## Abstract

Quantitative Phase Microscopy (QPM) by interferometric techniques can require a multiwavelength configuration to remove 2p ambiguity and improve accuracy. However, severe chromatic aberration can affect the resulting phase-contrast map. By means of classical interference microscope configuration it is quite unpractical to correct such aberration. We propose and demonstrate that by Digital Holography (DH) in a microscope configuration it is possible to clear out the QPM map from the chromatic aberration in a simpler and more effective way with respect to other approaches. The proposed method takes benefit of the unique feature of DH to record in a plane out-of-focus and subsequently reconstruct numerically at the right focal image plane. In fact, the main effect of the chromatic aberration is to shift differently the correct focal image plane at each wavelength and this can be readily compensated by adjusting the corresponding reconstruction distance for each wavelength. A procedure is described in order to determine easily the relative focal shift among different imaging wavelengths by performing a scanning of the numerical reconstruction along the optical axis, to find out the focus and to remove at the same time the chromatic aberration.

©2007 Optical Society of America

## 1. Introduction

Interferometry often requires an extended range of phase measurement without 2*π* ambiguity in the phase map due to the long optical path difference (OPD) in the tested sample. In order to overcome this limitation a typical approach is to use a multiwavelength configuration aimed at generating a longer synthetic wavelength for retrieving the phase without ambiguities in an extended range[1–7]. Of course, the use of multiple wavelengths in the same optical apparatus can lead to the occurrence of severe chromatic aberrations [8]. In standard Optical Coherent Microscopy (OCM) or even Interference Microscopy (IM) [9–12], the object is imaged by a microscope objective on the plane array of the detector. Different interferograms with different wavelengths can be recorded, but the optical components, in occurrence of chromatic aberration, will image the same object at different planes for each wavelength and, unavoidably, all but one images will result out-of-focus on the detector plane. Depending on the amount of chromatic aberration, the correctness of the phase map, obtained by subtracting two phases corresponding to two different wavelengths will be incorrect since at least one of them will result out-of-focus. Consequently, the final QPM map calculated with the synthetic wavelength will result incorrect. One simple possible solution to this problem would be the mechanical adjustment of the detector (i.e. the movement of the CCD array) or the displacement of the imaging optics to compensate the focal shift of the image plane. One more sophisticated solution will include adaptive focus optics to take into account and compensate the longitudinal shift.

Recently, various holographic techniques have been developed for various applications [13–17] based on digital holography or scanning holography [18]. In DH, the imaging procedure is conceptually different. In fact the object wave front, scattered by the sample, is recorded out-of-focus after it interferes with the reference wave. Subsequently, the in-focus image is obtained by a numerical reconstruction of the digital interferogram at the right focus image plane [19, 20].

This aspect constitutes a very important advantage of DH with respect to the OCM or IM. The flexibility, intrinsically embedded in the DH, consisting in the numerical re-focusing process, offers in fact a very important and useful opportunity to compensate aberrations and remove the errors in the QPM without mechanical adjustment or wavefront correction by means of active devices. Indeed, no mechanical movement is required in DH since the process of focus tuning is fully performed numerically.

Multi-Wavelengths operation in DH has been often required and adopted in various optical configuration for multiplicity of application from biology to MEMS of for 3D color display [22–32]. In recent paper it was demonstrated that chromatic aberration was removed by a sort of calibration method [26]. However some problems occur in using such method as will be described below.

We illustrate and demonstrate in this paper that a multi-wavelengths phase map can be recovered free of chromatic aberrations by using DH. A procedure is implemented to find the relative focal shift among the various wavelengths by a procedure that we believe can be in practice easily automated, even if investigating the feasibility of such potentiality is out of scope in this paper.

We will show the results obtained for two different types of samples: an *in-vitro* mouse cell fibroblast and an optical waveguide written by a femto-second laser [33].

## 2. Previous approach for removing chromatic aberration in DH and its limitations

In a previous paper we described a general approach to remove chromatic aberrations in Multiwavelength Digital Holography (MWDH) [25]. We removed 2*π* ambiguity reconstructing the phase maps at wavelengths *λ*
_{1} and *λ*
_{2}. The two maps were subtracted each other to obtain a new phase map equivalent to that of a longer beat wave *λ*
_{12} = *λ*
_{1}
*λ*
_{2}/∣*λ*
_{1} - *λ*
_{2}∣. To remove phase chromatic aberration from the wavefront, besides an *object hologram* we recorded a *reference hologram* (i.e. hologram without object) for each wavelength. All the holograms were numerically processed to calculate the complex wave field in a plane between the CCD and the microscope objective, this plane being the image plane for one wavelength. We evaluate the final phase map as a difference phase map, Δ*ϕ*(*x*,*y*) = *φ*
_{1}(*x*,*y*)-*φ*
_{2}(*x*,*y*)-*Arg*(*R*
_{1}(*x*,*y*)/*R*
_{2}(*x*,*y*)), where *φ*
_{1}(*x*,*y*) and *φ*
_{2}(*x*,*y*) are the phase maps corresponding to the object phase retardation for *λ*
_{1} and *λ*
_{2}, respectively, while the third term takes into account the chromatic aberration, being *R*
_{1}(*x*, *y*) and ** R_{2}(x, y)** the complex reconstructed waves from reference holograms at wavelengths

*λ*

_{1}and

*λ*

_{2}. The previous equation can be written as Δ

*φ*(

*x*,

*y*) =

*φ*

_{1}(

*x*,

*y*) -

*φ*

_{2}(

*x*,

*y*) -

*φ*

^{(12)}

*, to make it clear that we are actually accounting for the chromatic aberration by subtracting the corresponding phase contribution,*

_{R}*φ*

^{(12)}

*(*

_{R}*x*,

*y*) =

*Arg*(

*R*

_{1}(

*x*,

*y*)/

*R*

_{2}(

*x*,

*y*)), from the object phase difference

*φ*

_{1}(

*x*,

*y*)-

*φ*

_{2}(

*x*,

*y*).

In this way a sort of calibration of the holographic set-up allows to compensate for the chromatic aberration once and for all. However, even if this procedure in general works properly, it presents some drawbacks such as a fourfold recording that implies a higher computational load and higher noise, owing to multiple subtractions. Moreover, in some cases sample removal to record reference hologram (i.e. dynamic processes) is not possible or practical.

The procedure described above [25] has in addition a systematic error, since the QPM phase map at the synthetic wavelength has cleared out the chromatic phase-aberration term but, in any case, the holograms with different wavelengths have been reconstructed at the same distance even if they have different image planes. Assuming one hologram has been reconstructed at the right distance the other has not and for this reason the subtraction occurs between two phase maps, one in focus and the other out-of-focus. As demonstrated in a previous paper the phase map is strongly affected by out-focus reconstruction [31].

In the following we describe a novel approach that allows to retrieve the phase map with an extended OPD range using only two holograms at two different wavelengths overcoming all the limitations described above.

## 3. Description of the optical set-up

Figure 1 illustrates the DH off-axis set-up adopted for recording MWDH holograms. The setup is based on a Mach-Zehnder interferometer in transmission geometry. The off-axis configuration allows one to avoid the problem of the twin image and to eliminate the zero order diffraction. Two lasers with different wavelengths are used, a laser emitting in the green region at *λ*
_{1} = 532 *nm* and the other in the red region at *λ*
_{2} = 632.8 *nm*. The optical configuration is arranged to allow the two lasers to propagate almost along the same paths either for reference and object beams that are plane waves obtained by a beam expander (BE). The first beam splitter is a cube polarizing beam splitter (PBS) and a *λ*/2 wave plate is inserted in the reference beam to obtain equal polarization direction for the two beams. The microscope objective (MO) is an achromat objective (20× Olympus UMPlanFI) with N.A.=0.46. Two objects are investigated: an in-vitro mouse cell and an optical waveguide. The CCD array has 1024 × 1024 square pixels with a pixel size Δξ = 6.7 μm.

## 4. New approach based on correct distance tracking for removing chromatic aberration in DH

#### 4.1 Description and demonstration of the method

In order to obtain a good superposition of the reconstructed phase-contrast images at different wavelengths, one has to take into account the different curvatures of the wavefronts due to the different wavelengths and the presence of chromatic aberrations due to the optical elements in the setup. Chromatic aberration involves a difference in the magnification and consequently a longitudinal shift of the image plane position on the optical axis as shown in Fig. 2.

Let ** D** be the distance between the CCD and the imaging lens and

**the image plane distance for the green wavelength**

*d*_{G}*, then the reconstruction distance measured backward from the CCD plane is*

**λ**_{G}*=∣*

**r**_{G}*-*

**D***∣. After reconstruction of the complex field at distance*

**d**_{G}*we obtain the total phase*

**r**_{G}Where *x* and *y* are the coordinates at the image plane, **φ**_{0} is the phase retardation introduced by the object and * M_{G}* =

*/*

**d**_{G}*is the magnification at wavelength*

**p***λ*which is related to the corresponding focal length and the distance

_{G}*p*of the object to the lens plane (thin lens approximation). The phase

*(*

**φ**_{G}*x*,

*y*) at the image plane is the sum of the object phase, scaled according to

*M*and the quadratic term related to the curvature of the wavefront introduced by the magnifying lens. The reconstructed phase (by FFT method) is the discrete approximation of (1), at point

_{G}*n*,

*m*namely:

where the reconstruction pixel at wavelength *λ _{G}* is

where Δ*ξ* is the CCD pixel dimension and *N _{G}* the number of pixels employed in the reconstructions. For the red wavelength

*λ*we have correspondingly:

_{R}where *d _{R}* =

*d*+ Δ

_{G}*d*is the position of the image plane at red wavelength which differs form that of the green wavelength by a quantity Δ

_{G}*d*and the corresponding reconstruction pixel is

_{G}Both phases, in Eqs. (2) and (4), are expressed as the summation of two terms. The first one takes into account the phase retardation owing to the sample while the second one regards the optical wave propagation from the imaging lens to the image plane.

To get the phase map for the equivalent wavelength we subtract the foregoing phase maps:

The residual phase, Δ* φ_{r}*, is a parabolic term that never cancels out and invalidates the difference phase map by the presence of circular fringes. However, Δ

*can be minimized to get a difference phase map without circular fringes. The method is first applied on a test object of known magnifications and image plane positions at both wavelengths. Test object consists of a transparent glass plate with an opaque ruler on it, the distance between 5 ruler lines is of 50μm.*

**φ**_{r}We reconstruct the two phase maps in their own image plane and, before subtracting them, we make a padding operation taking into account different magnifications distances and wavelengths; in particular, we set

Equation (7) assures that the reconstruction pixel size is the same for both wavelengths and that the residual phase has a minimum. Figure 3(a) shows the difference phase map obtained recovering the complex wavefront for both wavelengths on the same plane, which is the image plane for the green wavelength (*d*= *105mm*); to subtract the phase maps a padding operation has been applied taking into account only the difference in wavelengths. The difference phase map is affected by circular fringes. Noise around the image is present because of the number of pixels in the in reconstruction plane is smaller than 512. The number of pixel in the reconstruction plane are 230 and, taking into account of the reconstruction pixel size of 16,3μm, the field of view is 3,7mm. In Fig. 3(b) red and green phase maps are reconstructed each in its own image plane and then subtracted taking into account Eq. (7). Circular fringes are not visible. Fig. 3(c) shows the difference phase map where the green QPM is reconstructed on the image plane while the red QPM is recovered in a plane beyond the red and green image planes. The padding operation made in this reconstruction takes into account the different wavelengths and distances but not the magnifications and therefore, even in this case, circular fringes are present.

A numerical simulation of the residual phase Δ* φ_{r}* has been carried out and in Fig. 3(d) the quantity Δ

*/*

**φ**_{r}*2π*is reported. This evaluation refers to the case shown in Fig. 3(b) according to Eqs. (2) and (4). The maximum value for Δ

*/*

**φ**_{r}*2π*is 20% on the edge of the matrix (500×500 pixels), while it is about 10% in the area taken by the object, which is about 300×300 pixels in the matrix center.

The method described can be used as a rapid, even if approximate, graphical technique to find the reconstruction distance for one or more wavelengths (i.e. green and red, in this case) provided that the reconstruction distance for another wavelength (blue, for example) is known. This graphical technique is applied to an *in-vitro* mouse cell and an optical waveguide, as illustrated in Figs. 4 and 5, respectively.

#### 4.2Experimental results: in-vitro mouse cell

The object in this case can be considered as pure phase object. The purpose of the investigation is the measurement of the QPM for *in-vitro* mouse cell. Physical dimension of the cell is about 25μm in the * x* -

*plane. In Fig. 4 is shown the QPM map of the*

**y***in-vitro*cell as it appears at

*λ*

_{12}=

*λ*

_{1}

*λ*

_{2}/∣

*λ*

_{1}-

*λ*

_{2}∣ when the two maps are reconstructed at the same distance of 110mm. It is clear, from the superimposed fringes, that a longitudinal shift exists between the two reconstructed images. These fringes are due to the residual phase factor that is the result of the chromatic aberration and different wave front curvature.

If the correct reconstruction distance is known for one wavelength the focus shift can be easily tracked and found automatically by scanning the reconstruction distance. By applying the proposed procedure by scanning the focus of the blue reconstruction across the red we found that the fringes due to the chromatic aberration can be effectively nulled as shown in the movie in Fig. 4(c), where the phase-difference is obtained by subtracting the phase map of the red hologram reconstructed at the fixed distance of 110 mm, while the blue hologram is reconstructed at various distances from 90mm to 120mm. The cell is cleared of aberration fringes at the distance of 114mm, i.e 4 mm further than the red hologram.

The phase retardation introduced by the cell is about 5.2 rad in the image plane, we have estimated that a slight difference (∼4mm) in the reconstruction distance for the red wavelength implies a difference in the phase map of about 0.5 rad across the edges of the cell where the defocus produces blurring Such difference is reported in Fig. 5. That demonstrates to have an accurate Quantitative Phase Map it is necessary to have reconstruction at right distance for each wavelength.

#### 4.3Experimental results: refractive index profile of an optical waveguide

We apply our method to an optical waveguide written using femtosecond laser pulses in a borosilicate glass substrate [31]. The optical waveguide has a diameter of about 10μm. The best focus was established for the green wavelength, and the red wavelength focus was scanned back and forth with a step size of 5mm. Two frames and a movie of the scanning procedure are reported, respectively, in Figs. 6(a), 6(b) and Fig. 6(c).

The purpose of the measurement is the retrieval of the refractive index change induced by
the femtosecond pulses. The expected refractive index change is about 10^{-2} and considering a thickness of the glass sample of *275μm*, it was necessary to use MWDH configuration to avoid 2π ambiguity. The laser wavelengths used are *λ*
_{1} = *532 nm* and *λ*
_{2} = *632.8 nm*.

Figure 7 shows 2D and 3D representation of the refractive index obtained using the equivalent synthetic wavelength corresponding to *3.3μm* after removal of the chromatic aberration.

The measured value for the refractive index positive change is Δ*n _{x}* = 5.5×10

^{-3}, in agreement with other previous measurements made on thinner samples by means of a single wavelength[32].

## Conclusions

DH makes it possible to obtain an extended OPD by means of QPM using an equivalent wavelength. The use of two or more wavelengths has some drawbacks in the reconstruction process, such as the presence of circular fringes on the phase maps owing to the combined effect of different wavefront curvatures and chromatic aberration in to the optical set-up. We have demonstrated a method that minimizes this combined effect and retrieves phase maps without circular fringes. The method can be easily applied thanks to the DH feature of numerical re-focusing. Our procedure can be employed to find automatically the right reconstruction distances for different wavelengths.

The procedure allows the application of MWDH in such cases where an extended measurement range is desired while maintaining interferometric resolution.

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