# A New Method to Retrieve the Three-Dimensional Refractive Index and Specimen Size Using the Transport Intensity Equation, Taking Diffraction into Account

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Set-Up

#### 2.2. Sample Preparation

#### 2.3. Measurement Procedure

- -
- image
**T**, the contribution of the background and the dark current in the detector; it was achieved with nothing in the microscope and with the source illumination turned off;_{d} - -
- image
**T**, the bright reference; it was achieved with an empty slide in the microscope;_{b} - -
- image
**T**, the raw image of the sample, it was achieved with the sample in the microscope._{s}

**T**of the sample is defined by Equation (1) (flat-field correction), as is commonly described by Tadrous [10], Brydegaard et al. [11], and Agnero et al. [12]:

#### 2.4. Analysis Methods

**B**models noise, and the symbol ⊗ denotes the convolution operation. BF microscopy is presented as an alternative to fluorescence in deconvolution image processing, because of the possibility for observing unstained objects. However, its corresponding PSF consists of two separate components, one for phase ($PS{F}_{p}$) and one for absorption ($PS{F}_{a}$). Therefore, the BF-3D image $\mathit{i}\left(x,y,z\right)$ is generally described as the sum of the convolutions of the real (

**P**) and imaginary (

**A**) parts of the object scattering potential with the corresponding PSFs [18]:

- -
- computed from the optical properties of the microscope system,
- -
- estimated from the measurements of the microspheres.

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Set-up of the microscope constructed for automated image acquisition. The red lines illustrate the beam path interacting with the sample.

**Figure 2.**X–Z cross section of the computed point spread function (PSF) with the acquisition parameters of the experimental device: the numeric aperture of the dry objective $N{A}_{ob}=0.75$ with working $WD=0.71\mathrm{mm}$; the refractive index of specimen layer ${n}_{m}=1.33$; the standard value of refractive index of the air ${n}_{{c}^{\ast}}=1$; the refractive index of the air during the experimentation ${n}_{c}=1.00024$, ${z}_{p}=0$; the pixel size in the object space is 0.1 µm.

**Figure 3.**Recorded images in the different planes of a polystyrene microsphere of diameter $1\mathsf{\mu}\mathrm{m}$ at the wavelength $\lambda =640\mathrm{nm}$ (

**a**–

**c**), and the corresponding restored images (

**d**–

**f**).

**Figure 4.**3D refractive index reconstruction at $640\mathrm{nm}$ for a polystyrene microsphere with a diameter of $1\mathsf{\mu}\mathrm{m}$, without PSF consideration (

**a**–

**g**), and with PSF consideration (

**h**–

**n**).

**Figure 5.**Size’s map for the sample. The intensity of each pixel at a point of the microsphere indicates in μm the thickness (at this point) of the microsphere. The intensity of the pixel in the middle of the microsphere represents its diameter, as indicated by the arrow.

**Figure 6.**Mean values of the refractive index of the microsphere as a function of the z-position along the optical axis. The full width at half maximum (FWHM) indicated by the double arrow corresponds to the diameter of the microsphere.

**Table 1.**Mean values of the refractive index of the microsphere at the wavelength 640 nm for each plane Z (μm).

Z = −0.4 | Z = −0.3 | Z = −0.2 | Z = −0.1 | Z = 0 | Z = 0.1 | Z = 0.2 | |
---|---|---|---|---|---|---|---|

Refractive index before deconvolution | 1.444 | 1.460 | 1.526 | 1.512 | 1.556 | 1.558 | 1.515 |

Refractive index after deconvolution | 1.584 | 1.585 | 1.587 | 1.586 | 1.587 | 1.585 | 1.586 |

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**MDPI and ACS Style**

Agnero, M.A.; Konan, K.; Kossonou, A.T.; Bagui, O.K.; Zoueu, J.T.
A New Method to Retrieve the Three-Dimensional Refractive Index and Specimen Size Using the Transport Intensity Equation, Taking Diffraction into Account. *Appl. Sci.* **2018**, *8*, 1649.
https://doi.org/10.3390/app8091649

**AMA Style**

Agnero MA, Konan K, Kossonou AT, Bagui OK, Zoueu JT.
A New Method to Retrieve the Three-Dimensional Refractive Index and Specimen Size Using the Transport Intensity Equation, Taking Diffraction into Account. *Applied Sciences*. 2018; 8(9):1649.
https://doi.org/10.3390/app8091649

**Chicago/Turabian Style**

Agnero, Marcel A., Kouakou Konan, Alvarez T. Kossonou, Olivier K. Bagui, and Jérémie T. Zoueu.
2018. "A New Method to Retrieve the Three-Dimensional Refractive Index and Specimen Size Using the Transport Intensity Equation, Taking Diffraction into Account" *Applied Sciences* 8, no. 9: 1649.
https://doi.org/10.3390/app8091649