Open Access
Add to BibSonomy
Abstract
We propose a numerical technique that analyses the imaging properties of optical scanning holography (OSH) constructed using real optical components in a real optical alignment situation. Using the proposing technique, we analyze the aberrations and the vulnerability of the OSH about the optical alignment. After that, we propose a digital filter that compensates the aberrations of the OSH. Finally, we show that the digital filter removes the aberrations of an experimentally recorded OSH.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recording the hologram of a real object as a form of electric signal has a long-standing history [1,2]. Optical scanning holography (OSH) was proposed to record a hologram of a real object using two-pupil heterodyne scanning [3,4]. Twin-image noise elimination in OSH was proposed by utilizing in (I)-phase and Quadrature (Q) -phase heterodyne detection scheme [5]. Speckle free recording of a complex hologram using OSH was demonstrated [6]. Recently full-color optical scanning holography (FC-OSH) has been proposed and shows that the full-color complex hologram of a real object is recorded by two-dimensional (2D) scanning [7]. A chromatic aberration compensation filter for FC-OSH has been proposed [8]. The imaging properties of OSH has been analyzed analytically with ideal Gaussian beam or quasi-spherical wave assumptions [10,11]. In this paper, we propose a numerical technique that analyses the imaging properties of OSH constructed using real optical components in a real optical alignment situation. In which, we choose commercially available real optical components for numerical analysis. Using the proposing technique, we analyze the degradation of a point spread function (PSF) and an optical transfer function (OTF) according to the misalignment of an optic axis. This shows the degree of the vulnerability of OSH about the misalignment. We also show that even for a perfect aligned case, the aberrations of the OSH are caused by the real optical components. This technique can be used for the evaluation and the design of OSH with real optical components in a real optical alignment situation. Virtually all aberrations of a hologram recording system can be compensated by convolving the recorded hologram with the complex conjugate of the PSF of the system. [9] The PSF of the hologram recording system can be obtained by recording a sub resolution pinhole. However, recording the hologram of a pinhole whose size is smaller than the resolution limit of the system is tedious and requires sub-resolution alignment. That is even worse in the hologram recordings of reflective objects, which requires the recording of the hologram of a sub-resolution reflective pinhole. In practice, the reflected light from the sub-resolution reflective pinhole is weak and vulnerable to the optical alignment of the pinhole. Based on the numerical analysis of OSH, we propose a digital filter that removes the aberrations of OSH. First, we numerically generate a PSF which contains the aberrations of OSH using the proposing numerical analysis technique. Second, we synthesize a digital filter using the numerically generated PSF. We call the digital filter, an aberration compensation filter (ACF). Finally, we show that the ACF removes the aberrations of an experimentally recorded OSH. In section 2, we briefly review the principles of OSH. In section 3, first, we analyze the aberrations of OSH constructed with real optical components with optical misalignment. Second, we propose a numerical technique that analyzes the imaging properties of OSH constructed using commercially available real optical components with optical misalignment using Zemax. Third, we analyze the imaging properties of OSH constructed using real optical components in a real optical alignment situation. Fourth, we propose an ACF that compensates the aberrations of the OSH. In section 4, we show that the ACF compensates the aberrations of the experimentally recorded OSH.
2. Optical scanning holography
Figure 1 shows OSH which consists of a modulation unit, a time-dependent Fresnel zone plate (TD-FZP) generation unit, a scanning unit, a space-integrating photo-detecting unit and a demodulation unit. In the modulation unit, the laser beam generated by a laser passes through a half-wave plate 1 (HW 1) and becomes a linearly polarized beam. This beam goes to an electro-optic modulator (EOM) which modulates the phase of the input beam with a carrier frequency, $\Omega$. Note that the polarization direction of the input beam is rotated by ${45^o}$ to the induced birefringence axis (IBA) of the EOM. This makes the input beam compose two orthogonally polarized parts. One part is polarized along the IBA and the other part is polarized orthogonal to the IBA. The phase of a beam polarized along the IBA is modulated by the EOM and the phase of the other part of the beam remains unchanged. The modulated beam goes to the TD-FZP unit. In The TD-FZP unit, the beam goes to the Mach-Zehnder Interferometer (MZI). The MZI generates the Fresnel zone plate (FZP) which is beating according to the modulation frequency of the EOM. We call the FZP with beating, a time dependent FZP (TD-FZP). The TD-FZP goes to a demodulation unit through a beam splitter (BS). In the demodulation unit, the TD-FZP goes to a photo detector 1 (PD1). The PD1 generates a reference beating signal for demodulation. Meanwhile, the TD-FZP reflected by the BS goes to the scanning unit. In the scanning unit, the x-y Galvano scanning mirror scans an object with the TD-FZP. The transmitted light from the object is collected by a lens 2 (L2) and the collected light goes to the photo detector 2 (PD 2). The PD 2 generates the electric current according to the intensity of the collected light. The electric current contains a beating signal which contains the information of the complex hologram of the object. The electric current is demodulated into the complex hologram by a dual-output lock-in amplifier (LIA).
Fig. 1. OSH system. (L, laser; HW 1,2, half wave plates; EOM, electro-optic modulator; M 1,2,3, mirrors; PBS, polarization beam splitter; BE 1,2, beam expanders; L 1,2, lens; BS, beam splitter; PD 1,2, photo detectors; LIA, lock-in amplifier;)
3. Imaging properties of OSH
3.1 Aberrations in OSH
In this section, we analyze the aberrations of OSH constructed with real optical components with optical misalignment. In OSH, the complex hologram is recorded by the convolution between an object’s intensity distribution and TD-FZP. This makes the aberrations of OSH are originated from the phase distortion of TD-FZP. TD-FZP is an interference pattern between a spherical wave and a plane wave with beating. As shown in Fig. 1, the spherical wave is generated by a lens, L1. Thus, the spherical wave is distorted by the aberrations of the lens. In addition to the distortion, the shifting and tilting of the optic axis of the lens in the alignment cause the phase distortion of the TD-FZP. This phase distortion due to the misalignment of the optic axis as well as the aberrations of the lens causes the aberrations in OSH. The TD-FZP with aberrations is given by:
3.2 Numerical analysis technique
In this section, we propose a numerical technique that analyzes the imaging properties of OSH constructed using commercially available real optical components with optical misalignment. Zemax is a computer added design (CAD) tool for optics design and evaluation using ray tracing technique. The recent version of Zemax gives an interferogram by calculating the path-length difference of rays. The PSF of OSH is determined by the FZP that is generated by a MZI with phase shift in TD-FZP generation unit. We synthesize the FZP using Zemax. First, we construct an optics layout for a MZI in Zemax. As shown in Fig. 2(a), the optics layout for the MZI consists beam splitters 1,2 (BS1,2) and mirrors 1,2 (M1,2), a bi-convex lens (KBX 139, Newport) and a flat glass (FG). We choose a commercially available biconvex lens and locate the lens with tilting and shift of the optic axis, which corresponds to a misalignment situation. The shifting distance and tilting angle of the optic axis of the lens are denoted by ${d_x},{d_y}$ and ${\theta _x},{\theta _y}$ in Fig. 2(b). We set a plane wave of 660 nm wavelength with a uniform beam profile which goes into the BS1. The BS1 splits the plane wave into upper and lower path beams. The upper path beam becomes a plane wave with phase delay that is introduced by a FG. The lower path beam becomes a spherical wave after being passed through the bi-convex lens. Here, we note that the spherical wave contains the aberrations caused by misalignment of the optic axis as well as the aberrations of the lens. Second, we calculate an interferogram at an imaging plane (IP) using Zemax. The optical path length difference between the upper and the lower path beams onto the IP gives an interferogram. Here, we introduce a phase delay for the interferogram by adjusting the thickness of the FG. This gives us the interferogram with arbitrary phase delay. We get simulated interferograms with I- and Q- phase by setting the thickness of the FG, zero and $\frac{\lambda }{2}$ respectively. Third, we synthesize a complex FZP by adding I-and Q-phase interferograms in a complex manner. That is given by:
Fig. 2. Top: ZEMAX simulation information. (a) ZMEAX generated MZI layout. The red path beam is a plane wave, the blue path beam is a spherical wave. (BS 1,2, beam splitters; M 1,2, mirrors; FG, a flat glass; IP, an image plane; L, a bi-convex), (b) magnification of a red dotted box in Fig. 2(a) (The $\mathop \theta \nolimits_x ,\mathop \theta \nolimits_y $ are tilting angles which are the angles of the transverse-axes of the lens to x- and y- axis respectively, $\mathop d\nolimits_x ,\mathop d\nolimits_y$ are the shifting distance between the optic axles of the lens and the MZI). Bottom: I- and Q- phase of the ZEMAX generated interferogram when the bi-convex lens is perfectly aligned. (c) I-phase of the interferogram. (d) Q-phase of the interferogram.
3.3 Imaging properties of OSH
In this section, we analyze the imaging properties of OSH constructed using real optical components in a real optical alignment situation. First, we calculate PSFs by reconstructing the FZPs according to Eq. (3). Figures 3(a) and 3(b) show the PSFs when the optic axis of the bi-convex lens is shifted and tilted respectively. Here, we can see that the shift of the optic axis broadens the PSFs along the shifted direction and the tilting of the optic axis generates coma tail in the PSFs along the tilted direction. As expected, we can also observe that more shifting and tilting cause more broadening and coma tail. Even for a perfectly aligned case as shown in Fig. 3 with ${d_x},{d_y}$ and ${\theta _x},{\theta _y}$ being all zeros, the PSF is not as sharp as the diffraction limited case due to the aberrations of the real optical components. Since the broadening and the coma tail of the PSFs occur along the shifting and tilting directions of the optic axis, the occupancy of the PSFs in the frequency domain becomes asymmetric. Figure 4(a) shows the contour map of an OTF when the optic axis of the lens is tilted by ${\theta _x},{\theta _y} = ({3^ \circ },{3^ \circ })$. Here, we can see that the OTF is distorted from circular to elliptical due to the aberrations introduced by the tilting angles. In Fig. 4(a), we call the largest length and the smallest length of the elliptical OTF, the major axis and the minor axis respectively. Figures 4(b) and 4(c) show the profiles of the major axis and the minor axis of the OTFs along the shift distances and the tilting angles. These results show that more shifting and tilting decrease the bandwidth of the OTFs along the directions of the shifting and tilting.
Fig. 3. PSFs of OSH. (a) PSFs of OSH according to the shifted distance of the optics axis of the bi-convex lens. (b) PSFs of OSH according to the tilting angle of the bi-convex lens.
Fig. 4. OTFs according to the shift distance and the tilting angle of optic axis. (a) Contour map of the OTF when the optic axis of the lens is tilted by ${\theta _x},{\theta _y} = ({3^ \circ },{3^ \circ })$. (b) Profiles of the major axes and the minor axes of OTFs according to the shifting of the optic axis, (c) Profiles of the major axes and the minor axes of OTFs according to the tilting of the optic axis.
The simulation results show that the imaging properties of OSH are rapidly deteriorated according to the optical misalignment. Especially in terms of the circular symmetricity of a PSF, we can see in Fig. 4 that the difference of the lengths between the major axis and the minor axis is larger in the tilting than the shifting and thus, the circular symmetric property of a PSF is more vulnerable to the tilting. In addition to the aberrations due to the optical misalignment, the aberrations of the lens also degrade the imaging properties of OSH. In the next section, we propose an aberration compensation filter that removes the aberrations caused by the lens as well as the optical misalignment.
3.4 Aberration compensation filter
In this section, we propose an aberration compensation filter (ACF) that removes the aberrations of OSH constructed using real optical components with a real optical alignment situation. The FZP that is synthesized in section 3.2 contains the aberrations introduced by the real optical components as well as the optical misalignment. The phase of the synthesized FZP contains the whole phase information of the aberrations. Using the phase, we construct the ACF that compensates the aberrations. First, we synthesize the FZP of OSH by using the digital analysis technique proposed in section 3.2. Second, we set the complex conjugate of the FZP as an impulse response function of an ACF. Finally, we filter the OSH using the ACF. The filtered output is given by:
where $ACF(x,y;z)$ is the impulse response function of the ACF filter, ${\otimes}$ represents convolution operation and $H(x,y)$ is the recorded hologram with aberrtions. After that, we reconstruct the complex hologram using the corresponding ACF according to Eq. (5). The simulation results are shown in Fig. 5. In Fig. 5, we can see that the lengths of the major axis and that of the minor axis become the same regardless of the shifted length and the tilted angle of the optic axis. This means that the ACF filters out the asymmetric aberration caused by the optical misalignment. In addition to this, we also can see that the amplitudes of the filtered OTFs are larger than that of the OTF in a perfectly aligned case that is shown as a blue line in Fig. 4. This means that the ACF filters out the aberrations introduced by the lens itself even in the perfectly aligned case.Fig. 5. Simulation of ACF. (a) Cross sections of compensated OTF when the shift distance changes (b) Cross sections of compensated OTF when the tilting angle changes
4. Experimental results
In this section, we experimentally record the complex hologram of a pinhole using OSH and show that the ACF removes the aberrations. In experiments, we use a red ($\lambda = 660\mbox{ }nm$) laser and modulate the EOM at 20 kHz and locate a 12.5-micron pinhole as an object. The collimated beam diameter is 5 mm and the focal length of the biconvex lens (KBX 139, Newport) is 38.1 mm. Here, the biconvex lens is tilted by ${\theta _x},{\theta _y} = ({10^ \circ },{0^ \circ })$ to the optics axis of the collimated beam. The scan area is about 10 mm by 10 mm in x-y direction and the size of the recorded hologram are 1000 by 1000 pixels. First, we record the complex hologram of the pinhole using OSH. Note that tilted bi-convex lens generates aberrated TD-FZP. This makes the recorded hologram is aberrated by the tilting as well as the aberration of the lens. Second, we synthesize the FZP using Zemax. Third, we construct an ACF using the FZP. Here, we determine the tilting and shift parameters of the ACF by searching the parameters which make that the occupation of the OTF of the filtered output is maximized along the spatial frequency axis. Figures 6(a) and 6(b) show the reconstructed image of the experimentally recorded complex hologram using a conventional digital back-propagation technique and the ACF respectively. Figures 6(c) and 6(d) shows the contour map of the Fig. 6(a) and 6(b). Figure 6(e) shows the profiles of the power spectrum of the conventionally reconstructed hologram and the reconstructed hologram using the ACF. Note that Fig. 6 shows that the ACF removes the aberrations of the recorded hologram. Note that Fig. 6 shows that the ACF removes the aberrations of the recorded hologram.
Fig. 6. Reconstructions of the recorded hologram. (a) reconstructed image using FSIRF, (b) reconstructed image using ACF, (c) contour map of the reconstructed image using FSIRF, (d) contour map of the reconstructed image using ACF (yellow, lime, cyan and blue colors represent the 90% of the peak intensity, the 3dB of the peak value, the full-width at half maximum and the 36% of the peak intensity respectively) (e) power spectrums of the reconstructed images using FSIRF and ACF. (red lines are major and minor axes of reconstructed image using FSIRF, blue lines are major and minor axes of the reconstructed image using ACF)
5. Conclusion
In this paper, we propose a numerical technique that analyses the imaging properties of OSH constructed using real optical components in a practical alignment situation. Computer simulations show that the broadening and the coma tail of the PSF of OSH occur along the shifted and the tilted directions of the optic axis, and even for a perfectly aligned case, the PSF is also contaminated by the aberrations of the real optical components used in OSH. Based on the computer simulations, we propose an ACF that removes the aberrations of OSH. The ACF reconstructs the complex hologram with canceling out the aberrations caused by the real optical components and the misalignment of an optic axis. Finally, we show that the ACF cancels out the aberrations of an experimentally recorded OSH.
Funding
Institute for Information and Communications Technology Promotion (ITTP) (2017-0-00417, Openholo library technology development for digital holographic contents and simulation); MSIT Ministry of Science, ICT and Future Planning.
References
1. C. Burckhardt and L. Enloe, “Television transmission of holograms with reduced resolution requirements on the camera tube,” Bell Syst. Tech. J. 48(5), 1529–1535 (1969). [CrossRef]
2. J. Berrang, “Television transmission of holograms using a narrow-band video signal,” Bell Syst. Tech. J. 49(5), 879–887 (1970). [CrossRef]
3. T. C. Poon and A. Korpel, “Optical transfer function of an acousto-optic heterodyning image processor,” Opt. Lett. 4(10), 317–319 (1979). [CrossRef]
4. T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A 2(4), 521 (1985). [CrossRef]
5. T. C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three dimensional images in optical scanning holography,” Opt. Lett. 25(4), 215–217 (2000). [CrossRef]
6. Y. S. Kim, T. Kim, S. S. Woo, H. Kang, T. C. Poon, and C. Zhou, “Speckle-free digital holographic recording of a diffusely reflecting object,” Opt. Express 21(7), 8183–8189 (2013). [CrossRef]
7. H. Kim, Y. S. Kim, and T. Kim, “Full-color optical scanning holography with common Red, Green and Blue channels,” Appl. Opt. 50(7), B81–B87 (2011). [CrossRef]
8. T. Kim, “Chromatic aberration issue on full-color optical scanning holography,” IEEE INDIN 2016, (Futuroscope-Poitiers, France, 18 - 21 July 2016), pp. 532–535.
9. G. Indebetouw and W. Zhong, “Scanning holographic microscopy of three-dimensional fluorescent specimens,” J. Opt. Soc. Am. A 23(7), 1699–1707 (2006). [CrossRef]
10. B. D. Duncan and T.-C. Poon, “Gaussian Beam Analysis of Optical Scanning Holography,” J. Opt. Soc. Am. A 9(2), 229–236 (1992). [CrossRef]
11. G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17(3), 380–390 (2000). [CrossRef]
