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Abstract
We present a single-shot wavelength-multiplexing technique for off-axis digital holography based on a spectral filter. Only a spectral filter is inserted between beam splitter and mirror in reflection off-axis digital holography (RODH). The spectral filter can transmit a well-defined wavelength band of light, while reject other unwanted radiation. By adjusting the filter and mirror separately, the propagation orientation of different reference beams of two wavelengths can be separated, and thus two off- axis holograms with different fringe directions are simultaneously captured by a monochrome camera. The wavefront interference analysis of using a spectral filter is discussed in detail. Our scheme is available for real-time wavelength-multiplexing but requires fewer optical elements and system modifications. Numerical simulation and experiment results of different types of spectral filters demonstrate the validity of proposed method.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, digital holography (DH) has drawn intense attention for non-contact biological studies [1,2] and high precise surface micro-topography measurement [3,4] because of its wide-field, low-cost, and quantitative imaging nature. It can simultaneously obtain the intensity and phase of specimen with a single-wavelength off-axis hologram. In addition, dual-wavelength or multi-wavelength illumination in DH can offer extra information for extending measurement range [5–9], expanding sampled area [10–13], retrieving object phase [14,15], realizing pixel super-resolution [16,17], etc.
One of the key issues in dual-wavelength DH is the acquisition of two holograms for different wavelengths. The simplest approach is to record two holograms in sequence [5–7]. But it is time-consuming and incapable of realizing dynamic phase retrieval. To achieve real-time wavelength-multiplexing and phase retrieval, one way to capture two holograms simultaneously is utilizing a color Bayer-mosaic camera [18–20]. However, additional measurement error would be introduced by the spectral crosstalk between different color channels. In contrast, a monochrome camera also can be used for single-shot recording two holograms based on spatially multiplexed operation [21–35]. The main idea of spatially multiplexed is separating the spectrum terms of two wavelengths in frequency domain by adjusting the propagation orientation of two reference beams for different wavelengths. Kim [21] et al. combined grating with pinhole array to distinguish different wavelengths components by the fringe frequencies of distinct diffraction orders. Zhao [22] et al. obtained a multiplexed dual-wavelength hologram based on common-path lateral shearing interferometry. In [23], a common-path off-axis self-interference dual-wavelength DH based on a cube beam splitter is presented for expanding phase range and measuring the refractive index of specimen. Tahara [24] et al. designed a reference arm with a 2D spatial carrier to simultaneously capture the quantitative phase distributions of waves containing multiple wavelengths. Due to the spectrum terms of two wavelengths located at the same direction in frequency domain [21–24], a cross-talk will exit between the two wavelengths especially for the DH system with lager phase parabolic aberrations. It will affect the accuracy of phase retrieval for different wavelengths. Though the larger difference of two wavelengths can reduce the cross-talk, the smaller synthetic wavelength would limit its application for measuring deep steps [5–9].
To avoid the cross-talk, many works on yielding a multiplexed hologram with orthogonal spatial frequency have been reported [25–36]. The multipath design [25–29] and polarization modulation [30–36] are the more general solution for recording orthogonal hologram simultaneously. The basic strategy of multipath schemes is designing different reference arms for different wavelengths with the aid of introducing additional mirrors and beam splitters [25–27], diffractive optical element [28], and dichroic mirror [29]. The core concept of polarization modulation is to build orthogonal polarization for different wavelengths by combining optical polarization components [30,31] and retro-reflector mirrors [32–34], long working distance objective [35], two-dimensional cross grating [36], which can realize single-shot recording of two holograms. Although these methods [21–36] work well for wavelength-multiplexing in a single-shot, complex system with multiple reference arms and more optical elements for polarization multiplexed will make them not suitable for compact DH system. Because these systems, such as lensless reflection DH based on Michelson interferometer [37,38] and transmission DH [38,39], are so compact that require no more optical elements and much modifications of system. Recently, a single-shot dual-wavelength off-axis DH without beam splitter and mirror is proposed, which implemented with only a single diffraction grating [13,40].
In this work, we propose a single-shot wavelength-multiplexed DH using a spectral filter. Only a spectral filter inserting between beam splitter and mirror in reflection off-axis digital holography (RODH). The spectral filter can transmit a well-defined wavelength band of light, while reject other unwanted radiation. Therefore, the propagation direction of two reference waves for different wavelengths is controlled by separately adjusting filter and mirror. So, a multiplexed hologram with different fringe orientations can be obtained with fewer optical element and system modification. The wavefront interference analysis of using a spectral filter is discussed in detail. Our proposal is available for real-time wavelength-multiplexing in DH. The validity of proposed scheme is illustrated by numerical simulation and experiment results of different types of spectral filters. Due to the great performance of fewer optical element and system modification for proposed method, it has special potential for the digital holographic recording of dynamic processes especially for the lensless DH system [37–39].
2. Basic theory
As presented in Fig. 1(a), the proposed RODH configuration is based on a reflection Michelson interferometer. A point source located inside beam splitter produced when the two wavelength sources are incident on a converging lens. The object wave is reflected by the tested sample. A microscope objective is used to obtain image hologram. To adjust the propagation direction of different reference waves for two wavelengths, a spectral filter is inserted between the beam splitter and mirror in proposed system. The spectral filter can transmit a well-defined wavelength band of light, while reject other unwanted radiation. If the source with ${\lambda _2}$ can pass through spectral filter but the source with ${\lambda _1}$ cannot. Thus, the reference wave of ${\lambda _1}$ and ${\lambda _2}$ is reflected by spectral filter and mirror, separately. By adjusting spectral filter and mirror, we can change the fringe direction of two holograms. Then a multiplexed hologram with two wavelengths captured in a single shot by a monochrome camera. Different positions of sample stage, spectral filter and mirror would induce different parabolic phase aberrations for different wavelengths.
Fig. 1. The schematic diagram of (a) RODH and (b) wavefront interference analysis using a spectral filter.
Here, we firstly consider a more general case that the spectral filter permits ${\lambda _2}$ but with low peak transmittance and blocks ${\lambda _1}$ . Because the source with ${\lambda _2}$ can pass through spectral filter but the source with ${\lambda _1}$ cannot, so the reference wave of ${\lambda _2}$ may not only reflected by mirror but also spectral filter. In addition, the reference wave of ${\lambda _2}$ reflected by mirror and spectral filter also will generate a hologram. Figure 1(b) is the schematic diagram of wavefront interference analysis for our system. We assume that the diverging object spherical wave is generated from the coordinates $({{F_{Ox}},{F_{Oy}},({d_{O{\lambda_i}}^2 - F_{Ox}^2 - F_{Oy}^2} )} ),\; i = 1,2$. The diverging reference spherical wave reflected from spectral filter and mirror is generated from the coordinates $({{F_{RFx}},{F_{RFy}},({d_{RF{\lambda_i}}^2 - F_{RFx}^2 - F_{RFy}^2} )} )$ and $({{F_{RMx}},{F_{RMy}},({d_{RM{\lambda_i}}^2 - F_{RMx}^2 - F_{RMy}^2} )} )$, respectively. Therefore, the corresponding object wave and reference wave can be separately expressed as
The off-axis angle for different interference terms in multiplexed hologram can be measured by [41]
where $\lambda $ is the wavelength, N is the number of pixels of CCD in the x direction and $\delta $ is the sample interval. D is the distance between the center of the +1 order and -1 order in hologram spectrum. As shown in Fig. 1(b), The off-axis angle ${\theta _{{\lambda _2}FM}}$ of ${|{{R_{F{\lambda_2}}} + {R_{M{\lambda_2}}}} |^2}$ will equal the sum of ${\theta _{{\lambda _1}F}}$ of ${|{{O_{{\lambda_1}}} + {R_{F{\lambda_1}}}} |^2}$ and ${\theta _{{\lambda _2}M}}$ of ${|{{O_{{\lambda_2}}} + {R_{M{\lambda_2}}}} |^2}$.Due to the separation of spectrum terms for different wavelengths, the corresponding unwrapped phase (+1 order) can be extracted by phase retrieval procedures [42], which can be expressed as
3. Numerical simulations
The following numerical simulation demonstrate the imaging analysis of using a spectral filter in proposed system. In this simulation, two wavelengths are ${\lambda _1} = 632.8\; nm$, ${\lambda _2} = 532\; nm$, respectively. The simulated hologram generated with pixel size $7.8\mu m \times 7.8\mu m$ and $512 \times 512$ pixels. The simulated spectral filter blocks the source with ${\lambda _1}$ but permits ${\lambda _2}$. Here, we also consider that the spectral filter with low peak transmittance for ${\lambda _2}$. To simulate hologram with phase aberrations for different wavelengths, we set ${d_{O{\lambda _1}}} = {d_{O{\lambda _2}}} = 0.69m$, ${d_{RF{\lambda _1}}} = 0.36m$, $ {d_{RM{\lambda _2}}} = 0.39m$.
First, only the source with ${\lambda _1}$ works in this simulation. Because ${\lambda _1}$ cannot pass through spectral filter, so the light reflected by spectral filter regarded as the reference wave of ${\lambda _1}$. By adjusting spectral filter, a high contrast hologram ${|{{O_{{\lambda_1}}} + {R_{F{\lambda_1}}}} |^2}$ and corresponding Fourier spectrum are shown in Fig. 2(a) and 2(d), respectively. Based on Eq. (5), the off-axis angle ${\theta _{{\lambda _1}F}}$ is $9.2mrad$.
Fig. 2. The simulation analysis of wavefront interference using a spectral filter. The simulated spectral filter permits ${\lambda _2}({532\; nm} )$ with low peak transmittance and blocks ${\lambda _1}({632.8\; nm} )$. (a)- (b) The hologram and (d)-(e) Fourier spectrum obtained with only ${\lambda _1}$ and ${\lambda _2}$ works, respectively. (c) The multiplexed hologram and (f) Fourier spectrum obtained with ${\lambda _1}$ and ${\lambda _2}$ simultaneously work.
Then, only the source of ${\lambda _2}$ works. The transmission light from beam splitter pass through spectral filter and reflected by mirror. By adjusting mirror, the propagation direction of reference wave of ${\lambda _2}$ can be different from ${\lambda _1}$. Figure 2(b) shows the hologram of ${\lambda _2}$ with multiple fringes and the corresponding Fourier spectrum is shown in Fig. 2(e). Due to the low peak transmittance of spectral filter for ${\lambda _2}$, the source of ${\lambda _2}$ not only can pass through spectral filter but also reflect from spectral filter. Therefore, the interference between the object wave and reflection wave of ${\lambda _2}$ from spectral filter will generate an interferogram ${|{{O_{{\lambda_2}}} + {R_{F{\lambda_2}}}} |^2}$ and its Fourier spectrum depicted in Fig. 2(e) by the green arrow. The off-axis angle ${\theta _{{\lambda _2}M}}$ calculated by Eq. (5) is $14.9mrad$. In addition, another interferogram ${|{{R_{F{\lambda_2}}} + {R_{M{\lambda_2}}}} |^2}$ also produced by the reflection wave of ${\lambda _2}$ from spectral filter and mirror. The corresponding Fourier spectrum is shown in Fig. 2(e) marked with white dotted circle which has smaller parabolic phase aberration because of shorter distance between object wave and reference wave. The correspond off-axis angle ${\theta _{{\lambda _2}FM}}$ calculated by Eq. (5) is $24.1mrad$ that equal the sum of ${\theta _{{\lambda _1}F}}$ and ${\theta _{{\lambda _2}M}}$. These results agree well with our theoretical analysis.
After the adjustment of spectral filter and mirror, the source of ${\lambda _1}$ and ${\lambda _2}$ works simultaneously. A multiplexed hologram with multiple fringes is obtained in a single-shot, as shown in Fig. 2(c). Due to the off-axis configuration, the zero order and the virtual image can be easily separated from the real image in the spectral domain for different wavelengths. Figure 2(f) describes the Fourier spectrum of multiplexed hologram. From Fig. 2(f), the spectrum (+1 order) of ${|{{O_{{\lambda_2}}} + {R_{F{\lambda_2}}}} |^2}$ and ${|{{O_{{\lambda_1}}} + {R_{F{\lambda_1}}}} |^2}$ are depicted in Fig. 2(e) by the green arrow and red arrow, respectively. They will overlap especially when the system has large phase aberrations. All holograms have parabolic phase aberration because of the unmatched curve between object wave and reference wave. Different frequency directions of two wavelength holograms can be obtained by adjusting spectral filter and mirror. It is worth noting that the higher peak transmittance of spectral filter for ${\lambda _2}$ can avoid the generation of additional spectrum terms. As mentioned in our theoretical analysis, the spectral filter blocks ${\lambda _2}$ also can be used for wavelength separation and the spectrum distributions of multiplexed hologram will be different from Fig. 2(f).
Furthermore, the unwrapped phase (+1 order) maps for different wavelengths in multiplexed hologram are analyzed. Figure 3 shows the phase retrieval results when using a spectral filter permits ${\lambda _2}({532\; nm} )$ with low peak transmittance and blocks ${\lambda _1}({632.8\; nm} )$. The tested reflection specimen is simulated micro-balloon as shown in Fig. 3(a). Due to the low peak transmittance of spectral filter for ${\lambda _2}$, some additional spectrum terms would be added as shown in Fig. 3(b). Figure 3(c) shows the unwrapped phase (+1 order) of ${\lambda _1}$ in multiplexed hologram. Because of the spectrum overlapping for ${\lambda _1}$ in multiplexed hologram, so the recovered phase turns worse. The phase profiles along the white dotted line in Figs. 3(a) and 3(c) are demonstrated in Fig. 3(e). The root means square error (RMSE) between Fig. 3(a) and 3(c) is $3.646\; rad$. In contrast, the unwrapped phase (+1 order) of ${\lambda _2}$ in multiplexed hologram, as shown in Fig. 3(d), would be correct phase which avoid the spectrum overlapping. The phase profiles along the white dotted line in Figs. 3(a) and 3(d) are demonstrated in Fig. 3(f). The RMSE between Fig. 3(a) and 3(d) is $0.0623\; rad$.
Fig. 3. The simulation results of phase retrieval for different wavelengths using a spectral filter with low peak transmittance for ${\lambda _2}$ . (a) The simulated sample. (b) The Fourier spectrum of multiplexed hologram. The unwrapped phase (+1 order) of (c) ${\lambda _1}$ and (d) ${\lambda _2}$ extracted from multiplexed hologram. (e) The phase profiles along the white dotted line in (a) and (c). (f) The phase profiles along the white dotted line in (a) and (d).
To obtain the correct phase for different wavelengths, we should choose a spectral filter with high peak transmittance for ${\lambda _2}$. Figure 4 shows the phase retrieval results when using a spectral filter permits ${\lambda _2}({532\; nm} )$ with high peak transmittance and blocks ${\lambda _1}({632.8\; nm} )$. The multiplexed hologram is shown in Fig. 4(a). Figure 4(b) describes the corresponding Fourier spectrum which avoid the spectrum overlapping due to the use of spectral filter with high peak transmittance for ${\lambda _2}$. The unwrapped phase (+1 order) of ${\lambda _1}$ and ${\lambda _2}$ in multiplexed hologram are separately shown in Fig. 4(c) and Fig. 4(d). Both of them demonstrate correct phase distribution. Figure 4(e) illustrates the phase profiles along the white dotted line in Figs. 3(a) and 4(c). The RMSE between Fig. 3(a) and 4(c) is $0.0688\; rad$.The phase profiles along the white dotted line in Figs. 3(a) and 4(d) are demonstrated in Fig. 4(f). The RMSE between Fig. 3(a) and 4(d) is $0.0617\; rad$. These results agree well with our theoretical analysis.
Fig. 4. The simulation results of phase retrieval for different wavelengths using a spectral filter with high peak transmittance for ${\lambda _2}$ . (a) The multiplexed hologram. (b) The Fourier spectrum of multiplexed hologram. The unwrapped phase (+1 order) of (c) ${\lambda _1}$ and (d) ${\lambda _2}$ extracted from multiplexed hologram. (e) The phase profiles along the white dotted line in Fig. 3(a) and (c). (f) The phase profiles along the white dotted line in Fig. 3(a) and (d).
4. Experimental results and discussion
To further illustrate the effectiveness of proposed method, experiments are conducted. Figure 5(a) shows the experiment setup for our proposal. The front view of orange dotted box in Fig. 5(a) is shown in Fig. 5(b). Two wavelengths are ${\lambda _1} = 632.8nm$, ${\lambda _2} = 532nm$, respectively. The distance between spectral filter and mirror is about $6mm$. Both spectral filter and mirror installed with a two-dimensional precision mirror frame (Daheng optics, GCT-080315), as shown in Fig. 5(b), which can flexibly change the reflection angle of reference wave for different wavelengths. The hologram under different wavelengths recorded by a monochrome CMOS camera with pixel size $2.2\mu m \times 2.2\mu m$ and $1944 \times 2592$ pixels. The tested sample is a standard step (Bruker, S/N:1703-655). The distance from sample to camera is about $260mm$. We use a microscope objective (MO2, Olympus, $4 \times $) to obtain the image hologram.
Fig. 5. (a) The photography of proposed experiment system. (b) The front view of orange dotted box area in (a). MO, microscope objective; BS, beam-splitter; F, spectral filter; M, flat mirror; S, sample.
The transmittance effect of different spectral filters is analyzed before the experiment. Based on the original product parameter data, it is easy to obtain the corresponding transmittance curve at different wavelengths for different types of spectral filters, as shown in Fig. 6. The transmittance curves of bandpass filter with different cut-off wavelengths are illustrated in Figs. 6(a)–6(c). These bandpass filters provide one of the simplest ways to transmit a well-defined wavelength band of light, while rejecting other unwanted radiation. Figures 6(d) and 6(e) are transmittance curves of edge-pass filters with different cut-off wavelengths. The longpass and shortpass filter are also very useful for isolating regions of a spectrum and eliminating any unwanted radiation. Take the shortpass filter as an example which shown in Fig. 6(d), this spectral filter (Thorlabs, FESH0550) transmits ${\lambda _2}({532nm} )$ with higher peak transmittance (about 98.5%), while transmits ${\lambda _1}({632.8nm} )$ with very low transmittance (about only 0.00028%). It means that the reference wave of ${\lambda _1}$ can be controlled by spectral filter, while the reference wave of ${\lambda _2}$ controlled by mirror. So, a multiplexed hologram with different spectrum directions for different wavelengths can be obtain using these spectral filters.
First, a spectral filter permits ${\lambda _2}({532\; nm} )$ with high peak transmittance was tested. A premium bandpass filter (Thorlabs, FLH532-4) with central wavelength of $532nm$ and peak transmittance of $91.4{\%}$ is inserted in proposed system. The full width at half maximum (FWHM) of filter $4\; nm$. First, only the source with ${\lambda _2}$ works. Before inserting the spectral filter in system, a hologram of ${\lambda _2}$ is obtained shown in Fig. 7(a). Figure 7(e) shows the corresponding Fourier spectrum. Figures 7(b) and 7(f) are the hologram and Fourier spectrum of ${\lambda _2}$ after inserting the spectral filter, respectively. It is clear shown that the use of spectral filter does not introduce additional spectrum terms for ${\lambda _2}$. Then, only the source with ${\lambda _1}$ works. Because ${\lambda _1}$ blocked by spectral filter (the transmittance is only about 0.000012%), as shown in Fig. 6(b), the reference wave of ${\lambda _1}$ reflected from spectral filter. By adjusting spectral filter, a hologram with different fringe direction from ${\lambda _2}$ can be obtained. Figures 7(c) and 7(g) show the hologram and Fourier spectrum of ${\lambda _1}$, separately. Both ${\lambda _1}$ and ${\lambda _2}$ work, the multiplexed hologram and corresponding Fourier spectrum are shown in Fig. 7(d) and 7(h), respectively. Due to the high peak transmittance (about 91.4%) of spectral filter for ${\lambda _2}$, as shown in Fig. 6(b), the multiplexed hologram avoids the additional interference terms such as ${|{{O_{{\lambda_2}}} + {R_{F{\lambda_2}}}} |^2}$ and ${|{{R_{F{\lambda_2}}} + {R_{M{\lambda_2}}}} |^2}$. Therefore, the spectral filter used also does not introduce additional spectrum terms for ${\lambda _1}$.
Fig. 7. The experiment analysis of wavefront interference using a bandpass filter (Thorlabs, FLH532-4) with high peak transmittance for ${\lambda _2}({532\; nm} )$.This bandpass filter blocks ${\lambda _1}({632.8\; nm} )$. (a)-(b) The hologram and (e)-(f) corresponding Fourier spectrum obtained with only ${\lambda _2}$ works before and after inserting the filter in system, respectively; (c) The hologram and (g) Fourier spectrum obtained with only ${\lambda _1}$ works; (d) The multiplexed hologram and (h) Fourier spectrum obtained with ${\lambda _1}$ and ${\lambda _2}$ simultaneously work.
According to phase retrieval procedures [42], the unwrapped phase (+1 order) of red box area, as shown in Fig. 7(d), can be reconstructed for all holograms. The unwrapped phase (+1 order) extracted from Fig. 7(e) and 7(f) are shown in Fig. 8(a) and 8(b), respectively. Figure 8(c) describes the phase profiles along the red dotted line in Figs. 8(a) and 8(b). The root means square error (RMSE) between Fig. 8(a) and 8(b) is $0.2327\; rad$. Figures 8(c) and 8(d) show the unwrapped phase (+1 order) of Fig. 7(g) and 7(h), respectively. The phase profiles along the red dotted line in Figs. 8(d) and 8(e) are depicted in Fig. 8(f). The RMSE between Fig. 8(d) and 8(e) is $0.1666\; rad$. One repeating experiment is applied to discuss the stability of using spectral filter for our proposal. We repeatedly capture 20 multiplexed holograms. One of them is shown in Fig. 7(d). Thus, the corresponding 20 unwrapped phase maps (+1 order) of ${\lambda _1}$ and ${\lambda _2}$ can be reconstructed separately. The RMSE between 20 unwrapped phase maps of ${\lambda _1}$ is $0.0318rad$. The RMSE between 20 unwrapped phase maps of ${\lambda _2}$ is $0.0602rad$.These results illustrate that the proposed method can obtain accurate wavelength separation and phase retrieval for different wavelengths in multiplexed hologram when using a spectral filter with high peak transmittance. In addition, the use of spectral filter also does not introduce additional phase error and influence the stability of system and change the polarization property of reflection light and transmission light from spectral filter.
Fig. 8. The experiment results of phase retrieval for different wavelengths using a bandpass filter (Thorlabs, FLH532-4) with high peak transmittance for ${\lambda _2}$. This bandpass filter blocks ${\lambda _1}$. The unwrapped phase (+1 order) extracted from (a) Fig. 3(e); (b) Fig. 3(f); (d) Fig. 3(g); (e) Fig. 3(e). (c) The phase profiles along the red dotted line in (a) and (b); (f) The phase profiles along the red dotted line in (d) and (e).
In addition, the effectiveness of our scheme for dynamic recording and imaging is further illustrated. The sample is manually translated and a sequence of interferograms are captured. Figure 9(a) and 9(b) show the multiplexed hologram recorded at different times ${T_1}$ and ${T_2}$. The Fourier spectrum of multiplexed hologram is similar with Fig. 7(h). Due to the use of spectral filter with high peak transmittance, the multiplexed hologram avoids the overlapping of Fourier spectrum terms for different wavelengths. Using the described method above, the phase maps of different wavelengths are retrieved. Figures 9(c) and 9(d) show the unwrapped phase (+1 order) of ${\lambda _1}$ extracted from Fig. 9(a) and 9(b), respectively. The unwrapped phase (+1 order) of ${\lambda _2}$ extracted from Fig. 9(a) and 9(b) are shown in Fig. 9(e) and 9(f), separately. The phase profiles along the red dotted line in Figs. 9(c) and 9(d) are depicted in Fig. 9(g). The phase profiles along the red dotted line in Figs. 9(e) and 9(f) are shown in Fig. 9(h). These results show that our scheme also can achieve accurate wavelength separation and phase retrieval for moving object.
Fig. 9. Experiment results of dynamic recording for moving sample. Multiplexed hologram recorded at different times (a) ${T_1}$ and (b) ${T_2}$. The unwrapped phase (+1 order) of (c) ${\lambda _1}$ and (e) ${\lambda _2}$ separately extracted from (a). The unwrapped phase (+1 order) of (d) ${\lambda _1}$ and (f) ${\lambda _2}$ separately extracted from (b). (g) The phase profiles along the red dotted line in (c) and (d); (h) The phase profiles along the red dotted line in (e) and (f).
Then, we verify the feasibility of proposed method when using a spectral filter with low peak transmittance. A bandpass filter (Hengyang Optics, HNIF-010-532-D25) with central wavelength of $532nm$ and peak transmittance of $74.7{\%}$ is inserted in proposed system. The FWHM of filter is $10\; nm$. After adjusting the spectral filter and mirror, a multiplexed hologram is obtained that shown in Fig. 10(a). Figure 10(b) is the Fourier spectrum. Due to the lower peak transmittance of spectral filter for ${\lambda _2}$, as shown in Fig. 6(a), so it will introduce another interference terms ${|{{O_{{\lambda_2}}} + {R_{F{\lambda_2}}}} |^2}$ and ${|{{R_{F{\lambda_2}}} + {R_{M{\lambda_2}}}} |^2}$ which marked in Fig. 10(b) by green narrow and white dotted circle, respectively. The white dotted box area is the amplification of white dotted circle area in Fig. 10(b). Figure 10(c) demonstrates the retrieved phase of red box area in Fig. 10 (a) which extracted from hologram under only ${\lambda _1}$ illuminate. In contrast, the unwrapped phase (+1 order) of ${\lambda _1}$ obtained from multiplexed hologram is shown in Fig. 10(d). The phase profiles along the red dotted line in Figs. 10(c) and 10(d) are demonstrated in Fig. 10(f). The RMSE between Fig. 10(c) and 10(d) is $2.548\; rad$. These results show that inserting spectral filter with lower peak transmittance cannot obtain accurate wavelength separation and phase retrieval for ${\lambda _1}$ . The distance of diffraction order for different interference terms is ${D_{F{\lambda _1}}} = 206$ pixels, ${D_{M{\lambda _2}}} = 568$ pixels, and ${D_{FM{\lambda _2}}} = 812$ pixels, respectively. Based on Eq. (5), the corresponding off-axis angle are ${\theta _{{\lambda _1}F}} = 15.2mrad$, ${\theta _{{\lambda _2}M}} = 35.3mrad$, ${\theta _{{\lambda _2}FM}} = 50.5mrad$. These results agree well with our theory and simulation analysis.
Fig. 10. Experiment results of using a bandpass filter (Hengyang Optics, HNIF-010-532-D25) with low peak transmittance for ${\lambda _2}$. This bandpass filter blocks ${\lambda _1}$. (a) The multiplexed hologram; (b) Fourier spectrum of multiplexed hologram; (c) The unwrapped phase (+1 order) of hologram obtained with only ${\lambda _1}$ works; (d) The unwrapped phase (+1 order) of hologram of ${\lambda _1}$ in multiplexed hologram; (e) The phase profiles along the red dotted line in (c) and (d).
In addition, we explore the feasibility of spectral filter with different cut-off wavelengths. A premium bandpass filter (Thorlabs, FLH633-5) with central wavelength of $633nm$ and peak transmittance of $95{\%}$ is tested. The FWHM of filter is $5\; nm$. By adjusting spectral filter and mirror, the multiplexed hologram and corresponding Fourier spectrum is shown in Fig. 11(a) and 11(b), separately. Because the spectral filter blocks ${\lambda _2} = 532\; nm$, the spectrum distributions of multiplexed hologram are contrary compared with Fig. 7(h). From Fig. 11(b), the spectral filter with high peak transmittance, as shown in Fig. 6(c), also does not introduce another interference terms for ${\lambda _2}$ in multiplexed hologram. Figure 11(c) demonstrates the retrieved phase of red box area in Fig. 11(a) which extracted from hologram under only ${\lambda _2}$ illuminate. In contrast, the unwrapped phase (+1 order) of ${\lambda _2}$ obtained from multiplexed hologram is shown in Fig. 11(d). The phase profiles along the red dotted line in Figs. 11(c) and 11(d) are demonstrated in Fig. 11(f). The RMSE between Fig. 11(c) and 11(d) is $0.1876\; rad$. These results show that a spectral filter with different cut-off wavelengths also can obtain accurate wavelength separation and phase retrieval if it has high peak transmittance.
Fig. 11. Experiment results of using a bandpass filter (Thorlabs, FLH633-5) with high peak transmittance for ${\lambda _1}$. This bandpass filter blocks ${\lambda _2}$. (a) The multiplexed hologram; (b) Fourier spectrum of multiplexed hologram; (c) The unwrapped phase (+1 order) of hologram obtained with only ${\lambda _2}$ works; (d) The unwrapped phase (+1 order) of hologram of ${\lambda _2}$ in multiplexed hologram; (e) The phase profiles along the red dotted line in (c) and (d).
Furthermore, we explore the effectiveness of our proposal with different types of spectral filters. A premium shortpass filter (Thorlabs, FESH0550) with cut-off wavelength of $550\; nm$ and peak transmittance of $98.5{\%}$ is tested. As shown in Fig. 6(d), the source with ${\lambda _2} = 532\; nm$ can pass thought spectral filter with higher peak transmittance, while ${\lambda _1}$ blocked by spectral filter. Figure 12 shows the experiment results of using this shortpass filter that similar with Fig. 8. The RMSE between Fig. 12(c) and 12(d) is $0.1426\; rad$. This shortpass filter used can realize accurate wavelength separation and phase retrieval. We also test another edge-pass filter (Thorlabs, FELH0600) with different cut-off wavelength of $660\; nm$. This premium longpass filter also has the high peak transmittance ($97{\%}$) for ${\lambda _1} = 632.8\; nm$. The transmittance of longpass filter for ${\lambda _2}$ is only about 0.00014% as shown in Fig. 6(e). Figure 13 shows the corresponding experiment results of using this longpass filter. It is also similar compared with Fig. 11. The RMSE between Fig. 13(c) and 13(d) is $0.2185\; rad$. Both premium edge-pass filters can obtain clear wavelength separation and accurate phase retrieval for different wavelengths. These results illustrate the flexibility and universality of our method.
Fig. 12. Experiment results of using a shortpass filter (Thorlabs, FESH0550) with high peak transmittance for ${\lambda _2}$. This shortpass filter blocks ${\lambda _1}$. (a) The multiplexed hologram; (b) Fourier spectrum of multiplexed hologram; (c) The unwrapped phase (+1 order) of hologram obtained with only ${\lambda _1}$ works; (d) The unwrapped phase (+1 order) of hologram of ${\lambda _1}$ in multiplexed hologram; (e) The phase profiles along the red dotted line in (c) and (d).
Fig. 13. Experiment results of using a longpass filter (Thorlabs, FELH0600) with high peak transmittance for ${\lambda _1}$. This longpass filter blocks ${\lambda _2}$. (a) The multiplexed hologram; (b) Fourier spectrum of multiplexed hologram; (c) The unwrapped phase (+1 order) of hologram obtained with only ${\lambda _2}$ works; (d) The unwrapped phase (+1 order) of hologram of ${\lambda _2}$ in multiplexed hologram; (e) The phase profiles along the red dotted line in (c) and (d).
5. Conclusion
In conclusion, we have presented a multiplexed off-axis DH scheme with a spectral filter, which is capable of wavelength multiplexing in one-shot. By inserting a spectral filter between beam splitter and mirror in RODH, the propagation direction of two reference waves for different wavelengths can be controlled separately by spectral filter and mirror. Thus, a multiplexed hologram with different fringe directions for two wavelengths can be simultaneously obtained. It is worth noting that the spectral filter should has higher peak transmittance for one of wavelengths. Normally, a spectral filter with peak transmittance greater than 90% can be used for realizing accurate wavelength separation and phase retrieval. Our proposal is suitable for real-time wavelength-multiplexing with minimum optical element and system modification. Numerical simulation results agree well with theorical analysis. The bandpass filter and edge-pass filter with different transmittance and cut-off wavelengths are tested in this work. The experiment results demonstrate the feasibility and universality of our scheme. The proposed method is also available for other interference systems, such as the lensless reflection DH [37,38]. In the further work, we will explore the effectiveness of spectral filter in transmission DH system [39]. In addition, the plane mirror would be replaced by a convex mirror [37] in our proposal which minimize the phase curve of hologram for different wavelengths.
Funding
Basic and Applied Basic Research Foundation of Guangdong Province (2020A1515110283); Provincial Key Platform and Scientific Research Projects of Universities in Guangdong Province (2020KQNCX078); Supported by the Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province (GD202105).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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