## Abstract

A diffractive optical element (DOE) is presented to simultaneously manipulate the coherence plane tilt of a beam containing a plurality of discrete wavelengths. The DOE is inserted into the reference arm of an off-axis dual wavelength low coherence digital holographic microscope (DHM) to provide a coherence plane tilt so that interference with the object beam generates fringes over the full detector area. The DOE maintains the propagation direction of the reference beam and thus it can be inserted in-line in existing DHM set-ups. We demonstrate full field imaging in a reflection commercial DHM with two wavelengths, 685 nm and 794 nm, resulting in an unambiguous range of 2.494 micrometers.

© 2011 OSA

## 1. Introduction

In digital holographic microscopy [1–5], the long coherence length of laser light causes parasitic interferences due to multiple reflections in and by optical components in the optical path of the microscope and thus degrades the image quality [6]. The parasitic effects are greatly reduced by using a short coherence length light source such as a femtosecond laser or a broadband continuous wave superluminescent laser diode. In addition to reducing the parasitic effects, a short coherence length enables providing optical sectioning in the axial direction, which enables to exclusively sample the signal from the depth of interest [6].

The main drawback of using a short coherence light source in off-axis DHM [7, 8] is the reduction of the interference fringe contrast occurring in the field of view because the reference beam interferes with the object beam only in a diamond shaped area smaller than the field of view as Fig. 1 illustrates.

If two beams interfere with an angle $\alpha $, the interference occurs over a distance of

where $\lambda $and$\Delta \lambda $ are the center wavelength and the bandwidth of the source, respectively. For example, if $\lambda =700nm$, $\Delta \lambda =16nm$ and $\alpha =2\xb0$, the fringes occur over a distance of $L=877\mu m$, which is approximately 1/10th the lateral size of a typical detector.There have been several attempts to manipulate short coherence length light beams so that off-axis interference occurs over the whole physical overlap area. These techniques rely on manipulating the pulse front tilt of ultrafast pulsed sources [9], or equivalently the coherence plane tilt of continuous wave broadband sources [10]. In the remaining of this paper, we refer to coherence plane tilt, although the results also apply for pulses.

In [11], a diffractive beam-splitter and a pair of relaying lenses provide the coherence plane tilt required for full interference overlap. The concept was applied to digital holography [7] by combining grating and lens into one imaging diffractive lens placed after the interferometer. The imaging diffraction lens provides full field interference. A full field reflection off-axis DHM is presented in [12], that uses a transmission diffraction grating in the reference arm to provide full field interference overlap. In Refs [11, 12] full field imaging requires an imaging condition between DOE and the camera plane and occurs at a specific wavelength.

In this paper, we present a diffractive optical element placed in the reference arm of the DHM whose unique features are: 1) to maintain the propagation direction of the reference beam, 2) to provide the coherence plane tilt at several wavelengths simultaneously, 3) no imaging condition is required between DOE and camera image plane. The multi-wavelength capability is essential in a DHM to increase the unambiguous axial range. In a single wavelength reflective DHM, the axial unambiguous range is equal to half the wavelength in the medium i.e. $0.35\mu m$in air when using 685 nm light. By using a DHM with two distinct wavelengths${\lambda}_{1}$and${\lambda}_{2}$, the effective synthetic wavelength is equal to ${\lambda}_{1}{\lambda}_{2}/\left|{\lambda}_{1}-{\lambda}_{2}\right|$. For example, if ${\lambda}_{1}=685nm$ and${\lambda}_{2}=794nm$, the synthetic wavelength is $4.9898\mu m$ which results in an unambiguous range of $2.494\mu m$. In this paper, we experimentally demonstrate single-shot full field imaging and dual wavelength operation with two low coherence sources (685nm (FWHM = 8.3 nm) and 794nm (FWHM = 16.7 nm)) in a commercial reflection DHM, by inserting the developed DOE in the reference arm without any modification/adjustment of the instrument.

The proposed DOE is based on transmission volume phase gratings recorded holographically into a thin ($17\mu m$) photopolymer. In section 2, we present two DOE designs for single wavelength operation. In the first design, the holographic grating is laminated on the back of a wedge prism and works in the Raman-Nath regime (multiple diffraction orders). In the second design, both facets of the wedge prism are laminated with a holographic grating. Although the second design is more complex to realize, it provides the advantage of higher effective efficiency than the first design because both gratings operate in the Bragg regime. In section 3, we extend the single wavelength operation of the dual grating design to dual wavelength by adding multiplexing. Experimental results in a DHM are presented in section 4 and show unambiguous depth recovery up to $2.494\mu m$ over the full field of view.

## 2. Single wavelength DOE

#### 2. 1. Background: coherence plane tilt

It is well known that any angularly dispersive system introduces a coherence plane tilt [13–16]. Bor introduced a general relation between the tilt angle and the angular dispersion, which is device-independent [14, 15]:

Where $\lambda $ is the wavelength, $d\delta /d\lambda $ is the angular dispersion, and ${\gamma}_{t}$is the angle between the coherence plane and the phase front. It is intuitively simpler to visualize the delay experienced by a short pulse in the time domain. Figure 2 illustrates graphically the tilt angle between the phase fronts (propagation direction) and the pulse front in an angularly dispersive system.Prisms and diffraction gratings are dispersive elements that provide a convenient mean for producing a coherence plane tilt. We propose to use the combination of prisms and volume phase gratings fabricated in a holographic polymer to engineer the functional coherence plane tilt device.

#### 2.2 DOE in Raman-Nath regime

The structure of the device is shown in Fig. 3(a) . The wedge prism refracts the incident beam and the volume phase grating diffracts the beam so as to orient the first order diffraction to propagate in the same direction as the incident beam. Figure 3(b) illustrates the parameters of the device. The following analysis computes the phase grating parameters i.e. grating period and slant angle to obtain a specific plane tilt angle for a beam output collinear with the input beam (Fig. 3(a)).

From Fig. 3, the deviation of the output beam direction from the input direction is given by:

where ε is the wedge angle and ${\theta}_{0}$the angle of incidence. The incident beam direction is fixed and independent of wavelength. Derivation of Eq. (3) with respect to wavelength λ gives the angular dispersion of the device:At each interface, refraction occurs according to Snell’s law:

Inside the volume phase grating, the vectorial grating equation is ${K}_{i}+{K}_{G}={K}_{d}$, which yields the following two relations:

In Eq. (8) and (9), $\Lambda $ and $\varphi $ are the grating period and slant angle, respectively. Taking the derivative of Eq. (8) with respect to wavelength, and using Eq. (2), (4) provides the following relation:

The system of Eqs. (2-9) and the collinearity condition between input and output provide 8 equations for 9 unknowns (${\theta}_{0},{\theta}_{0}{}^{\prime},{\theta}_{2},{\theta}_{4}{}^{\prime},{\theta}_{4},\epsilon ,\delta ,\Lambda ,\varphi $). To remove the underdetermination, the incident angle ${\theta}_{0}$is set so that the input beam is at normal incidence to the backfacet of the wedge. It implies that ${\theta}_{0}$ = ε. This choice forces the output angle ${\theta}_{4}=0$ in order to obtain collinear input and output (the output being in the direction of the first order diffraction). For a coherence plane tilt ${\gamma}_{t}$ of 4 degrees and using a wavelength of $\lambda =685nm$and a wedge angle of $\epsilon =7.41\xb0$with dispersion $d{n}_{w}/d\lambda =-2.8022\times {10}^{4}$, the grating period is calculated by summing the square of Eq. (8) and (9) which yields $\Lambda =10.284\mu m$. The slant angle $\varphi =1.2778\xb0$ is obtained from Eq. (10).

The unitless factor $\rho ={\lambda}^{2}/\Delta n\cdot n\cdot {\left(\Lambda /\mathrm{cos}\varphi \right)}^{2}$ is smaller than unity for the parameters above. This indicates that the grating is operating in the Raman-Nath regime [17]. In this regime, there are multiple diffraction orders. The diffraction efficiency in the first order is a Bessel function of the first kind [18],

where $d$ is the grating thickness and $\theta $ is the incident angle. According to Eq. (11), the maximum obtainable diffraction efficiency in the first order is 33.85%.#### 2.3 Holographic phase grating fabrication

The phase grating was recorded on a BAYFOL^{®} HX photopolymer from Bayer MaterialScience AG which was laminated prior to recording on a round BK7 optical wedge. A continuous-wave, single frequency green laser (532 nm Coherent Compass 315M) is coupled to a single mode polarization maintaining fiber. The output of the fiber is collimated and split by a beam splitter to generate a plane wave reference and signal beams with intensity respectively $2mW/c{m}^{2}$ and $1.7mW/c{m}^{2}$. The reference and signal beams interfere on the photopolymer, thereby recording the phase grating. The angle between the reference and signal beam and the slant angle are controlled by rotation stages with 0.2 milli-degree accuracy (Newport Micro-controle stepper motor URS150BPP).

Figure 4 shows a picture of the fabricated phase grating laminated on the wedge prism. The diffraction orders are shown in this figure. The measured efficiency of the first order diffraction (the output of the DOE) is 5%. This value was not optimized further, although higher efficiency close to the maximum is achievable in the photopolymer given that its maximum refractive index change $\Delta n$ is of the order of 2%.

The DOE device was inserted in the reference arm of the DHM R2101(LynceeTec). No modification or alignment was necessary except a low pass filter (hole in a plate) was used to filter out the unwanted orders.

The optical set-up is a DHM R2101, shown in Fig. 5 . Two low coherence sources of different wavelengths are combined and split into two beams by a first beam splitter. One beam (reference beam) is split again in its two monochromatic components (reference beam 1 and 2). The other one illuminates the object through a microscope objective. It is reflected from the sample and recombined with the two reference beams on the camera. Two delay lines equalize the optical path difference between the two references and the object paths. The object and reference beams interfere in off-axis geometry, meaning that the object beam is normal to the camera, whereas an angle is introduced for the reference beams. The planes defined by the object beam and reference 1 is orthogonal to the one defined by the object beam and reference 2. This geometry enables the filtering of the frequencies for each wavelength separately and allows real-time measurements. The hologram records the interference at both wavelengths simultaneously.

For this first part, the DHM is used with a single wavelength. The inline DOE is inserted in the 685nm wavelength reference arm only. It generates a 4 degrees tilt angle in the coherence plane of the reference beam. The inset of Fig. 5 illustrates schematically, for one wavelength, that the tilted coherence plane created by the DOE results in interference throughout the CCD plane.

Figure 6(a) left panel shows the interferogram without the DOE in the reference arm. Figure 6(a) right panel shows the corresponding fringe modulation depth as a function of coordinate along the white line shown in left panel. Figure 6(b) shows the interferogram and the corresponding fringe modulation with the DOE incorporated in the reference beam path. Figure 6(b) clearly demonstrates the increase in the usable field of view. The fringe spacing is measured for both graphs and yielded $20.45\mu m\pm 2\mu m$, which demonstrates that the insertion of the DOE does not alter the propagation direction of the reference beam.

A drawback of this DOE design is the low efficiency in the first order diffraction (max. 33.85%). The spectral bandwidth of the grating at full width half maximum can be calculated according to the approximate relation: $\frac{\Delta \lambda}{\lambda}=\frac{\Lambda}{d}\cdot \mathrm{cot}\theta \cdot \mathrm{cos}\varphi $ [19]. With the parameters of the fabricated DOE, the spectral bandwidth is equal to 440 nm. This DOE design can be extended to operate in dual wavelength by multiplexing gratings. However it is not practical because the wavelength separation of a dual wavelength DHM is smaller than the bandwidth of the DOE which would lead to a further decrease of the diffraction efficiency due to cross-talk.

In the next section, we propose a DOE concept that provides high efficiency (up to 100%) and provides control of the spectral bandwidth for dual wavelength operation in a DHM.

#### 2.4 DOE in Bragg regime

A phase grating operating in the Bragg regime has the advantage of producing a single diffraction order which can reach 100% efficiency. The parameter $\rho $introduced in section 2.2 needs to be greater than unity in order to obtain the Bragg regime. For example, if the ratio between the wavelength and an unslanted ($\varphi =0$) grating period is equal to unity, the parameter $\rho $ is 66 for a photopolymer with a refractive index change of 1% and index of refraction $n=1.5$. In this case, the Bragg condition is well fulfilled. However, the coherence plane tilt generated by a phase grating in the Bragg regime is, according the Eq. (2) given by $\mathrm{tan}{\gamma}_{t}=\frac{\lambda \mathrm{cos}\varphi}{\Lambda \mathrm{cos}\theta}\sim 1$, which gives a tilt angle much larger than the 4 degrees required in the DHM. By introducing a second phase grating in series that provides a negative tilt angle of similar magnitude, the difference between the two tilt angles can generate any arbitrary tilt angle while maintaining operation in the Bragg regime.

The structure of the device operating the Bragg regime is shown in Fig. 7 .

Similar to the analysis in section 2.1, the slant angle and grating period for each grating are determined by solving a set of two grating equations, Snell’s law at four interfaces, the relation between tilt angle and the device dispersion as well as a condition to maintain collinearity between input and output. Details of the calculation are given in the appendix. There are a total of 11 equations and 13 variables.

After taking the derivative of the tilt angle with respect to wavelength, we obtain the following relation:

Figure 8 shows a picture of the fabricated two phase gratings laminated on each side of the wedge prism. The Bragg diffraction order is shown in the figure.

## 3. Dual wavelength DOE

By multiplexing gratings in each photopolymer laminated on both sides of the wedge, the tilt angle can be tailored at several distinct wavelengths.

The two wavelength of the DHM are${\lambda}_{1}=685nm$and${\lambda}_{2}=794nm$. By repeating the procedure outlined in section 2.4, the period and slant angle for both wavelengths are determined. For the second wavelength$\lambda =794nm$, we obtain ${\Lambda}_{1}=1.2\mu m$, ${\varphi}_{1}=7.8361\xb0$for the top grating and${\Lambda}_{2}=1.123\mu m$, ${\varphi}_{2}=-13.6772\xb0$ for the bottom grating. The measured diffraction efficiency of the fabricated DOE resulted in 68.18% efficiency at 685nm and 30.77% at 794nm. The recording wavelength and set-up is the same as described in section 2.3.

Figure 9(a) and Fig. 9(b) show the simulated diffraction efficiency versus wavelength for the top and bottom phase gratings respectively. The simulated and measured diffraction efficiency of the combined top and bottom phase gratings for each wavelength is shown in Fig. 9(c).

At the output of the DOE, there are in total 8 parasitic beams resulting from zero orders and cross-talk diffraction (e.g. top grating period for ${\lambda}_{1}$ diffracting ${\lambda}_{2}$). Figure 10 illustrates the direction of the parasitic beams. In practice, in a commercial DHM it is difficult to spatially filter beams angularly separated by less than 1 degree. Out of the 8 parasitic beams, beam #7 in Fig. 10 makes an angle of 0.97 degrees with the desired output beam. This beam cannot be filtered in the experiment. Beam #4 makes an angle of 2 degrees with the desired output beam and then can be spatially filtered out. The other parasitic beams can also be easily spatially filtered. Figure 11 shows a plot of the diffraction efficiency ratio between the unfiltered cross talk beam #7 and the desired output beam. The effect of the unfiltered parasitic beam on the phase measurement and beam walk-off has not been quantified in this work. Quantitative phase measurement sensitivity due to the cross-talk beam will be performed in a separate study.

The output beam of the DOE propagates in the same direction as the input beam, albeit with a lateral shift, so-called walk-off beam. Beam walk-off is inversely related to the bandwidth. As Fig. 12 illustrates, the walk-off increases quickly below 110 nm bandwidth for the material thickness used in the experiment ($17\mu m$). The measured experimental walk-off of the fabricated DOE is 2.4 mm, which provides a bandwidth of 113 nm in agreement with the theoretical and experimental bandwidth in Fig. 9(c). With thicker photopolymers (e.g. $50\mu m$), a lower bandwidth is achievable and thus a lower cross-talk. From Fig. 11, a bandwidth of 113 nm yields less than 2% of the power in the cross-talk beam with respect to the power in the desired output beam of the DOE.

## 4. Experiments on DHM

In this second experiment, the DHM is used with both wavelengths simultaneously (685nm and 794nm). The DOE with multiplexed gratings (685nm and 794nm) operating in the Bragg regime was placed in the 794nm reference arm, and a single wavelength DOE (685nm) was inserted into the 685nm reference arm. The object used in the experiment is a reflective staircase with heights of 10; 100; 1000; 2000 and 4000 nm.

Holograms are taken in one shot (i.e. simultaneously) with both wavelengths turned on. The orientations of the coherence plane tilt in each arm are orthogonal to each to match the off-axis geometry. Since the multiplexed DOE is placed in the 794 nm arm, cross-talk effects with the 685 nm phase grating are present.

Figure 13 shows the intensity and phase profiles for both wavelengths without the DOEs. Figure 14 shows the intensity and phase for both wavelengths with the DOEs inserted in the reference arms of the DHM, after subtracting the reference intensity (extracted from the reference hologram taken with a flat mirror) and numerically filtering out the parasitic interference pattern in the Fourier plane. Comparing Fig. 13 and Fig. 14, it is evident that the field of view is severely limited without the DOEs.

From the intensity and phase images at 685 nm and 794 nm in Fig. 14, a height map can be calculated. Figure 15 shows the resulting 3D image.

Stair case heights of 10 nm; 100 nm; 1000 nm and 2000 nm can be unambiguously determined. However, the 4000 nm staircase height cannot be unambiguously determined since the maximum unambiguous range is $2.494\mu m$. The staircase image is clearly shown in its entirety demonstrating full field imaging with low coherence illumination and increased axial range up to $2.494\mu m$ due to dual wavelength. Although, this image is taken in single shot, the DOE did not require dual wavelength operation because of the availability of two reference arms in the DHM. However, this experiment validates the concept of multiple wavelength operation in a single DOE. The cross-talk effects are present but manageable (digital fourier filtering) and can be further reduced by using thicker holographic substrates (e.g. $50\mu m$vs $17\mu m$), which would decrease the bandwidth and thus the cross-talk.

## 5. Conclusion

We have designed and experimentally demonstrated a DOE based on a combination of holographic phase gratings for introducing a desired coherence plane tilt, with high throughput transmission (68%) while maintaining the original propagation direction. This DOE enables full field off-axis digital holographic microscopy with a short coherence illumination. In addition to the intrinsic benefits of short coherence illumination which are optical sectioning and reduced parasitic interferences, the demonstrated DOE enables multi-wavelength operation while maintaining full-field imaging.

We have shown single-shot full-field imaging with two low coherence sources (685nm and 794nm) in a digital holographic microscope. The dual wavelength provides an axial (sample depth) unambiguous measuring range of $2.494\mu m$. To our knowledge, it is the first time that full field imaging with a dual low coherence wavelength source has been demonstrated.

## Appendix

In this appendix we provide the formulas and detailed derivation of Eq. (12).

Snell’s law at four interfaces in Fig. 7 requires that

The phase grating equation, for the top grating is:

And for the bottom phase grating:

After taking derivative of Eq. (A8) with respect to wavelength and substituting (A1) and (A6), we obtain:

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