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An interferometry based method to achieve resolution beyond the diffraction barrier is proposed. Object is illuminated with different tilted beams, generated by using a Spatial Light Modulator (SLM). In addition, some constant phases are also assigned to each tilted beam with the SLM display. Then, the object is simultaneously illuminated with all tilted beams, producing an on-axis interferometry scheme. An interferogram at the image plane is formed for each set of constant phases added to the tilted beams. Using proper selection of constant phases for each of the interferograms, the synthetic aperture can be calculated. During the post processing, we take the Fourier transforms of the each image and the portions of the spectrum are spatially shifted and combined to obtain synthesized spectrum whose inverse Fourier transform gives high resolution image.
©2013 Optical Society of America
1. Introduction
Resolution criteria are defined in different ways: Rayleigh resolution, Abbe number, etc. The two main factors affecting the lateral resolution are wavelength of light and numerical aperture of the imaging system. Further, the CCD sampling and noise are considered resolution limiting sources. Different optical components were used to enhance the resolution of the optical system. In 1952, G.Toraldo used optical mask [1] to improve the resolution of the imaging system, which is the first attempt towards the resolution problem.
Different researchers have attempted to overcome resolution limitations, but resolution is still a problem in imaging world. Every mind had tried to present some suitable solution using different optical components. Nearly after fifteen years of Toraldo work [1], W. Lukoz used a diffraction grating [2,3] to reduce the diffraction limit of the band limited system. After few years of this idea, Ueda Mitsuhiro [4,5] used holography for super resolution, which was actually used for 3D imaging of data storage systems. The usage of grating in holography [6–9] to enhance the resolution remains attractive for few years and produce meaningful results. In these techniques amplitude and phase grating were used to increase the resolution. Later on the fringe illumination [10,11] and tilted beam illumination [12,13] were tested to shift the spectrum of the object into the pass band at Fourier plane in 4f optical system.
Other relevant attempts to recover the missing information are related to the use of structured illumination microscopy [14–17] and to the time multiplexing super resolution [18–20] techniques, leading to significant improvements for super resolution imaging. In many of the above mentioned techniques, a given reference beam is required in the setup. This reference beam may lead to a loss of stability, when an extra reference beam arm is required in the set-up, or may lead to a field of view loss because half of the light beam is being used as reference [19]. Other works [16,18], required of different wavelengths to transport different regions of the spectrum.
In this work a new technique able to achieve super resolution imaging is proposed using tilted beams produced by spatial light modulator (SLM). The interference occurs at the image plane between these tilted beams, which travelled along the common path of the system. The novelty of our proposed technique is twofold. First, the simultaneous illumination of the object is achieved in a simple way, by means of different tilted beams of a single wavelength which are generated by using an SLM. Second, unlike many of the previously described techniques, the necessity of a reference beam is avoided, leading to a higher stability of the system and simplifying the complexity of the set-up.
The outline of this paper is as follows. In section 2, the working principles of our set-up are given. Next, the whole technique is justified in three different ways. In section 3 the mathematical proof is described. Afterwards, the simulation results and the experimental results are presented in section 4 and in section 5 respectively. Finally, the conclusions of the work are given in section 6.
2. Working principles of the experimental setup
In Ref [13], a setup based on sequentially tilted beams illumination of the object in off-axis interferometery was used to achieve superresolution imaging. Here, with the will of going a step further, two main improvements have been performed on the setup, which is sketched in Fig. 1. First, reference beam has been excluded of such previous setup, allowing a significant increase of the stability of system. Second, instead of sequential illumination of the object, it is illuminated by using different tilted beams simultaneously to make the system more comprehensive.
Fig. 1 (a) Experimental optical system for structure illumination using Spatial light modulator for on-axis interferometry. (b) Equivalent and simplified sketch in transmission showing the expanded beams.
The SLM is the main component to create tilted beams with added constant phases, which illuminate an object placed at the entrance of a 4f optical system. A quadratic phase is added at the beam exiting from the SLM plane to separate the different diffraction orders produced by the nonlinearity and phase fluctuations of the SLM. Collimating lens is placed after the SLM to produce tilted plane waves to illuminate the object. As imaging system we are using a 4f optical system (although the proposed technique can be applied with many imaging systems). In this way, we can easily control the aperture size. This aperture is intentionally placed at the Fourier transform plane in 4f optical system to measure the resolution enhancement of the system. Usually this aperture allows only low spatial frequencies to pass and blocks high spatial frequencies of the object. As a result, the decrease of resolution at image plane is produced. To retrieve these high spatial frequencies, tilted beams with simultaneous illumination procedure are used.
In the setup sketched in Fig. 1(a), the optical system used to illuminate the object with the different tilted beams is based on the SLM and the collimating lens, which is placed at the proper distance of the SLM. To illustrate how these tilted beams allow us to recover some of the high frequencies filtered at the aperture plane, a simplified sketch in transmission, being analogous to the one given in Fig. 1(a), is drawn at Fig. 1(b). In Fig. 1(b) we see as a collimated beam reaches the SLM device. By addressing three different quadratic phases at three different sections of the SLM, three different beams are generated (i.e. red, blue and green rays in Fig. 1(b)). By means of a convergent lens with a focal equal to distance between its position and the virtual focus of the rays, tilted illumination is achieved at its exit. Finally, Fourier spectrum of object is obtained at the aperture plane, but for each light beam, with a given displacement at the transversal direction. With this process, an aperture 3 times larger than the real one is achieved.
We have tested the technique by using computer simulation and experimentally in 1D. Note that even we have restricted our experiments to 1D, this set-up can be also applied to extend the experiment for 2D. The SLM surface is horizontally divided into three equal sections each one having different linear phases. Here, if the object is sequentially illuminated by each part of the SLM then different parts of the spectrum of the object passes through the aperture at Fourier plane. When central part illuminates the object, only the band-limited image is recorded. When the other two shifted sections having linear phases illuminate the object then the spectrum at Fourier plane is shifted allowing the pass of higher frequencies. As said before, all the three sections illuminate the object simultaneously and result into an interferogram at image plane due to interference of all the three beams. During this process, the central part of object spectrum and also the other two shifted spectra are passing simultaneously through imaging system. To recover all the correlation terms, some constant phases (i.e. 0, π/2, π) are added to each section of SLM plane. Regarding these three values of the constant phase, a total of seven sets are formed. For each image, one set of constant phases is applied and never repeat again, as shown in Table 1. These constant phases produce correlation between the shifted parts of the SLM and result into seven interferograms at the image plane.
Table 1. Set of constant phases used to separate the shifted terms
The final step is to reconstruct these recovered sections of object spectrum, which is totally based on computer calculations. The recovered parts are shifted to their original position using the data of aperture size and SLM shift of tilted beams.
In this work we have used a particular array of phases (i.e. 0, π/2, π) in order to illustrate the validity of the proposed technique. However, different arrays of phases may be used to retrieve the required information and the array of phases selected will have some impact on the final noise propagated at the reconstructed image. To minimize this noise propagation, an optimization process based on some quality indicator, as for instance the condition number (CN) [21], can be applied. In our particular case, the CN value calculated for the phases (i.e. 0, π/2, π) is of 3.86, being not optimal (the theoretical minimum value for the CN is equal to 1). However, a CN of 3.86 is a reasonable value to accomplish our illustrative proposes.
To a better understanding of the recovering procedure, we us use the following example. Let us imagine that the Fourier spectrum of certain object is described by the function illustrated in Fig. 2(a). In this case, the whole object spectrum is defined from –3a/2 to 3a/2. However, due to the transfer function of the system, high frequencies are filtered and only the object information related to spatial frequencies into the range from –a/2 to a/2 can be obtained. The same object information given in Fig. 2(a), can be obtained by properly performing a summation of different parts of the object spectrum with a size equal to a, as it is illustrated in Fig. 2(b).
Fig. 2 (a) Example of an object Fourier spectrum; (b) Summation of different parts of the object spectrum displaced to this original position; (c) Summation of different parts of the object spectrum centered at the origin.
However, what we are able to recover at this stage with the technique proposed above is the addition of the different parts of the object spectrum displaced to the origin, as shown in Fig. 2(c). So, in order to recover the object given in Fig. 2(a), proper displacements for the different shifted object parts obtained must be imposed. Therefore, a data post processing is applied, leading to the final super resolved image. The whole procedure is mathematically detailed in the next section of this article.
3. Mathematical modeling of the proposed technique
Our technique is based on off-axis illumination diffracted toward on axis, leading to an on-axis scheme. In this technique, the intensity images are captured with different tilted illuminations. First, let us assume that the system shown in Fig. 1 is illuminated with a plane wave making a zero angle with optical axis. In this situation, the intensity at the exit of the 4f system (i.e. the intensity recorded by the camera) is given by,
in which O(x) and h(x) are object transmission and the coherent Point Spread Function (PSF) of the system respectively. The sign denotes convolution. The Fourier transform of Eq. (1) is given as,where and are the Fourier spectrum of the object and the coherent transfer function respectively. The symbol * represents the correlation operator. To shift the spectrum of the object, the incident plane waves are tilted. Moreover, constant phases are also added. Thus, the Fourier transform of the recorded image isIf the object is simultaneously illuminated with n tilted plane waves, each of them having different constant phase, the FT of the recorded image is given as,
In order to understand the model, it is verified for one dimension and for three different tilted plane waves (i.e. n = 3). Let us assign the following relations to simplify calculations,
By using the definitions given in Eq. (6) and (7) and for n = 3, the Eq. (5) becomes,
At this stage, we obtain an expression for the Fourier transform of the recorded image that only depends of the summation of the following seven terms:
whereAfterwards, by selecting three particular phases (0, π, π/2) for each constant phase used (i.e. δ1, δ2 and δ3) the following linear system can be written:
Note that from intensity vector A and the inverse of matrix M, the vector X is calculated, which gives the coefficients Ci (i.e. images) of Eq. (9). By using the correlation properties [22],
the Ci coefficients are back shifted to their original position to obtain the synthesized spectrum (see example given in Fig. 2). Finally, these displaced coefficients, which actually give the reconstructed spectrum of the object, are added to obtain the following synthetized spectrum,
The value of obtained from Eq. (14) can be written similar to Eq. (2), but now, the coherent transfer function is related to a larger aperture size,
The inverse Fourier transform of this equation gives the super-resolved image,
In Eq. (16) the PSF of the system is narrower compared to the PSF in Eq. (1), which is the proof of the enhancement of resolution.
4. Simulated results
In this section, the proposed technique is tested using computer simulation for one dimensional USAF resolution chart. For one dimension a total of three shifted illumination along x-axis are used to image the object. The shift of each illumination from the central one is related to the dimension of aperture placed at imaging system (see Fig. 1) and also explained in [12,13]. These shifted beams illuminate the object simultaneously such that they shift the object spectrum at aperture plane, thus permitting different spatial frequencies to pass through. To find the cross correlation terms each illumination having different constant phases are used as explained in section 3. Simulated results for the different recovered parts of the object spectrum are shown in Fig. 3.
Fig. 3 Different spectra reconstruction obtained after solving the linear equation system composed of 7th correlation terms.
The next step is to combine the different parts of the object spectrum as shown in the Fig. 4. During the reconstruction the different parts of the spectrum are shifted to their original position and then stitched together. As a result, the whole reconstructed spectrum is extended along x-axis, as shown in Fig. 4(a). This extension clearly shows that extra spatial frequencies, which are discarded by aperture when using single illumination (see Fig. 4(b)), are now recovered. The inverse Fourier transform of the reconstructed spectrum gives an image which is highly resolved.
Afterwards, Fig. 5 shows the comparison between three different images. First, the original object, which is used as target object, is shown in Fig. 5(a). Second, in Fig. 5(b) we show the image obtained without our proposed technique (let us call it low resolution image). Finally, the image given in Fig. 5(c) corresponds to the super-resolved image which is obtained by using the tilted simultaneous beam illumination. From this comparison, it is clear that some missing frequencies are achieved with our technique, enhancing the resolution of the final image along x-axis. The right column with vertical lines is for comparison. In Fig. 5 (b) is difficult to identify the number of lines for objects in that column, as the corresponding frequencies are not resolved. Unlike, lines in the same column for Fig. 5(c) are clearly resolved, as we can see that each part is composed of three lines, similarly to Fig. 5(a), which is original object.
Fig. 5 Simulated results. a) Original image; b) low resolution image obtained with the system; c) higher resolution image obtained by applying the proposed method.
5. Experimental results
To demonstrate the validity of our technique, an experiment is performed by using the optical system shown in Fig. 1. The coherent illumination source is a He-Ne laser working at λ = 632.8 nm followed by objective lens of 0.85 numerical aperture and 60x. It is followed by a spatial filter to clean the beam. As SLM we use a Parallel Aligned (PAL) Liquid Crystal on Silicon (LCoS) spatial light modulator model PLUTO from HoloEye. This device has a resolution of 1920x1080 pixels, a pixel pitch of 8 µm, and fill factor of 87%. As the PA LCoS display is a reflective device, a beam splitter is used to provide 90 degrees between the incident and the reflected beams. This configuration is selected because it provides larger phase modulation than oblique incidence, as proved in Ref [23].
The camera is a charge coupled device (CCD) model piA1000-60gm from Basler. The camera resolution is 1000x1000 pixels, and its pixel pitch is 7.4 µm. In addition, to control the intensities and assure the linear polarization, quarter wave plates and polarizers distributed by Thorlabs are used as well.
In our experiment, one dimensional (x-axis) experimental results are presented to verify the technique. Figure 6(a) shows the low resolution image and Fig. 6(b) shows the high resolution image in 1D. In this figure, the second column from right side (i.e. vertical lines) is presented for comparison, where the resolution is improved along x-axis. Note as from the top to bottom of this column initial three sections are resolved in Fig. 6(b), which are blurred in Fig. 5(a). Therefore, even experimental results (Fig. 6(b)) are less resolved than simulated results (Fig. 5(c)), due to the expected experimental error sources (SLM limited resolution, time-fluctuations of the phase [24], etc.), we want to emphasize that experimental results obtained by using our proposed technique (Fig. 6(b)) provide higher resolution than those related to low resolution image (Fig. 6(a)).
Fig. 6 Experimental results: (a) Low resolution image obtained with the system,(c) higher resolution image obtained by applying the proposed method.
6. Conclusion
Tilted beams illumination brings higher spatial frequencies into the pass-band sequentially at aperture plane to increase the resolution of the resulting image. In this work, simultaneous illumination of all the tilted beams is used to increase the image resolution. To retrieve the cross correlation terms, some constant phases are assigned to each illumination. As a result of these constant phases, seven interferograms are recorded experimentally for one dimensional object. Simulation results are also shown for 1D in which seven interferograms are recorded to retrieve the cross correlation terms along x-axis. In post processing these recorded interferograms are processed such that to shift the side bands of the object spectrum. By taking the inverse Fourier transform, the super-resolved image has been obtained.
Acknowledgments
We acknowledge financial support from Spanish Ministry of Science and Education and FEDER (FIS2009-13955-C02-01). Anwar Hussain acknowledges Higher education Commission (HEC) of Pakistan for funding the grant.
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