Theoretical analysis of digital in-line holography through an arbitrary incoherent aperture and its implications for on-chip lensfree microscopy

Holography is all about recording the optical phase information in the form of amplitude oscillations. To be able to read or make use of this phase information for microscopy, most existing lensfree in-line holography systems are hungry for spatial coherence and therefore use a laser source that is filtered through a small aperture (e.g., 1–2 µm). Utilizing a completely incoherent light source that is filtered through a large aperture (e.g., >100λ–200λ in diameter) should provide orders-of-magnitude better transmission throughput as well as a much simpler, inexpensive and more robust optical set-up. Here we aim to provide a theoretical analysis of this opportunity and its implications for compact lensless microscopy as we illustrated in this manuscript.

To record cell holograms that contain useful digital information with a spatially incoherent source emanating from a large aperture, one of the key steps is to bring the cell plane close to the detector array by ensuring z

_{2}z

_{1}, where z

_{1} defines the distance between the incoherently illuminated aperture plane and the cell plane, and z

_{2} defines the distance between the cell plane and the sensor array (see ). In conventional lensless in-line holography approaches, this choice is reversed such that z

_{1}z

_{2} is utilized, while the total aperture-to-detector distance (z

_{1}+z

_{2}) remains comparable in both cases, leaving the overall device length almost unchanged. Therefore, apart from using

*an incoherent source through a large aperture*, our choice of z

_{2}z

_{1} is also quite different from the main stream lensfree holographic imaging approaches and thus deserves more attention.

To better understand the quantified impact of this choice on incoherent on-chip microscopy, let us assume two point scatterers (separated by *2a*) that are located at the cell plane (z=z_{1}) with a field transmission of the form *t*(*x, y*) = 1 + *c*_{1} δ(*x − a, y*) + *c*_{2} δ(*x + a, y*) where *c*_{1} and *c*_{2} can be negative and their intensity denotes the strength of the scattering process, and δ(*x,y*) defines a Dirac-delta function in space. These point scatterers can be considered to represent sub-cellular elements that make up the cell volume. For the same imaging system let us assume that *a large aperture of arbitrary shape* is positioned at z=0 with a transmission function of *p*(*x,y*) and that the digital recording screen (e.g., a CCD or a CMOS array) is positioned at z=z_{1}+z_{2}, where typically z_{1} ~ 2–5 cm and z_{2} ~ 0.5–2 mm.

Assuming that the aperture,

*p*(

*x,y*) is

*uniformly* illuminated with a

*spatially incoherent light source*, the cross-spectral density at the aperture plane can be written as:

where (

*x*_{1},

*y*_{1}) and (

*x*_{2},

*y*_{2}) represents two arbitrary points on the aperture plane and

*S*(γ) denotes the power spectrum of the incoherent source with a center wavelength (frequency) of λ

_{0} (γ

_{0}).

We should note that in our experimental scheme (), the incoherent light source (the LED) was butt-coupled to the pinhole with a small amount of unavoidable distance between its active area and the pinhole plane. This remaining small distance between the source and the pinhole plane also generates some correlation for the input field at the aperture plane. In this theoretical analysis, we ignore this effect and investigate the imaging behavior of a *completely* incoherent field hitting the aperture plane. The impact of such an unavoidable gap between pinhole and the incoherent source is an “effective” reduction of the pinhole size in terms of spatial coherence (without affecting the light throughput), which we will not consider in this analysis.

Based on these assumptions, after free space propagation over a distance of z

_{1}, the cross-spectral density just before interacting with the cells can be written as

^{24}:

where

represent two arbitrary points on the cell plane. After interacting with the cells i.e., with

*t*(

*x,y*), the cross-spectral density, right behind the cell plane, can be written as:

This cross-spectral density function will effectively propagate another distance of z

_{2} before reaching the detector plane. Therefore, one can write the cross-spectral density at the detector plane as:

where (

*x*_{D1},

*y*_{D1}) and (

*x*_{D2},

*y*_{D2}) define arbitrary points on the detector plane (i.e., within the hologram region of each cell); and

At the detector plane (

*x*_{D}, y_{D}), the

*optical intensity* *i*(

*x*_{D}, y_{D}) can then be written as:

Assuming

*t*(

*x, y*) = 1 +

*c*_{1} δ(

*x − a, y*) +

*c*_{2} δ(

*x + a, y*), this last equation can be expanded into 4 physical terms, i.e.,

where:

In these Equations “

*c.c.*” and “*” refer to the complex conjugate and convolution operations, respectively,

, and

is the 2D spatial Fourier Transform of the aperture function

*p*(

*x, y*). It should be emphasized that (

*x*_{D}, y_{D}) in these equations refers to the cell hologram extent,

*not* to the entire field-of-view of the detector array.

Further,

which effectively represents the 2D

*coherent impulse response* of free space over Δ

*z* =

*F* ·

*z*_{2}. For the incoherent source, we have assumed a center frequency (wavelength) of γ

_{0}(λ

_{0}), where the spectral bandwidth was assumed to be much smaller than λ

_{0} with a power spectrum of

*S*(γ)

*S*_{0}δ(γ − γ

_{0}). This is a valid approximation since in this work we have used an LED source at λ

_{0} ~591 nm with a spectral FWHM of ~18 nm.

Note that in these derivations we have also assumed paraxial approximation to simplify the results, which is a valid assumption since for this work z_{1} and z_{2} are typically much longer than the extend of each cell hologram (L_{H}). However for the digital microscopic reconstruction of the cell images from their raw holograms, *no such assumptions* were made as also emphasized in the Experimental Methods Section.

Furthermore,

*D*_{0} of

Eq. 1 can further be expanded as:

which simply represents the background illumination and has no spatial information regarding the cells’ structure or distribution. Although this last term,

*D*_{0} can further be simplified, for most illumination schemes it constitutes a uniform background and therefore can be easily subtracted out.

Equations (1–

4) are rather important to understand the key parameters in lensfree on-chip microscopy with spatially incoherent light emanating from a large aperture.

Equation 1 describes the

*classical diffraction* that occurs from the cell plane to the detector under the paraxial approximation. In other words, it includes both the background illumination (term

*D*_{0}) and also the

*self-interference of the scattered waves* (terms that are proportional to |

*c*_{1}|

^{2} and |

*c*_{2}|

^{2}). It is quite intuitive that the self interference terms representing the classical diffraction in

Eq. (1) are scaled with

(0,0) as the extent of the spatial coherence at the cell plane is not a determining factor for self interference.

Equation 2, however, contains the information of the interference between the scatterers located at the cell plane. Similar to the self-interference term, the cross-interference term,

*I*(

*x*_{D}, y_{D}), also does not contain any useful information as far as holographic reconstruction of the cell image is concerned. This interference term is proportional to the amplitude of

, which implies that for a small aperture size (hence wide

) two scatterers that are located far from each other can also interfere. Based on the term

, one can estimate that if

(where

*D* is roughly the aperture width) the scattered fields can quite effectively interfere at the detector plane giving rise to the interference term

*I*(

*x*_{D}, y_{D}). This result is not entirely surprising since the coherence diameter at the cell plane is proportional to

, as also predicted by the van Cittert-Zernike theorem.

*It is another advantage of the incoherent holography approach presented here that the cross-interference term, I*(

*x*_{D}, y_{D}),

*will only contain the contributions of a limited number of cells within the imaging field-of-view since* *will rapidly decay to zero for a large aperture.* This cross-interference term will be stronger for coherent in-line holography due to much better spatial coherence. This difference can especially make an impact in favor of incoherent large aperture illumination for imaging of a dense cell solution such as

*whole blood samples* where

*I*(

*x*_{D}, y_{D}) can no longer be ignored.

The final two terms (

Eqs. (3–

4)) describe the

*holographic diffraction* phenomenon and they are of central interest in all forms of digital holographic imaging systems, including the one presented here. Physically these terms dominate the information content of the detected intensity, especially for weakly scattering objects, and they represent the interference of the scattered light from each object with the background light, i.e.,

*H*_{1} (

*x*_{D}, y_{D}) represents the holographic diffraction of the first scatterer

*c*_{1} δ(

*x − a, y*), whereas

*H*_{2} (

*x*_{D}, y_{D}) represents the holographic diffraction of the second scatterer,

*c*_{2} δ(

*x + a, y*). Note that the complex conjugate (

*c.c.*) terms in

Eqs. 3 and

4 represent the source of the

*twin images* of the scatterers since

*h*_{c}^{*}(

*x*_{D}, y_{D}) implies propagation in the reverse direction creating the twin image artifact at the reconstruction plane. Elimination of such twin images in our cell reconstruction results is discussed in the Experimental Methods Section.

A careful inspection of the terms inside the curly brackets in

Eqs. (3–

4) indicates that,

*for each scatterer position, a scaled and shifted version of the aperture function p (x, y) appears to be coherently diffracting with the free space impulse response h*_{c}(x_{D}, y_{D}). In other words, as far as holographic diffraction is concerned,

*each point scatterer at the cell plane can be replaced by a scaled version of the aperture function (i.e., p*(

*−x*_{D} · M, −y_{D} · M)

*) that is shifted by F fold from origin, and the distance between the cell plane and the sensor plane can now be effectively replaced by* Δ

*z* =

*F* ·

*z*_{2}. Quite importantly this scaling factor is

, which implies that the large aperture size that is illuminated incoherently is effectively narrowed down by M fold at the cell plane (typically M≈40–100).

*Therefore, for M1, incoherent illumination through a large aperture is approximately equivalent (for each cell’s holographic signature) to coherent illumination of each cell individually, where the wave propagation over Δz determines the detected holographic intensity of each cell*. This is valid as long as the cell’s diameter is smaller than the coherence diameter (

, see

Eq. 2) at the cell plane, where

*D* defines the width of the illumination aperture and typically

*D*_{coh}~400λ

_{0} – 1000λ

_{0}, which is quite appropriate for most cells of interest. Accordingly, for a completely incoherent source and a sensor area of

*A, d=D/M* defines the effective width of each point scatterer on the cell plane and

*f* =

*A/F*^{2} determines the effective imaging field-of-view. Assuming some typical numbers for z

_{1} (~3.5 cm) and z

_{2} (~0.7 mm), the scaling factor (M) becomes ~50 with

*F* ≈ 1, which means that even a D=50 µm wide pinhole would be scaled down to ~1 µm at the cell plane, which can now quite efficiently be mapped to the entire active area of the sensor array, i.e.,

*f* ≈

*A*.

*To conclude: for M1 the spatial features of the cells over the entire active area of the sensor array will not be affected by the large incoherent aperture, which permits recording of coherent hologram of each cell individually*.

Even though the entire derivation above was made using the formalism of wave theory, the end result is quite interesting as it predicts a geometrical scaling factor of

*M* =

*z*_{1}/

*z*_{2} (see ). Further, because M

1, each cell hologram only occupies a tiny fraction of the entire field-of-view and therefore behaves independent of most other cells within the imaging field-of-view.

*That is the same reason why (unlike conventional lensfree in-line holography) there is no longer a Fourier transform relationship between the detector plane and the cell plane. Such a Fourier transform relationship only exists between each cell hologram and the corresponding cell.*Notice also that in

Eqs. (3–

4) the shift of the scaled aperture function

*p*(−

*x*_{D} ·

*M* *a* ·

*M* ·

*F*, −

*y*_{D} ·

*M*) from origin can be written as

*x*_{D} =

*a* ·

*F*, which is in perfect agreement with the choice of the word “

*fringe magnification factor*” to describe the function of

for the holographic diffraction term. This also explains the reduction in the imaging field-of-view by

*F*^{2} fold for in-line digital holography. Assuming M

1, Δ

*z* approaches to z

_{2} and the shift terms in

Eqs. (3–

4), i.e.,

*a* ·

*F* also approach to

*a*, which makes sense since it corresponds to the shift of the scatterers at the cell plane from origin.

According to

Eqs. (3–

4), for a narrow enough

*p*(−

*x*_{D} M, −y_{D} M) (such that the spatial features of the cells are not washed out), the modulation of the holographic term at the detector plane can be expressed as

. This modulation term of the holographic signature at the detector plane implies that for a large fringe magnification (

*F*), the pixel size of the sensor array will have an easier time to record the rapidly oscillating fringes of the cell hologram, which effectively increases the numerical aperture of the sampling as much as the sensor width permits. However, there are penalties to be paid for this large

*F* choice: (1) a large

*F* does not permit the use of an incoherent source emanating through a large aperture, which makes it more demanding on the optics and alignment, also increasing the relative cost and complexity; and (2) the effective imaging field-of-view is also reduced by factor proportional to

*F*^{2}.

*More analysis on this topic is provided in the Supplementary Text S1.*The derivation discussed above was made for 2 point scatterers separated by 2

*a*, such that

*c*_{1} δ(

*x − a, y*) +

*c*_{2} δ(

*x + a, y*). The more general form of the incoherent holographic term (equivalent of

Eqs. 3 and

4 for a continuous distribution of scatterers - as in a real cell) can be expressed as:

where

*s*(

*x*_{D}, y_{D}) refers to the transmission image of the sample/cell of interest, which represents the 2D map of all the scatterers located within the sample/cell volume. The above derivation assumed a narrow enough

*p*(

*−x*_{D} M, −y_{D} M) such that M

1, which is characteristic of the approach discussed in this manuscript. The physical effect of the fringe magnification factor (

*F*) on the object hologram can also be visualized in this final equation, in harmony with our discussions in the previous paragraphs.

Finally, the supplementary text provides further discussion on the spatial sampling requirements at the detector array, as well the space-bandwidth product of the presented technique.^{36}^{–}^{38}