Second harmonic generation microscopy with tightly focused linearly and radially polarized beams

14 1.4 Electric field of a tightly focused linearly polarized beam.. 49 3.3 Induced SHG polarization from collagen for a tightly focused linearly polarized beam.. 52 3.4 Induced SHG pola

SECOND HARMONIC GENERATION

MICROSCOPY WITH TIGHTLY FOCUSED LINEARLY AND RADIALLY POLARIZED BEAMS

Yew Yan Seng Elijah

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAMME IN BIOENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

1.1 Motivation 1

1.2 Overview 3

1.3 Nonlinear interaction of light with matter 5

1.4 The nonlinear susceptibility tensor 7

1.5 Factors affecting SHG efficiency 9

1.6 Focusing with high NA objectives 12

1.6.1 Linearly polarized beams 13

1.6.2 Radially polarized beams 15

2 Literature review 19 2.1 SHG 19

2.1.1 SHG from biological materials 20

2.1.2 Collagen and myosin 22

2.1.3 Nonlinear microscopy 27

2.2 Polarization and focusing of light 30

2.2.1 Radial polarization 31

2.2.2 Linearly to radially polarized beams 33

3 Theory of SHG with tight focusing 38 3.1 Tensorial nature of SHG 38

3.1.1 Effects of focusing on SHG 41

3.2 SHG far-field radiation 46

3.3 Induced SHG polarization with tight focusing 49

3.3.1 Induced SHG polarization 50

3.4 Far field radiation of SHG 54

3.4.1 Linear objects 54

4 Materials and methods 64 4.1 Preparation of samples 64

4.2 SHG and TPF microscopy 65

4.3 Back aperture of objective 69

4.4 Filters and imaging 71

4.5 Generation of radial polarization 73

4.6 Pupil functions for radial polarization 82

5 Results and discussion 91 5.1 SHG and TPF microscopy of skin 91

5.2 SHG with radially and linearly polarized beams 95

5.3 Longitudinally sectioned collagen 97

5.3.1 Low NA focusing 97

5.3.2 High NA focusing 101

5.3.3 Resolution of SHG 104

5.4 Transversely sectioned collagen 108

5.4.1 Low and high NA focusing 108

5.4.2 High NA with radially polarized beams 110

6 Recommendations 121

List of publications 127

Appendix: Accepted publications 143

List of Figures

1.1 Two-photon excitation and SHG 7

1.2 Phase matching factor against confocal parameter 12

1.3 Vectorial effects of tight focusing 14

1.4 Electric field of a tightly focused linearly polarized beam 16

1.5 Linearly and radially polarized beams 17

2.1 Illustration of a sarcomere 25

2.2 Cross-section of a sarcomere 26

3.1 Schematic of the illumination scheme 43

3.2 Dipole array of SHG scatterers 49

3.3 Induced SHG polarization from collagen for a tightly focused linearly polarized beam 52

3.4 Induced SHG polarization from collagen for a tightly focused radially polarized beam 53

3.5 Induced SHG polarization from myosin for a tightly focused linearly polarized beam 55

3.6 Induced SHG polarization from myosin for a tightly focused radially polarized beam 56

3.7 Far-field radiation of SHG (A) 58

3.8 Far-field radiation of SHG (A) 58

3.9 Far-field radiation of SHG (B) 59

3.10 Far-field radiation of SHG (B) 60

3.11 Far-field radiation of SHG (A) 61

3.12 Far-field radiation of SHG (A) 61

3.13 Far-field radiation of SHG (B) 62

3.14 Far-field radiation of SHG (B) 63

4.1 General layout of a SHG microscope 66

4.2 Beam path through the scan unit and microscope 69

4.3 Schematic setup of spatial light modulator mode converter 74

4.4 Phase patterns for radial polarization 77

4.5 spatial light modulator generated radial polarization 79

4.6 Profile of the radially polarized beam 81

4.7 Bessel-Gauss and Gaussian beam pupil functions 84

4.8 Ratio of |E2 z|/|E2 x| for different pupil functions 86

4.9 Intensity profiles 88

4.10 The variation of |E2 z|/|E2 x| with increasing annulus 89

5.1 SHG and autofluorescence TPF images of skin 93

5.2 SHG and autofluorescence TPF images of normal skin 94

5.3 SHG and autofluorescence TPF images of normal skin 96

5.4 SHG image of collagen with low NA objective (I) 98

5.5 SHG image of collagen with low NA objective (II) 100

5.6 SHG image of collagen with high NA objective 102

5.7 Polarization analyzed images of SHG images of collagen 105

5.8 Resolution obtained with a 60x NA 1.4 oil immersion objective 107 5.9 SHG images with low and high NA objective 111

5.10 Hollow-tubed structure of collagen fibres 112

5.11 SHG imaging with azimuthal, radial polarized beams on trans-versely sectioned tendon 113

5.12 High NA SHG images with radially polarized light 115

5.13 Polarization analysis of SHG signals 117

5.14 SHG images with an analyzer 119

List of Tables

4.1 Back focal plane diameters 715.1 Theoretical and experimental FWHM from SHG 108

This thesis is the result of a long journey first undertaken in 2002 Manyproblems have I met and yet many more people who were kind and helpful

I would like to thank:

Prof Sheppard for his kind guidance and his incredible patience Hehas been a veritable store of information and always ready to explain con-cepts, even basic ones with care and detail I also appreciate and value theimpromptu discussions we have had even when he had a full schedule.Prof Malini for her guidance and support as my co-supervisor on thisproject She has made her time and resources available to me and I amgrateful for that

Prof Raghunath for his support and guidance He has allowed me easyaccess to materials and resources that I would otherwise have had to travelfar and wide to obtain

My parents as they have encouraged me in every way, always ready tosacrifice that one bit more for me to pursue my dream My heartfelt thanks

to them for having given me the grounding and the foundations of lovingbooks

My wife, Karen, for her steadfast belief in me all these 8 years You havebeen supportive of me all this while, never rushing me nor pushing me thoughyou could have had so much more (and those yellow fluff balls that insist oninhabiting my bed)

Last but not least, I also thank God for His unfailing love for me Where

I have fallen and despaired, He has always provided a way for me

I feel glad now that this journey has come to a close But like the traveller

in Frost’s poem I may not tarry long for: “The woods are lonely, dark anddeep,/ But I have promises to keep,/ And miles to go before I sleep,/ Andmiles to go before I sleep.”

AbstractThe polarization dependence of second-harmonic generation (SHG) as a result

of focusing linearly and radially polarized beams with objectives of high andlow numerical apertures (NA) has been investigated

It was found that the axial and transverse components of the electric fieldpresent in a tightly focused beam interact through the nonlinear susceptibilitytensor resulting in SHG signals that have different polarizations In particularSHG from rat-tail tendons sectioned transverse to the long axis of the tendonwas imaged with tightly focused linearly and radially polarized beams Itwas found that an x or y polarized beam generated SHG signals that had acos2 or sin2 dependency on the orientation of the analyzer It was also foundthat a tightly focused radially polarized fundamental beam generated radiallypolarized SHG that was not dependent on the orientation of the analyzer

A scalar approximation that assumed a homogeneous electric field at thefocus did not explain the observed SHG polarization qualitatively Ratherthe vectorial nature of the electric field at the focus had to be assumed so as

to explain the experimental observations qualitatively

It was also observed that a higher NA objective gives better resolution

in general and that the resolution was better for a linearly polarized beamcompared to a radially polarized beam Resolution was also dependent on thepupil function For a tightly focused linearly and radially polarized beam with

a Gaussian and Bessel-Gauss pupil function respectively, the resolution (theFWHM) of the SHG image was found to be 220 nm and 350 nm respectivelyand matches the calculated value of 270 nm and 370 nm

The ability of tightly focused light of different polarizations to excite thedifferent tensor components as well as the dependency of the SHG polarization

on the tensor indicates that it is possible to identify different types of collagenthrough SHG imaging The use of radially and linearly polarized beamstherefore provides a novel way of exciting SHG efficiently when it is notpossible to orient the sample to the position for effective SHG This is a likelysituation in a clinical setting manipulating the polarization of the beam ratherthan the patient is easier This will have implications in the use of SHG andnonlinear optical methods in clinical diagnostics since the polarization state

of SHG is dependent upon the χ(2) tensor, minor changes in the tensor can beanalyzed and detected thereby extending the use and functionality of SHG

While there are many diseases worthy of attention two diseases- hepatitisinfection and myocardial infarction- in the Asia region stand out The firstbecause hepatitis is endemic to Asia and many are infected with the hepatitis

B virus (HBV) Infection with the hepatitis virus can lead to a hardening(cirrhosis) of the liver where it eventually loses its function Individuals in-fected with the hepatitis virus are also more prone to develop liver cancer andmake up 82% of those who are diagnosed with liver cancer annually [1].Thesecond since the rapid development as well as changes in diet and lifestylemark a shift of the types of diseases encountered towards those characteristic

of a developed nation Myocardial infarction (MI) is also potentially fatalboth during and after its occurrence A recent study indicates that differentethnic groups suffer different rates and outcomes In particular Indians have

a higher MI rate but Malays suffer a greater fatality [2]

In both diseases collagen is found to be deposited in excess For the case

of liver cirrhosis collagen type I is deposited till it constitutes up to 50% ofthe total protein, as compared to 1% for a healthy liver The excess collagenprevents the liver from regenerating as type I collagen is not the normalmatrix in which the hepatocytes grow in For the case of a patient who hassurvived, a ‘scar tissue’ in the form of collagen is prevalent in the region wherethe infarction has occurred

This scar tissue poses a potentially lethal problem as an infarct may ture Infarct rupture accounts for up to 30% of deaths within a week of aninitial MI [3] The mechanical properties of the scar tissue is also important

rup-as a stiff infarct results in limited expansion of the heart cavity and fore affects the blood circulation, wastes energy in overcoming the additionalstiffness [4] thereby leading to a either a heart failure or reduced quality oflife

there-Collagen is an important role in the post-infarct time course there-Collagen isdegraded immediately after a MI and the birefringence of collagen changes [5]possibly due to the release of collagenase [4] The amount of collagen laterallyconnecting myocytes also decrease Collagen type IV is observed after 4 daysfor rats along with type III mRNA and type I collagen procollagenase [5]

At the same time type III collagen with its characteristic thinner diameter

is laid down as a mesh for the second highly aligned type I collagen to grow

and provide the structural integrity [6] The type III collagen at this stage

is susceptible to stretch [7] At the same time, cross-linking of collagen underdifferent perfusion conditions [8] have been found to provide different tensilestrengths of scar tissue in rabbits

Since collagen, especially type I collagen, generates a very strong harmonic signal it offers great potential as a minimally invasive diagnostictool both in the clincal and laboratory setting At the same time, SHG hasthe ability to discriminate between different various types of collagen espe-cially types III and type IV, which is important in MI The second harmonicsignal form collagen is based upon the highly ordered internal structure ofcollagen and changes in the SHG signal in terms of intensity and polariza-tion can be exploited to good advantage When used in complement withcurrent single and multi-photon imaging techniques the changes in collagenthat occur throughout the time course of these diseases can be studied withSHG imaging At the same time, the sensitivity of SHG to collagen can beexploited and be incorporated into a multi-photon endoscopic device for earlydetection as well as real-time evaluation of the healing infarct or progression

second-of cirrhosis in the liver

This thesis investigates the effects of focusing linearly and radially polarizedbeams with high numerical aperture (NA) objectives on second-harmonic gen-eration (SHG) Among the nonlinear optical methods utilized in microscopy,SHG and two-photon fluorescence are among the most advantageous This

is because these methods offer the advantage of reduced photodamage as aresult of the nonlinear process, the excitation of endogenous fluorochromesleading to less dependence on toxic fluorescent dyes, typically improved pen-etration with the wavelengths utilized, as well as intrinsic resolution/opticalsectioning as a result of the highly localized excitation volume over which itoccurs over Additionally, processes such as SHG are dependent upon thenonlinear susceptibility of the sample that is indicative of the material andoptical properties It is thus possible to extract further information aboutthe structure of the sample that changes with disease or aging.

Much of the work currently done with SHG microscopy is based on the sumption of a linearly polarized input being focused with a low NA objective.The resolution of an objective is proportional to the excitation wavelengthand inversely proportional to the NA It is therefore apparent that working

as-in the near as-infrared (NIR) spectrum to as-increase the penetration reduces theresolution It is therefore of advantage to increase the NA of the objectivesince that is easier to change than changing the wavelength of the fundamen-tal beam Increasing the NA of the objectives to improve resolution is not

as straight forward as rotating the objective turret It is known that as the

NA increases, the paraxial approximation– which states that the electric field

at the focus is in the same direction as the input– is no longer valid and theelectric field at the focus is no longer homogeneous For tight focusing withhigh NA objectives the electric field at the focus thus consists of components

in the x, y and z directions Since the nonlinear susceptibility of a material

is described as a tensor, the induced SHG polarization is dependent on theincident polarization

We therefore propose that under high NA focusing conditions it is essary to take into account the other components of the electric field otherthan the dominant one as they are able to interact through the second-ordernonlinear susceptibility tensor (χ(2)) We investigate this through the use oftightly focused linearly and radially polarized beams on a common biologicalsample, collagen.

Shine red light through a piece of glass or, if you have it, quartz or some exoticcrystal Now observe the light exiting on the other end You would probablynot need to look at it to be able to tell that the output is certainly nothingmuch to rhapsodize about What has happened is an example of the linearity

of light In all these commonly experienced effects we find that whatever thewavelength we use the wavelength of the light as it passes through mediumtends to remain unchanged unless the material absorbs or reflects certainwavelengths This is a result of the linear assumption of conventional opticsand this dominates at the intensities commonly encountered and used

On the other hand if we pass a beam from an intense laser through aquartz crystal we find that the beam exiting is now composed of two fre-quencies One is at the fundamental (excitation wavelength) and another is afainter frequency at exactly twice that of the fundamental If we could adjustthe wavelength of the fundamental, we will find that the second frequencydetected will move along at exactly twice the frequency of the fundamental.This phenomenon, called SHG, is an example of how the linear assumption of

optics breaking down at high intensities and was demonstrated in 1961 soonafter the invention of the laser [9].

In the linear regime light interacts with the medium that it passes through

in a well defined manner Commonly encountered phenomena such as tion, refraction and dispersion are described by considering the relationshipbetween the electric polarization induced in the medium as given in equa-tion (1.1), where ǫ0 is the free space permittivity and χ is the susceptibility

reflec-of the medium:

P= ǫ0χE (1.1)

It was, however, subsequently found that when an intense beam of lightfrom a ruby laser passed through a crystal of quartz, an optical harmonic atexactly twice the frequency of the excitation wavelength was detected Thiswas the first demonstration of optical second harmonic generation in 1961 [9].The high intensities resulted in two photons of a lower energy combining tocreate a photon of a higher energy This happens through virtual energy levelsand no energy is lost, resulting in the output wavelength being exactly halfthat of the fundamental as shown in figure 1.3 Since then nonlinear optics hasprogressed rapidly and the invention of pulsed lasers has made multi-photonand harmonic generation microscopy accessible for imaging applications.With the invention of the laser other nonlinear optical phenomenon werequickly verified It was not possible to explain all the phenomena throughequation (1.1) and it was observed that higher order terms had to be added

to equation (1.1) that would describe the second, third and higher order

Figure 1.1: The left panel illustrates the two-photon excitation process Twoincoming photons combine to excite the atom to a higher state Some energy

is lost and the emitted photon is of a frequency < 2ω The right panelillustrates the process of SHG The atom is excited to a virtual energy leveland all the energy is emitted at exactly 2ω

phenomena as shown in equation (1.2)

P= ǫ0 χ(1)E+ χ(2)EE+ χ(3)EEE
(1.2)

As seen from equation (1.2) the higher order processes are dependent onthe second (χ(2)), third (χ(3)) and higher order susceptibilities which are,themselves, tensors These tensors govern the way in which different lightcomponents passing through the sample can interact and couple with eachother generating the observed nonlinear phenomenon

The nonlinear susceptibility (χ(n)) is a material property and is described

as an n- th order tensor For SHG we are interested in the second-order

nonlinear susceptibility tensor (χ(2)ijk) The χ(2)ijk tensor is a 3 × 3 × 3 tensorwith 27 elements These elements are constants of proportionality relatingthe amplitude of the induced nonlinear polarization to the product of the fieldamplitudes incident on the medium given by:

χ(2)ijk(2ω = ω1+ ω2) = χ(2)ikj(2ω = ω2+ ω1) (1.4)

The result is that the 27-term tensor reduces to 18 terms The χ(2) tensor can

be then written in a more compact form common in crystallography (crystalaxes are given as X-Y-Z ) as seen in equation (1.5):

P2ω Y

P2ω Z

Another important symmetry is that of Kleinman’s symmetry condition

[11] Briefly, Kleinman’s symmetry states that whenever the frequency is faraway from the resonant frequency of the medium and if dispersion is negligiblethen the indices may be permuted freely without rearranging the respectivefrequencies as shown in equation (1.6).

SHG from bulk material is predominantly forward directed This can beunderstood from phase matching considerations for an efficient SHG process

Being a coherent process, SHG requires the wave vector mismatch ∆k =

k1 + k2 − 2k to be minimized The intensity (I ) of the SHG produced by

a plane wave propagating has been found to vary as a function of the wavevector (k ) and the coherence length L (which is the distance over whichboth the fundamental and second harmonic beams are essentially propagatingtogether) and is given by equation (1.8):

I = Imax

sin2(∆kL/2)(∆kL/2)2 (1.8)where Imax is the maximum second harmonic intensity generated (∆k = 0)

As most crystals that produce SHG are highly birefringent, this wavevector mismatch can quite severely affect the efficiency of the generated har-monic The birefringence difference between the fundamental and harmonicgives rise to the situation where the harmonic eventually decouples with thefundamental resulting in the a decreasing signal One way to overcome this

is by rotating the crystal such that the fundamental beam makes a particularangle θ with the optical axis thereby controlling the degree of the wave vectormismatch [12]

Another method of improving the efficiency is by focusing the beam insidethe crystal [13, 14] The effect of focusing the beam is to reduce the size

of the beam at the focus, resulting in a region of high intensity which isnecessary for nonlinear effects to occur In general the intensity of the qthharmonic with focusing is dependent on the phase matching factor |Jq| given

2π (q−2)!

b∆k 2

From equation (1.11) figure 1.2 indicates that the case of q = 3 (thirdharmonic) is interesting when compared to SHG in the fact that a positivewave vector mismatch is required for maximum intensity to occur In contrast,SHG is more efficient when |∆k| is minimized

−100 −5 0 5 10 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

b ∆ k

Jq

SHG THG

Figure 1.2: Normalized plots illustrating the effect of the phase matchingfactor Jq on the normalized confocal parameter b∆k

In the previous section focusing was mentioned as a factor in improving SHGefficiency It is fortunate that the focusing necessary in a scanning microscopethus also improves the SHG signal By convention it is assumed that focusing

a linearly polarized beam results in an electric field that is homogeneous atthe focus i.e a dominant electric field at the focus in the same direction as theincident beam This is a result of a paraxial approximation The width of thepoint spread function for a scanning microscope that uses a focused opticalprobe is directly proportional to the wavelength λ and inversely proportional

to the NA of the illuminating objective For a SHG microscope the inputwavelength is typically 780 nm upwards in order that the resulting second-harmonic is in the visible range It is thus advantageous to increase the

NA to ‘compensate’ for the reduction in resolution as a result of the longerwavelengths used

As the focusing angle and hence the NA increases, the paraxial mation used to calculate the electric field at the focus becomes less accurate.For objectives at numerical apertures greater than 0.5 the electric field at thefocus is no longer homogeneous and comprises of electric field components ineach of the x-y-z directions [15] The distribution of the electric field in suchcases can be described with the Debye-Wolf integral [15] The effects of high

approxi-NA focusing are illustrated in figure 1.3

For a tightly focused beam the electric field components in the focal regioncan be expressed as equation (1.12) [15]:

−ik2π

Ex(u, v) = −i(I0+ I2cos 2φ), (1.13a)

Ey(u, v) = −iI2sin 2φ, (1.13b)

Ez(u, v) = −2I1cos φ, (1.13c)

Figure 1.3: Effects of focusing in the paraxial limit on (a) linearly, and (b)radially polarized beams The low angles over which the rays bend after thelens mean that the polarization at the focus can be considered as being thesame as the polarization before the lens Effects of high NA focusing for (c)linearly, and (d) radially polarized beams For a high numerical aperturelens, the rays (especially the outer rays) bend over a large angle resulting insignificant axial and transverse components (blue arrows) Away from the axis

of propagation, the linearly polarized beam has an axial component that is πout of phase on either side of the beam This cancels out at the origin Thetransverse component (in the x direction) add up at the origin.The radiallypolarized beam has an electric field that is directed axially at the origin whilethe transverse components cancel out at the origin but are non-zero at pointsaway from the axis of propagation The z axis here coincides with the axis

of symmetry of our collagen sample in some of our experiments

cos1/2θ sin θ (1 − cos θ) J2(kρ sin θ) exp (ikz cos θ) dθ (1.14c)

In equation (1.14) ρ = (x2 + y2)1/2, α the numerical aperture half-angle and

Jn(•) being a Bessel function of the first type and order n

The field at the focus of a high numerical aperture lens is calculated byintegrating over the numerical aperture θ in equations (1.12) and (1.13) It isseen from figure 1.4 (d) that when focusing with a high NA lens, the electricfield has components in the x-y-z directions Of note is the strength of theaxial component for a tightly focused radially polarized beam These effectsare more pronounced at NA> 0.5 for a dry objective [15] Figure 1.4 (a)illustrates that |Ex| at the focal plane is elongated in the x direction and ismaximum on the axis of propagation Figure 1.4 (b) shows |Ey| that is by farthe weakest component of the electric field at the focus Figure 1.4 (c) shows

|Ez| at the focal plane, which has two lobes on either side of the y axis

The presence of a significant electric field component in the axial direction can

be utilized as a means of observing molecular orientation both in fluorescence

as well as in SHG [16–21]

The vectorial nature of high NA focusing as seen in figure 1.4 indicates

Figure 1.4: The electric field (normalized) at the focal plane for a focused xpolarized ( (a) to (c) ) and radially polarized beam ( (d) to (f) ) (a) |Ex|2,(b) |Ey|2, and (c) |Ez|2 respectively at the focal plane (d) |Ex|2, (e) |Ey|2,and (f) |Ez|2.

that it is possible to control the relative strengths of the various components

of the electric fields present The use of various polarizations of illumination,one of which is the radially polarized beam, can be used to give more controlover the polarization in the focal region The radially polarized beam consists

of polarization vectors that are radiating out from the centre of the beam facelike spokes in a bicycle wheel as shown in figure 1.5

Figure 1.5: (a) Shows a linearly polarized beam, (b) shows the radially larized beam, and (c) shows an azimuthally polarized beam

po-The focused radially polarized beam has a strong axial component whentightly focused as seen in figure 1.3 (d) The electric fields in the radial and

axial directions are given by equation (1.15) [22]:

Eρ(u, v) =

Z α 0

cos1/2θ sin θ cos θJ1(kρ sin θ) exp (ikz cos θ) dθ, (1.15a)

Ez(u, v) = 2i

Z α 0

cos1/2θ sin2θJ0(kρ sin θ) exp (ikz cos θ) dθ (1.15b)

The field in the focal region for the radially polarized case depends strongly

on the tightness of focus In the paraxial limit there is no axial component, but

at high NA the axial component dominates In this it differs from focusing of alinearly polarized beam, where the principal transverse component dominatesfor both high and low NAs At the same time, the azimuthally polarized beamhas no axial component both in the paraxial limit as well as the high NA case.The unfocused radially polarized beam has an electric field in the trans-verse plane that is axially symmetric as shown in figure 1.5 The axiallysymmetric electric field gives the radially polarized beam the ability to focus

to a tighter spot according to some measures of focusing performance [22–24] This is not the case for a linearly polarized beam where the focal spot iselongated in the direction of the input polarization In the former case, thebeam can be focused to an even tighter spot by introducing an annulus, whichpotentially increases the strength of the axial component and decreases theside lobes at the same time [23] The methods of generating radially polarizedbeams will be covered later in section 2.2.1

Chapter 2

Literature review

SHG was first observed soon after the invention of the laser In 1961, Franken

et al [9] passed a beam from a ruby laser into a quartz crystal and recordedthe first SHG signal Early theoretical treatments dealt mainly with theinteraction in large crystals or interfaces [11, 25, 26] These early experimentsutilized CW lasers together with long crystals [9, 12, 27–30] Often theseearly experiments were limited to low efficiency in the second-harmonic as aresult of dispersion within the crystals so that the second-harmonic ‘walkedoff’ the fundamental thereby reducing the amount of available energy to thesecond harmonic signal This phenomenon is also thus related to wave vectormismatch (or index mismatch) [9, 12, 27–30] It was found that it was possible

to ‘tune’ the efficiency of the second-harmonic signal by simply rotating thecrystal and making use of the crystal birefringence to minimize the walk-offeffects [12] The effects of birefringence on SHG in uni-axial crystals has beentheoretically dealt with by Midwinter and Warner [31]

Another method of increasing the efficiency of SHG was to focus the beam

into the crystal [13, 14, 32, 33] It was found that significant SHG could

be produced, even under non matching conditions, if the beam was focused[33] In these experiments it was found that it was possible to neglect doublerefraction when l << lawhere l is the interaction length and lais the aperturelength (where the SHG beam separates from the fundamental This condition

is valid in high NA microscopy The position of the focus within the crystalsand the confocal parameter also have a part to play in improving the efficiency

of the SHG as well as the increase of intensity within the focus The increasedefficiency as a result of focusing means that it is possible to reduce the power

of the laser incident upon the sample while still maintaining a high intensity

at the focus

One of the earliest instances of SHG being performed on a biological material

is a study conducted by Fine and Hansen [34], in which they first observedSHG from collagen In 1979 Roth and Freund [35] conducted a study using aQ-switched, Nd-YAG laser focused by a low NA lens on excised rat-tail tendonsoaked in physiological saline In this study they found the tensor coefficient

dzzz of the collagen molecule to be opposite in sign but approximately equal

in magnitude in relation to the other coefficients The ratio of dzzz to theother coefficients varied with age as a result of the increased cross-linkingand changes to the collagen Later studies by Freund and co-workers [36, 37]with SHG also indicated a sharp, intense SHG signal indicating a long-rangepolarity to the fibres over a much broader SHG ‘background’, which was

hypothesized to belong to some fibrils randomly directed in the up/downdirection [36, 37] These findings were interesting as the results could beexplained with the assumption of a C∞ tensor for tendon, as established byexperiments on the piezoelectric effect in tendons [38] This study was one

of the first to suggest nonlinear optics as a means of studying the intrinsicstructure of biological tissue More papers followed describing also the issue ofcollagen polarity in tendon [35–37, 39–41] At this point, it is still undecided

if the polarity in collagen serves any biological purpose [35–37, 42]

The high power and long dwell times required to build up a sufficientlystrong signal consistently made SHG a method with limited applications

Up till then SHG had been primarily used as a method for experimentalverification of the theoretical behaviour of light in the nonlinear regime or forgeneration of new frequencies with an initial longer wavelength beam, ratherthan as an imaging tool The first instance of incorporating SHG as animaging tool in microscopy was in 1974 when Hellwarth and Christensen [43,44] performed the first SHG imaging with a microscope The availability ofthe laser scanning microscope in 1977 made imaging with lasers and large scanareas more efficient and Sheppard and co-workers consequently demonstratedthe first images from an SHG scanning microscope [45, 46] As a result,the scanning technique is highly preferable for SHG as well as other nonlinearmicroscopy techniques The high power required from a CW laser still limitedthe use of SHG in microscopy This eventually changed with the introduction

of the ultrafast pulsed laser, where the short pulses allowed for a lower overalltime-averaged power while still delivering a high power per pulse The SHGintensity and efficiency is increased through the use of focusing, allowing

for a very localized focal volume in which the nonlinear interactions occur.Rapid development in the application of SHG and other nonlinear methods formicroscopy and imaging followed after this and the first report of the ultrafastpulsed laser for SHG in imaging was by Guo et al [47, 48] With the pulsedlaser, SHG microscopy quickly became viable as a tool for biologists andmicroscopists alike since the short pulses could ensure high power deliveredper pulse while maintaining a low incident average power thereby limitinglaser damage Combined with conventional or two-photon fluorescence, thisdramatically improved the information that could be extracted from the data.

So far SHG microscopy has proved to be a useful tool for imaging intrinsicsignals of endogenous proteins such as collagen These proteins are highlyordered on the molecular level and have large birefringence Examples of suchproteins are myosin, collagen and tubulin As with conventional inorganiccrystals, SHG up-converting biological materials cannot be centrosymmetric

One important biological material that is an efficient up-converter of SHG iscollagen Collagen is an almost ubiquitous protein that exists in over twentydifferent variants and practically everywhere in the body The basic function

of collagen varies from load-bearing structures in the bones and tendons toforming the delicate extra-cellular matrix on which cells sit on The basicstructure of collagen is a triple helix structure and has been experimentallyverified through X-ray diffraction studies [49–51] Collagen is first produced incells (fibroblasts) and is secreted from the cells where they form tropocollagen,

the basic monomer of collagen The simplest form of the monomer is Gly The monomers at this stage are soluble in water or physiological saline.These monomers subsequently join end-to-end to form simple chains, and anindividual chain joins to another chain through intermolecular cross-links toform the triple helix chain At this stage the triple helix collagen chains areknown as collagen fibrils Collagen fibrils at this point are more stable than

Gly-Pro-in the tropocollagen form and only dissolve Gly-Pro-in weak acid but not Gly-Pro-in water.Experiments with polarization microscopy and side illumination also revealsthat collagen in its crystalline form is birefringent

Of the over twenty variants Type I collagen is one of the most widespread

It exists in the skin, bone as well as tendons In tendons the collagen fibrilsform a coiled-coil and, as a result, are highly ordered in the direction of theapplied stress This in contrast to skin where they are less ordered sinceskin, unlike tendon, functions as a protective layer rather than being load-bearing The orientation and structure of collagen is thus important Collagen

is known to ‘age’ as it matures from monomers, through tropocollagen tocollagen

In studies involving collagen and SHG researchers have typically looked

at rat-tail tendon [35–37, 39–41, 52, 53], skin [54–58], teeth [59], tissue neering constructs [60], arteries [53, 61], colon [53, 62] and tumours [63] This

engi-is because collagen engi-is present in almost all parts of the body and also because

it generates second harmonic signals very efficiently

The alignment of collagen in tissue has also been investigated by Stoller

et al [39–41] through polarization analysis of the emitted SHG signal,demonstrating the sensitivity of SHG to changes in collagen orientation By

measuring the polarization dependence of the SHG signal relative to the anglethat the incident beam polarization makes with the collagen fibril axis theywere able to discriminate between different patterns of the collagen orienta-tion Similar experiments have also been conducted on human skin and theresults indicate that skin is oriented differently at different depths [57, 58].The results from these experiments indicate the usefulness of SHG as a toolfor identifying the orientation dependence of the fibrils in relation to the func-tion of the organ Brown et al [63] have also shown the usefulness of SHG

as a diagnostic tool for early detection of cancer based on the sensitivity ofSHG to collagen that is upregulated with the development of cancer SHGhas also been noted to distinguish between various types of collagen, notablyType I and Type III [62] as well as Type IV [64]

Another protein that forms a coiled-coil structure is the actin/myosinstructure found in muscle The smallest active unit of a muscle is the sar-comere A sarcomere consists of thick filaments, thin filaments and proteinsthat serve to hold the thick and thin filaments in position or to regulate thecontractile motion of the muscle The sarcomere therefore has a characteristic

‘banded’ pattern that is visible in polarized light microscopy as a result of themixture of thick and thin filaments as shown in 2.1 The banding that occurs

in a sarcomere can be categorized in dark (A-bands) and light (I-bands) TheA-band itself is subdivided into: the M-line, which is the centre of the sarcom-ere and consists of proteins that stabilize and join thick filaments together;the H-zone, which consists only of thick filaments and the zone of overlap,which contains both thick and thin filaments The I-band contains only thinfilaments and no thick filaments, and overlaps with the adjacent sarcomere

as shown in figure 2.1 Thin filaments are 5-6 nm in diameter and about 1micron in length, and are made up of actin filaments that are twisted around.

In contrast, thick filaments are primarily myosin molecules with heads ure 2.2 illustrates the cross-sectional structure of the thick and thin filaments.The arrangements of the thick and thin filaments within the sarcomere results

Fig-in a hexagonal packed structure that also generates SHG [54, 65–68]

Figure 2.1: Illustration of a sarcomere indicating the different zones.The detected SHG from actin/myosin structures indicates the overall or-der and structure on the micro and macro scale of myosin Campagnola et

al [68] performed SHG microscopy on mutants with a missing gene that rupted sarcomeres, and found that the SHG images indicated a loss of orderand a weaker SHG signal overall More recently, advantage has been taken

dis-of the highly localized region in which SHG occurs Several groups [66–68]have looked at the differences in muscle filaments both in the resting and ac-tive stage and found that they were able to discriminate between both statesthrough measuring the gap between sarcomeres It was also found that SHG

Figure 2.2: Cross-sectional view of a sarcomere along the various zones.Cross-section across the (a) Z-line with red circles representing the thin fila-ments, black circles are the titin attachments and the red lines are the actininfilaments; (b) M-line with the thick filaments represented by the blue circles;and (c) zone of overlap Note the hexagonal symmetry formed by 6 thinfilaments around a thick filament.

was able to discriminate to a 20nm accuracy through SHG microscopy [67].The polarization dependence of SHG from muscle with respect to the funda-mental polarization has been examined and used to identify the various tensorcomponents [65] This sensitivity to coiled-coil structures has also identifiedmicrotubules as another protein that is an upconverter of SHG.

Microtubules including those found in the spindles during the interphase

of cell division [53, 54, 69] or in the brain [69] have been imaged clearly withSHG microscopy An added advantage is that the nature of SHG has made itpossible to detect a directionality in the imaged microtubules In interphasecells or during cell division, the interdigitating microtubules exhibit a darksection that does not emit SHG as a result of the opposite polarity of themicrotubules [54] More recently, action potentials have also been recordedusing SHG generated from special dyes that react to action potentials byaligning themselves in the cell membrane thereby disrupting the interfacesymmetry [70–73]

One of the advantages of nonlinear optical methods in imaging is the factthat repeated scanning results in comparatively less photobleaching whencompared with single-photon fluorescence as a result of the highly localizedvolume of excitation At the same time, the longer wavelengths typicallyused allow a deeper penetration into thick samples Being a nonlinear process,there is also very little energy deposition in the case of two-photon microscopyand virtually none for SHG, since it involves transitions to virtual energy

levels It is thus possible to image for extended periods of time withoutthe associated photobleaching for single photon microscopy [74, 75] Theprocess is also highly localized due to the dependency on the intensity of theincident beam In SHG, for example, the process drops off as the square ofthe intensity involved This makes optical sectioning an intrinsic property ofnonlinear optical methods As a result it is not necessary to employ a pinhole

to perform z -stack imaging as in confocal microscopy, although the pinholecan help with improving resolution [76]

There are, however disadvantages to SHG microscopy It turns out that,unlike fluorescence, SHG is predominantly forward directed [39, 62, 77–79]making it difficult to implement as an in-vivo diagnostic tool The coherentnature of the process also means that SHG is dependent on phase matching[62, 80] The propagation of the SHG is thus found to propagate as a cone[77, 78, 81, 82] and requires a condenser of equal or higher NA to that of thefocusing objective in order to collect the signals efficiently

Arising from a process that is dependent on the macro and microscopic ternal structure of the media, the detected SHG reflects this property throughits polarization This polarization is a direct result of the nonlinear suscep-tibility tensor χ(2) that describes the optical properties and is similar to thepiezoelectric tensor The polarization behaviour is simplified with the use ofbeams focused with low NA objectives The effect of such low NA focusing oflinearly polarized light is to approximate the electric field in the focal region

in-to having only one dominant component (in the direction of polarization).The χ(2) tensor is then very simply reduced to a few terms and the rela-tive magnitudes of the tensor coefficients can be found through polarization

analysis [35, 39, 40, 65, 65, 68, 79] As the NA of the objective increases itbecomes necessary to use a vectorial theory to describe the electric field at thefocus [15] Using such a theory it becomes apparent that with high numericalaperture objectives the axial component also becomes significant for the case

of a linearly polarized beam In certain media (e.g collagen) the χ(2) tensorcontains terms that we call ‘cross-components’ i.e the ‘off the principal diag-onal’ terms in the tensor These terms basically describe the induced SHGpolarization as a result of two orthogonal electric field components e.g EXEY

It will be thus possible to determine, experimentally, certain components ofthe tensor coefficients or identify new ones that would be difficult to detectwithout having to rotate the sample [19, 35] At the same time the use of high

NA focusing can also ensure optimal or efficient generation of SHG by havingthe fundamental polarization coincide with the large terms of the tensor whenorientation of the sample is not possible

In order to do so it is necessary to be able to control the strengths of thevarious components of the electric field at the focus For example, it might be

of advantage to have an electric field distribution at the focus with a strongaxial component as the strong axial component will help in the resolution

of the components of the tensor coefficients along the axis One possibleway to achieve such polarization control at the focus is through the use ofvarious optical elements like an annulus [83–85] or parabolic mirrors [84],

or more recently, through the direct control of the polarization of the beamwith liquid-crystal spatial light modulators (SLMs) in order to generate novelpolarization states (e.g radial polarziation) [19, 86–90] This will be dealtwith in the following section 2.2

2.2 Polarization and focusing of light

Under a low numerical aperture condition, the electric field at the focus can

be approximated to be homogeneous Thus a linearly polarized beam remainslinearly polarized at the focus when focused with a low NA objective Onthe other hand tight focusing of light leads to a depolarization of the electricfield at the focus [15] As a result the electric field at the focus is ‘vectorial’with electric fields in the transverse and axial directions The difference inexploiting polarization in the two transverse directions, together with therelatively strong axial component of tightly focused light, has allowed thesevectorial properties to be used for three-dimensional orientation mapping ofmolecules [16–20, 91]

For linearly polarized light, the axial component of the electric field is still

an order of magnitude lower than that of the dominant transverse component

In order to obtain a case where the axial field is much stronger than that ofthe transverse fields, it is necessary to control the polarization state of thelight One of the more interesting types of polarization is that of the radialpolarization The polarization state is best described as the ‘spokes of awheel’ where the polarization points outward at all points in the beam face

as seen in figure 1.5 The advantage of such a beam is a potentially smallerfocal spot as well as a strong axial component [22–24, 92, 93] A smallerfocal spot means that is be possible to improve resolution [23, 94] A strongaxial component has been proposed to be useful in a range of applicationsincluding laser cutting of metals [95], particle acceleration [96, 97] and as anoptical trap [98, 99] In the field of microscopy, the strong axial component

of the electric field can be used for three-dimensional orientation mapping ofparticles/dipoles [16–20, 91], increased SHG from metal surfaces [100, 101]and step-discontinuity imaging in a dark-field imaging [102].

Recently, several groups have been investigating tight focusing of radiallypolarized beams [22, 23, 92, 115] Tightly focused radially polarized fieldsexhibit a strong longitudinal field component Cicchitelli et al [97] discussedthe use of the longitudinal field component for particle acceleration as didFontana and Pantell [112] who focused radially polarized light through anaxicon Xie and Dunn [116] have used the different field components to excitedifferent orientations of fluorescent molecules Imaging with three beams withdifferent polarization states can be used to investigate the three-dimensionalorientation of molecules [117]