1. Introduction

Spiral phase plates are elements with a helically increasing optical thickness. Depending on the design wavelength λ_d the step height is engineered such that the phase distribution of incident radiation is azimuthally shifted in the interval [0, 2πℓ], where ℓ is the helical charge, a signed nonzero integer. Along the central axis the vortex phase distribution of an incident plane wave of wavelength λ_d contains all shifted phase values ranging from 0 to 2πℓ, giving rise to destructive interference resulting in on-axis cancellation of the intensity distribution and the formation of a ring-shaped beam in the far-field with a radius proportional to ℓ [1]. These so called doughnut beams approximate a Laguerre-Gauss mode of helical index (or helical charge) ℓ, although, due to the lack of the corresponding amplitude modulation, they are actually a superposition of various LG-modes, but dominated by the mode with helical index ℓ. In the following we call these beams ”LG-like” beams. An extended discussion can be found in [2]. An LG-like beam with helical index ℓ carries an orbital angular momentum L = ℓħ per photon, exerting a torque proportional to the intensity on illuminated objects [3–5].

In the visible regime, spiral phase or vortex plates are used for micro manipulation of trapped particles by orbital angular momentum exchange [6]. In stimulated emission depletion (STED) microscopy spiral phase plates are regularly used for the creation of a ring-shaped de-excitation beam [7,8]. A different application of spiral phase plates is Fourier filtering in imaging systems for contrast enhancement. Spiral phase contrast offers an easy to implement contrast channel in an optical microscope where a vortex structure is inserted in the back focal plane of the microscope objective to achieve isotropic [9] or anisotropic edge enhancement [10,11]. In extrasolar planet detection vortex plates with a helicity of ℓ = 2 can serve as a filter to cancel the image of a star in a coronagraph [12, 13]. Vortex structures, as present in mica or some crystallized organic substances, can be located and highlighted by taking the intensity difference of two images spiral phase filtered with vortices of opposite handedness, e.g. ℓ = ±1 [14]. A combination of the two main benefits of spiral phase plates was reported in STED microscopy where the same vortex plate for generation of the doughnut-shaped de-excitation beam is also utilized for contrast enhancement of the probe image [15]. In optical communication systems light in multiple orbital angular momentum states created with optical vortices of different helicities ℓ can be used for data multiplexing at the same frequency [16]. A recent interest of vortex beams arises in ultra-intense laser field physics. For example, intense ultra-short vortex pulses are expected to excite otherwise forbidden molecular transitions [17], to produce relativistic twisted light pulses which exert a strong torque on materials [18], and to produce terawatt vortex laser beams for filamentation studies in gases [19].

A common alternative to spiral phase plates for the creation of vortex phase distributions is the use of a spatial light modulator (SLM), where any desired phase distribution can be generated. However, high resolution SLMs are expensive and are only available as reflective devices resulting in a less compact experimental setup. Furthermore, liquid crystal SLMs are polarization sensitive, which reduces their total efficiency and may be an issue in low signal applications. Other methods to generate vortex beams use computer generated holograms [20], or Dammann gratings [21, 22], which, however have a limited efficiency when generating a vortex beam with a defined helical index.

Here we report on the design and fabrication of high efficiency spiral phase plates in fused silica with an adjustable helicity ℓ for a fixed wavelength of λ_d = 532 nm. The proposed optical unit consists of two axially stacked diffractive elements with conjugate diffractive patterns exploiting the optical Moiré effect [23]. The combined transmission function of the superposed elements corresponds to that of a plain glass substrate, while mutual rotation about the central axis by an amount θ is designed to transform the homogeneous optical thickness to a spiral phase plate with a helicity depending on the rotation angle. This enables the creation of LG-like beams with an orbital angular momentum L = ℓħ, tunable from negative to positive values. We present experimental results of the transformation of a focused Gaussian beam to focused LG-like beams with helicities of ℓ = +1, +2, +3, +4, +5 and +10.

2. Moiré diffractive spiral phase plates

Rotational Moiré diffractive optical elements (MDOEs) were proposed in an earlier publication [23] and experimentally demonstrated as focus tunable Fresnel lenses [24]. The working principle is based on the optical Moiré effect, where two superposed gratings of a similar spacing can lead to a structure with a much larger beating period. The joint transmission function of two stacked optical phase elements is given by T_joint = exp{i[Φ₁ + Φ₂]}, where Φ_1,2 are the spatially modulated phase profiles of the two elements, respectively.

The desired transmission function of a spiral phase plate is T_spiral = exp{iℓϕ}, where ϕ is the angle in polar coordinates ranging from 0 to 2π. This transmission function can be obtained by stacking two phase elements of the form

The joint transmission function of the compound optical unit T_joint(ϕ) = exp{i[Φ₁(ϕ) + Φ₂(ϕ)]} is therefore:

All phase contributions in the exponent of Eq. (3) which are multiplied with ϕ correspond to the transmission functions of a spiral phase plate, whereas the other contributions are just uniform phase offsets (for a fixed rotation angle θ) which are of no significance for most applications. Thus two different spiral phase plates are obtained in the two sectors, with helicities of

In order to produce the phase elements as diffractive optics components, the actual phase functions Φ₁ and Φ₂ in Eq. (1) are wrapped modulo 2π, i.e. they become

Figure 1(a) and 1(b) show the phase functions mod_2π{Φ₁(ϕ)} and mod_2π{Φ₂(ϕ)}, respectively, with a design parameter a = 90. For θ = 0° the vortex MDOE only introduces a uniform phase shift related to the optical thickness of the glass substrates, while a mutual rotation by θ = 2° leads to the phase profile of a first order spiral phase plate, i.e. with a helical index of ℓ = 1. In Fig. 1(c) the simulated combined phase function of the two phase plates is shown for an exemplary angle of mutual rotation of θ = 8°, which results in a vortex with a helicity of ℓ = 4. As expected, a second sector which contains a spiral phase functions with a different helicity appears in a small angular range corresponding to the rotation angle θ. As the sector formation is undesired for practical applications the design parameter a should be chosen sufficiently large to enable a wide tuning range of the dominant helical phase front with only a small sector containing the undesired phase contribution.

 

Fig. 1 Phase functions for a design parameter a = 90 of (a) Φ₁(ϕ), (b) Φ₂(ϕ) and (c) of the compound unit for an exemplary angle of mutual rotation ϕ = 8° resulting in a helicity of ℓ = 4. The grey levels in the figure correspond to phase shifts in an interval between −π and π.

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3. Experiments

Two phase plates with transmission functions according to Eq. (5) (with parameter a = 90 and a maximal radius of 0.5 cm), were fabricated by grey scale lithography and subsequent selective reactive ion etching. The parameter a is limited by the condition that the steepest phase gradient in the phase functions of Eq. (5) has to be lower than the lithographic resolution (i.e. the resolvable pixel size p) divided by π. But actually the phase gradient along a circle with radius R around the center of the phase plates becomes infinite with decreasing radius, i.e. the phase pattern can only be resolved outside of a circular disk with a radius of R_min. This leads to the condition:

The transmission functions were realized as two grey scale patterns in a pixel raster with 200 nm resolution, and 128 grey values corresponding to 128 different exposure dose values. Each pattern was exposed four times to average out the effect of dose fluctuations during the exposure.

A fused silica plate was coated with diluted photoresist (AZ 4562) resulting in a resist layer of 3.2 μm. The photoresist was exposed by a direct writing lithography system based on an Ar-ion laser with wavelength λ = 364 nm. After development with slightly diluted developer (AZ 351B : H₂O = 1 : 2) the desired pattern was reproduced in the height profile of the resist. In a preceding calibration procedure the height of the resist (after development) was determined as a function of the exposure dose. Applying this calibration during the exposure, a linear dependence between the grey values in the pattern and the final resist height was achieved.

The resist pattern was afterwards transferred into the fused silica substrate by selective reactive ion etching (rf power 150 W, DC bias 600 V, pressure 15 mtorr). CF₄ with a flow rate of 45 sccm and O₂ with a flow rate of 2.5 sccm were used to etch the fused silica and the resist, respectively. The ratio of the etch rates between fused silica and resist was 0.452. Height measurements of the finished phase plates are shown in Fig. 2.

 

Fig. 2 (a): Photograph of one of the fabricated MDOEs. (b): Height measurement of its phase profile in the central region by white light interferometry. (c): Phase profile along the vertical line indicated in B.

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The helicity of the phase distribution of a plane wave passing the assembled vortex MDOE was measured with phase stepping interferometry [26]. The vortex MDOE was placed in the probe beam of a self assembled Mach-Zehnder interferometer operated with a frequency-doubled Nd:Yag laser at a wavelength of λ = 532 nm, as indicated in the upper row of Fig. 3.

 

Fig. 3 Experimentally measured phase distribution of a plane wave (λ = 532 nm) after passing the vortex MDOE at different angles of mutual rotation θ. Upper row: Sketch of the interferometric setup, and 3 exemplary interferograms measured at different phase shifts α (indicated in the figure). Below: Phase behind the vortex plate, reconstructed from the corresponding interferograms, as a function of the rotation angle θ. The corresponding helicity of the combined vortex plate corresponds to the number of radially appearing 2π-phase jumps. Note that only positive helicities are shown, but the opposite sign can be obtained by reversing the mutual rotation direction of the two DOEs in the combined MDOE.

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Phase stepping was done with a mirror mounted on a piezo translation stage, and the corresponding interferograms were captured on a 14 bit digital sensor with a pixel size of 9×9 μm² (pco.4000s). An example of 3 phase-stepped interferograms (with mutual phase offsets of 2π/3) is shown in the figure. From a set of 3 phase shifted interferograms the complex amplitude is computed and then numerically back-propagated by a distance of d (indicated in the Figure) into the plane of the vortex plate. The corresponding phase distributions directly behind the vortex plate are shown in Figure 3 for different angles of mutual rotation θ. Gray-scale values between black and white correspond to a phase ranging from 0 to 2π, respectively. The obtained phase patterns agree with those shown in Fig. 1. Particularly, the two sectors containing different helical phase profiles are reconstructed as expected. The observable deviations from the perfect ”star-like” spiral phase patterns correspond to an spherical phase curvature on the order of a quarter wavelength over the extension of the DOEs, which is probably due to the production process. For applications, as e.g. the generation of vortex beams, this effect is in most practical cases negligible.

Vortex MDOEs can be utilized to transform a Gaussian beam into a LG-like beam with adjustable helical charge ℓ, where the radius of the ring-shaped transverse intensity distribution is proportional to ℓ. We demonstrate the mode converting performance by imaging the Fourier transform of an extended laser beam at a wavelength λ = 532 nm passing the vortex MDOE. A digital sensor (mvBlueFOX-224G) with a pixel size of 4.4 μm was placed in the focal plane of the Fourier transforming lens of focal length f = 0.7 m. Figure 4 shows the normalized transverse intensity distributions of 4(a) the TEM₀₀ mode for θ = 0° and 4(b)–4(g) LG-like beams for helicities ranging from ℓ = 1 to 10 corresponding to an angle θ ranging from 0° to 20°. By further increasing the angle θ, the influence of the complementary sector becomes notable, resulting in a gap in the ring-shaped intensity distribution and a non-zero intensity at the center. In Fig. 5(a) a quantitative comparison of the intensity distribution of a focused unmodified TEM₀₀ beam (θ = 0°) and a focused Laguerre-Gaussian beam with ℓ = 1 (θ = 2°) is given. The peak intensity of the TEM₀₀ beam is normalized to 1 and the doughnut beam intensity distribution (captured with the same exposure time) is proportionally scaled (see inset). The ratio of the two-dimensional integration over the LG-like beam and TEM₀₀ beam intensity yields a 94% efficiency for the transformation of a TEM₀₀ beam to an LG-like beam with ℓ = 1. For a perfect pair of Moiré spiral phase plates, an intensity distribution of 99.5% and 0.5% in the ℓ = 1 and ℓ = −179 LG-like beams would be expected, respectively, which corresponds to the relative areas of the two sectors within the combined DOE. The missing beam intensity in the experiment is probably due to tolerances in the DOE production and produces a marginal diffuse background by scattering. The green and blue curves show the corresponding cross-sections. From the zeros of the Airy disk (green curve) and the focal length of the Fourier transformation lens, the effective radius Σ of the aperture of the imaging system can be determined to be Σ = 3 mm.

 

Fig. 4 Normalized intensity distribution of a TEM₀₀ beam at λ = 532 nm, passing an MDOE spiral phase plate set at different angles of mutual rotation, focused on a digital sensor (mvBlueFOX-224G). (a) Unmodified TEM₀₀ beam (θ = 0°). (b)–(g) TEM₀₀ beam converted to LG-like beams of various helical charges ℓ.

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Fig. 5 (a) Line profiles, as indicated in the insets, comparison of an unmodified TEM₀₀ beam with θ = 0° (green curve) and a LG-like beam with helical charge ℓ = 1 (blue curve). The transformation efficiency is 94%. (b) Experimental results (blue circles) for the linear scaling of the radius of the ring-shaped intensity distribution with ℓ.

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The experimentally determined radii R_ℓ of the ring-shaped intensity distributions for ℓ = 1 to ℓ = 10 are plotted in Fig. 5(b) (blue circles). Here we define the radius of a LG-like beam according to [1] as the distance from the center to the intensity maximum within the ring of light, which does, however, not correspond to the dark core width of the LG-like beam [2]. The linear scaling in ℓ can be approximated by [1]

We also qualitatively investigated the dependence of the vortex beams on a mutual misalignment (decentering) of the two DOE plates. It turned out that small misalignments on the order of 10 μm still allow to produce almost undistorted vortex beams, if the relative rotation angle of the two plates is also adapted. Since the structure of the vortex plates is finer in their center, misalignments affect particularly the central area of the combined element. Therefore it turns out that higher vortex modes, which have a lower intensity in the central area, are less affected from a misalignment, than modes with lower helicities.

4. Conclusion

Using two DOEs with specially designed, conjugate phase profiles, an optical element is realized which produces vortex phase profiles with a helical index adjustable by a mutual rotation of the two plates. The experimental results agree with the theoretical expectations, and show a good quality of the obtained phase structures, which makes them practically employable in the typical field of application of vortex plates, as e.g. for the creation of donut beams with adjustable helical charge, or for Fourier filtering in microscopy. The vortex plates are not sensitive to polarization, yield a high conversion efficiency (typically more than 90%), and can potentially be used in high power applications.