Self-interference fluorescence microscopy: three dimensional fluorescence imaging without depth scanning

Mattijs de Groot; Conor L. Evans; Johannes F. de Boer

doi:10.1364/OE.20.015253

1. Introduction

Fluorescence microscopy has revolutionized biology and biomedicine during the past decades through the ability to specifically label and image cell structures or proteins of interest. With the development of clinically approved near-infrared fluorescent probes conjugated to therapeutic or diagnostic agents, a similar development is expected in the coming years in clinical medicine [1–5]. To bring these translational advances in fluorescence labeling to clinical practice, however, requires a new class of miniature endoscopic imaging systems that will enable clinicians to rapidly visualize the three-dimensional distribution of fluorescence labels in situ.

Self-interference fluorescence microscopy (SIFM) extends the capabilities of confocal microscopy by providing depth sensitivity much better than the axial spot size. Fluorescence self-interference has been explored before [6–10], but these approaches were based on a transmission configuration and therefore could not easily be incorporated into conventional or miniature imaging systems. OurSIFM method addresses this challenge by collecting the fluorescence in the backward direction and phase-encoding the depth information through the use of a phase plate (e.g., a glass plate with an opening) placed into the beam path. This plate presents the photons emitted by a fluorophore with two alternative paths that then interfere upon coupling into a single mode fiber that acts as a pinhole. The path length difference between the two paths gives rise to depth-dependent interference as a function of wavelength, where moving the fluorescent source axially through the beam waist causes the wavelength-dependent interference pattern to undergo a phase shift. The lateral resolution in SIFM is determined by both the beam waist and the fiber core aperture size, similar to confocal microscopy.

2. Theory

2.1 Principle of SIFM

Figure 1 illustrates the principle of self-interference fluorescence microscopy. The SIFM spectrum as a function of wave number k is given by:

I_{sifm} (k) = C \cdot I_{source} (k) (1 - \frac{A}{2} + \frac{A}{2} \cos (k (d + Δ_{opl} (δ))))

where

I_{SIFM} (k)

is the intensity detected at wave number k, C is the total collection efficiency of the confocal imaging system, and

I_{source} (k)

is the intensity emitted by the source at wave number k. A is the modulation depth of the self-interference, which varies between 0 and 1. It is based on the ratio of the field amplitudes integrated over the inner and outer sections of the phase plate. d is the optical thickness of the phase plate. Finally,

Δ_{opl} (δ)

is the average extra path length difference (on the order of a fraction of the wavelength) that is induced by the wavefront curvature between the photons passing through the inner and outer sections of the phase plate (see Fig. 1).

Δ_{opl} (δ)

is a function of the defocus distance δ, the distance between the source and the focal plane of the objective. A negative value forδ indicates that the source is further away from the objective than the focal plane, whereas a positive value indicates that that the source is closer to the objective. When the phase plate radius is optimal (

A = 1

), on average only half of the photons get accepted by the core of the single mode fiber because of constructive and destructive interference (the other half couples into the cladding). The cosine term in Eq. (1) describes the modulation, indicating that depth information is carried by half of the collected photons. In the shot noise limit the signal to noise ratio (SNR) of the interferometric signal is therefore given by

\frac{1}{2} N / \sqrt{N} = \frac{1}{2} \sqrt{N}

, where N is the total number of collected photons. Experimentally, the SNR is determined by the ratio of the length of the complex Fourier component at the modulation frequency

k d / 2 π

and the background level in the vicinity of the modulation frequency. The phase is determined by the arctangent of the imaginary over the real part of the Fourier component at the modulation frequency. This phase encodes a single depth position. In the presence of multiple fluorescent sources at different depths, the intensity weighted mean depth position is calculated instead. The minimum detectable phase difference and therefore the depth sensitivity of SIFM is fundamentally limited by the phase noise

σ_{ϕ}

, which is a function of the signal to noise ratio via_{$σ_{ϕ} = 1 / \sqrt{2} S N R$} [11].SIFM will detect the intensity weighted average depth location when multiple fluorescent sources are present at different depths within the depth-of-field. The technique therefore works best for relatively sparse samples.

Fig. 1 Principle of SIFM.Light from a fluorescent source in the sample is collimated by the objective lens and passes through a phase plate. The phase plate consists of an inner and an outer ring of different optical thickness, introducing two alternative optical paths. The wave front before and after the phase plate is drawn as a solid red line. The light passing through the thicker outer ring of the phase plate is retarded with respect to the light passing through the hole in the center. The light is focused on a single mode optical fiber, acting as a pinhole. The interference due to the optical path difference leads to a modulation on the detected fluorescence spectrum: for certain wavelengths the interference is destructive while for others it is constructive (top graph). The period of the spectral modulation is determined by the thickness of the plate. The modulation depth depends on the ratio of the integrated amplitude of the source field over the central disk of the phase plate and the outer ring, respectively. When the source is exactly in focus (δ = 0 µm) the wavefronts are flat. When the source is out of focus (δ = 100 µm) the wavefronts are curved (shown exaggerated). This leads to a small extra path length difference $Δ_{opl} (δ)$ and to a shift in the phase of the self-interference spectrum (bottom graph). The phase therefore directly encodes the axial position of the fluorescent source.

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2.2 Ray optics model

We describe the phase response of the self-interference spectrum as a function of defocus with the following numerical model. This simple model gives an intuitive, geometrical description for the phase shift as a function of depth and agrees very well with the experimental data up to a defocus of twice the Rayleigh length Z_R.

Figure 2 displays the optical layout that is the basis for the ray optics calculations of the phase of the self interference as a function of defocus d. The source of the fluorescence is located a distance δ from the focal plane of the objective L₀ with a focal length f₀. A telescope consisting of two lenses L₁ and L₂ with focal lengths f₁ and f₂ respectively magnifies the beam by a factor m = − f₂/ f₁. The telescope images the back focal plane of the objective onto the phase plate PP. The phase plate consists of an inner disk (or hole) with a diameter 2r and an outer ring (annulus) that has a different optical thickness. For the purpose of this model we neglect the refraction in the phase plate and consider it as an element that adds a fixed optical delay to all rays passing through the outer ring.

Fig. 2 Schematic representation of the ray matrix model. The rays v_inemitted by a source at a defocus distance δ from the focal plane of the objective L₀ are traced by multiplication with the free space propagation matrices S_i and the lens matrices L_i. From the position v_out and the angle θ_out we can calculate the radius of curvature R of the wave front at the phase plate. This allows us to calculate the optical path delay opd(y,δ) which we define as the difference in optical path to the phase plate between a ray intersecting the phase plate at y and the chief ray for which y = 0. Averaging of opd(y,δ) over the inner and outer sections of the phase plate now yields Δ_opl(δ)the extra path length difference between the field passing through the center and the field passing through the edge of the plate that is caused by the wave front curvature (see Fig. 1).

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The propagation of a ray through this optical system from the source to the phase plate can be described by the following ray transfer matrices:

S_{i} = (\begin{matrix} 1 & d_{i} \\ 0 & 1 \end{matrix}), L_{i} = (\begin{matrix} 1 & 0 \\ - \frac{1}{f_{i}} & 1 \end{matrix})

and the ray vector

v_{i} = (\begin{matrix} y_{i} \\ θ_{i} \end{matrix}) .

Here

S_{i}

is the i’th free space propagation matrix with

d_{i}

the propagation distance.

L_{i}

is the thin lens matrix with

f_{i}

the focal length.

v_{i}

is a ray with

y_{i}

the position and

θ_{i}

the angle of the ray. We set

d_{0} = f_{0} - δ

,

d_{1} = f_{0}

,

d_{2} = d_{3} = f_{1}

,

d_{4} = d_{5} = f_{2}

, (i.e. all optical elements are telecentrically aligned and δ describes the distance of the source from the focal plane of the objective). We need to find the average optical path difference introduced by the defocus δ between the sections of the beam that pass through the inner and outer rings of the phase plate respectively. The full ray propagation is described by the following equation:

v_{o u t} = S_{5} L_{2} S_{4} S_{3} L_{1} S_{2} S_{1} L_{0} S_{0} v_{i n} = (\begin{matrix} 0 & - \frac{f_{0} f_{2}}{f_{1}} \\ \frac{f_{1}}{f_{0} f_{2}} & \frac{f_{1} δ}{f_{0} f_{2}} \end{matrix}) v_{i n} .

The telescope

S_{5} L_{2} S_{4} S_{3} L_{1} S_{2}

can be described as a magnification

M = (\begin{matrix} m & 0 \\ 0 & \frac{1}{m} \end{matrix})

where

m = - \frac{f_{2}}{f_{1}}

, simplifying Eq. (4) to

v_{o u t} = M S_{1} L_{0} S_{0} v_{i n} = (\begin{matrix} 0 & m f_{0} \\ - \frac{1}{m f_{0}} & - \frac{δ}{m f_{0}} \end{matrix}) v_{i n} = (\begin{matrix} m f_{0} θ_{i n} \\ - \frac{y_{i n} + δ θ_{i n}}{m f_{0}} \end{matrix}) .

When

v_{i n} = (\begin{matrix} 0 \\ θ_{i n} \end{matrix})

, i.e. when a ray comes from the lateral focus,

v_{o u t}

becomes

v_{o u t} = (\begin{matrix} m f_{0} θ_{i n} \\ - \frac{δ θ_{i n}}{m f_{0}} \end{matrix}) .

From Fig. 2 we can see that in the paraxial approximation, i.e. $\sin (θ) = θ$ ,

R (δ) = \frac{m f θ_{i n}}{- \frac{δ θ_{i n}}{m f}} = - \frac{m^{2} f^{2}}{δ} .

Here R(δ) is the radius of curvature of the wave front at the phase plate. The optical path from the emitter to the wave front is equal for all rays. f, now dropping the index, is the focal length of the objective. In order to calculate the phase difference between the fields that pass through the inner and outer sections of the phase plate we first calculate the optical path delay as a function of the positiony along the phase plate. We define the optical path delay opd(y,δ) as the difference between the optical path lengths to the phase plate for a ray intersecting the phase plate at y and for the chief ray for which y = 0:

o p d (y, δ) = \sqrt{R {(δ)}^{2} + y^{2}} - R (δ) .

For large R(δ), i.e. for small δ,

o p d (y, δ) \approx \frac{1}{2} \frac{y^{2}}{R} = \frac{1}{2} \frac{y^{2} δ}{m^{2} f^{2}} .

The average optical path delay over a section of the phase plate from r = a to r = b, where r is the radius of a disk on the phase plate, is then given by the 2D integral

o p d_{a v g 2 D} (a, b, δ) = \frac{\int_{0}^{2 π} \int_{a}^{b} o p d (r; δ) e^{- \frac{r^{2}}{w^{2}}} r d r d ϕ}{\int_{0}^{2 π} \int_{a}^{b} e^{- \frac{r^{2}}{w^{2}}} r d r d ϕ},

wherew is the width of the Gaussian profile of the scaled fiber mode field at the phase plate. This weighting of the average with the Gaussian mode field is necessary because the single mode fiber will accept only the embedded Gaussian beam that matches the fiber mode field.The phase difference (in radians) of the interference spectra obtained with source locations δ = 0 and δ = z for a set-up with a phase plate of radius ρ is now given by:

Δ ϕ = \frac{2 π}{λ} (o p d_{a v g 2 D} (ρ, \infty, δ) - o p d_{a v g 2 D} (0, ρ, δ))

whereλ is the wavelength of the light. Using Eq. (12) and substituting

z_{R} = λ f^{2} / π w^{2}

for the Rayleigh length

z_{R}

we find

Δ ϕ / z_{R} = 1.33 rad / z_{R}

. Under these conditions, phase wrapping occurs only at ± 2.4 Z_R, where the confocal system rejects virtually all light; therefore, phase wrappingdoes not cause depth ambiguity.

3. Self-interference fluorescence microscope

To demonstrate the capabilities of this new imaging technique, a commercial microscope platform (Olympus IX71) was equipped with a home-built galvanometric x/y-scanner. A fiber coupled 635 nm Fabry-Perot diode laser (Thorlabs, S1FC635) was used as an excitation source (Fig. 3 ).All measurements reported were accomplished with a power of 650 µW at the sample. A dichroic mirror based wavelength division multiplexer (OZ Optics) was used to separate excitation and emission light. A home-built spectrometer based on a holographic transmission grating (1200 lpmm, Wasatch Photonics) and a CMOS line scan camera (spL4096-140km, Basler) was used to detect the fluorescence spectra. All measurements were performed at a line rate of 1 kHz (1000 spectra/s).

Fig. 3 Schematic representation of the SIFM microscope. Excitation light from a fiber coupled laser is sent through a wavelength division multiplexer (wdm) and a collimator (col) before passing through a phase plate (PP). The beam is sent into the microscope (IX71) via a home-built galvanometric x/y-scanner (xy) and is focused onto the sample by the objective (obj). The fluorescence is collected by the same objective and follows the same light path in the opposite direction to the wdm where it is sent to a home-built spectrometer that detects the interference spectra.

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The limited sensitivity of the spectrometer camera used currently restricts the imaging speed because sufficient photons need to be collected for the signal to dominate the read noise of the camera. One way to improve this is to use a phase plate with a much smaller optical path difference. With an optical path difference that results in only one modulation period over the spectral bandwidth of the fluorophore the phase of the modulation can be detected with only three detector elements. If one uses highly sensitive photomultiplier tubes or avalanche photodiodes the sensitivity, and therefore the imaging speed, of SIFM is comparable to standard confocal microscopy.

The phase plate consisted of a standard 1 mm microscope slide with a 4 mm diameter hole. This was placed into the collimated beam in front of the x/y-scanner in the virtual back focal plane of the objective. The Gaussian beam waist (radius of 1/e² intensity drop-off) at the phase plate was 2.59 mm. This value was calculated based on the fiber numerical aperture reported by the manufacturer and the focal length of the collimator. The 4 mm diameter hole for the phase plate was close to the optimal value of 4.14 mm for the 2.59 mm Gaussian beam waist. The optical path delay introduced by the 1 mm phase plate with a refractive index of 1.5 was 500 µm. This led to a modulation period of $2 π / 5 \cdot 10^{- 4} = 12560 m^{- 1}$ according to Eq. (1). At the center wavelength of the fluorescence (755 nm) this corresponded to a period of 1.1 nm⁻¹ which was easily resolved by the current spectrometer. An interpolation of the spectra was performed to linearize the data in k-space before the Fourier transformation.

4. Results and discussion

4.1 Depth sensitivity

Figure 4(a) shows the axial position dependence of the phase of the self-interference spectrum.A thin layer of fluorescent polymer beads was prepared by filling a 50 µl µ-Slide (Ibidi) with fluorescent polymer beads (fluospheres 715/755, Invitrogen, 100 nm, 0.2% by weight) and drying slowly to the air. This thin homogeneous fluorescent layer was moved through the focus of the SIFM microscope. We used a 4x objective (UPLSAPO 4X, Olympus) with a nominal NA of 0.16. The objective was underfilled by a Gaussian beam with a waist of 1.94 mm leading to an effective NA of 0.086. For each datapoint, 1024 fluorescence spectra were recorded at the same lateral position with an integration time of 1 ms per spectrum.

Fig. 4 Phase and SNR. (a) Dependence of the phase of the self-interference spectrum on the axial position of a thin homogenous fluorescent layer with respect to the focal plane of the objective. Each datapoint is the mean value from 1024 spectra. The error bars indicate ± 1 standard deviation. The imaging NA for this measurement was 0.086 (using the approximate definition for the NA of Gaussian beams:NA ≈2λ / πw₀) and the Rayleigh lengthz_Rwas 128 µm. A linear fit of the central section between −2 and 2z_Ryields a slope dϕ/dz = −1.31 rad/z_R. A simple ray optics analysis of the system predicts a slope of −1.33 rad/z_Rindicated by the solid gray line. (b) Dependence of the standard deviation of a phase measurement on the signal to noise ratio. Here the SNR was varied by scanning the fluorescent layer through the focus of the objective. The dots are the measured standard deviations of 1024 phase measurements. The solid line is the theoretical curve $σ_{ϕ} = 1 / \sqrt{2} \cdot S N R$ .

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From these spectra, the mean and standard deviation were calculated of the total intensity and the phase of the fringes. The central section of Fig. 4(a), from −250 to 250 μm, clearly shows a linear dependence of the phase on the depth position z, with a slope of .Using the ray transfer matrix model described above the dependence of the phase on the axial position of the source was calculated. The predicted slope was $1.33 rad/ z_{R}$ with $z_{R}$ the Rayleigh length (128 μm) of the optical system, based on the numerical aperture ( $N A = 0.086$ ), the phase plate geometry, and the mode field diameter as reported by the fiber manufacturer. The solid gray line in Fig. 4(a) indicates the slope of the phase predicted by our theoretical model of the optical system, which is in excellent agreement with the measured data. The smallest and largest standard deviation measured in the linear regime is 6.7 and 39 mrad, with corresponding SNRs of 114 and 30, respectively. These standard deviations were within 1.1 to 1.7 times the theoretically predicted standard deviations based on the SNR. The smallest and largest standard deviations using this SIFM system correspond to a depth sensitivity of 0.7 to 4 μm, respectively, compared to the large (500 µm) depth of field. Here, we define thedepth of field as four times the Rayleigh length, which is determined by the numerical aperture.The depth-of-field scales with $1 / N A^{2}$ while the lateral resolution scales with $1 / N A$ , similar to standard confocal microscopy. In order to achieve a large depth-of-field one needs to use a low NA and therefore sacrifice some lateral resolution.

As the depth position accuracy is defined by the system SNR and objective NA, far higher localization accuracy can be obtained by either simply collecting more photons or by choosing a higher NA objective. Figure 4(b) shows the excellent agreement between measured $σ_{ϕ}$ as a function of the SNR and theory. The solid gray line is the theoretical curve $σ_{ϕ} = 1 / (\sqrt{2} \cdot S N R)$ . In the shot noise limit, with only 100 collected photons per spectrum (SNR = 5), the phase noise $σ_{ϕ}$ is predicted to be 0.14.Thistranslates to a depth position standard deviation of 13 µm when using a low NA of 0.09. If one instead uses a high 1.2 NA water immersion objective, this depth position error shrinks to an accuracy of just 73 nm. It is worth noting that these resolutions can be achieved even when imaging in the near infrared.

4.2 Three dimensional imaging

To illustrate the 3D imaging capabilities of SIFM and to validate the depth information collected in a single 2D SIFM scan, a thick sample was imaged using both SIFM and standard confocal microscopy.

400 mg agarose was dissolved in 20 ml filtered water (milli-Q) by boiling. This hot solution was quickly mixed with 50 µl bead solution (1 µm fluospheres 715/755, Invitrogen). The warm solution was quickly pipetted into a 150 µm high microfluidic channel made with a microscope slide, a coverslip and double sided tape. The channel was sealed with a 1:1:1 mixture of vaseline, lanoline and paraffin.

For SIFM, the focus was placed 75 µm into the gel using a 20x objective (Olympus UPlanSApo 20X, underfilled to an effective NA of 0.43)and scanned over a single 125 × 125 μm lateral plane The resulting intensity image, Fig. 5(a) , clearly shows the beads distributed in the gel matrix. In the corresponding phase image, Fig. 5(b), each pixel is colored according to its own unique phase. From these phase values, the depth of each emission source was assigned to a corresponding voxel in a 125 × 125 × 30 μm 3D volume.

Fig. 5 Comparison of SIFM and confocal microscopy on a three-dimensional distribution of fluorescent microspheres. (a) SIFM intensity image (100 × 100 µm). The red square marks the area that was compared to a standard confocal stack. (b) SIFM phase image. (c) Representative slice from the SIFM 3D reconstruction. (d) Corresponding slice from the confocal stack. (e) SIFM and confocal slices overlapped. SIFM data displayed in green and confocal in red. Yellow indicates the overlap of both data sets. A video that shows the overlap of the data sets for the whole volume is available online (Media1).

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After the SIFM volume was collected, the 20x objective was replaced by a high-NA 60x objective (Olympus, UPlanSApo 60XW) mounted on a piezoelectric z-scanner (PI, P-725.2CD). The phase plate was then removed from the setup so as to collect a standard confocal image stack. Thirty images over a lateral area of 65 × 65 µm (red square in Fig. 5(a)) were collected with a depth spacing of 1 µm each. Figures 5(c) and 5(d) show a representative slice through the reconstructed 3D SIFM volume and the corresponding slice from the confocal stack, respectively. The axial point spread function using the 60x objective was found to be 2.1 µm (FWHM) for 755 nm emission. In order to facilitate the comparison of the SIFM and confocal data, the SIFM stack was convolved with a Gaussian with a width (FWHM) of 2 µm in the z direction. An erode/dilate filter with a radius of 190 nm was subsequently applied. Figure 5(e) shows a composite image of the same optical slice acquired using SIFM (green) and confocal microscopy (red). The co-localization of beads in the combined images is excellent, demonstrating that SIFM volumes of comparable quality to confocal imaging can be collected in just a single 2D scan over a much wider area in a much shorter period of time.

4.3 Mouse heart vasculature

In tissue, fluorescent emission can undergo multiple scattering events while propagating to the objective due to the turbid microenvironment. One could expect that multiple scattering events distort the spherical wave fronts emitted by the fluorophore, posing a significant challenge to this technique or affecting the self-interference pattern. To demonstrate that SIFM works even in the presence of tissue scattering, the vasculature of an ex vivo mouse heart was imaged.

A C57BL/6 mouse was anesthetized and subsequently sacrificed with avertin according to standard procedure. A cut was then made in the right ventricle and saline perfusion solution was injected into the left ventricle until all blood was removed from the circulation. The mouse was then perfused with 0.5 ml of fluospheres (2% solids). The heart was removed and imaged fresh using a 10x objective (Olympus UPlanSApo 10X, underfilled to an effective NA of 0.22). Animal housing, breeding and experiments were carried out in accordance to Dutch law and the Animal User Care Committee of the VU University.

Figure 6 shows a color-coded, three-dimensional map of the myocardial microvasculature obtained with a single 512 × 512 pixel raster scan without depth scanning. The highly oriented capillaries follow a pattern that is tightly associated with individual myocites (not visible in this image) [12, 13]. The SIFM volume also shows larger vessels connecting to the capillary bed.

Fig. 6 SIFM image of microvasculature in a mouse heart. The image represents a 500 x 500 x 60 µm volume of tissue, starting at 15 µm below the tissue surface. (A) The depth of the vessels is color-coded: the deepest layers are displayed in violet and the top layers are displayed in red. (B) A movie that shows a three-dimensional rendering of the data is available online (Media 2).

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5. Conclusions

In summary, we have introduced self interference fluorescent microscopy, a novel interferometric method to detect the depth position of a (fluorescent) emitter without depth scanning. By introducing a phase plate into the optical path of a standard confocal imaging system, we present the fluorescence photons with two alternative optical paths, thereby forcing them to self-interfere. The phase information of the spectrally resolved self-interference can be exploited to precisely measure the depth location of a fluorescent signal without any need for depth scanning. Self interference fluorescent microscopycan easily be integrated in any confocal imaging system and is especially suited for applications such as miniature endoscopy where space constraints pose a challenge to obtain depth information.

Acknowledgments

We would like to thank Joke Wortel and Ruud Toonen from the Center for Neurogenomics and Cognitive Research, Neuroscience Campus Amsterdam for their help with the animal experiments. This work was supported by the National Institutes of Health (project R21 RR023139). This work is part of the research program Vernieuwingsimpuls which is (partly) financed by the Netherlands Organization for Scientific Research (NWO).

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Self-interference fluorescence microscopy: three dimensional fluorescence imaging without depth scanning

Abstract

1. Introduction

2. Theory

2.1 Principle of SIFM

2.2 Ray optics model

3. Self-interference fluorescence microscope

4. Results and discussion

4.1 Depth sensitivity

4.2 Three dimensional imaging

4.3 Mouse heart vasculature

5. Conclusions

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (6)

Equations (12)

Optics Express