# Multispectral incoherent holography based on measurement of differential wavefront curvature

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## Abstract

We propose a technique of multispectral incoherent holography. The differential wavefront curvature is measured, and the principle of Fourier transform spectrometry is applied to provide a set of spectral components of three-dimensional images and continuous spectra for spatially incoherent, polychromatic objects. This paper presents the mathematical formulation of the principle and the experimental results. Three-dimensional imaging properties are investigated based on an analytical impulse response function. The experimental and theoretical results agree well.

## Keywords

Incoherent holography Spectroscopy Multispectral imaging 3D imaging property Impulse response function Depth resolution## 1 Introduction

Various techniques have been developed to expand the applications of incoherent holography [1, 2, 3, 4]. These techniques include using a spatial light modulator [5, 6], a concave mirror [7], radial shearing [8], and rotational shearing [9] in the interferometric systems, and are based on diffraction theory and the self-interference principle [10].

We previously developed a method of interferometric spectral imaging for three-dimensional (3D) objects illuminated by a natural light source [11]. Subsequently, we introduced the synthetic aperture technique to advance digital holographic 3D imaging spectrometry [12, 13]. This method is also called the spherical-type method, because the fringe patterns recorded in the measured volume interferogram are arranged in the same way as spherical wavefronts propagating from the object. Variations of the method have been developed, including the hyperbolic-type method (H-type) [14] and rotated hyperbolic-type method [15]. Variations have also been extended to single-pixel imaging, in which an H-type volume interferogram can be measured directly without using a synthetic aperture technique [16]. Each variation has its own advantages; however, these methods generally have a long measurement time.

In this paper, we propose a method, called multispectral incoherent holography, that requires a simple system and a short measurement time compared with previous methods, and thus is better suited for unstable objects, such as biological samples. A set of spectral components of the 3D images and continuous spectra for spatially incoherent polychromatic objects are obtained from the measured volume interferogram. Because the method is based on measuring differential wavefront curvature, it is a generalization of Fresnel incoherent correlation holography [4], combined with the Fourier transform spectrometry.

We present our experimental results and mathematical analysis of the method. To characterize the spatial imaging properties in the lateral and depth directions and the spectral resolution, a new analytical solution of the impulse response function (IRF) under the paraxial approximation is derived. This IRF is defined over four-dimensional (4D) \((x, \, y, \, z, \, \omega )\) space. Based on this function, it is possible to estimate both 3D spatial and spectral resolutions.

In Sect. 2, we summarize the procedure for reconstructing the multispectral components of 3D images. This reconstruction process is based on the Wiener–Khinchin theorem and the propagation law of optical coherence. Section 3 is divided into two parts. The first part shows the experimental setup and objects that we measured. The second part describes the experimental results that demonstrate the performance of the method for reconstructing multispectral images. Section 4 consists of two subsections. First, we derive the analytical IRF solution defined over 4D space. Next, we examine the validity of the derived IRF solution by comparing the experimental results using a monochromatic point source. The summary is given in Sect. 5. Part of this work has been presented elsewhere in the literature [17].

## 2 Description of the method

In this section, we present the concept of multispectral incoherent holography. We begin by introducing the measurement system, which measures the differential wavefront curvature between two split wavefronts. The system obtains a volume (3D) interferogram that records a 3D spatial correlation function. We show the signal processing procedure for spectral decomposition to obtain a set of complex incoherent holograms for different spectral components. The 3D image at each spectrum can be reconstructed from the complex incoherent hologram by applying the usual inverse propagation techniques.

### 2.1 Measurement of 3D spatial correlation function

*Z*, between the two wavefronts is introduced by moving the piezoelectric translator (PZT) along the optical axis stepwise. The 2D interferogram at each position of the PZT is recorded. This procedure is like that of Fourier transform spectrometry. The recorded data create a 3D volume interferogram. We write \(\varvec{\rho}= (\varvec{\rho}_{\bot },\, Z) = (X,\,Y,\,Z)\) as the coordinate system taken in the volume interferogram. Let the distance between \({\text{M1}}\) and D be \(d_{1}\) and let the distance between \({\text{M2}}\) and the origin of Cartesian coordinate system be \(d_{0} + {Z \mathord{\left/ {\vphantom {Z 2}} \right. \kern-0pt} 2}\), and defining the distance between D and the origin of Cartesian coordinate system as \(z_{0} = d_{0} + d_{1}\), the optical intensity of volume interferogram, \(I(\varvec{\rho})\), is given by:

### 2.2 Recovery of 3D images for many spectral components

*c*is the speed of light in free space, and \(k = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }\) is the wavenumber of wavelength \(\lambda\). Equation (2) is a special case of the Wiener–Khinchin theorem, in which temporal difference is set to be zero. This equation means that for a stationary optical filed, spectral components of the optical fields for different frequencies are mutually uncorrelated. The cross-spectral density function on the right-hand side of Eq. (2) is defined as the cross correlation of the monochromatic components \(U_{1}\) and \(U_{2}\) of \(V_{1}\) and \(V_{2}\) as:

In Eq. (7), the integrand has taken over the actual extension of the interferogram with respect to \(Z\).

From Eq. (1), the 3D interferogram being recorded includes two intensity distribution terms and two interference terms. These interference terms can be separated from the intensity distribution terms during the retrieval of the cross-spectral density functions, because the interference term \(\varGamma {\kern 1pt}_{12} (\varvec{\rho})\) contains only positive-frequency spectral components, as shown in the integral region in Eq. (6), and \(\varGamma {\kern 1pt}_{12}^{*} (\varvec{\rho})\) contains only negative-frequency components. On the other hand, the intensity distribution terms \(\varGamma {\kern 1pt}_{11}\) and \(\varGamma {\kern 1pt}_{22}\) do not change rapidly within the volume interferogram. This means that the spectra of \(\varGamma {\kern 1pt}_{11}\) and \(\varGamma {\kern 1pt}_{22}\) appear close to zero spatial-frequency region, separated from those of \(\varGamma {\kern 1pt}_{12}\) and \(\varGamma {\kern 1pt}_{12}^{*}\). By choosing the positive-frequency components, we obtain the information of \(\varGamma {\kern 1pt}_{12}\), separated from other terms.

Here, \(\kappa\) is a coefficient, \(\varvec{r}_{s} = (\varvec{r}_{s \bot } ,\,z_{s} ) = (x_{s} ,\,y_{s} ,\,z_{s} )\) is a point on the polychromatic object, \(\gamma (z_{s} )\) is the radius of the differential wavefront curvature as a function of \(z_{s}\), and *m* is the lateral magnification.

Here, \(\otimes\) stands for the convolution integral. The objects that are located across plane \(z = z_{s}\) are in focused and other objects are defocused.

## 3 Experiment

### 3.1 Experimental conditions

System specifications

Parameter | |
---|---|

\(d_{0}\) | 160 mm |

\(d_{1}\) | 120 mm |

\(f_{1}\) | 524 mm |

\(z_{1}\) | 5 mm |

\(z_{2}\) | − 5 mm |

PZT | |

Number of steps | 256 |

Step interval | 80 nm |

D | |

Number of pixels | 1024 square |

Pixel size | 6.9 μm |

### 3.2 Experimental results

*Z*, at particular point on (

*X*,

*Y*) space. Figure 6 shows the continuous spectral profile over the observation plane that was obtained by taking the Fourier transform of the intensity distribution in Fig. 5. The number of data points is 52, which covers the spectral range from 400 to 800 nm. The spectral resolution is limited by the step interval and number of steps of the PZT. According as the spectral profiles of the MHL and blue LED in Fig. 3, we focus on two spectral peaks located at 470.8 and 553.5 nm. The spectral resolutions at these wavelengths are 5.43 and 7.50 nm, respectively.

*x*–

*y*plane, where the reconstruction distance is \(z = 4\;{\text{mm}}\). The images of K and 2 are separated clearly. Because the wavefront shape of the 2 mask screen, recorded in the phase distribution of the complex incoherent hologram, has been eliminated, this reconstruction distance specifies the in-focus plane of K. Figure 7e, f shows the intensity profiles along the

*x*- and

*y*-directions in the object position of Fig. 7d. From these intensity profiles, the size of the reconstructed object is \(0. 7 {\text{ mm }} \times { 0} . 8 {\text{ mm}} .\) The shape of the measured object was reconstructed and the size of the letter K was close to the original size of the K in Fig. 4a. Figure 8a, b shows the intensity distributions over the

*x*–

*z*and

*y*–

*z*planes, and Fig. 8c shows the intensity profile along the

*z*-direction at \(\lambda = 553.5\;{\text{nm}}\). The intensity peak is close to the \(z = 4\;{\text{mm}}\), which is in agreement with object position, \(z_{1} = 5\;{\text{mm}}\).

*x*- and

*y*-directions in the object position of Fig. 9d. From these intensity profiles, the sizes of the reconstructed object are \(0.7\;{\text{mm}} \times 0.9\;{\text{mm}}\). The shape of the measured object was reconstructed. Figure 10a, b shows the intensity distributions over the

*x*–

*z*and

*y*–

*z*planes, and Fig. 10c shows the intensity profile along

*z*-direction at \(\lambda = 470.8\;{\text{nm}}\). The intensity peak is close to \(z = - 6\;{\text{mm}}\), which is in agreement with object position, \(z_{2} = - 5\;{\text{mm}}\).

Finally, we note that the measurement time of this experiment is 256 s for 256 frames. This is about 295 times shorter than our previous work [16] that takes 21 h, because the present method uses single-axis PZT scan instead of the 3D scan by a single-axis PZT and independent two-axes stages. Measurement time of the previous method is quite long, because each stage stops at every sampling point for measuring. In principle, measurement time of the present method may be realized as short as that of Fourier transform spectroscopy.

## 4 Comparison of imaging properties predicted by the IRF and experimental results

In this section, we derive an analytical solution of the 4D IRF of the present method. The derivation is performed under the paraxial approximation. To validate the IRF solution, the imaging properties predicted by the IRF were compared with the experimental results.

### 4.1 Mathematical analysis of 4D IRF

*x*-,

*y*-, and

*z*-axes are measured within baseline lengths of \(l_{x}\), \(l_{y}\), and \(l_{z}\). We may write:

*i*indicates that the parameters are used for the reconstruction, so that the angular frequency for reconstruction is \(\omega_{i} = ck_{i} = {{2\pi c} \mathord{\left/ {\vphantom {{2\pi c} {\lambda_{i} }}} \right. \kern-0pt} {\lambda_{i} }}\). In Eq. (15), the product of coefficient \({{l_{z} } \mathord{\left/ {\vphantom {{l_{z} } {2\pi c}}} \right. \kern-0pt} {2\pi c}}\) and the sinc function represents the spectral IRF that is characterized by the limited baseline length, \(l_{z}\). Using Eqs. (8) and (13), we may rewrite Eq. (15) as:

*h*, is expressed as:

*M*, as:

*M*is defined as the ratio of the product of wavelength and differential curvature radius of the object and the reconstructed image. The in-focus condition is realized if

*M*= 1, because the quadratic phase factors in the integrations by

*X*and

*Y*on the right-hand side of Eq. (18) vanish. For \(M \ne 1\), these integrations are carried out and lead to the following expression of IRF [18]:

*O*, is generally expressed as the superposition integral of the input spectral density function and IRF,

### 4.2 Comparison of imaging properties predicted by IRF and experimental results

Specifications in the section in which the imaging properties predicted by IRF and the experimental results were compared

Parameter | |
---|---|

\(d_{0}\) | 375 mm |

\(d_{1}\) | 120 mm |

\(f_{1}\) | 524 mm |

\(z_{s}\) | 0 mm |

PZT | |

Number of steps | 256 |

Step interval | 80 nm |

D | |

Number of pixels | 1024 square |

Pixel size | 6.9 μm |

For \(\lambda = 640\;{\text{nm}}\), we find \(\Delta \lambda = 9.77\;{\text{nm}}\). This value agrees with the intervals of spectral channels around the peak, as shown circles in Fig. 13. These expressions of spectral resolution are common in the field of Fourier transform spectrometry.

*x*-axis in Fig. 15; the solid curve shows the experimental results and the dotted curve shows the theoretical results based on the 4D IRF. Figures 15, 16 correspond to the intensity profile of a diffraction-limited image of a point source. The experimental and theoretical results both show that for a hologram with a rectangular aperture, the 2D point spread function is represented by the second and third sinc functions of the IRF in Eq. (26). Figure 17a shows the intensity distribution over the \(x_{i}\)–\(z_{i}\) plane calculated from an analytical solution of the 4D IRF and Fig. 17b shows the corresponding image obtained from the experimental complex incoherent hologram (Fig. 14b). Figure 18 compares the experimental intensity profile (solid curve) with the analytical solution of 4D IRF (dotted curve) along the

*z*-axis across the object position of Fig. 17. The peak positions and the distribution shapes agree well. We conclude from these results that the 4D IRF in Eq. (20) specifies the spectral resolution and 3D imaging properties in multispectral incoherent holography.

## 5 Conclusion

We presented experimental and theoretical studies of multispectral incoherent holography, which is based on measuring differential wavefront curvature. The experimental results showed that 3D spatial information at every spectral component of the measured object was acquired properly by this method. A paraxial IRF defined over the space–frequency domain was derived. Based on this IRF solution, we investigated the imaging properties of multispectral incoherent holography.

By comparing the theoretical prediction of the 4D IRF and the experimental results, we have shown that the spectral resolution and 3D imaging properties observed experimentally agree well with the theoretical prediction.

The measurement time of the present method is considerably smaller than those of our previous methods. This simplified lensless optical system is expected to be useful in wide range of applications, such as biological observations via spectrally resolved 3D images.

## Notes

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