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Grating deployed total-shear 3-beam interference microscopy with reduced temporal coherence

Krzysztof Patorski, Piotr Zdańkowski, and Maciej Trusiak

Open Access Open Access

Abstract

Interference microscopy is a powerful optical imaging technique providing quantitative phase distribution information to characterize various type technical and biomedical objects. Static and dynamic objects and processes can be investigated. In this paper we propose very compact, common-path and partially coherent diffraction grating-based interference microscopy system for studying small objects like single cells with low densities being sparsely distributed in the field of view. Simple binary amplitude diffraction grating is the only additional element to be introduced into a conventional microscope optical system. By placing it at a proper distance in front of the microscope image plane the total-shear operation mode is deployed resulting in interferograms of the object-reference beam type. Depending on the grating to image plane separation distance two or three-beam interferograms are generated. The latter ones are advantageous since they contain achromatic second harmonics in the interferogram intensity distributions. This feature enables to use reduced temporal coherence light sources for the microscope to reduce coherent noise and parasitic interference patterns. For this purpose we employ the laser diode with driving current below the threshold one. Results of conducted experiments including automatic computer processing of interferograms fully corroborate analytical description of the proposed method and illustrate its capabilities for studying static and dynamic phase objects.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Attractivity in interferometry with diffraction gratings used as beam splitters and recombiners follows from the optical system compactness, implementation simplicity and common-path architecture providing immunity from environmental instabilities [1,2]. Diffraction gratings can be deployed in two most popular configurations, i.e., object-reference and shear type interferometry [1]. The first one provides the object phase distribution whereas the second one renders its spatial derivative (gradient) when two basic experimental conditions are met (slow object phase variations and small displacement between the object beam and its replica).

In this paper we develop a new object-reference beam type interference method using a simple binary amplitude grating as beam splitter. Common-path total-shear approach is implemented. Two-shot recording enables obtaining the specimen gross phase and the microscope system reference phase distributions. Their subtraction provides pure specimen phase mapping. Our literature review starts, chronologically, with the two-aperture common-path phase shifting interferometer (TACPI) using a simple binary grating [3–5]. TACPI is deployed in the 4f-Fourier optical imaging system. Two windows in the object plane accommodate the object and reference beams with some lateral separation between them. By placing a diffraction grating with appropriately selected period at the system spatial frequency plane or its vicinity the object and reference beams are brought to overlap at the output plane. In this way the conventional object-reference two-beam interferogram is generated. Grating lateral displacement can be applied for accurate phase measurements using the phase-shifting method. Similar idea has been recently proposed in interference microscopy under the name of spatially-multiplexed interferometric microscopy (SMIM) [6–9]. Both phase-shifting and single-shot operation capabilities in a standard microscope converted to holographic one were experimentally corroborated.

Binary amplitude diffraction grating acting as the beam splitter was applied in so-called diffraction phase microscopy [10–13]. The grating was placed in the image plane of the inverted microscope and was followed by a conventional 4f imaging and spatial filtering optics. Two filtered diffraction orders at its output, mutually inclined, served as object and reference beams. The 4f system added to a microscope makes the configuration rather bulky.

An extensive list of references describing object-reference two-beam interference methods using various implementations of the Twyman-Green (Michelson) and Mach-Zehnder architectures for interference microscopy can be found in [14]. In those systems, however, the object and reference beams are distinctly spatially separated and this fact leads to interferogram instabilities caused by environmental conditions. Simplified approach avoiding a separately generated reference beam was described in [14,15]. It bases on the assumption that the microscope field of view (FOV) contains the object and object-free regions (specimen free reference areas, SFRA). They can be superimposed using the concept of the object beam total lateral shearing. A compact Twyman-Green (Michelson) interferometer with bulk optics might be used for that purpose. This simplified approach is only an approximate since the beam parts passing through the specimen and SFRA do not coincide in space and might suffer from diversified system and substrate aberrations together with light diffraction effects. The latter errors might be reduced by applying light sources of reduced coherence length. In general the solutions including additional operation of non-numeric calibration of the optical measurement system are to be considered as more complete ones [14–18]. Although Refs. [14–18] concern digital holographic microscopy the concept of recording two interference patterns, one of a sample substrate (for pre-calibration purpose) and the second with specimen on a substrate can be extended to conventional interference microscopy. Subtraction of phase distributions demodulated from the two interferograms provides information on the sample only induced phase delays overlaid with the instrument noise [17,18].

This paper presents the principle and implementation of common-path, grating total-shear (object-reference beam type) interference microscopy for studying small phase specimens well separated in the instrument input plane. A simple binary amplitude (Ronchi) diffraction grating acting as the input field divider is used instead of conventional Twyman-Green (Michelson) type interferometer arrangements [14,15]. Advantageous feature of using gratings as beam splitters is their achromatic operation [19–21] which allows for coherent noise reduction by deploying light sources with reduced temporal coherence. We describe two total-shear operation modes of our grating interferometric system. In both of them the microscope output plane is located in the grating Fresnel diffraction field.

In the first configuration the specimen under test occupies one-third of the system input plane (dimensions considered in the direction perpendicular to grating lines) and is centrally located within it. These two conditions require precise specimen-field stop adjustments (greatly simplified in the second operation mode). Three lowest diffraction orders of the grating in its Fresnel field form three interference patterns in the microscope image plane (higher grating orders walk-off the interference patterns of interest). The interferograms are of the object-reference beam type. The central one is generated by all three grating diffraction orders. It encodes the specimen gross phase in its amplitude (contrast) modulation. Other two fringe patterns, situated symmetrically on both sides of the on-axis one, are two-beam interferograms. They encode the object gross phase in the shape of fringes which is advantageous as compared with amplitude modulation encoding (central pattern) from the point of view of further fringe pattern processing. The reference phase distribution to be subtracted from the object gross phase can be retrieved from side interferograms obtained for the specimen substrate without object. In fact only one side interferogram is needed since the second one carries conjugate phase information.

The second operation mode is also destined for testing small isolated objects like single cells with low densities laterally separated within the FOV. The object-free regions on both specimen sides in the input plane should be now at least two times larger than the object size (in the direction perpendicular to grating lines). For this condition satisfied there is no strict requirement for the size of the total FOV, i.e, the field stop might not be required in the object plane. This property significantly simplifies assembling a laboratory microscope system. We prove that similar phase information can be obtained from side interferograms as in the first operation mode but now all three interferograms are generated by grating three diffraction orders. This fact enables achromatic recording of the object phase in side interference patterns. Both solutions are robust with respect to the environmental instabilities because of the common-path setup.

2. Theoretical description and analysis

Figure 1 shows a general layout of our grating total-shear interference microscopy. The input plane IP is imaged by the microscope objective MO and the tube lens TL at the output plane OP. A diffraction grating G is placed at a distance $z$ in front of the image plane OP and serves as the beam splitter to obtain three interferograms carrying object information. The formation of interferograms depends on the operation modes described below.

 figure: Fig. 1.

Fig. 1. Schematic representation of the grating total-shear 3-beam interference microscopy: LD – laser diode; SMF – single mode fiber; CL – collimator objective; IP – input (sample) plane; MO – objective; TL – tube lens; G – diffraction grating; OP – output plane.

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2.1 First operation mode

Schematic representation of the input plane IP for this mode is shown in Fig. 2(a). The specimen $S$ should be placed at the system optical axis and the field stop diameter adjusted to be three times the specimen size. Object and object-free areas are indicated as $S$, $R1$ and $R2$, respectively. The situation in the output plane OP is shown in Fig. 2(b). Three images of input plane are presented by solid black, dashed red and dashed green lines. The beams interfering in the component parts (vertical rectangles) are indicated with the lower index relating to the beam diffraction order number at the grating. The formation of interferograms in the vertical cross-section passing through the optical axis is schematically shown in Fig. 2(c).

 figure: Fig. 2.

Fig. 2. Schematic representation of interferences exploited in the grating total-shear first operation mode: (a) situation in the specimen (microscope input) plane IP with empty regions $R1$ and $R2$ and specimen region $S$; (b) three interference areas in the OP with indicated interfering grating diffraction orders; (c) formation of interferograms shown in the vertical cross-section plane passing through the system optical axis.

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It will be shown in detail below that:

  • • the central interferogram (interference of beams $S^\prime _0$, $R1^\prime _{-1}$, $R2^\prime _{+1}$) encodes the object phase information in the amplitude modulation of its intensity distribution;
  • • two side interferograms (two beam interferences $S^\prime _{+1}$, $R1^\prime _0$ and $S^\prime _{-1}, R2^\prime _0$) deliver the specimen gross phase.
It follows from Fig. 2(c) that spatial separation of interferograms is readily obtained by properly selecting the grating spatial period and the separation distance $z$ between G and OP. It is easily adjustable to the specimen lateral extent in the image plane. The spatial separation of fringe patterns corresponds to grating total-shear value (specimen images do not overlap with themselves but with images of empty regions $R1$ and $R2$) [14]. This requirement is common to previously reported solutions with spatial multiplexing [3–9].

We will present now the mathematical description of the first operation mode. To keep it clear as possible the instrumental errors (system aberrations including sample substrate and light scattering noise) are not taken into account. They will be addressed in the paper experimental section. The expression describing the complex amplitude of the grating three-beam field in the central image ($S^\prime _0$, $R1^\prime _{-1}$ and $R2^\prime _{+1}$), Fig. 2, has the form

$$E_0(x,y,z) = a_0\exp{\left[ ik \cdot g(x,y)\right]} + a_{+1}\exp{\left[ ik \left( \frac{\lambda}{d} x - \frac{\lambda^2z}{2d^2} \right) \right]} + a_{-1}\exp{\left[ ik \left( -\frac{\lambda}{d} x - \frac{\lambda^2z}{2d^2} \right) \right] };$$
where $a_0$ and $a_{+1} = a_{-1}$ are the amplitudes of orders 0 and +/-1, respectively, $d$ denotes the grating period, $\lambda$ is the light wavelength, $k = 2\pi /\lambda$, $g(x,y)$ designates the optical path across the specimen under test and the term ($\lambda ^2z/2d^2$) denotes the optical path difference between side orders +/-1 and the zero order beam [22]. Spatial overlapping of on-axis located specimen with undisturbed parts of the illuminating beam (situated on both sides of the specimen) results in the central segment of the interferogram formed by the axial object beam $S^\prime _0$ and two symmetrically impinging object-free reference beams $R1^\prime _{-1}$ and $R2^\prime _{+1}$. These three beams are expressed by the terms in Eq. (1). By multiplying Eq. (1) by its complex conjugate the intensity distribution is calculated as
$$\begin{aligned} I_0(x,y,z) = & a_0^2 + 2a_1^2 + 2a_0a_1cos \left\{ k\left[\frac{\lambda}{d}x + g(x,y) + \frac{\lambda^2z}{2d^2}\right]\right\} + \\ & +2a_0a_1cos\left\{k\left[-\frac{\lambda}{d}x + g(x,y) + \frac{\lambda^2z}{2d^2}\right]\right\} + 2a_1^2cos\left(\frac{4\pi}{d}x\right) = \\ & = a_0^2 + 2a_1^2 + 4a_0a_1cos\left[kg(x,y) + \frac{\pi\lambda z}{d^2}\right]cos\left(\frac{2\pi}{d}x\right) + 2a_1^2cos\left[\frac{4\pi}{d}x\right]; \end{aligned} $$
where $a_0^2$ + $2a_1^2$ denotes the interferogram bias. The term given by the product of two cosine functions in the latter form of Eq. (2) describes the first harmonic of the central interference pattern intensity distribution. Straight cosinusoidal fringes, represented by the second cosine $cos\left (2\pi x/d\right )$ of the product, are amplitude modulated by the first cosine carrying the specimen phase $kg(x,y)$ in its argument. The amplitude of the first harmonic attains maximum value for $z$ equal to an integer multiple of $d^2/\lambda$ (i.e., in the self-image planes) [22,23]. This fact can be explained as follows. Denoting the modulation cosine by $M_c$ we can write
$$M_c = cos \left[ kg(x,y) + \frac{\pi\lambda z}{d^2} \right] = cos\left(\frac{\pi \lambda z}{d^2} \right) cos[kg(x,y)] + sin \left( \frac{\pi \lambda z}{d^2} \right) sin \left[ kg(x,y) \right].$$
It follows that for $z$ equal to even and odd integer multiples of $d^2/\lambda$ the first harmonic of intensity distribution expressed by Eq. (2) attains its maximum value proportional to $cos[kg(x,y)]$. Correspondingly, by calculating the modulation distribution of the first harmonic of the recorded interferogram in one of these planes the specimen phase distribution can be obtained. The cases of even and odd multiples differ in sign. On the other hand for the distances $z$ corresponding to the planes lying at odd multiples of the distance equal to $d^2/2\lambda$ (planes located in the middle between the self-image planes) the amplitude of the intensity first harmonic is given by $sin[kg(x,y)]$. In the case of testing very weak phase objects this value is close to zero. In the extreme, i.e., $g(x,y)=0$ (no phase object) the first harmonic vanishes in those planes. This situation is well known from the theory of three-beam interference [22,23]. Note that the axial localization of maxima and minima of the first harmonic is chromatic; it depends on the wavelength $\lambda$. This fact results in decreased contrast of first harmonic fringes due to reduced coherence length.

The above presented possibility of the amplitude demodulation of the first harmonic of intensity distribution to quantitatively retrieve the specimen phase is rather complex in practice especially when small specimen phase modulations are investigated. Additionally, the instrumental errors enter both cosines in their product and decrease the demodulation accuracy. For example, the first harmonic carrier fringes might deviate from straightness since the two reference beams on both sides of the object one might be influenced by diversified wave front aberrations.

It was shown in our previous papers [20,21] that the phase information retrieval from the second harmonic of the recorded three-beam interference field is much more straightforward than processing the first harmonic. The last cosine term of Eq. (2) describes the second harmonic of the intensity distribution of the central interferogram ($S^\prime _0$, $R1^\prime _{-1}$, $R2^\prime _{+1}$). It is formed by two symmetrically impinging object-free beams with theoretically plane wave fronts and zero path difference. The latter property results in achromatic character of second harmonic fringes and permits to use light sources with reduced temporal coherence. Any departure of second harmonic fringes from straightness can be attributed to the difference between wave fronts of the two object-free parts of the illuminating beam. Unfortunately, the central image second harmonic does not carry the specimen phase information.

Let us derive and interpret intensity distributions of side interferograms $S^\prime _{+1}$, $R1^\prime _{0}$ and $S^\prime _{-1}$, $R2^\prime _{0}$ obtained under the first operation mode, Fig. 2. The complex amplitude of the first pattern is given by

$$E_{+1}(x,y,z) = a_0 + a_{+1}exp\left\{ik\left[\frac{\lambda}{d}x + g(x - \Delta,y) - \frac{\lambda^2z}{2d^2}\right]\right\};$$
where $\Delta$ represents the lateral displacement of the specimen image from the optical axis. The object information is carried by the +1 diffraction order of the grating, Fig. 2(c). The zeroth order $R1^\prime_0$ serves as the reference beam. The intensity distribution is calculated as the modulus square of Eq. (4) and described by
$$I_{+1}(x,y,z) = a_0^2 + a_1^2 + 2a_0a_1cos \left\{ k \left[ \frac{\lambda}{d}x + g(x - \Delta,y) - \frac{\lambda^2z}{2d^2} \right] \right\}.$$
The interferogram obtained here is a two-beam interference pattern with carrier fringes deformed proportionally to the object phase $kg(x,y)$. Its phase demodulation is straightforward, therefore. The last term in the argument of cosine function indicates the lateral displacement of fringes upon changing the grating to the output plane distance $z$. It depends on $\lambda$, hence it is chromatic. Additionally, the required information on the system reference phase can be obtained by recording the side interference pattern with phase specimen removed from the microscope FOV and the specimen substrate present. For the second side interferogram ($S^\prime _{-1}$ and $R2^\prime _{0}$) with specimen information carried by the -1 diffraction order of the grating, Fig. 2(c), the complex amplitude distribution is given by the following equation
$$E_{-1}(x,y,z) = a_0 + a_{-1}exp\left\{ik\left[-\frac{\lambda}{d}x + g(x + \Delta,y) - \frac{\lambda^2z}{2d^2}\right]\right\}.$$
The intensity distribution is expressed as
$$I_{-1}(x,y,z) = a_0^2 + a_1^2 + 2a_0a_1cos \left\{ k \left[ \frac{\lambda}{d}x - g(x + \Delta,y) + \frac{\lambda^2z}{2d^2} \right] \right\}.$$
Note similar expressions in Eqs. (5) and (7) except for their mutually conjugate character, i.e., carrier fringes depart from straightness in opposite directions. This property follows from the fact that object beams (side diffraction orders) are located on opposite sides of the reference beam (grating zeroth diffraction order) in the two interferograms [24, and references therein]. The object information carried by the two-beam interferogram was preliminarily experimentally investigated in [9]. The properties of the first operation mode of our three-beam total-shear method can be summarized as follows:
  • • specimen gross phase can be estimated by demodulating one of the side two-beam interference patterns. This approach is straightforward and more effective than calculating the object gross phase from the amplitude modulation of the first harmonic of the central interference pattern;
  • • the reference phase is obtained by demodulating the above mentioned side pattern obtained with the specimen substrate only present in the microscope input plane;
  • • the first operation mode requires precise mutual adjustments of an object and the microscope field stop.
Both first and second harmonics explained are readily separable from the fringe pattern due to the presence of carrier fringes. For example, the Fourier [25], continuous wavelet [26] or Hilbert-Huang [8,27] transforms can be used. We will return to this issue in the experimental part of the paper.

2.2 Second operation mode

The second operation mode, requiring two times larger object-free regions on each specimen side includes two experimental possibilities:

  • 1. for the same specimen size as in the first mode discussed above the field stop in the object plane IP should be 5 times larger or more than the object lateral dimension (vs. 3 times in the first operation mode). In practice, therefore, for small and well separated objects under test the field stop is not needed. This configuration is quite attractive because of the experiment simplicity,
  • 2. for the field stop of size used in the first mode appropriately smaller size objects can be tested.
Before introducing the mathematical description of the second operation mode we show schematic representation of its input and output planes in Fig. 3 (compare with Fig. 2). In each rectangular shape region of the output plane the interfering beams are indicated. Specimen ($S$) and reference ($R1, R2, R3, R4$) beams have subindices corresponding to the diffraction order number at the grating. The central interference pattern generated by beams $S^\prime _{0}$, $R1^\prime _{-1}$, $R2^\prime _{+1}$ and its side interference images generated by beams $S^\prime _{+1}$, $R1^\prime _{0}$, $R3^\prime _{-1}$ and $S^\prime _{-1}$, $R2^\prime _{0}$, $R4^\prime _{+1}$ are all now three-beam interferograms. Adjacent outer regions are formed by two-beam interferences $R1^\prime _{+1}$, $R3^\prime _{0}$ and $R2^\prime _{-1}$, $R4^\prime _{0}$, respectively.

 figure: Fig. 3.

Fig. 3. Schematic representation of the input plane IP (a) with empty regions $R1$, $R3$ and $R2$, $R4$ on specimen sides being two times larger (or more) than the specimen $S$ size; (b) and (c) show interference fields in the system output plane OP. Their intensity distributions are calculated in the paper. Formation of the central pattern generated by the beams $S^\prime _{0}$, $R1^\prime _{-1}$, $R2^\prime _{+1}$ is the same for the first and second operation modes. Interference patterns adjacent to it are now 3-beam interferograms. Specimen ($S^\prime$) and reference ($R1^\prime$, $R2^\prime$, $R3^\prime$, $R4^\prime$) beams indicated in each interference region have subindices relating to the diffraction order number at the grating.

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We are ready to express, in mathematical terms, the three 3-beam interference patterns. It follows from Figs. 2 and 3 that the description of the central pattern generated by beams $S^\prime _{0}$, $R1^\prime _{-1}$, $R2^\prime _{+1}$, Eq. (2), is common to both operation modes so it does not have to be repeated here. The situation is different for two side interference images as in the second mode they are generated by three-beam interference.

The complex amplitude in the region of interference of the beams $S^\prime _{+1}$, $R1^\prime _0$, $R3^\prime _{-1}$ can be expressed in the form

$$E_{+1}(x,y,z) = a_0 + a_{+1}exp \left\{ ik \left[ \frac{\lambda}{d}x + g(x - \Delta,y) - \frac{\lambda^2z}{2d^2}\right]\right\} + a_{-1}exp \left\{ ik \left[ -\frac{\lambda}{d}x - \frac{\lambda^2z}{2d^2}\right]\right\};$$
where $\Delta$ represents, as before, the specimen image lateral displacement from the optical axis. The object information is carried by the grating +1 diffraction order. Two object-free beams propagating along the 0 and -1 order directions are situated on the same side of the object beam. The intensity distribution is calculated as the modulus square of Eq. (8) and described by
$$ \begin{aligned} I_{+1}(x,y,z) & = a_0^2 + 2a_1^2 + \\ & + 2a_0a_1cos \left\{ k \left[ \frac{\lambda}{d}x + g(x - \Delta,y) - \frac{\lambda^2z}{2d^2} \right] \right\} + 2a_0a_1cos \left\{ k \left[ \frac{\lambda}{d}x + \frac{\lambda^2z}{2d^2} \right] \right\} + \\ & + 2a_1^2cos \left\{ k \left[ 2\frac{\lambda}{d}x + g(x - \Delta,y) \right] \right\} = a_0^2 + 2a_1^2 + \\ & + 4a_0a_1cos \left\{ \frac{k}{2}g(x - \Delta,y) - \frac{\pi \lambda z}{d^2} \right\} cos \left[ \frac{2\pi}{d}x + \frac{k}{2}g(x - \Delta,y) \right] + \\ & +2a_1^2cos \left[ \frac{4\pi}{d}x + kg(x - \Delta,y) \right]; \end{aligned}$$

The properties of the first harmonic of the intensity distribution expressed by the product of two cosines in the latter form of Eq. (9) are different from the ones established from Eq. (2). Carrier fringes described by the second cosine term in the product are now deformed proportionally to the object phase by $(k/2)g(x-\Delta , y)$. Simultaneously they are amplitude modulated by the first cosine. The influence of this modulation cosine can be interpreted using the same argumentation as used for discussing Eq. (3). The only difference is the factor $(k/2)$, instead of $k$, which multiplies the object generated optical path $g(x-\Delta ,y)$ in the argument of the modulation cosine. First harmonic fringes are chromatic.

It is more straightforward to obtain the object phase from the second harmonic of the side pattern intensity distribution, see the last term of Eq. (9). Besides doubled sensitivity in comparison with the case of demodulating the first harmonic we have now the freedom to choose the grating axial location, i.e., the distance $z$ since the amplitude of the second harmonic does not depend on this parameter (self-imaging free).

For the second side interferogram with specimen information carried by the grating -1 diffraction order, Fig. 3(c), the complex amplitude distribution can be describes as

$$E_{-1}(x,y,z) = a_0 + a_{+1}exp \left\{ ik \left[ \frac{\lambda}{d}x - \frac{\lambda^2z}{2d^2}\right]\right\} + a_{-1}exp \left\{ ik \left[ -\frac{\lambda}{d}x + g(x + \Delta,y) - \frac{\lambda^2z}{2d^2}\right]\right\};$$
Its intensity distribution is derived as
$$ \begin{aligned} I_{-1}(x,y,z) & = a_0^2 + 2a_1^2 + 2a_0a_1cos \left\{ k \left[ \frac{\lambda}{d}x - g(x + \Delta,y) + \frac{\lambda^2z}{2d^2} \right] \right\} +\\ & + 2a_0a_1cos \left\{ k \left[ \frac{\lambda}{d}x - \frac{\lambda^2z}{2d^2} \right] \right\} \\ & + 2a_1^2cos \left\{ k \left[ 2\frac{\lambda}{d}x - g(x + \Delta,y) \right] \right\} = a_0^2 + 2a_1^2+\\ & +4a_0a_1cos \left\{ \frac{k}{2}g(x + \Delta,y) - \frac{\pi \lambda z}{d^2} \right\} cos \left[ \frac{2\pi}{d}x - \frac{k}{2}g(x + \Delta,y) \right] + 2a_1^2cos \left[ \frac{4\pi}{d}x - kg(x + \Delta,y) \right]. \end{aligned} $$
Note similar expressions in Eqs. (9) and (11) except for their conjugate relationship, i.e., carrier fringes in their both first and second harmonics depart from straightness in opposite directions. This property follows from the fact that the reference beam pairs are located on opposite sides of the specimen beam [24]. The three-beam side interferogram expressed by Eq. (11) was preliminarily experimentally investigated in [9].

It can be readily proved that the derived second operation mode properties apply to its extended version, i.e., with two empty regions on both object sides larger than the doubled object dimension (in the direction perpendicular to grating lines). In practice, therefore, for small objects located approximately on axis no field stop is required neither its size adjustment.

Summarizing we conclude similar, to some extent, demodulation properties of our three-order interference method for both operation modes. Information required for the object and the system reference phase estimation is encoded in one of two side interferograms recorded for the specimen and its substrate, respectively. The two phase distributions are calculated by demodulating its first harmonic in the first operation mode and second harmonic in the second mode, respectively. Note that object and system reference phases are determined from the same regions of the object plane. This fact is essential for obtaining high measurement accuracy. Achromatic generation of second harmonic fringes allows for deploying light sources with limited temporal coherence to minimize speckle noise and spurious interferences.

3. Experimental works

We used the experimental system assembled according to schematic representation shown in Fig. 1. As the light source the laser diode Thorlabs LPS-635-FC was used. To shorten its coherence length to reduce coherent noise artifacts in the recorded fringe patterns we performed experiments for variable drive current values and choose the value below the threshold one [28,29]. Interferograms presented in this paper were recorded with the driving current of 35 mA resulting in the spectral width FWHM (full width at the half maximum) equal to 11 nm. According to the catalog values the current threshold value is equal to 39.9 mA and the operating current is 51.7 mA. The second advantage following from the above mentioned LD operation is substantially diminished or eliminated interference between the zero and +/- grating orders in the recording plane. The optical path difference between them, see Eq. (1), is equal to $\lambda ^2z/2d^2$. It can be made equal or larger than the LD coherence length for the appropriate choice of the distance $z$ between the grating G and the detection plane OP. Please note that this condition is congruent with selecting the distance $z$ providing the walk-off of higher Ronchi grating diffraction orders, i.e., +/-3, with respect to the three-beam interferograms analyzed in Section 2. At the same time the object replicas are mutually separated in space and the camera matrix is placed to detect one side interference pattern.

Before proceeding to further description of our experiments we would like to shortly compare the operation properties of grating based microscope systems working with an SLD source [8,9] and the LD source operated below the threshold current (our system). Main differences concern the coherence length. It is fixed in the first system [8,9] and equal to 6 nm (SLD from Exales, Model EXS6501-B001). In our solution it is adjustable and depends on the driving current. For 35 mA the spectral bandwidth of the laser diode LPS-635-FC from Thorlabs is 11 nm, i.e., almost two times the one obtained from the SLD source. The price we have to pay for conducting experiments with low laser diode driving current is low LD output power dictating longer recording times of interferograms. The considerable advantage we gain, however, is very clear spatial frequency spectrum of recorded interferograms containing only the zero and +/-2 orders (second harmonic terms). This fact enables comfortable second harmonic filtration (see Figs. 4(b) and 6(b) below) and high accuracy object phase demodulation (no parasitic interferences and noise from possible other interferogram diffraction orders which appear for the source longer coherence lengths). Last but not least the use of a pigtailed diode in our experimental solution minimize radiation reflection to the source. Otherwise the laser diode spectrum changes (multimode operation), output intensity might fluctuate and in the extreme the diode failure might occur. As for the interference pattern formation and its processing we would like to highlight that in our solution we exploit three-beam interference instead of the two-beam one [8,9]. In result truly achromatic interferogram, formed by symmetrical diffraction orders and encoded in the second spatial harmonic of the recorded intensity pattern is generated and processed.

 figure: Fig. 4.

Fig. 4. Three-beam interferogram ($S^\prime _{-1}$, $R2^\prime _0$, $R4^\prime _{+1}$, as described by Eq. (11)) for the (a) object-less case (sample substrate only present), and (b) its spectrum (intensity logarithmic scale), for determining the optical system reference phase distribution. The microscope system (OB plus TL, see Fig. 1) magnification is 20x. The period of the first harmonic (vertical carrier fringes) is equal to the grating period $d = 16.67\mu m$; this harmonic is absent in the recorded image (a) and the second harmonic diffraction orders are clearly seen in the spectrum (b).

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Our experimental work focuses on benchmarking the novel interference microscopy technique, therefore as specimen we choose phase test target - Sample No. 3 from Lyncée Tec "Transmission Sample Kit". Declared height of etched elements is equal to 150 nm, however reference measurements on WLI Veeco suggest height around 125+/-5 nm. The phase targets were created in cooperation between Warsaw University of Technology (Institute of Micromechanics and Photonics) and Lyncée Tec Company under User Project financed by ACTMOST. One of the elements of that sample was introduced into the FOV of our microscope system, occupying approximately one fifth of it. This experimental condition corresponds to the second operation mode of the proposed novel 3-beam interference microscopy technique, described in details in the theoretical part 2.2. Employed microscopic objective MO was Nikon Plan Apo $\lambda$ 20x, NA 0.75; the tube lens TL focal length was 200 mm. Ronchi type binary amplitude grating G with spatial frequency of 60 lines/mm was deployed, and the CMOS photodetector was Flir Backfly S BFS-U3-120S4M-CS (without objective).

It follows from the theoretical description of the three-beam interference patterns and the discussion of their intensity pattern harmonics that the most advantageous demodulation of the object and optical system reference phases is obtained by demodulating the interferogram second harmonic. It is formed by two-beam interference, hence bypasses the influence of the self-imaging effect: it is free of amplitude modulations and independent of the distance $z$ between the beam-splitting grating G and the object image plane S’. For this reason we present here the demodulation results of the second harmonic only. It is also interesting to emphasize that the second harmonic of interference pattern is achromatically generated in zero optical path difference regime (between two interfering beams). Thus, it is not affected by the limited temporal coherence of the light source used. Coherent artifacts are minimized without altering the quality of information carrying interference pattern formation.

Figure 4 shows a three-beam interference pattern ($S^\prime _{-1}$, $R2^\prime _0$, $R4^\prime _{+1}$; see Fig. 3(b)) of the system reference phase (object-free space, i.e., sample substrate only present) recorded for the distance $z$ of the grating G to the image plane S’ equal to 185.3 mm. Interference pattern contains information about system/instrument reference phase defined within the object-free space (substrate only).We recall that the experiment conditions correspond to the second system operation mode discussed above, i.e., the lateral size of empty (object-free) regions on both sides of the object used were larger than object doubled lateral dimension (accordingly, in our experiment no field stop was necessary which is indispensable in the spatial multiplexing approach SMIM). The selected distance $z$ corresponds to the CMOS matrix recording plane located approx. in the middle between the three-beam interference self-image planes. Specifically, in our system the grating period $d$ is equal to 16.67 $\mu$m and light wavelength is $\lambda$ = 0.635 $\mu$m; the distance between neighboring self-image planes is equal to $d^2/\lambda$ = 0.438 mm. Correspondingly $z/(d^2/\lambda ) \approx 423.5$. As mentioned above for demodulating the second harmonic of recorded Fresnel field intensity distributions the distance $z$ is irrelevant, however. Additionally, for selected value of the distance z equal to 185.3 mm and reduced coherence length of the laser diode radiation the interferogram first harmonic is not generated. Specifically the optical path difference (OPD) between side orders +/-1 and the zero order beam is equal to $\lambda ^2z/2d^2$, see Eq. (1), and in our experimental case OPD = 134 $\mu$m. The LD radiation temporal coherence length calculated from the approximate formula $l_c \approx \lambda _0^2/\Delta \lambda$ is equal to 36.7 $\mu$m. We see that for the experimental data considered OPD > $l_c$ and this explains the absence of the first harmonic in the image spectrum in Fig. 4(b).

As we can see the second harmonic is well spectrally separated and can be comfortably filtered for the Fourier transform based phase retrieval [25].

Second harmonic phase demodulation results are presented in Fig. 5. 2D and 3D representations of the phase distribution over the $S^\prime _{-1}$, $R2^\prime _0$, $R4^\prime _{+1}$ image area are shown in Figs. 5(a) and 5(b). We consider it as the system reference phase to be subtracted from the demodulated phase of the specimen under test, see below.

 figure: Fig. 5.

Fig. 5. Results of the second harmonic phase demodulation of the side interference pattern, $S^\prime _{-1}$, $R2^\prime _0$, $R4^\prime _{+1}$ (substrate only present); (a) 2D and (b) 3D phase distributions.

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According to the method theoretical description, the second harmonic of one of the side interferograms, Eqs. (9) and (11), can be used to obtain the specimen gross phase distribution. Figure 6 shows the side image $S^\prime _{-1}$, $R2^\prime _0$, $R4^\prime _{+1}$ for the specimen present in the microscope input plane and its spectrum. Figure 7 presents its demodulation results.

 figure: Fig. 6.

Fig. 6. (a) Side three-beam interferogram $S^\prime _{-1}$, $R2^\prime _0$, $R4^\prime _{+1}$, described by Eq. (11), and (b) its spectrum (intensity logarithmic scale). Experiment details – see the Caption of Fig. 3. The second harmonic diffraction orders are clearly separated in the spectrum.

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 figure: Fig. 7.

Fig. 7. Results of the second harmonic demodulation of the side interference image $S^\prime _{-1}$, $R2^\prime _0$, $R4^\prime _{+1}$; (a) 2D and (b) 3D phase distribution representations. Figures 7(c) and 7(d) show 2D and 3D representation of the difference between phase maps shown Fig. 7(b) (gross specimen phase) and Fig. 5(b) (system reference phase).

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Qualitatively (visually) evaluating demodulated phase maps it is evident that the phase correction scheme improves the final phase imaging results. Phase values are directly linked with height measure, one needs to account for wavelength (635nm) and refractive index (division by 4$\pi$(n-1)) of the glass (Borofloat 33, n=1.4699). To quantitatively evaluate the phase imaging quality and phase correction serviceability we propose to deploy standard deviation (STD) measure and calculate it within the object area and in background region (object free zone). Considering the case of uncorrected phase imaging, Fig. 7(b), the STD for object area reads 34.41 nm. It is reduced upon phase correction to around 8.61 nm, Fig. 7(d). The STD minimization is similarly noticeable for object-free area, where it decreased from 32.35 nm to 6.45 nm. Peak-to-valley error measure was not feasible in this case due to manufacturing imperfections resulting in highly uncorrelated scattered phase errors. Summing up, subtraction of the system reference phase reduces the STD by approx. 75%. Calculating the average height of the imaged object, we obtained perfectly plausible value of approx. 113 nm +/-1 nm.

We would like to additionally highlight the capabilities of our proposed common-path total-shear grating-based 3-beam interference microscope system in biological specimen imaging. Imaged sample was a freshly prepared human cheek cell. Figure 8 shows the captured side interferogram $S^\prime _{-1}$, $R2^\prime _0$, $R4^\prime _{+1}$ of the cheek cell (Fig. 8(a)) and the side interferogram $S^\prime _{-1}$, $R2^\prime _0$, $R4^\prime _{+1}$ of the object-less case (Fig. 8(c)). Figures 8(b) and 8(d) present the second harmonic phase demodulation of the object and the reference phase, respectively. Figure 8(e) shows the phase subtraction of the object gross phase and reference phase, resulting in the object only phase distribution. We can clearly distinguish the nucleus and cytoplasm, as they exhibit different refractive index resulting in different phase in the demodulated image. Notably, cell structure is visible without the need for staining the sample. Timelapse movies Visualization 1 and Visualization 2 show the movement of the cheek cell from Fig. 8, proving the system capabilities of recording of dynamic events and the ability to correct the demodulated phase.

 figure: Fig. 8.

Fig. 8. (a) Side interference image $S^\prime _{-1}$, $R2^\prime _0$, $R4^\prime _{+1}$ of the human cheek cell and (b) second harmonic demodulation of its phase; (c) side interferogram of the object-less case, (d) second harmonic demodulation of the reference phase and (e) difference between phase maps shown in (b) and (d). Please see Visualization 1 and Visualization 2 for full dynamic sequence of uncorrected (b) and corrected (e) phase imaging, respectively.

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4. Conclusions

A simple, common-path, object-reference beam type interference microscopy system has been devised to study sparsely distributed phase objects, e.g., single cells with low density. The object beam total-shear approach with splitting the beam (after its passage through the object) into the object and object-free parts and their subsequent recombination has been implemented in the Fresnel diffraction field of a simple binary amplitude (Ronchi) grating. The beam-splitting grating is placed in a conventional, infinity corrected microscope optical system behind the tube lens, at a proper distance in front of the instrument image plane. In this way the required object-reference beam interferograms with carrier fringes are obtained. This solution is to be classified, therefore, as an off-axis configuration and is pixel less efficient in comparison with the on-axis configuration. However, given the size of camera pixels and the NA of the objective used, resolution loss in the off-axis and on-axis cases is negligible. The optics simplicity and straightforward carrier fringe interferogram processing make our solution worth of interest.

The Fresnel diffraction field deployed for interferometric imaging is generated by grating three lowest diffraction orders from a binary amplitude Ronchi grating. For sufficiently large distance between the grating and the microscope image plane higher grating orders walk-off the interferograms of interest. Simultaneously, three object replicas are appropriately separated in space. The diode operates with the driving current below the threshold one. At the same time the partially coherent illumination enables substantial reduction of coherent noise and parasitic diffraction/interference patterns.

Detailed analyses of two basic operation modes of our system are presented. They depend on the microscope input plane conditions and the grating to image plane separation distance. In the first mode the specimen-free regions on both specimen sides are of lateral extent equal to the specimen diameter (for this mode the field stop in the object plane is necessary). Three adjacent interferograms are generated in the microscope output plane. The central one is a 3-beam interference pattern and the two side ones located symmetrically with respect to the central one are two-beam interferograms. The latter ones carry the object phase information. Two-beam interferograms, however, are dependent on the wavelength.

Achromatic interferograms are generated in the second operation mode which requires two-times (or more) larger specimen surrounding regions as compared with the first mode. Now all three interferograms generated in the image plane are three-beam interference ones. Their second harmonics are generated by two-beam interferences with the zero path difference. They are achromatic and ideally suited for object information encoding under partially coherent illumination (with reduced temporal coherence). Since the formation of the interferogram achromatic second harmonic is independent of the distance $z$ between the beam-splitter grating and the recording plane we can eliminate the formation of the chromatic first harmonic in the image plane. Appropriate selection of the distance $z$ makes the OPD between the +/-1 and 0 grating diffraction orders longer than the laser diode coherence length (depending on the diode driving current). Another great advantage of the second operation mode is that there is no need to use the field stop in the microscope input plane. This feature enables very simple microscope laboratory assembly and studying non-stationary objects and dynamic processes.

In our studies we determine the object gross phase including the optical system errors, i.e., the so-called reference phase which is subtracted from the same regions of the FOV. This approach is essential for obtaining high measurement accuracy of the object phase by subtracting the two phase maps. It corresponds to non-numeric instrument calibration [17,18] and is superior to self-referencing solutions [14–16] exploiting specimen free areas adjacent to the specimen.

The technique of three-beam interferometry for precise measurements of very small phase differences introduced by plane parallel plates was initiated by Zernike in 1950 [30]. Subsequent three-beam interference studies dealing with wave front aberration testing and generation of microlens arrays, optical vortex lattices and edge dislocations are referenced in [23]. The present contribution deals with full-field studies of phase micro-objects and extends theoretical interpretation and applications of Zernike 3-beam interferometry.

Funding

Narodowe Centrum Nauki (2017/25/B/ST7/02049); National Agency for Academic Exchange; Politechnika Warszawska (Faculty of Mechatronics Dean's Grant).

Acknowledgments

We would like to thank Prof. Vicente Micó for fruitful discussions. We acknowledge the support of Faculty of Mechatronics Warsaw University of Technology statutory funds (Dean’s grant).

Disclosures

The authors declare no conflicts of interest.

References

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Supplementary Material (2)

NameDescription
Visualization 1       Fig. 8(b) - uncorrected phase imaging dynamic sequence.
Visualization 2       Fig. 8(e) - corrected phase imaging dynamic sequence.

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Figures (8)
Fig. 1.
Fig. 1. Schematic representation of the grating total-shear 3-beam interference microscopy: LD – laser diode; SMF – single mode fiber; CL – collimator objective; IP – input (sample) plane; MO – objective; TL – tube lens; G – diffraction grating; OP – output plane.

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Fig. 2.
Fig. 2. Schematic representation of interferences exploited in the grating total-shear first operation mode: (a) situation in the specimen (microscope input) plane IP with empty regions $R1$ and $R2$ and specimen region $S$ ; (b) three interference areas in the OP with indicated interfering grating diffraction orders; (c) formation of interferograms shown in the vertical cross-section plane passing through the system optical axis.

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Fig. 3.
Fig. 3. Schematic representation of the input plane IP (a) with empty regions $R1$ , $R3$ and $R2$ , $R4$ on specimen sides being two times larger (or more) than the specimen $S$ size; (b) and (c) show interference fields in the system output plane OP. Their intensity distributions are calculated in the paper. Formation of the central pattern generated by the beams $S^\prime _{0}$ , $R1^\prime _{-1}$ , $R2^\prime _{+1}$ is the same for the first and second operation modes. Interference patterns adjacent to it are now 3-beam interferograms. Specimen ( $S^\prime$ ) and reference ( $R1^\prime$ , $R2^\prime$ , $R3^\prime$ , $R4^\prime$ ) beams indicated in each interference region have subindices relating to the diffraction order number at the grating.

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Fig. 4.
Fig. 4. Three-beam interferogram ( $S^\prime _{-1}$ , $R2^\prime _0$ , $R4^\prime _{+1}$ , as described by Eq. (11)) for the (a) object-less case (sample substrate only present), and (b) its spectrum (intensity logarithmic scale), for determining the optical system reference phase distribution. The microscope system (OB plus TL, see Fig. 1) magnification is 20x. The period of the first harmonic (vertical carrier fringes) is equal to the grating period $d = 16.67\mu m$ ; this harmonic is absent in the recorded image (a) and the second harmonic diffraction orders are clearly seen in the spectrum (b).

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Fig. 5.
Fig. 5. Results of the second harmonic phase demodulation of the side interference pattern, $S^\prime _{-1}$ , $R2^\prime _0$ , $R4^\prime _{+1}$ (substrate only present); (a) 2D and (b) 3D phase distributions.

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Fig. 6.
Fig. 6. (a) Side three-beam interferogram $S^\prime _{-1}$ , $R2^\prime _0$ , $R4^\prime _{+1}$ , described by Eq. (11), and (b) its spectrum (intensity logarithmic scale). Experiment details – see the Caption of Fig. 3. The second harmonic diffraction orders are clearly separated in the spectrum.

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Fig. 7.
Fig. 7. Results of the second harmonic demodulation of the side interference image $S^\prime _{-1}$ , $R2^\prime _0$ , $R4^\prime _{+1}$ ; (a) 2D and (b) 3D phase distribution representations. Figures 7(c) and 7(d) show 2D and 3D representation of the difference between phase maps shown Fig. 7(b) (gross specimen phase) and Fig. 5(b) (system reference phase).

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Fig. 8.
Fig. 8. (a) Side interference image $S^\prime _{-1}$ , $R2^\prime _0$ , $R4^\prime _{+1}$ of the human cheek cell and (b) second harmonic demodulation of its phase; (c) side interferogram of the object-less case, (d) second harmonic demodulation of the reference phase and (e) difference between phase maps shown in (b) and (d). Please see Visualization 1 and Visualization 2 for full dynamic sequence of uncorrected (b) and corrected (e) phase imaging, respectively.

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Equations (11)

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E 0 ( x , y , z ) = a 0 exp ⁡ [ i k ⋅ g ( x , y ) ] + a + 1 exp ⁡ [ i k ( λ d x − λ 2 z 2 d 2 ) ] + a − 1 exp ⁡ [ i k ( − λ d x − λ 2 z 2 d 2 ) ] ;
I 0 ( x , y , z ) = a 0 2 + 2 a 1 2 + 2 a 0 a 1 c o s { k [ λ d x + g ( x , y ) + λ 2 z 2 d 2 ] } + + 2 a 0 a 1 c o s { k [ − λ d x + g ( x , y ) + λ 2 z 2 d 2 ] } + 2 a 1 2 c o s ( 4 π d x ) = = a 0 2 + 2 a 1 2 + 4 a 0 a 1 c o s [ k g ( x , y ) + π λ z d 2 ] c o s ( 2 π d x ) + 2 a 1 2 c o s [ 4 π d x ] ;
M c = c o s [ k g ( x , y ) + π λ z d 2 ] = c o s ( π λ z d 2 ) c o s [ k g ( x , y ) ] + s i n ( π λ z d 2 ) s i n [ k g ( x , y ) ] .
E + 1 ( x , y , z ) = a 0 + a + 1 e x p { i k [ λ d x + g ( x − Δ , y ) − λ 2 z 2 d 2 ] } ;
I + 1 ( x , y , z ) = a 0 2 + a 1 2 + 2 a 0 a 1 c o s { k [ λ d x + g ( x − Δ , y ) − λ 2 z 2 d 2 ] } .
E − 1 ( x , y , z ) = a 0 + a − 1 e x p { i k [ − λ d x + g ( x + Δ , y ) − λ 2 z 2 d 2 ] } .
I − 1 ( x , y , z ) = a 0 2 + a 1 2 + 2 a 0 a 1 c o s { k [ λ d x − g ( x + Δ , y ) + λ 2 z 2 d 2 ] } .
E + 1 ( x , y , z ) = a 0 + a + 1 e x p { i k [ λ d x + g ( x − Δ , y ) − λ 2 z 2 d 2 ] } + a − 1 e x p { i k [ − λ d x − λ 2 z 2 d 2 ] } ;
I + 1 ( x , y , z ) = a 0 2 + 2 a 1 2 + + 2 a 0 a 1 c o s { k [ λ d x + g ( x − Δ , y ) − λ 2 z 2 d 2 ] } + 2 a 0 a 1 c o s { k [ λ d x + λ 2 z 2 d 2 ] } + + 2 a 1 2 c o s { k [ 2 λ d x + g ( x − Δ , y ) ] } = a 0 2 + 2 a 1 2 + + 4 a 0 a 1 c o s { k 2 g ( x − Δ , y ) − π λ z d 2 } c o s [ 2 π d x + k 2 g ( x − Δ , y ) ] + + 2 a 1 2 c o s [ 4 π d x + k g ( x − Δ , y ) ] ;
E − 1 ( x , y , z ) = a 0 + a + 1 e x p { i k [ λ d x − λ 2 z 2 d 2 ] } + a − 1 e x p { i k [ − λ d x + g ( x + Δ , y ) − λ 2 z 2 d 2 ] } ;
I − 1 ( x , y , z ) = a 0 2 + 2 a 1 2 + 2 a 0 a 1 c o s { k [ λ d x − g ( x + Δ , y ) + λ 2 z 2 d 2 ] } + + 2 a 0 a 1 c o s { k [ λ d x − λ 2 z 2 d 2 ] } + 2 a 1 2 c o s { k [ 2 λ d x − g ( x + Δ , y ) ] } = a 0 2 + 2 a 1 2 + + 4 a 0 a 1 c o s { k 2 g ( x + Δ , y ) − π λ z d 2 } c o s [ 2 π d x − k 2 g ( x + Δ , y ) ] + 2 a 1 2 c o s [ 4 π d x − k g ( x + Δ , y ) ] .
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