Modeling the Effect of Wave-front Aberrations in Fiber-based Scanning Optical Microscopy

(1)

JTu4A.13.pdf Imaging and Applied Optics © OSA 2013

Modeling the Effect of Wave-front Aberrations in

Fiber-based Scanning Optical Microscopy

Hans R.G.W. Verstraete, Michel Verhaegen, and Jeroen Kalkman

Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, Delft 2628CD, The Netherlands h.r.g.w.verstraete@tudelft.nl

Abstract: In scanning microscopy and optical coherence tomography, aberrations of the

wave-front cause a loss in intensity and resolution. Intensity and resolution are quantified using Fresnel propagation, Fraunhofer diffraction, and the calculation of overlap integrals.

OCIS codes: (180.0180) Microscopy; (110.0110) Imaging systems; (110.1080) Adaptive Optics

1. Introduction

High resolution optical imaging in biomedicine is of paramount importance in the study of biological processes and in medical diagnosis. Scanning optical microscopy and optical coherence tomography are two high resolution optical imaging techniques that are based on point scanning of an optical beam over a sample. A scanning optical microscope, such as a confocal microscope, is a sequential imaging system which scans an optical beam over the sample to obtain a full image of the sample. In theory, a diffraction-limited spot is used for scanning.

Optical aberrations, either in tissue or in the imaging system, can significantly decrease the spatial resolution and imaging depth. Adaptive optics (AO) can improve the performance of such systems through correction of the wave-front [1]. Thorough knowledge of the effect of aberrations on the system performance can improve the performance of AO through more efficient aberration correction and/or the development of accurate image quality metrics.

Here we model the effect of system aberrations in fiber-based scanning optical microscopes to identify their influence on the measured intensity and spatial resolution. The influence of aberrations on the intensity is calculated using the back coupling efficiency of light from a single mode fiber (SMF) to a mirror reflector. The RMS radius of the intensity distribution in the focal plane quantifies the image spatial resolution. Good agreement is observed between calculations and analytical theory. For some aberrations a maximum in the intensity is not a sufficient criterion for optimal system performance.

2. Theory and model

The scanning optical microscope in reflection geometry is modeled using the geometry shown in Fig. 1, with reflection caused by a mirror oriented perpendicular to the optical axis. A collimated aberrated wave-front at 1 is incident on an imaging lens with focal length f = 17 mm. After passing through the lens, the wave propagates over a distance zT = 2(f+d) and passes through the same imperfect lens again, resulting in a disturbed wave-front at 2. The

propagation through the optical system is calculated using Fresnel propagation [2]. Since Fresnel propagation is not well suited for high numerical aperture (NA) systems, we limit ourselves to low NA. It is assumed that the incident wave-front originates from the SMF. The disturbed wave-front at 2 is coupled back into the fiber [3]. The total intensity is numerically and in some cases analytically calculated using the overlap integral between the incident wave-front and the disturbed wave-front Eq. (1). Aberrations in the lens are modeled using Zernike polynomials and are normalized to have a variance equal to one on the unit disk. The Zernike coefficients are noted as_nm, where n and m are the radial and the azimuthal degree, respectively. Aberrations are odd if n is odd. The proposed model allows the reflector to be at an arbitrary distance d from the focal point and with an input field having an arbitrary number of aberrations. The calculations are performed at a wavelength of 850 nm.

2 * 1 2 2 1 2 1

( , ) ( , )

|

( , ) |

u x y u x y dxdy

a

u x y

dxdy

   





(1)

(2)

To calculate the RMS radius of the spot in the focal plane, the spatial intensity in the focal plane (distance f of the lens) is calculated using Fraunhofer diffraction. From this intensity image the RMS radius of the intensity is calculated. The calculations are performed for a Gaussian distributed incident wave-front with beam waist 20.5 mm

and a full circular pupil with radius 3 mm.

3. Results

As a first step we calculate the influence of the defocus aberration. Figure 1 shows the numeric result for the coupling efficiency and a comparison with the analytical result. As can be observed the calculation matches the Lorentzian distribution very well. The function computed for the defocus introduced by the distance d matches the analytical Lorentzian function for a Gaussian distributed incident wave-front [4].

Fig. 2. Intensity overlap vs. distance zT. The numerical model (open dots) matches the analytical model (solid line)

Figure 3 shows calculations for all aberrations with radial n=2 and n=3. The dashed line is the result for a Gaussian distributed incident wave-front and the full line is the result of a fully filled circular pupil. The intensity vs. aberrations for a Gaussian distributed pupil is smoother than for a circular pupil. As mentioned in various other literature, small odd Zernike aberrations cancel, because of the double pass effect [5]. This makes them harder to quantify in terms of loss of coupling efficiency or intensity. This fact is clearly demonstrated in Fig. 3 for the third order aberrations, coma and trefoil. For even aberrations, the phase aberrations add and further deteriorate the coupling efficiency. For the second order aberrations, astigmatism and defocus, the functions are very similar, but the main lobe of the defocus is thinner than the main lobe of the astigmatism.

(3)

The resolution, determined by the spot size in the focal plane, is affected by all aberrations. In an attempt to quantify the resolution of a disturbed spot, the RMS of the radius of the spot size in the focal plane is compared with the minimal RMS radius of the analytically known, aberration free, diffraction limited focal plane images for a circular and Gaussian pupil (Airy disk, Gaussian beam, respectively), these are shown with the dashed and full line, respectively. It can be seen in Fig. 4 that the RMS radius is affected similarly by both odd and even aberrations. This leads to conclusion that odd aberrations are hard to quantify from the image intensity, but clearly have a deteriorating influence on the obtainable resolution. Note that the aberrations considered for the RMS radius are smaller than those for the coupling efficiency.

Fig. 4. Effect of Zernike Aberrations on the RMS Radius of Spot Size in the Focal Plane

4. Discussion

The calculation of the effect of aberrations on the coupling efficiency and the RMS radius has shown that odd aberrations are harder to detect in the intensity, but deteriorate the resolution in the same order as even aberrations. Furthermore, a Gaussian pupil results in smoother dependence of the intensity on aberrations than the full pupil.The proposed model matches the known analytical results for Gaussian beams and circular pupils for both the intensity and the RMS radius of the spot size. We conclude that a useful image quality metric in AO should encompass both an intensity component and a resolution component, as for example used in [6].

5. References

[1] Martin J. Booth, "Adaptive optics in microscopy, " Phil. Trans R. Soc. A 365, 2829-2843 (2007).

[2] J. W. Goodman, "Introduction to Fourier Optics", 3rd ed. (Roberts & Company, 2004).

[3] M. Gu, C. Sheppard, and X. Gan, "Image formation in a fiber-optical confocal scanning microscope," J. Opt. Soc. Am. A 8, 1755-1761 (1991).

[4] T.G van Leeuwen, D.J. Faber, M.C. Aalders, "Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography," Selected Topics in Quantum Electronics, IEEE Journal of , vol.9, no.2, pp. 227- 233, March-April 2003.

[5] P. Artal, S. Marcos, R. Navarro, and D. Williams, "Odd aberrations and double-pass measurements of retinal image quality," J. Opt. Soc. Am. A 12, 195-201 (1995).