1. Introduction

High resolution liquid crystal SLMs are currently used in various applications like holographic image projection [1, 2], adaptive optics [3, 4], spatial Fourier filtering of images in microscopy [5, 6], or projecting structured light fields for microscopy and optical trapping [7, 8, 9]. In most of these applications the SLMs are used to display phase-only holograms, which diffract a readout beam in a controlled way. More precisely, the phase structures displayed on the SLMs typically correspond to diffractive optical elements (DOEs) with blazed structures (unlike real holograms), which can achieve a diffraction efficiency in the desired first diffraction order, which is near 100%. The remaining light intensity is mostly located in the zero-diffraction order, corresponding to non-diffracted light, which focuses in a Fourier plane behind the SLM. For the case of so-called Fourier holograms, the desired reconstructed hologram is thus superposed by a disturbing spot of focused zero order light, with can have a high peak intensity. This zero order spot arises even in ”ideal” DOEs, which have a 100% diffraction efficiency (at a certain ”design” wavelength), if the wavelength (λ) is changed. The reason is that the DOE corresponds to a spatially modulated surface profile h(x, y), which translates into an optical path length profile h_opt (x, y) = h(x, y)[n_D(λ) − n₀(λ)], where n_D(λ) and n₀(λ) are the wavelength dependent refractive indices of the DOE material and the surrounding, respectively. This path length modulation corresponds to a phase modulation Φ(x, y) = 2πh_opt (x, y)/λ. Obviously, the resulting phase profile magnitude ΔΦ_max depends reciprocally on the readout wavelength λ, and can also vary if the refractive index of the DOE material (or its surrounding) is dispersive. The optimal diffraction efficiency is typically achieved, if the phase profile magnitude corresponds to 2π, which is thus only possible at a certain ”design” wavelength. Changing the wavelength leads to a decrease of the diffraction efficiency, and an increase of the zero order beam.

In order to avoid this zero-order beam, which is disturbing in hologram projection and spatial filtering applications, different approaches have been demonstrated. One of the most common methods is to transform the original on-axis DOE into an off-axis DOE. This spatially separates the zero order from the desired first diffraction order. The focused zero order beam can then be removed by a beam stop. Practically this is done by adding a blazed phase grating to the original on-axis DOE, followed by a subsequent modulo 2π operation, such that the resulting off-axis DOE ranges between 0 and 2π. This modulo 2π operation ”entangles” the DOE with the grating, such that all photons which are diffracted into the first diffraction order of the grating, are also diffracted into the first order of the original DOE, whereas the zero-order beam travels along the original optical axis and focuses in the Fourier plane. The light field in the first diffraction order thus ”perfectly” corresponds to the desired field, even for the case that the diffraction efficiency of the DOE is low, e.g. due to fabrication tolerances or a wavelength mismatch.

Unfortunately this method to spatially separate the undesired zero diffraction order from the first order has some disadvantages. One is that it violates a normally useful design concept of optical systems, namely to keep the light beams close to the optical axis [10]. Non-specular beam deflections like the diffraction at an off-axis hologram produce a geometric shortening along the direction of diffraction, which in turn can be the cause of aberrations, such as astigmatism. A second disadvantage is the strong dispersion of the fine grating, which causes broadband light to be fanned out like a rainbow along the diffraction direction.

Therefore methods have been developed which suppress the zero-order beam for on-axis DOEs [11, 12, 13]. The diffraction efficiency η_m into the order m-th diffraction order depends on the extension Φ_max of the phase modulation range [14]:

If Φ_max equals 2π, which may be accomplished for DOEs at a certain ”design” wavelength, the zero diffraction order (m = 0) is completely suppressed, and the first order maximal. But note that Φ_max shows a strong wavelength dependence. Even if the refractive index of the DOE material would be constant, corresponding to a constant optical path length, the corresponding phase shift Φ_max would decrease linearly with increasing wavelength. However, in actual DOEs (including the liquid crystals of an SLM) this trend is even increased, due to linear, and -approaching the blue wavelength range - also quadratic dispersion of the refractive index. For example, the phase range extension Φ_max of our SLM (Hamamatsu LCOS) increases by a factor of 2.4 in a wavelength range between 700 nm and 400 nm. Thus, if a DOE is optimized for suppressing the zero diffraction order at a certain wavelength, the zero order will reappear if the wavelength is changed. One method to avoid this was demonstrated in [15]. There, an achromatic π-phase shift of a ferroelectric SLM was achieved by using the phase associated with a polarization change, which is independent of the incident wavelength. However, this method could only be used to control binary holograms, which suppress the zero diffraction order, but have only a diffraction efficiency limited to 40%.

One interesting principle to get rid of the zero diffraction order in an on-axis DOE was originally demonstrated in [16]. The idea was to transform the ”standard” on-axis DOE into an off-axis DOE by adding a blazed grating and performing a subsequent modulo 2π operation. Afterwards the off-axis diffraction is compensated by a back-diffraction process at a subsequent conjugate blazed grating, such that the reconstructed image now appears again on the optical axis, whereas the zero-order beam is diffracted to an off-axis position, where it can be blocked. A similar method was used for performing broadband Fourier image filtering operations in [17] and [18]. Furthermore a method to use an extended phase modulation range of 3π for contrast enhancement of binary DOEs was suggested in [11]. In the original version demonstrated in [16] the operation principle is demonstrated for a double-sided on-axis beam shaper, where the front side contains the off-axis DOE, and the back-side the back-diffracting blazed grating. As an example the authors assume that the diffraction efficiency of each of the two individual DOEs (i.e. the off-axis hologram and the blazed grating) is η₁ =95%, and that the remaining light intensity appears in the zero order, i.e. η₀ =5%. If a single DOE with these properties were used to display an on-axis image, the contrast V₀ between the reconstructed first-order hologram and the zero order beam would be given by:

On the other hand, the double-sided on-axis DOE produces a zero order intensity of only ${\eta }_{00}={\eta }_0^2=0.25\%$, corresponding to the amount of light that passes through both of the two subsequent DOEs as a zero-order beam, whereas the efficiency of the reconstructed on-axis hologram is still ${\eta }_{11}={\eta }_1^2=90\%$, which corresponds to the amount of light that is diffracted at both of the two subsequent DOEs into the first diffraction order. The overall contrast V_g between the on-axis reconstructed hologram and the remaining zero-order beam (which focuses on-axis) is then given by:

Here we show that the same idea (i.e. the ”grating method”) can be straightforwardly implemented by an SLM with a phase shifting capability of at least 4π, which is often achievable with modern SLMs. The principle is based on the fact that a stack of two individual DOEs placed behind each other, which are both modulated in a phase range between 0 and 2π, has the same total transmission function as a single DOE whose phase profile is the sum of the two individual DOE phases, and which thus extends in a range between 0 and 4π. Thus the phase profiles of the two individual DOEs which are embossed at the front and rear surfaces of the double-sided on-axis DOE in [16] (i.e. the off-axis hologram and the back-diffracting grating) can be added to form a single DOE with a phase range between 0 and 4π, which is then displayed at an SLM with a 4π-phase range.

An advantageous generalisation of the ”grating-method” is to superpose the original DOE with other phase structures, followed by the addition of the respective conjugate structures. For example, superposing an on-axis DOE with a random phase mask (modulo 2π), followed by the addition of the conjugate random phase mask, leads to the same level of zero-order suppression as the ”grating method”, but in this case the higher diffraction orders (which appear in the ”grating method”) are dispersed, i.e. diffusely scattered, thus diluting their intensity in the camera plane. This allows to use the entire diffraction angle range of the SLM to project the hologram, whereas in the original ”grating method” about half of the image field-of-view has to be sacrificed for the segregation of the undesired diffraction orders.

There is an interesting variant of this ”random mask method”, which is mainly suited for the projection of holograms which are calculated as ”kinoforms” [19], or with an iterative method such as a Gerchberg-Saxton (GS) algorithm [20]. In this case two independent on-axis DOE structures (DOE₁ and DOE₂) are calculated which project nevertheless the same image, for example by performing two runs of a GS algorithm to produce the same image, but with different random starting conditions. The phase profile is then constructed by superposing DOE₁ with the conjugate of DOE₂ (modulo 2π), and then adding DOE₂ (without the modulo operation). In principle the conjugate of DOE₂ acts like a random phase mask for DOE₁, i.e. the sum of the two structures produces just a randomly scattering phase profile, similar to the ”random mask method”. The subsequent addition of DOE₂ then compensates for this and produces the reconstructed original on-axis image. The advantage of this method is that the zero order component which passes through the first part of the 4π phase profile (i.e. through mod_2π [DOE₁+conj(DOE₂)]) is ”recycled”, since it afterwards passes through the second part of the 4π DOE, namely through DOE₂. There it can be again diffracted to produce the desired image. Assuming that the diffraction efficiency is only distributed between the zero and first diffraction orders, the total reconstruction efficiency η_d of this ”double DOE method” is given by:

Compared to the efficiency of the ”grating method”, ${\eta }_1^2$, one now has an additional contribution η₀η₁, which is due to ”zero-order recycling”, and which becomes prominent if the zero-order intensity increases, which happens particularly for a readout wavelength mismatch to longer wavelengths. Note that the remaining zero order efficiency is still given by the contribution which passes through both of the two subsequent phase structures in the zero order, i.e. by ${\eta }_0^2$. In the following experiments, we compare the performance of the ”grating-method” with that of the ”double DOE method”, finding the expected increase in diffraction efficiency, whereas the zero orders remain equal.

2. Experiments

Our experimental setup (sketched in Fig. 1) consists of a wavelength tunable, fiber coupled thermal light source (Till Photonics Monochromator Polychrome IV), which delivers monochromatic light with a bandwidth of 8 nm to 15 nm in a wavelength range between 400 nm and 700 nm to the output of a multi mode fiber. There the light is expanded by a microscope objective (60x, NA 0.85) and subsequently collimated by an achromatic lens (f=200 mm). The beam then illuminates a liquid crystal phase-only reflective SLM (Hamamatsu LCOS SLM X10468-01, 792 x 600 pixels, pixel size: 20 x 20 μm²) under a small tilt angle (15°). A linear polarisation filter in front of the SLM optimizes the incident light polarisation for maximal diffraction efficiency. A second polariser after the SLM takes out light whose polarization changed due to inter pixel fringing effects. The SLM displays our test DOEs, which are modulated in a phase range up to 4π at our ”design” wavelength of 500 nm. Due to the tilt angle the light which is reflected by the SLM includes an angle of about 330° with respect to the incident beam. It passes through a second achromatic lens L2 (f=200 mm) and focuses at a CMOS camera (Matrix Vision mvBlueFox 221G, without objective lens). Thus the image recorded by the camera is the Fourier transform of the field diffracted off the SLM.

 

Fig. 1 Standard setup for the reconstruction of a Fourier DOE. Light from a monochromator passes through a multimode fiber and is collimated by an objective (Obj.) and an achromatic lens L1 (f=200 mm). The light passes through a linear polariser (Pol1) to the surface of a reflective, phase-only SLM. The light diffracted off the SLM passes through a second linear polariser (Pol2) and is focused with lens L2 (f=200 mm) at a CMOS camera.

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For comparison a set of three DOEs was calculated, all of them designed to project a test object, namely an image of the letter ”R” (see Fig. 2(A)) with three different methods. All of the DOEs were calculated using an image resolution of 600 x 600 pixels, which could be displayed on the SLM. The first DOE (D₀) was calculated as a ”standard” on-axis DOE using an iterative GS algorithm [20]. The corresponding phase profile, extends over a range between 0 and 2π, is displayed as a gray level image in Fig. 2(B)).

 

Fig. 2 A: Intensity image to be reconstructed by the DOEs displayed on the SLM. B: On-axis phase-only DOE calculated with a GS algorithm which reconstructs image (A) in its Fourier plane. The gray levels in the picture correspond to phase values in the DOE (the colorbar displays the phase values in radians). C: DOE with zero-order suppression calculated according to the ”grating method” (Eq. (5)). D: DOE calculated according to the ”double DOE method” (Eq. (6)). Note that the phases of the DOEs displayed in (C) and (D) range from 0 to 4π. All DOEs consist of a 600 x 600 pixel array.

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The second DOE (D_g) was calculated using the ”grating-method” for suppression of the zero diffraction order. Its phase profile was calculated according to:

The third DOE (D_d) was constructed according to the ”double DOE method”. For this purpose, a second on-axis DOE (D₁) for the reconstruction of the test image (letter ”R”) was calculated using the GS algorithm, similar to the calculation of D₀, but with another random starting condition. The resulting D₁ is different from D₀, although the two DOEs reconstruct the same intensity image (but with a different phase distribution). Thus the sum or difference between D₀ and D₁ results in a random phase distribution, which would diffusely scatter an incident beam. Using these two on-axis DOEs, D_d is calculated according to

The first term describes the sum of D₀ and the conjugate of D₁, namely of (2π − D₁), entangled by the modulo operation. Afterwards the second DOE (D₁) is added. The resulting phase D_d, which is thus modulated in a range between 0 and 4π, is displayed in Fig. 2(D). As explained above, an incident beam which is first order diffracted by the first part of D_d, namely by mod_2π [D₀ + (2π − D₁)], and then again first order diffracted by the second part (D₁), reconstructs the Fourier transform of D₀ in the camera plane, i.e. the test image ”R”. However, the remaining fraction of the incident beam which passes through the first part of D_d as a non-diffracted zero order beam then arrives at the second part of D_d, namely at D₁, where it can be again first order diffracted, which reconstructs the Fourier transform of D₁ in the camera plane, which is again the test image ”R”. Thus the total efficiency for the reconstruction of the test image is increased with respect to the ”grating method”, while in both cases the residual zero order beam is given by the light which passes undiffracted through both 2π slides of the 4π-DOEs (D_g and D_d). Furthermore, a diffuse background is generated by light which is first order diffracted by the first phase term of D_d, and then passes undiffracted through its second part. This background actually consists in a convolution of the image (”R”) with its conjugate, which has another random phase distribution, such that the resulting intensity is diffusely distributed in a circular area with approximately twice the image diameter.

The DOEs D₀ (Fig. 2(B)), D_g (Fig. 2(C)) and D_d (Fig. 2(D)) were then displayed at the SLM, using a so-called look-up table (i.e. a wavelength-dependent ”translation table” provided by the manufacturer, which translates the intended phase shifts to the corresponding voltage levels which have to be applied at the SLM pixels) which was optimized for a readout wavelength of 500 nm. Thus, at that wavelength the phase range of a DOE displayed at the SLM corresponds optimally to the calculated phases, whereas for larger wavelength the SLM phase range decreases (i.e. the DOEs are ”undermodulated”), and vice versa. The DOEs were then reconstructed using different wavelengths of the readout beam (from 420 to 700 nm with a step size of 20 nm), which could be adjusted at the illumination source (monochromator). Finally, in order to determine absolute diffraction efficiencies, an intensity spectrum of the illumination source was recorded at all readout wavelength positions. Care was taken to avoid saturation of the camera, i.e. each exposure was controlled by a histogram, and in case of saturation the exposure time was decreased. In a post image processing step, the image intensities were then normalized according to their different exposure times, and according to the intensity spectrum of the illumination source. Some exemplary reconstructed DOEs are shown in Fig. 3 as a function of the readout wavelength (indicated in the upper row).

 

Fig. 3 Reconstructed images of a standard DOE D₀ (upper row), a DOE optimized with the ”grating method” D_g (middle row), and a DOE optimized with the ”double DOE method” D_d (bottom row) at different readout wavelengths, quoted at the top. The focused spot of the zero order beam is indicated in each image by an arrow. All images are normalized to the peak intensity of the indicated zero order spot.

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The top row shows the images reconstructed from a standard DOE, D₀ (which spans only over a phase range of 2π), the second row the corresponding images of a DOE constructed by the ”grating-method”, D_g (Eq. (5)), and the bottom row the reconstructed images of a DOE calculated according to the ”double DOE method” (Eq. (6)). All images were normalized to the peak intensities of their corresponding zero order beams which focus at the optical axis, indicated by arrows. Since the SLM lookup table is optimized for a readout wavelength of 500 nm, there the contrast between first order and zero order spot is approximately equal for all DOE types. For an ideal SLM one would even expect in this case a complete suppression of the zero order beam. The residual intensity seems to be due to reflections at the glass cover plate of the SLM surface as well as from hologram imperfections at phase wrapping zones. For larger wavelengths, the ratio between the intensities of the first order image and the zero order peak of the standard DOE (upper row) strongly decreases. On the other hand the images reconstructed by the ”grating method” (middle row) show a much less pronounced contrast reduction with increasing wavelength. Note that in this case a second focused spot with higher intensity appears at the right side of the on-axis zero-order spot. It corresponds to the original zero-order beam (of the ”standard” DOE), which is diffracted away from the optical axis by the subsequent grating, where it could be removed by a beam stop. At lower wavelengths (particularly at 420 nm) also higher orders of diffraction at the subsequent grating become visible: This is due to the fact that there the SLM phase range becomes ”over modulated”, i.e. it extends over more than 4π, which according to Eq. (1) leads to diffraction into higher orders.

The last row of the figure shows the images reconstructed by the ”double DOE method”. The intensities of the first order reconstructed test images ”R” are at a first glance similar to those of the ”grating method” (middle), but a quantitative evaluation (next figure) shows that they are actually more efficient, due to the ”zero-order recycling” explained above. A second difference with respect to the ”grating method” is that higher diffraction orders are suppressed. This is due to the fact that the ”double DOE method” disperses all diffraction orders (including the zero order), which end up in a slightly increased uniform background, except for the desired first order which is reconstructed. Compared to the ”grating method”, the ”double DOE method” allows to utilize the whole field-of-view of the reconstructed image, since no higher diffraction orders or off-axis focused spots have to be removed by additional apertures.

A quantitative comparison of the three DOE types is shown in Fig. 4, and compared to a numerical simulation of the expected first order and zero order efficiencies. For this purpose the integrated energies in the first order images (i.e. the letter ”R”) and the integrated energies in the on-axis zero order spots were determined in all reconstructed holograms, and normalized to the spectral energy of the illumination source, which was measured independently for all reconstruction wavelengths. Thus the plotted efficiencies correspond to the relative diffraction efficiencies of the SLM in the respective orders, compared to the total light deflected by the SLM. For the simulations the DOE structures displayed on the SLM were numerically Fourier transformed and the energies in the zero and first diffraction orders were integrated as in the experimental images. The wavelength dependence could be calculated by using the look-up table set of the SLM manufacturer, which was provided for the whole range between 400 nm and 700 nm. The simulations show that the first order efficiencies (upper curves) of all DOE types should achieve a maximum at the design wavelength 500 nm, for which the look-up table of the SLM was optimized. For lower and higher wavelengths there is an efficiency decrease, which is strongest for the ”grating method” DOE, due to the double diffraction processes at the two subsequent phase profiles. As expected, the efficiency decrease is less pronounced for the ”double DOE method”, where the zero diffraction order of the first part of the 4π phase profile is recycled by its second part. Considering the zero order efficiencies of the respective DOEs it turns out that both, the ”grating method” and the ”double DOE” method show the same level of zero order suppression as compared to the standard DOE.

 

Fig. 4 Numerical simulation (left) and experimental results (right) of the first order (upper curves) and zero-order (lower curves) efficiencies of three DOE types, namely a standard DOE (in a phase range of 2π, green), a DOE calculated according to the grating method (Eq. (5), blue) and a DOE calculated according to the ”double DOE method” (Eq. (6), red). The first order efficiencies (upper curves) of the 4π-DOEs are slightly lower than those of the standard DOE, but the corresponding zero order efficiencies (lower curves) are significantly suppressed for both the ”grating method” and the ”double DOE method”, resulting in a strong contrast improvement. The numerical simulations at the left were performed using the look-up tables of the employed SLM, which are provided by the producer.

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The experimental data shows a similar qualitative behaviour as the numerical simulations. However it turns out that the zero order suppression is not ideal, as in the simulations. Instead the zero order has an offset, which is probably due to a reflection from the cover glass of the SLM, and due to phase imperfections in wrapping zones of the DOE. Moreover the whole curve corresponding to the first order efficiency of the ”grating method” (blue) seems to be shifted to longer wavelength, as compared to the simulations. This might be due to an ”overmodulation” of the SLM look-up table in the extended phase range (i.e. the range between 2π and 4π), for which the ”grating method” is more sensitive than the other DOE types. Nevertheless the suppression of the zero diffraction orders of the 4π-DOEs as compared to the standard DOE is qualitatively reproduced, leading to a significant contrast enhancement for projection or imaging applications, and increasing the wavelength range in which such a DOE can be used.

3. Conclusion

Modern liquid crystal based SLMs often offer the possibility to modulate the phase in a much wider range than 2π. This feature becomes more and more interesting for various applications, i.e. in [21] it was used to increase the diffraction efficiency of SLMs by reducing the density of phase wrapping contours, and in [22] higher order diffraction was demonstrated for various wavelengths. Here we suggest and demonstrate a method generating DOEs in a 4π phase range which suppresses the disturbing zero order intensity of the diffracted beams in a broad wavelength range. This allows to read out the corresponding on-axis DOEs with a broadband light source, such as a thermal source or an ultrafast laser, without generating an intense spot of zero order focused light. We presented a ”grating method”, where the disturbing light is diffracted to an off-axis position, where it can there be removed with an aperture. This method is particularly useful for cases where the DOEs have uniquely defined transfer functions, e.g. for (Fresnel-)lenses, axicons, or vortex plates, which can now be used in an extended wavelength band. On the other hand, for image projection purposes only the projected intensity is important, and there are manifold realizations of the corresponding DOEs, which can be calculated with iterative algorithms, like the GS algorithm. In these cases the ”double DOE method” can be applied, leading to a higher image intensity as compared to the ”grating method”, and to a suppression of higher diffraction orders, such that the whole image field can be used. The principles of zero-order suppression by using an extended phase modulation range are not only limited to SLM-based DOEs, but can be applied as well to standard etched DOEs, if these can be manufactured in a correspondingly extended etching depth range.