourier-transforming imaging systems made of the lenses labled “R”, creating an uninverted image in the output plane. Meanwhile, left-circular polarization sees a cascade of two Fourier-transforming imaging systems made of the lenses labled “L”, creating an inverted image in the output plane. A polarization analyzer (not shown) generates the sum and difference of the two images, thereby separating the even and odd spatial parities of the input image.
The system is operated as an image-inversion interferometer by using a horizontally polarized optical field with spatial distribution f(x) in the input plane:
where êR and êL are unit vectors corresponding to right and left circular polarization, respectively.Transmission through the imaging system transforms this field into
In a linear polarization basis with horizontal and vertical unit vectors eH, eV, this is equal to where and are the odd and even parts of f(x), respectively. Hence, simply analyzing the polarization components of E2(x) by use of a polarizing beam splitter will also provide separation of the parity components of the input spatial distribution f(x).Since this entire process occurs collinearly, acting on a single beam, the issues of alignment and path-length stabilization that have plagued previous implementations of image-inversion interferometers can be virtually alleviated.
4. Experimental verification
4.1. Diffractive waveplate
The diffractive waveplate (DW) lens (Pancharatnam-Berry phase lens) is constructed by depositing a liquid crystal (LC) polymer on a plastic substrate and exposing the polymer to polarized UV light with a special spatial pattern. This is accomplished in a multi-step process. First, the substrate is coated with photoalignment layer PAAD-72 (BEAM Co.) [37–39]. Next, the spatial pattern is created by exposing the photoalignment layer for 30 minutes with a power density of 8.8 mW/cm2. Finally, a liquid-crystal monomer solution is spin coated onto the photoalignment layer, and is photopolymerized using unpolarized ultraviolet light of 365-nm wavelength generated by a He-Cd laser. To form the lens, a parabolic phase distribution is created by giving each liquid crystal an orientation rotation that is proportional to the square of the radial coordinate, as measured from the center of the lens.
For a DW with diameter D, and grating period Δ defined as the period of the parabolic phase modulation at the edge of the DW, light of wavelength λ will experience the effect of a lens with focal length f = DΔ/2λ. The final thickness of the DW is typically ∼ 1 μm. The lenses we use are designed for 100 mm and 200 mm focal length for 532 nm wavelength. The diffraction efficiency spectrum of the DW lenses can typically reach 99.7% efficiency at 532 nm.
The ease of this process has led to a number of manufacturing breakthroughs, allowing for broadband, highly efficient, and tunable DW lenses [40,41]. Most importantly in the context of future implementation of our image-inversion interferometer, it is possible to coat the DW directly onto a refractive element, allowing for more robust, compact performance. The compactness of the DW offers an attractive advantage over implementing this system with bulkier optics such as spatial-light-modulators.
4.2. One-dimensional parity analysis
In the first experiment, we test the operation of the common-path image-inversion interferometer and assess its ability to faithfully separate the even and odd components of an optical beam with a one-dimensional spatial distribution. We use a spatial pattern
that is a linear superposition of even and odd spatial functions, Φe(x) and Φo(x), of equal power. The optical powers PV and PH measured at the outputs of the polarization analyzer should be proportional to and , respectively.We found it convenient to use odd and even functions related by the equation Φo(x) = [2H(x) − 1]Φe(x), where H(x) is the Heaviside step function centered at x = 0, i.e., the odd function is simply obtained from the even function by introducing a phase shift of ϕ = π for x < 0. It follows that , i.e. f (x) is obtained from the even function by adding a phase shift of in the right-half plane and in the left-half plane.
The experiment was conducted by the use of a coherent Gaussian beam generated by 532-nm diode laser. Hence, Φe(x) is a Gaussian envelope. Phase modulation was implemented by reflecting the beam off the surface of a liquid-crystal-on-Silicon spatial light modulator (SLM). By having a fixed gray scale value on the left half of the SLM face and a tunable gray scale value on the right half, we were able to tune the phase ϕ and thereby vary the even-odd composition of the spatial distribution.
Measured values of PV and PH are plotted in Fig. 4 as functions of ϕ in the [0, π] range. The visibility of this interferogram, which is a measure of the efficacy of the interferometer as a spatial parity analyzer, is 0.86. We believe that the imperfect visibility comes from the diffraction efficiency of our anisotropic doublets. Each DW acts on circularly polarized light as either a positive or negative lens, but the process is not perfect. Along with the focused and diffracted beam paths, there is a small amount of light that remains unmodulated. Although the fraction of light that is not diffracted is small, this effect occurs at each of the six DWs used in our system. We believe that this effect can be improved in the manufacturing process, as depositing the DW substrate directly onto the refractive element would allow for the use of DWs with better diffraction efficiency.
Fig. 4 Powers PV and PH of the vertically and horizontally polarized outputs of the polarization analyzer for an input spatial distribution containing a superposition of even and odd functions with amplitudes and as a function of ϕ, the phase difference between the faces of the SLM. At each value of ϕ, PV and PH are divided by their sum to normalize for the effect of variations of the SLM reflectance at different values of ϕ.
4.3. Two-dimensional parity analysis
As an example of the operation of the system for two-dimensional spatial fields, we consider an optical beam in a superposition of orbital-angular-momentum (OAM) distributions (Laguerre-Gauss modes). The mode of order ℓ has the distribution A(ℓ)(r) exp (−iℓϕ), where (r, ϕ) are the polar coordinates and A(ℓ)(r) is the radial distribution, so that the image inversion operation f(x) → f(−x) in this case is equivalent to multiplication by a phase factor eiℓπ = (−1)ℓ. Thus, the even-order OAM modes have even distributions and the odd-order modes have odd distributions. The interferometer will therefore serve as an OAM-parity analyzer.
We have experimentally tested the operation of the interferometer in this context by modulating our Gaussian beam Φe(x) with a phase profile exp (−iℓϕ) by use of a spatial light modulator (SLM) with vortex phase patterns. We used phase profiles with OAM values ranging between ℓ = −4 and ℓ = 4. The optical powers PV and PH at the outputs of the interferometer should hence be proportional to the powers in the even and odd modes of the input beam, respectively. Ideally, PV should alternate between a high value and zero as ℓ alternates from even to odd, and vice-versa for PH.
The results of this measurement are plotted in Fig. 5. The ratio PV /PH for the even modes, or PH/PV for the odd modes, is a measure of selectivity of this mode-parity classifier. This ratio equals 6.66 for ℓ = 0 and drops to 3.51 and 3.11 for ℓ = ±4, respectively. The fact that this ‘eye diagram’ is less open for higher order modes may be attributed to either the creation of the spatial distributions, or the alignment and diffraction efficiency of the interferometer. For higher order modes, misalignment in either the SLM center or the image-inversion interferometer’s optic axis greatly affects the measurement, since the variation in phase due to small deflections off the beam’s center becomes greater.
Fig. 5 Powers PV and PH of the vertically and horizontally polarized outputs of the polarization analyzer for OAM modes of order ℓ. As ℓ alternates between even and odd, the power switches between the vertical- and horizontal-polarization channels. For each mode, the sum of these powers have been normalized to unity in order to account for variation in the strength of the vortex singularity on the face of the SLM that generates these modes.
5. Conclusion
Image-inversion interferometry is a versatile tool with potential utility in several areas of optics, including spatial-mode analysis, high-resolution microscopy, and optical image processing. The lack of its widespread use in research and commercial applications may be attributed to the stringent requirements on balancing and pathlength stabilization of the conventional two-path interferometer, which often makes the advantages offered by the device not worth the trouble. The new configuration presented in this paper obviates these challenging requirements since it uses interference of the polarization modes of a single-path optical beam.
The principal challenge in the polarization-based interferometer is the implementation of image inversion in one polarization mode and not the other. We addressed this requirement by means of an anisotropic imaging system that provides upright imaging for one polarization, and inverted imaging for the orthogonal polarization, at the same magnification. This required the use of anisotropic lenses, which we implemented by use of conventional lenses coated with specially designed anisotropic diffractive waveplates. Our design employed a set of six such lenses in two groups: two arranged as a 4-f system acting on one polarization to produce an inverted image; and four, each of half the refractive power, arranged as an 8-f system acting on the other polarization to provide an uninverted image. The lenses are concatenated such that the two images coincide. Evidently, other configurations, possibly with fewer lenses, may also be used to implement the requisite anisotropic imaging. Also, reflective optical elements may substitute refractive components.
We have tested the operation of the common-path interferometer as a spatial-parity analyzer—a demultiplexer of one-dimensional even and odd spatial modes. We have also demonstrated the use of the interferometer to classify orbital angular momentum modes of oven and odd order up to ±4. Based on this demonstration, we expect the system to be developed further and to find home in support of many optical toolboxes.
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