Common path in-line holography using enhanced joint object reference digital interferometers

Joint object reference digital interferometer (JORDI) is a recently developed system capable of recording holograms of various types [Opt. Lett. 38(22), 4719 (2013)]. Presented here is a new enhanced system design that is based on the previous JORDI. While the previous JORDI has been based purely on diffractive optical elements, displayed on spatial light modulators, the present design incorporates an additional refractive objective lens, thus enabling hologram recording with improved resolution and increased system applicability. Experimental results demonstrate successful hologram recording for various types of objects, including transmissive, reflective, three-dimensional, phase and highly scattering objects. The resolution limit of the system is analyzed and experimentally validated. Finally, the suitability of JORDI for microscopic applications is verified as a microscope objective based configuration of the system is demonstrated. ©2014 Optical Society of America OCIS codes: (090.0090) Holography; (090.1995) Digital holography; (090.1760) Computer holography; (110.6880) Three-dimensional image acquisition; (100.3010) Image reconstruction techniques; (070.6120) Spatial light modulators; (180.6900) Three-dimensional microscopy; (090.2880) Holographic interferometry; (090.1970) Diffractive optics. References and links 1. Y. Rivenson, B. Katz, R. Kelner, and J. Rosen, “Single channel in-line multimodal digital holography,” Opt. Lett. 38(22), 4719–4722 (2013). 2. B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM),” Opt. Express 20(8), 9109–9121 (2012). 3. D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948). 4. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52(10), 1123–1128 (1962). 5. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). 6. V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Lett. 34(10), 1492–1494 (2009). 7. V. Linnik, “Simple interferometer for the investigation of optical systems,” Proc. Acad. Sci. USSR 1, 208–210 (1933). 8. R. N. Smartt and W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–357 (1975). 9. H. Kadono, N. Takai, and T. Asakura, “New common-path phase shifting interferometer using a polarization technique,” Appl. Opt. 26(5), 898–904 (1987). 10. C. Ramírez, E. Otón, C. Iemmi, I. Moreno, N. Bennis, J. M. Otón, and J. Campos, “Point diffraction interferometer with a liquid crystal monopixel,” Opt. Express 21(7), 8116–8125 (2013). 11. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45(5), 836–850 (2006). #204375 $15.00 USD Received 8 Jan 2014; revised 11 Feb 2014; accepted 13 Feb 2014; published 24 Feb 2014 (C) 2014 OSA 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.004995 | OPTICS EXPRESS 4995 12. O. Mudanyali, D. Tseng, C. Oh, S. O. Isikman, I. Sencan, W. Bishara, C. Oztoprak, S. Seo, B. Khademhosseini, and A. Ozcan, “Compact, light-weight and cost-effective microscope based on lensless incoherent holography for telemedicine applications,” Lab Chip 10(11), 1417–1428 (2010). 13. V. Arrizón and D. Sánchez-de-la-Llave, “Common-path interferometry with one-dimensional periodic filters,” Opt. Lett. 29(2), 141–143 (2004). 14. V. Mico, Z. Zalevsky, and J. García, “Superresolution optical system by common-path interferometry,” Opt. Express 14(12), 5168–5177 (2006). 15. P. Gao, G. Pedrini, and W. Osten, “Structured illumination for resolution enhancement and autofocusing in digital holographic microscopy,” Opt. Lett. 38(8), 1328–1330 (2013). 16. R. Kelner, J. Rosen, and G. Brooker, “Enhanced resolution in Fourier incoherent single channel holography (FISCH) with reduced optical path difference,” Opt. Express 21(17), 20131–20144 (2013). 17. R. Kelner and J. Rosen, “Spatially incoherent single channel digital Fourier holography,” Opt. Lett. 37(17), 3723–3725 (2012). 18. O. Bryngdahl and A. Lohmann, “Variable magnification in incoherent holography,” Appl. Opt. 9(1), 231–232 (1970). 19. D. N. Naik, G. Pedrini, and W. Osten, “Recording of incoherent-object hologram as complex spatial coherence function using Sagnac radial shearing interferometer and a Pockels cell,” Opt. Express 21(4), 3990–3995 (2013). 20. B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18(2), 962–972 (2010). 21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 8, p. 273 and Chap. 5, p. 97. 22. M. Born and E. Wolf, Principles of Optics (Cambridge, 1999), Chap. 8.6.3, p. 471.


Introduction
Recently, the joint object reference digital interferometer (JORDI) has been introduced as a system for recording digital holograms of various types under coherent illumination [1].In JORDI, hologram formation is accomplished through the interference of two distinctively different parts of the same input beam: one that contains the object (or scene) scattered information, and another unscattered part.The latter does not contain any object related information, and can thus be used as a source for the reference beam.The actual interference between the object and reference parts is made possible through the utilization of the birefringence properties of liquid crystal spatial light modulators (SLMs), by which orthogonal polarization components of the input beam can be optically manipulated in different ways.These manipulations enable hologram recording using a digital camera, upon which an imaging of the object scattered information is made to overlap with a reference beam.The use of SLMs offers flexibility, and holograms of several modalities (including offaxis and on-axis phase-shifting Fresnel holograms, and image holograms) can easily be recorded [1].The combination of the polarization dependent control and the presence of both object and reference information at the same input wave (i.e., joint object reference) enables the realization of JORDI in a single-channel, common-path configuration.This holds the advantages of offering system robustness due to inherent resistance to vibrations, easier assembly and alignment of the optical system, and can possibly allow the recording of temporally incoherent holograms, as long as the optical path difference (OPD) between interfering beams is small enough [2].
In retrospect, it is not surprising that the first ever holographic setup, invented by Gabor [3], was realized by a single channel configuration.However, two significant issues restrict the usability of Gabor's original implementation.The first is the well-known twin image problem, where two conjugated images are formed for a single recorded object and together with an additional 0th diffraction order term undermine the quality and the visibility of the holographic reconstruction.The second is the weak-scattering condition which restricts the hologram recording to objects that must emit two mutually coherent waves: one that is much more intense than the other, but does not carry any spatial information, and another that contains the information in a form of a scattering, low intensity beam.Luckily, solutions for solving these issues have been proposed throughout the years; the most famous one is the offaxis holography suggested by Leith and Upatnieks [4].Other approaches maintain the single channel configuration and adopt a phase-shifting procedure [5] for the removal of the twinimage and bias (0th order) term.For example, Mico et al. [6] propose the "phase-shifting Gabor holography" method where an SLM replaces the usual semi-transparent pinhole slide found in the point-diffraction interferometer (PDI) [7,8], and the central pixel of the SLM is used for phase-shifting the non-scattering light of the input target .This non-scattering term, in turn, serves as the reference of the recorded hologram.A similar approach for phaseshifting had been suggested before [9], but required a mechanical intervention between exposures.Recently, a more economical implementation of the phase-shifting PDI using a mono-pixel SLM has been proposed [10].
Though the twin-image problem in PDI is easily resolved via phase-shifting, under high scattering conditions the non-scattering term might prove too weak relative to other diffracted terms, limiting hologram recording due to an insufficient visibility of the interference fringes.Alternative approaches to single-channel holography remedy this issue by ensuring that a large enough portion of the illumination freely passes around the object (see, e.g., [11] where coherent illumination is used, and [12] where a very compact design of the optical apparatus enables the use of a light source of a lesser degree of spatial and temporal coherence), or demand either a separation [13] or coexistence [1] of the object and reference within the system input plane.In [13] the interference between the reference and object originated waves is achieved through the use of a diffraction grating, a concept that has successfully been used later for achieving super-resolution holography via either off-axis [14] or structured [15] illumination.
In JORDI, one may consider the hologram recording as a result of the combination of lateral and radial shearing, where the object part of the input beam is made to overlap with the reference part of the same input beam.The object and reference parts undergo different magnifications (attributed to the radial shearing), and additional lateral shearing is applied (whenever necessary) to the reference part so that the two parts overlap at the hologram plane.Interestingly, similar concepts were used for the successful realization of spatially incoherent holography.For example, in a recent publication [16] we have shown how the Fourier incoherent single channel holography system [17] can be considered as a combination of both rotational and radial shearing interferometers.Other examples are described in [18,19] where incoherent Fourier hologram are recorded using a radially shearing Sagnac interferometer, which is an example of a dual-channel common-path configuration.Note that the systems in [16][17][18][19] are self-referencing in nature, and unlike JORDI do not require separate object and reference parts.In such systems, the recorded hologram represents the summation over all point source contributions, as many self-interferences occur in parallel.In the former JORDI implementation [1], the above mentioned radial and lateral shearing was achieved by the use of each of the two SLMs as a single lens imaging system, offering simplicity and flexibility at the same time.However, the pixel size of currently available SLMs limits the achievable numerical aperture (NA) of the SLM displayed diffractive lens to values well below those that can be reached by a refractive lens, implying a strict limit on the resolutions that can be achieved by the prior design.Following, we introduce a new JORDI system that incorporates an additional objective lens.Hence, this system is not limited by the low NA of the SLM.

System design and theoretical analysis
The proposed JORDI design is schematically presented in Fig. 1(a).The system consists of an objective lens o L , two phase-only SLMs located between two polarizers, and a digital camera [e.g., a charge-coupled device (CCD)].The two SLMs are positioned with the ordinary/extraordinary axis of SLM1 in parallel to the extraordinary/ordinary axis of SLM2, meaning that their active axes are perpendicular to each other.The two polarizers, P1 and P2, are set in parallel to each other, and at a 45° angle to the active axes of the SLMs.This setup can be analyzed by considering each of the two SLMs as affecting orthogonal components of the input wave, as described in Figs.1(b) and 1(c).A collimated laser beam is used to illuminate a target object (e.g., a resolution test chart) and also serves as a reference, where the requirement is that some part of the collimated beam arrives to the system input plane unaffected by the target object, thereby forming the joint object reference input plane.For the object scattered part of the input wave, polarized perpendicularly to the plane of the page, the system acts as an afocal imaging system [Fig.1(b)] built from two converging lenses: the refractive lens o L with a focal length of o f and an additional diffractive lens with a focal length of 2 f displayed on SLM2.Consider an arbitrary input wave ( , ) in u x y arriving to the joint object reference input plane of the optical system.Upon arrival to the CCD plane, this wave is: where is the quadratic phase function, in which  is defined as the central wavelength of the coherent laser illumination, propagation of a wave to a distance d z (mathematically, the  symbol denotes a two- dimensional convolution), ( denotes the influence of a converging lens of a focal length of f and 1 c is a complex constant.For the reference part of the input wave, polarized in parallel to the plane of the page, the system acts as a two-lens imaging system [Fig.1(c)], realized by the refractive lens o L and an additional diffractive lens with a focal length of 1 f , displayed on SLM1.Unlike the imaging system of Fig. 1(b), this system is not afocal, and the wave reaching the CCD after passing through it is: where is the linear phase, with ( , ) c is a complex constant, and exp( ) i denotes an additional phase term that enables the phase-shifting procedure [1], where / 2 3 . Note that SLM1 is used for the realization of the expression , which includes the phase-shifting term, a diffractive lens, and an additional linear phase that can be used to achieve lateral displacement of the reference part, controlled by the two dimensional parameter ( ) , .Based on a mathematical justification presented in [20], it is easy to show that, for this case, Eq. ( 1) results with: , where where 2 b is a complex constant.The two right-most terms in Eq. ( 5) indicate a transverse magnification of 2) where 2 a is a complex constant.Assume that the object and reference parts of the input wave occupy the regions f is chosen to be equal twice the focal length 2 f (i.e., 12 2 ff  ).A reasoning based on Eqs. ( 4)-( 6), which is detailed in the Appendix, shows that the intensity inside the region , out obj


, recorded by the CCD, is: .
Following a phase-shifting procedure described in [1], a complex-valued digital hologram is formed: is a conjugated phase term attributed to the reference.Note that this last term does not contain any object related information, and its parameters are determined by the physical properties of the optical setup.As such, it can be digitally eliminated, although this step is usually not essential.Further note that in order to achieve balance between the object and the reference waves it is possible to set the two polarizers, P1 and P2, at angles which are different from the previously mentioned 45°.
Consider an object located at the joint object reference plane of the imaging system presented in Fig. 1(b) (i.e., at the front focal plane of the objective lens o L ).According to the Abbe resolution criterion [22], the minimum resolvable distance between a pair of grating lines at the input plane of a coherent imaging system is 0.82 / L .The resolution of the imaging system presented in Fig. 1(c) is similar, but of lesser importance, since the spatial bandwidth of the reference part of the input wave is much smaller than the bandwidth of the object wave.However, the fringe patterns formed by the interference of these two waves should be faithfully recorded by the CCD.The maximal spatial frequency of such a pattern can be found based on geometrical considerations.Under the assumption that both the CCD and SLM2 are limited to an aperture of the size D , the maximal possible angle of an object wave reaching the CCD is given by    

Experiments and results
We implemented a JORDI system as shown in Fig. 2.This implementation is based on Fig. 1(a), but since the two SLMs are reflective, beam splitters were placed in front of each of them, aligning their planes perpendicularly to the optical axis.We note that for light demanding applications one can avoid using the beam splitters, and instead, position the SLMs at an angle to the optical axis while applying corrections to the SLM displayed elements (see, for example [1,2]).For imaging of samples that might be damaged by an excessive amount of light it may be beneficial to position the polarizer P1 before the sample, or simply use a weaker linearly polarized laser, assuming the influence of the sample on the  1951 USAF resolution test target (Thorlabs R3L3S1N) was placed 18.5cm away from o L .Resolution groups 2 (elements 2 to 4) and 3 to 7 (all elements) served as the target, whereas the empty square located above these two groups served as a free passing region for the reference [see Fig. 2(a)].The amplitude and phase of the recorded JORDI hologram are presented in Figs.3(a) and 3(b), respectively.Its reconstruction is presented in Fig. 3(c), where a Fresnel back propagation to a distance of 1.34 r m z c was used to achieve best focus.Note that the reconstruction distance is affected by the reference term respectively, and indicate that the resolution of the holography system is better than 40.3 line pairs per millimeter (lp/mm) (for element 3 of group 5), where the distance between the centers two bright lines (i.e., the width of two single lines) is Δ = 24.81µm.However, at 45.3 lp/mm (for element 4 of group 5), with Δ = 22.08µm, the visibility is significantly lower and the resolution of the system is barely sufficient to resolve details.Comparing this result to the resolution limit indicated by the Abbe criterion [Eq.( 9 , indicates that indeed, the resolution of the proposed system matches the theoretical limit.Note that in the presented case, the pixel size of the digital camera does not limit the resolution of the system [Eq.( 9)].In the second experiment, the test target was replaced with a positive 1951 USAF resolution test target (Thorlabs R3L3S1P) in order to demonstrate the system operation with a highly reflective object.Similarly to the previous experiment, resolution groups 2 (elements 2 to 4) and 3 to 7 (all elements) served as the target, whereas the square located above these two groups served as a mirror for the reference.This time, the collimated laser beam was used to  3), shows that in terms of resolution, both cases lead to similar results.However, background artifacts are much more prominent in the case of the reflective object, and are the result of the reflectivity of the non-mirrored parts of the glass made 1951 USAF target.In the previous experiment, light had been able to pass only through clear parts of the target, while here there was a strong reflection from the lines and figures and an additional weak reflection elsewhere.The fourth experiment demonstrates the system capability of recording holograms of reflective, highly scattering objects.An Israeli 5 Agorot coin (withdrawn from circulation in 2008) has been used as the target object.The coin was placed in front of a reflective neutral density filter that allowed part of the collimated laser to serve as a reference beam of controllable intensity.To extend the field of view, an afocal system of 1/4x magnification was inserted between the object and the system entrance lens o L , with 30  Note that Eq. ( 9) indicates an achievable resolution of 2.0 μm 8 min  .In order to further justify the suitability of the proposed system for microscopy, the system capability of recording holograms of phase objects was verified.The configuration of the first experiment was used together with a plano-convex cylindrical lens (Newport CKX300, N-BK7 made, effective focal length 30 ( 1) ac r m f n  [21].Some variations are evident between the two curves, but once an average phase profile is considered, as in Fig. 8(e), the two curves become very similar.It should be also noted that the hologram presented in Fig. 8 is the only image hologram in this study, whereas the other holograms herein are Fresnel holograms.

Summary and conclusions
A new holographic system based on the JORDI concept has been presented.In the previous JORDI implementation [1], each of the two SLMs functioned as a single lens imaging system, and so the resolution of the system was limited by the relatively low numerical apertures that can be achieved using currently available SLMs.The proposed enhanced system, however, is not limited by this factor and a higher resolution can be achieved using an objective lens of a higher numerical aperture.
We have successfully demonstrated the recording of various holograms using transmissive, reflective, three-dimensional, highly scattering and phase objects.The analysis of the experimental results indicates that the achieved resolutions are similar to those implied by the Abbe resolution criterion for coherent imaging systems.In addition, by incorporating a microscope objective into the proposed setup, a hologram of a transmissive resolution test target capable of distinguish details as small as 4.38μm was recorded, indicating the applicability of JORDI in the microscopy regime.

Fig. 1 .
Fig. 1.Schematics of the proposed JORDI design: Lo, objective lens; P1 and P2, polarizers; SLM1 and SLM2, spatial light modulators; CCD, charge-coupled device.In (a) the complete system is presented, whereas in (b) and (c) the imaging systems creating the object and reference waves are presented separately and respectively.The symbols , , and are polarization orientations perpendicular, parallel and at a 45° angle to the plane of the page, respectively.

Fig. 1 (
Fig. 1(c).Without loss of generality, assume an input wave of ( , )(1/ z ( ) )/ in s s s u L z x y Q r   due to


inside the joint object reference input plane, respectively.Based on Eq. (of the object and reference parts on the output plane of the system, respectively, can easily be identified.Successful hologram recording requires the object region , condition which can be satisfied by a proper selection of the parameters A are indeed chosen so that the object and reference parts of the same input wave overlap each other on the CCD plane [Figs.1(b)and 1(c)].For simplicity, further assume that the focal length 1 is the numerical aperture of the system.The parameter D represents the system aperture size [Fig.1(b)],determined either by the diameter of objective lens o L or by the height of the SLM that is being used (the smaller of these two sizes), and o f represents the focal length of the objective lens o

Fig. 2 .
Fig. 2. Experimental setup of JORDI: Lo, objective lens; P1 and P2, polarizers; BS1 and BS2, beam splitters; SLM1 and SLM2, spatial light modulators; CCD, charge-coupled device.The symbols , , and are polarization orientations perpendicular, parallel and at a 45° angle to the plane of the page, respectively.The setup in (a) is suitable for recording transmissive objects, whereas the alternative illumination configuration demonstrated in (b) enables recording reflective objects.In the first experiment, a plano-convex refractive lens with a focal length of 17.5 o m f c  cross-sections of elements 1 to 4 of resolution group 5, which are highlighted in Fig. 3(c), are shown in Figs.3(d)-3(g), #204375 -$15.00USD Received 8 Jan 2014; revised 11 Feb 2014; accepted 13 Feb 2014; published 24 Feb 2014 (C) 2014 OSA

Fig. 3 .
Fig. 3. JORDI recording of a transmissive target: (a) amplitude and (b) phase of the recorded hologram; (c) hologram reconstruction at the plane of best focus; (d)-(g) cross sections of elements 1 to 4 of group 5 (highlighted) of the resolution targets, respectively.

#
204375 -$15.00USD Received 8 Jan 2014; revised 11 Feb 2014; accepted 13 Feb 2014; published 24 Feb 2014 (C) 2014 OSA illuminate the target from its front, instead of its back, using a beam-splitter, as shown in Fig. 2(b).Other parameters of the experimental setup were left unchanged.The amplitude and phase of the recorded hologram are presented in Figs.4(a) and 4(b), respectively, with its reconstruction in Fig. 4(c), where a Fresnel back propagation was used to achieve the best focus.An enlarged portion of the reconstruction, for resolution groups 4 and above, is shown in Fig. 4(d).Comparing Fig. 4(d) to the transmissive target results (Fig.

Fig. 4 .
Fig. 4. JORDI results for a reflective target: (a) amplitude and (b) phase of the recorded hologram; (c) hologram reconstruction at the plane of best focus; (d) enlarged portion of (c) shows details of resolution groups 4 and 5.The third experiment demonstrates the system capability of recording holograms of a three-dimensional scene.Two transmissive targets were placed roughly 1cm away from each other: a 4.0 cycles/mm group of a positive NBS 1963A resolution test target and a 'X1' inscription from a positive 1951 USAF resolution test target.The amplitude and phase of the recorded hologram are presented in Figs.5(a) and 5(b), respectively.Figures 5(c) and 5(d) demonstrate the preservation of the three-dimensional information of the scene, with the 4.0 cycles/mm and 'X1' inscriptions in the best focus, respectively.Note that in Fig. 5(b) the phase of the reference beam was digitally eliminated based on Eq. (8), in order to emphasize phase details from the actual objects.This manipulation does not affect the quality of the hologram reconstructions [Figs.5(c)and 5(d)] and nor is essential.However, it does influence the reconstruction distance to a small degree.

Fig. 5 .
Fig. 5. JORDI recording of a three-dimensional scene: (a) amplitude and (b) phase (shown after eliminating the quadratic phase term of the reference) of the recorded Fresnel hologram; (c) hologram reconstruction at the plane of best focus of the 4.0 cycles/mm inscription; (d) hologram reconstruction at the plane of best focus of the 'X1' inscription.
. 6(a)].Experimental results are presented in Figs.6(b)-6(d).While the amplitude of the recorded hologram [Fig.6(b)] exposes only few details of the out-of-focus coin, the additional phase information [Fig.6(c)] enables focusing into the coin [Fig.6(d)],where much more details are visible, including the digit '5.

Fig. 6 .
Fig. 6.JORDI results for a highly scattering object: (a) modification of the experimental setup (Fig. 2), where a relay system of 1/4x magnification, realized by the lenses La and Lb, extends the field of view of the system; (b) amplitude and (c) phase of the recorded hologram, where out-of-focus details of the Israeli 5 Agorot coin are barely visible; (d) hologram reconstruction at the plane of best focus, exposing details of the 5 Agorot coin.#204375 -$15.00USD Received 8 Jan 2014; revised 11 Feb 2014; accepted 13 Feb 2014; published 24 Feb 2014 (C) 2014 OSA 10 March 2014 | Vol.22, No. 5 | DOI:10.1364/OE.22.004995| OPTICS EXPRESS 5005

Fig. 7 .
Fig. 7. JORDI based holographic microscopy: (a) amplitude and (b) phase of the recorded hologram; (c) hologram reconstruction at the plane of best focus, showing complete details of resolution groups 6 and 7, up to 228 lp/mm.
test target, located 17.5cm away from o L .The amplitude of the recorded hologram is shown in Fig. 8(a).In Figs. 8(b) and 8(c) the wrapped and unwrapped phase profiles of the cylindrical lens [magnified by the imaging system of Fig. 1(b)] are shown, respectively, after the phase of the reference beam has been digitally eliminated according to Eq. (8).In Fig. 8(d) the measured phase profile of the lens at a fixed x-axis position of 1 x mm  along the y-axis is shown together with the theoretical profile calculated