[Advances in Imaging and Electron Physics] Volume 175 || Point Spread Function Engineering for...

CHAPTER FOUR

Point Spread FunctionEngineering for Super-ResolutionSingle-Photon and MultiphotonFluorescence MicroscopyPartha Pratim Mondal* and Alberto Diasproy,z*Nanobioimaging Laboratory, Department of Instrumentation and Applied Physics, Indian Institute ofScience, Bangalore 560012, IndiayDepartment of Nanophysics, Istituto Italiano di Tecnologia, 16163 Genova, ItalyzDepartment of Physics, Universit�a degli Studi di Genova, 16153 Genova, Italy

Contents

1. Introduction 2012. Theory 2033. Results and Discussions 205

3.1. Single-Photon and Two-Photon Excitation 2063.2. Aperture Engineering for PSF Modeling 2073.3. Aperture Engineering for High-Resolution Imaging 2093.4. Aperture Engineering In 4Pi Geometry 2113.5. Multiple Excitation Spot Optical Microscopy and Theta Detection 214

4. Conclusions 217Acknowledgments 218References 218

1. INTRODUCTION

Visualization of biological processes at the molecular level requiresboth high resolution and flexible point spread function (PSF) structure (shapeand size). This facilitates many interesting biophysical studies. For example,receptor-mediated endosomes migrate from membrane to nucleus (Salazaret al., 2007) and mitochondria (Bohnert et al., 2007) depending on the tar-geting molecule. Deciphering the protein trafficking is a challenging task thatwould benefit from resolution tenability. Single-photon and multiphotonfluorescence microscopy provide a detailed view of key processes both at thecellular andmolecular level.Recently developedmicroscopy techniques suchas stimulated emission depletion (STED; Hell et al., 1994, 2007),

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202 Partha Pratim Mondal and Alberto Diaspro

photoactivated localization microscopy (PALM; Betzig et al., 2006), fluo-rescence photoactivation localization microscopy (fPALM; Hess et al., 2006),stochastic optical reconstructionmicroscopy (STORM;Rust et al., 2006), 4Pi(Hell et al., 1992, 2007), spatially structured light illumination (Gustafsson2000; Kner et al., 2009; Gustafsson et al., 2008), photoactivated localizationmicroscopy (Betzig et al., 2006), nonlinear patterned excitation microscopy(Heintzmann et al., 2002), aperture engineering (Mondal andDiaspro, 2008),individual molecule localization-selective plane illumination microscopy(IML-SPIM; Cella Zanacchi et al., 2011), single-wavelength two-photonexcitation–stimulated emission depletion (SW-2PE-STED; Bianchini et al.,2012) coupled with image reconstruction methods (Mondal et al., 2008) arebringing far-field optical microscopy to the nanoscale. These techniques haveincredible potential of defying the classical limits imposed by diffraction andare capable of surpassing it. On the flip side, most of these techniques areoptically complicated, computationally time consuming, not always suitablefor live cell imaging, and mostly depend on the photophysical properties ofthe target fluorescent molecules to achieve super-resolution.

Even if in different ways, at the heart of super-resolution imagingmodalities is the system PSF. This is quite obvious since the PSF defines thecharacteristics of an imaging system and determines the system resolution.Control over the PSF allows researchers to obtain fruitful results and can giverise to promising new perspectives, both on the biological and optical sides. Itmay be noted that the best spatial resolution of an imaging system is deter-mined by Abbe’s diffraction limit (Abbe, 1884). The shape and size of the PSFin light microscopy is determined by the excitation wavelength and numericalaperture of the overall imaging system. The system PSF projection in thelateral xy-plane is essentially a Airy disk intensity pattern (central disk withconcentric dark rings). Half the diameter of the first dark ring sets the limit onthe resolution (the smallest possible resolvable distance between two objectsin the specimen before they can be confused as a single spot). So far, PSFmodeling is attractive for achieving super-resolution as well as for a variety ofinteresting applications in nanoscale imaging. Toraldo di Francia (1952)showed that the diffraction pattern of a variety of shapes can be obtained bysuitably subdividing the pupil plane. This advance was followed by excitingdevelopments by Neil et al. (2000), Bothcherby et al. (2006), Martinez-Corralet al. (2003), and Mondal (2009, 2010). For example, lateral resolutionimprovement in two-photon microscopy can be achieved by apertureengineering techniques (Mondal and Diaspro, 2008; Ronzitti et al., 2009),and high-resolution stereoscopic image pairing was successfully obtained by

Point Spread Function Engineering 203

using depth-of-focus PSF (Bothcherby et al., 2006). This chapter demon-strates that application-specific tailor-made system PSFs can be generated byaperture engineering techniques for a variety of applications, including depthimaging, multifocal imaging, and super-resolution microscopy.

2. THEORY

From the fluorescence microscopy perspective it is important tocharacterize the field structure and its interaction with the molecules in thegeometrical focus. For example, there is immense potential if the PSF shapeand size can be altered for optical trapping applications. Similarly, multilayerfluorescence correlation spectroscopy can be realized if several nanospots canbe generated. Hence, it is the size and structure of the field that matters themost for various applications. In this regard, the first complete vectorial theoryof the field at the focus was reported by Richards and Wolf (Richards et al.,1959). For linearly polarized light illumination in a single-photon excitationimaging system, the field is given by (Richards et al., 1959; Biovin et al., 1965)

h1PEexc ¼ A�jI0j2þ 4jI1j2cos2

�f�þ jI2j2þ 2ReI0I�2 cos

�2f

��: (1a)

For randomly polarized light, the intensity distribution becomes

h1PEexc ¼ A�jI0j2þ 2jI1j2þ jI2j2

�; (1b)

where f is the angle between the incident electric field and direction ofobservation, A is the proportionality constant., and I0, I1, and I2 are thediffraction integrals over the semi-aperture angle (a) given by

I0�u; v

� ¼ Raq¼0

ð1þ cos qÞsin q J0ðvsin q=sinaÞ

� ffiffiffiffiffiffiffiffiffifficos q

peiðusin q=sin2aÞdq

(2a)

I1�u; v

� ¼ Raq¼0

sin2 q J1ðvsin q=sinaÞ


peiðusin q=sin2aÞdq

(2b)

I2�u; v

� ¼ Raq¼0

ð1� cos qÞsin q J2ðvsin q=sinaÞ


peiðusin q=sin2aÞdq;

(2c)


where J0, J1, and J2 are the Bessel functions of the first kind and k ¼ 2p/l.
The diffraction theory [Eqs. (1) and (2)] defines the electric field distri-
bution at the focus, which is essentially an Airy disk pattern. The radius ofthe first dark ring of the Airy pattern sets the classical diffraction limit(termed Abbe’s diffraction limit) for resolution. The lateral (rXY) and axial (rZ)resolution is (Hell, 2007; Abbe, 1884)

rXY ¼ lill

2n sina(3a)

rZ ¼ lill

n sin2a; (3b)

where n is the refractive index of the objective immersion medium and lill isthe wavelength of the illuminated light.

The development of the multiphoton excitation process has shown hugepotential for nanoscale imaging of biological specimens. Such a rare processwas first predicted by G€oppert-Mayer (1931), and the first bioimagingapplication was realized by the W.W. Webb group (Denk et al., 1990). Theabsorption cross section for such a rare event is w1 GM (10–50 cm4s) formost dyes used for fluorescence microscopy (the cross sections for a largenumber of commonly used fluorescent dyes are reported in Diaspro et al.,2006). Despite a small cross section, the intrinsic localization propertyof multiphoton absorption is precisely what makes it a natural choice foroptical sectioning. The localization is due to the high probability ofmultiphoton absorption at the geometrical focus where the photon densityis maximum. For n-photon excitation, the intensity distribution followsInPEðx; y; zÞ f In1PEðx; y; zÞ, where the proportionality factor is theabsorption cross section. Specifically, for two-photon excitation (n¼ 2), theexcitation PSF is given by

h2PEexc ¼ A�jI0j2þ 2jI1j2þ jI2j2

�2: (4)

In this chapter, our objective is to explore ways of achieving highresolution for both single-photon excitation (1PE) and two-photon exci-tation (2PE) microscopy. The dimension of the PSF determines the three-dimensional resolution of the imaging system. We (Mondal et al., 2010)recently proposed a technique for achieving super-resolution, termed aper-ture engineering (AE), for fluorescence microscopy capable of surpassing theclassical diffraction limit. It must be noted that other sophisticated techniquesfor super-resoution exist, but they are very complicated and depend heavilyon the photophysics of the target fluorescent molecule. On the contrary,


AE is an optical technique and does not depend on the photophysical natureof fluorescent molecules. The AE technique involves structuring the inci-dent wave front by using a spatial filter at the back aperture of the objectivelens and the rest is determined by diffraction. In the case of AE, vectorialdiffraction theory for randomly polarized light illumination gives thefollowing excitation PSF:

hAEexc ¼ A�jI 00j2þ 2jI 01j2þ jI 02j2

�; (5)

where the modified diffraction integrals are obtained by multiplying theoptical mask (spatial filter) function to the diffraction integrals. The modifiedintegrals I00(u,v), I10(u,v), and I20(u,v) over the semi-aperture angle a aregiven by

I 0mðu; vÞ ¼Za

q¼0

BmðqÞT ½q�ffiffiffiffiffiffiffiffiffifficos q

pei

ucosqsin2adq (6)

and the vector

B ¼

2664ð1þ cos qÞsin q J0ðvsin q=sinaÞsin2q J1

�vsin q=sina

�ð1� cos qÞsin q J2ðvsin q=sinaÞ

3775:

ðu ¼ k z sin2a; v ¼ kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

psinaÞ are, respectively, the longitudinal

and transverse coordinates (Richards et al., 1959; Biovin et al., 1965). T [q] isthe amplitude transmission function of the mask, q1 is the stop angle thatscales the amplitude transmission, and a is the semi-aperture angle of theobjective lens. It should be noted that the system generates a plane wavefront for H ¼ 1.

3. RESULTS AND DISCUSSIONS

We plan to use AE techniques to realize the desired shape and size ofthe system PSF. Both linearly and randomly polarized light is used forexcitation. The excitation wavelengths for 1PE and 2PE are 488 nm and980 nm, respectively. The aperture angle is 60� unless otherwisementioned. We have chosen to work with 128 optical layers with each


layer of size 128 � 128, which spans over a physical dimension of3:84� 3:84 mm2.

3.1. Single-Photon and Two-Photon ExcitationThe basic structure of the field at the geometrical focus of a single objectivelens system is shown in Figure 1a. Plane waves are expanded by the beamexpander to fully cover the back aperture of the focusing lens. Corre-sponding field distribution shows an enlarged field along the optical axis, anindication of poor axial resolution compared with lateral resolution.Furthermore, compared with 1PE-PSF, 2PE-PSF is highly localized aboutthe focus due to the intensity-squared dependence of the 2PE process.Figure 2 shows the polarization-dependent characteristics of the excitationPSF along both the lateral and optical axes for 1PE. The corresponding full

Figure 1 (A) Schematic diagram for 1PE and2PEmicroscopy. (B) Resultant PSF fora¼ 60�.See the color plate.

Figure 4.2 Lateral (XY) and axial (XZ) PSF profile for randomly polarized (RP) andlinearly polarized (LP) light for 1PE. See the color plate.

Table 1 FWHM Values for 1PE and 2PE

Type of Light 1PE 2PE

Randomly polarized 360 nm 1470 nm 510 nm 1980 nmLinearly polarized 420 nm 1470 nm 510 nm 2010 nm


width half maximum (FWHM) values for both linearly and randomlypolarized light are tabulated in Table 1.

3.2. Aperture Engineering for PSF ModelingMany applications in fluorescence microscopy and bioimaging require thedesired shape and size of PSF. Careful control of these parameters (shape andsize) has many applications ranging from nanoscale imaging to opticalmicroscopy. Axially extended PSF has been successfully used in super-resolution imaging (Mondal and Diaspro, 2008; Toraldo di Francia, 1952;Neil et al., 2000; Martinez-Corral et al., 2003), multiple excitation spotoptical (MESO)-optics (Soroko, 1996), and others. Botcherby et al. (2006)have produced Bessel beams for scanning microscopy using only the binary

Figure 3 (A) Schematic diagram for AE-based depth-of-field optical system.(B) Resulting PSF for depth imaging. See the color plate.


phase. Such a Bessel-like beam can be produced in many ways. One of thesimplest techniques is to use an optical mask just before the back aperture ofthe objective lens. The optical mask allows the light to pass through a pre-defined aperture angle (the outermost annular ring) as shown in Figure 3a.This results in a structured wave front that upon diffraction through theobjective lens generates a Bessel-like beam. The optical mask acts as a spatialfilter for which the transmission function is given by

TðqÞ ¼ H ½q� b� �H ½q� a�; (7)

where H[•] is the Heaviside function and b and a are, respectively, the fieldtruncation parameter and aperture angle.Corresponding single and multiphoton PSFs along both the lateral and axialaxes are shown in Figure 3b. Ring artifacts are clearly visible in the lateralplane (XY) for 1PE PSF, whereas 2PE PSF is free from such artifacts becauseof intensity-squared dependence. This further shows the importance of theproposed technique, especially for multiphoton imaging applications. Onthe other hand, the system PSF (XZ) shows an elongated depth-of-focusPSF particularly useful for depth imaging applications.

Figure 4 Stop angle b–dependent resultant PSF for variable-depth imaging. See thecolor plate.


In addition, the parameter b describing the stop angle (see Figure 3) canbe tuned accordingly to achieve variable-depth imaging and PSF resizing.Figure 4 shows this for varying b-parameters (0�, 20�, 40�, and 57�) for anaperture angle of 60

�. Controlled elongation of the PSF along both the axial

(XZ) and lateral (XY) axes is achieved. The corresponding FWHM valuesalong the optical axis are 330 nm, 990 nm, 1590 nm, and >2.5 mm. Thiscan be used for variable-depth imaging for both in vivo cell and tissueimaging.

3.3. Aperture Engineering for High-Resolution ImagingThe other application of AE is in high-resolution imaging. This requiresa different optical mask. The mask acts as a spatial filter that is capable ofenhancing the higher frequencies. It can be shown that such a filter cansurpass Abbe’s classical diffraction limit. The filter consists of two trans-mission windows: a central circular window and an outermost annularwindow (Figure 5a). The transmission function for such a spatial filter isgiven by

TðqÞ ¼ H ½q� b1� �H ½q� b2� þH ½q� b3��H ½q� a�:

(8)

Figure 5 (A) Schematic diagram for AE-based super-resolution system. (B) Resultingsystem PSF. See the color plate.


Figure 5 shows the spatial filter along with the resultant excitation PSF.
We have chosen a window size of 3�dthat is, a circular transmissionwindow (0�–3�) and an annular window (57�–60�). Such a spatial filterresults in a structured wave front that upon focusing produces a compactcentral spot accompanied by a concentric ring pattern (see Figure 5b).This is predominately due to the diffraction effects of the new opticalmask. Study of 2PE also shows a compact central lobe but without ringartifacts, which is again because of the intrinsic intensity-squareddependence of the 2PE process. Compared with classical lateral resolutionfor 1PE (FWHM ¼ 270 nm) and 2PE (FWHM ¼ 390 nm), this tech-nique has a reduced FWHM of 210 nm and 330 nm, respectively. Thisshows that the classical lateral resolution is surpassed by 60 nm for both1PE and 2PE.Resolution improvement is even better at a low aperture angle (a ¼ 45�).The 1PE and 2PE resolution is surpassed beyond the classical resolution limitby approximately 180 nm and 60 nm, respectively (Figure 6); this issubstantial considering the simplicity of the proposed approach and poorresolution of multiphoton excitation microscopy.

Figure 6 Comparison of classical and AE PSF for 1PE and 2PE microscopy at a ¼ 45�.See the color plate.


Line plots (not shown) are used to characterize the ring artifacts, whichare prominent in the 1PE case, whereas these artifacts are absent in AE-2PEPSF. The side lobes are found to be well within 8% of the central lobe,which is acceptable for most imaging applications. Next we study thepolarization effect of the incident radiation field on the system PSF.Figure 7 shows the system PSF (XY- and XZ- planes) for both linearlypolarized (LP) and randomly polarized (RP) light for 1PE and 2PEmicroscopy. Elongation of PSF (along the x-axis) is observed for LP lightillumination and the corresponding PSF bears a characteristic structurecompared with RP light.

3.4. Aperture Engineering In 4Pi Geometry4Pi geometry is the simplest way to increase the effective aperture angle torealize high resolution along the optical axis. This geometry was firstproposed by Cremer et al. (1978) for 4Pi holography and high-resolutionlaser scanning microscopy. Their approach was later vastly improved by Hellet al. (Hell, 2007; Hell et al., 2009) for high-resolution fluorescencemicroscopy. 4Pi microscopy has promising applications in biophysics andnanoscale imaging (Hell et al., 2009).

Figure 7 Polarization effect (with linearly polarized [LP] and randomly polarized [RP]light illumination) of AE PSF (a ¼ 45�) for both 1PE and 2PE microscopy. See the colorplate.


In this section, we present an advanced imaging technique for fastimaging and the capability to produce several excitation spots. Thistechnique has the advantage of simultaneous visualization of multiplelayers of a biological specimen, thereby overcoming slice-by-slice z-axisscanning used in state-of-art confocal microscopy. The proposed tech-nique is achieved by integrating AE (Mondal, 2009, 2010) in a 4Pi-geometry (Cremer et al., 1978), resulting in several high-resolution


excitation spots. This technique is termed multiple excitation spot optical(MESO) microscopy.

The phase-matching condition is maintained between both the coherentbeams. The optical mask truncates the field in the central annular region tocreate a Bessel-like beam. The phase-matched counterpropagating beamsresult in constructive and destructive interference, thereby generatingmultiple spots along the optical axis. The emitted fluorescence light iscollected and subsequently deflected by dichroic mirrors followed byfocusing onto the detector. Figure 8a shows the schematic diagram of theoptical setup; the corresponding PSF is shown in Figure 8b for a semiaperture angle of a ¼ 6�. Light is allowed to pass through both the trans-mission windows (central circular window (0

�- b1) and the outermost

annular window (b2 - a), where b1 ¼ 3� and b2 ¼ 57�. The diffractionintegrals were carried out over the transmission windows defined by b1, b2.

Figure 8 (A) Schematic diagram for AE in a 4Pi geometry. (B) Resultant multispotexcitation PSF. See the color plate.


The excitation wavelengths for both 1PE and 2PE scheme are 488 nm and910 nm, respectively. Figure 8b shows an improvement in axial resolutionthat is about fourfold and fivefold compared with existing 1PE and 2PEmicroscopy systems. Reduced artifact (along both the lateral and axial axes)in 2PE microscopy is evident from the system PSF (see Figure 8b).

Intensity plots were used to exemplify the characteristics of the systemPSF. Comparisons are drawn between the AE and non-AE approach in a 4Pigeometry. It must be noted that the non-AE case is essentially the state-of-art4Pi type-A PSF. In Figure 9, the top and bottom panels are, respectively, theline plots for 1PE and 2PEmicroscopy.Multiple equi-intense excitation spotsare generated by the proposed AE technique, whereas the non-AE technique(plane wave front illumination technique) gives a PSF resembling an inter-ference pattern. For 1PE, the lateral and axial resolution is 210 nm, whereas2PE gives a lateral and axial resolution of about 330 nm and 270 nm,respectively. Axial resolution of the non-AE technique is slightly bettercompared with the AE technique, which is due to the combined effect of themodifiedwave front and its diffraction. On the other hand, the AE techniquehas the advantage of generating multiple excitation spots that are highlylocalized and distinct. The proposed AE technique exhibits simultaneousmultilayer scanning capability and in-principle fast imaging.

3.5. Multiple Excitation Spot Optical Microscopy and ThetaDetectionThis section explores MESO microscopy in a theta-detection geometry forrealizing a complete imaging system. It is worth noting that such a system iscapable of spatiotemporal super-resolution (i.e., super-resolution both intime and space; Partha et al., 2011).

In the previous section, we showed that the generation of multispotexcitation is possible using MESO microscopy. The detection is equallyimportant for realizing a complete imaging system. We use a theta-detectionsystem to collect photons emitted from the individual nanospots (Figure 10).The 4Pi optical arrangement ensures multiple excitation nanospots, whereasorthogonal theta detection by a third objective lens collects the output fluo-rescence light. The spatial filters (SF1 and SF2) produce Bessel-like beamscentered about the focus and traveling opposite to each other. Under thephase-matching condition, the beams interfere, thereby resulting in an inter-ference pattern that consists of a chain of nanospots (Partha et al., 2011).Orthogonal detection helps to minimize artifacts by eliminating the incidentlight altogether and facilitating fluorescence detection from individual

Figure 9 Intensity plots of 1PE (top) and 2PE (bottom) PSF shown in Figure 8. For colorversion of this figure, the reader is referred to the online version of this book.


nanospots by simply scanning along the optical axis. The overall system PSF isthe product of dot-like excitation PSF multiplied by the detection PSF.

The study is performed with an excitation aperture angle ofaill ¼ 30�and detection angle of adet ¼ 45�: To show the multilayer

Figure 10 Schematic of the optical setup for MESO imaging system. Excitation isperformed in a 4Pi geometry and orthogonal theta-detection geometry is used fordetection. For color version of this figure, the reader is referred to the online version ofthis book.


imaging capability, we have carried out detection for nanodots situated atvarying depths (z-layers). Figure 11 shows the detection of a target nanodotsituated at depths of 540 nm and 1.56 mm. Detection of the central nanodot(at z ¼ 0) is also shown alongside for reference. The detection system isscanned and fluorescence from each nanospot is obtained. In Figure 11, thetarget nanospot is indicated by the red arrow within the excitation PSF.Practically, simultaneous detection from all nanospots can be obtained byusing a diffraction grating in the detection path of the imaging system(Dalgarno et al., 2010).

Finally, a comparison with a 2PE and 2PE-4pi imaging system is carriedout to elucidate the benefit of the proposed system. The excitation,detection, and the system PSF for all three imaging systems are shown inFigure 12. The 2PE system with orthogonal detection shows a resolution of390 nm, whereas both the 2PE-4Pi and 2PE-MESO systems have a reso-lution of about 120 nm and 150 nm respectively.

Both 2PE and 2PE-4pi systems are capable of single point illumination,whereas 2PE-MESO microscopy can produce several nanospots with theextra advantage of selectivity (selectively obtaining fluorescence fromindividual nanospots.). This combination of spatial super-resolution andsimultaneous multiplane excitation (high temporal resolution) can result ina spatiotemporal super-resolution imaging system.

Figure 11 Orthogonal theta-detection–based nanospot (of sizez360 nm) detection ata depth of 0 nm, 540 nm, and 1560 nm. The red arrow indicates the target nanospot inthe excitation PSF. See the color plate.

Figure 12 Comparison of state-of-art 2PE, 2PE-4Pi, and 2PE-MESO PSF. Illumination,detection, and total PSF are shown for all the imaging modalities. See the color plate.


4. CONCLUSIONS

This chapters presents techniques for application-specific fluorescenceimaging. The capability of a spatial filter and its advantages are described ina variety of excitation geometry (such as single-lens and 4Pi geometry); thesetechniques are possible using a specially designed optical mask to generatea Bessel-like beam. The optical mask acts as a spatial filter that modifies theplane wave front to obtain the desired PSF characteristics. PSF shape and sizeare scaled and altered for a variety of applications, such as depth imaging,multifocal imaging, and super-resolution imaging. Especially in a 4Pigeometry, spatial filtering results in several excitation nanospots. In principle,this can improve both spatial and temporal resolution. Moreover, thetailoring of filter parameters allow the tuning of spatial resolution as required


by the specific application. Potential applications range from nanoscaleimaging to fluorescence microscopy.

ACKNOWLEDGMENTSWe acknowledge funding from the Indian Institute of Science under start-up grant, DAE andDST. In addition, Alberto Diaspro received funding from Istituto Italiano di Tecnologia.

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[Advances in Imaging and Electron Physics] Volume 175 || Point Spread Function Engineering for...

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