This paper examines the use of Gauss-Laguerre beams in STED microscopy. These types of beams are shown to have beneficial properties that can be utilised to generate stable, high quality STED beams resulting in an aberration-resilient generation volume. In this paper we obtain general expressions for Gauss-Laguerre beams being focused through a stratified medium and describe their optimization for STED microscopy purposes. Our results show that the circularly polarised, lowest order “dark” beam is the most beneficial for STED purposes.
©2004 Optical Society of America
The idea of STED microscopy was suggested by Hell and Wichmann in 1994  well before the first experimental demonstration  of the method was published. In essence the technique is based on ordinary fluorescent excitation of molecules and their quenching by stimulated emission outside the probe volume . With the use of this method 28nm optical resolution has been demonstrated  experimentally.
A typical experimental set-up, based on a confocal scanning fluorescence microscope, realizing STED operation would consist of excitation by ≈100fs laser pulse at 558nm wavelength . This linearly polarized radiation is incident upon a microscope objective lens which creates a tightly focused laser beam. The numerical aperture of such a lens is usually as high as possible – typically 1.4 at oil immersion or 1.2 at water immersion. As a result of illumination by this lens the usual excitation volume forms. In order to reduce this volume, and hence create the generation volume, the sample is double-illuminated; this time with an ≈40ps laser pulse at 775nm wavelength (the STED beam). Ideally the intensity distribution of the STED beam is such that it is uniformly high everywhere apart from the focus point, where it is zero. The STED beam depletes the original excitation volume everywhere apart from those locations where it is dark and so a small generation volume results.
There seem to be a number of STED beams suggested in the literature  that can be divided into two broad categories: one based on introducing aberrations in a well-controlled manner and the other one based on placing two, semi-circular phase transparencies into the STED beam. Although the latter type beam, that was called a “donut” beam by the authors of Ref. , is loosely related to what the literature refers to as a “dark” or “singular” beam, no publications seem to be present in the literature to analyse the general behaviour of a class of vector beams called Gauss-Laguerre beams , and in particular their use in conjunction with STED microscopy. The purpose of this paper is to fill this gap.
The paper is constructed as follows. First we derive general expressions for vectorial Gauss-Laguerre beams focused by high numerical aperture lenses. The expressions are of general applicability, including the case of focusing through stratified media. Then we examine how these beams behave under various conditions of focusing via numerical examples. Finally we specialize our findings to STED microscopy.
Consider a right handed Cartesian co-ordinate system and a high numerical aperture, high Fresnel-number lens placed such that its Gaussian focus coincides with the origin of the coordinate system, as shown in Fig. 1.
On the left-hand-side of the lens, light emerging from a point source, which in practice may be modelled as an harmonically oscillating electric dipole, is collimated by the collimating lens. The plane wavefront emerging from this lens is then incident upon a diffractive optical element (DOE) which generates the required order, or possibly a sum of, Gauss-Laguerre beam(s). To realize this in practice one might choose to follow the generation method described by Paterson and Smith  or Khonina et al. . The DOE is placed at a distance f from the principal plane of the high numerical aperture lens. This in turn means that the physical aperture of the DOE also becomes the aperture stop of the objective lens and hence the objective lens is telecentric from the image side. The electromagnetic field is to be computed at P(r p ). In this work we use both cylindrical and spherical co-ordinates for which the (ρ,ϕ,z) and (r,ϕ,θ) generic notation will be applied, respectively.
The general form of an in-focus Gauss-Laguerre beam is given by :
where w 0 is the beam diameter at waist and (x) are the Laguerre polynomials that, for numerical purposes, can be efficiently computed by upward recurrence.
It is clear from Fig. 1 that ρ=f sinθ which means that the ratio ρ/w 0 can readily be written as
where a is the aperture radius. The quantity F shall be referred to as the fill parameter and it effectively tells us what fraction of the beam is inside the physical aperture. When F<1 the aperture is said to be overfilled, that is a large diameter Gaussian does not fit the aperture. On the other hand, when F>1 the aperture is underfilled, which means that a small diameter Gaussian fits well the aperture possibly including its side lobe structure.
Focusing by the high numerical aperture lens is described with the help of the Debye-Wolf integral :
where s=(sx , sy , sz )=(cosϕ sinθ, sinϕ sinθ,cosθ) is the (unit) ray vector (note that this is not the one shown in Fig. 1), E(sx , sy ) is the electric field of the geometric optics approximation in the aperture plane, and the constant multiplier -ik/2 has been omitted. We now approximate the field emerging from the DOE by assuming that whatever the polarization of the incident plane wave was before incidence is maintained and the field distribution is perturbed as Eq. (1). Note that this approximation only holds in certain cases and it will fail most notably when the z component of the electromagnetic field emerging from the DOE is significant when compared with the transverse components. However, a nonvanishing z component in the aperture plane E(sx , sy ) would result in a not fully transverse electric field vector on the surface of the Gaussian reference sphere. Hence, such an input field would violate the basic conditions for the applicability of the Debye-Wolf formula .
Keeping this approximation in mind we may write E(ϕ,θ) with the help of the generalised Jones matrices  and tracing the ray corresponding to the lower half of the optical system as:
with E in =(1,0,0) for an x-polarized beam and E in =(0,1,0) for a y-polarized beam. When the light incident on the DOE is circularly polarised we may write:
where R, L are rotation matrices corresponding to the given lens and
is the generalised Jones matrix corresponding to a left-handed quarter waveplate. For x-polarized incident illumination the above formulae give the following expression for the electric field in the vicinity of the focus:
and for left-handed circularly polarized incident illumination:
where we have used the identities:
where Jm (.) is the Bessel function of the first kind, order m and
We note that the term appears to account for the aplanatic projection due to a lens satisfying Abbe’s sine condition. A perhaps simpler notation may be used with the introduction of
so now Eq. (5) is given by:
and Eq. (6) is given by:
For completeness the right-hand polarized field is given by:
Clearly the above expressions only apply in the case of a homogeneous image space. We now give extended formulae for the case of focusing into stratified medium when the last medium is semi-infinite. The principle of the solution is discussed elsewhere  so we shall not detail how they are obtained. Let is suffice that in case of a stratified medium we need to replace the functions , etc. by the following integrals:
being the initial aberration function,
with βl =kl (hl -1-hl )cosθl and D given in Ref. . In the above formulae we denoted the axial locations of the subsequent interfaces by -h 1,-h 2, …,-hN -1, the refractive indices by k 1,k 2,…,kN of the N media, and the Fresnel transmission coefficients by for transition from the j to the j+1 medium. We also used the convention of subscript j to denote directions and co-ordinates in the j-th medium. Of course, on setting all wavenumbers k or the number of interfaces to zero these formulae reverts to Eq.(8).
3. Numerical results
In order to reduce the available degrees of freedom in the theory we initially confine our attention to Gauss-Laguerre beams focused into water by a high numerical aperture lens corrected for the nominal focusing depth. Note that there is no need to complicate our present calculations by considering that the illumination is of pulsed nature because above ≈40fs pulse width this would not make a difference in terms of determining the generation volume  anyway due to the narrow width of the corresponding spectrum. As stated before, our intention is to examine the practical applicability of the Gauss-Laguerre beams in conjunction with STED microscopy.
Before we embark upon a detailed analysis we wish to point out the strong dependence of the STED process on polarisation  meaning that the STED beam needs to be co-polarised with the excitation beam in order to achieve maximum depletion efficiency.
Let us denote the electric vector field of the STED beam by E STED and that of the excitation by E exc. We start by analyzing the Cartesian components of E exc, as shown in Fig. 2. These distributions were computed from the corresponding equations of Ref.  and they represent the modulus square of the field amplitude in the focal plane. The significance of this graph is that fluorescent excitation is proportional to the plotted quantities. This means that, for example, a fluorescent molecule situated in the focal point would get excited such that the resulting dipole moment is fully x-oriented. Away from the focal point the fluorescent excitation decreases and the resulting dipole moment is no longer purely x oriented. For example, by moving away from the focal point in the x lateral direction, the orientation of the induced dipole moment swiftly becomes 45° measured from the focal plane with strength apparently lower than in focus.
In order to effectively reduce the generation volume we need to devise a STED beam whose electric field is zero in the focal point, but swiftly becoming non-zero when one moves away from the focus. In addition, its polarisation should coincide with that of the excitation beam. Klar et al.  have suggested such a beam by imposing a pupil-plane phase distribution. An alternative approach is using an x-polarized Gauss-Laguerre beam with m>0.
When such a beam is focused by a high numerical aperture lens, the Cartesian components of the electric field can be computed from Eq. (10) and the resulting field distributions are shown in Fig. 3. The figures reveal that both the x- and z-polarized components of the STED beam could efficiently reduce the x-polarized component of the excitation illumination. However, it is also apparent from the figures, and from observations that can be confirmed by additional computations, that the generation volume produced by focusing an x-polarized Gauss-Laguerre beam is not rotationally symmetrical.
A rotationally asymmetrical generation volume is sometimes disadvantageous in microscopy and so a possible further aim could be to make the volume rotationally symmetric. This can be achieved if, instead of a linearly polarized light, circularly polarised Gauss-Laguerre beam is focused. Equation 11 readily evaluates to give the Cartesian components of the electric field vector in the vicinity of the focus as shown in Fig. 4. Note that because of the time-evolution of the circularly polarized light in this case the electric field vector does not coincide at all times with that of the excitation beam. However, the 40ps STED pulse contains ≈1.6×104 cycles of the transverse component of the electric field vector turning a full circle about the z axis as shown in Fig. 5. This in turn means that dot product E STED ·E exc reaches its maximum ≈1.6×104 times within the 40ps light pulse, i.e. it can clearly interact with the fluorescent molecules to cause stimulated emission.
Having established that a circularly polarised Gauss-Laguerre beam is the better choice for the STED beam, we now turn to optimize this beam in order to give the smallest generation volume. First we examine the behaviour of the focused beam as a function of the fill parameter F. We note that the role of F can readily be understood from the following comparison of the present case to e.g. when a Gaussian beam is focused. In the latter case, when the aperture is underfilled vignetting occurs that results in a reduction of the effective numerical aperture. Overfilling the aperture results in reduction of the full width half maximum (FWHM) of the lateral distribution as the aperture function reverts from a Gaussian to a uniform pupil, and also some reduction in peak power. In case of focusing a Gauss-Laguerre beam the expectation is that underfilling the aperture will result in the same behaviour as for a Gaussian beam.
Overfilling the aperture, on the other hand should have a more severe effect. The reason for this is that the beam has a lateral maximum when ρ≠0 so once the beam is sufficiently wide, even though the FWHM of the focal distribution improves the light power in the vicinity of focus will severely deteriorate. This reasoning is confirmed by the results shown in Fig. 6. The figure reveals that in order to achieve the maximum intensity the fill factor needs to be set at F=0.94 regardless the lens convergence angle. Clearly, the best FWHM would correspond to a fill factor that is almost zero. However, for a Gauss-Laguerre beam for which m>0 this is an impractical proposition because this in turn would lead to vanishing focused intensity. It remains to be noted that our consideration is based on the fact that the laser power remains constant. However, under certain experimental conditions when the laser power is not a limiting factor it is always possible achieve better resolution by increasing the laser power to compensate for the loss.
We now examine the axial behaviour of a focused Gauss-Laguerre beam. Figure 7 shows the evolution of the x-z distribution of the transverse component of the time-averaged electric field when the fill factor is varied from 0.1 to 2.0. The movie shows that with increasing fill factor F, the axial distribution of the intensity becomes less confined. However, this behaviour is advantageous in STED microscopy in case of an axially less confined excitation beam.
It is interesting to consider focusing of higher order Gauss-Laguerre beams. It is quite apparent when graphical representations  of the unfocused Gauss-Laguerre beams are inspected that in general the beams seem to be the most confined when both m and n have the lowest value. Upon increasing either m or n the energy contained in the beam spreads. Let us define the encicled energy Wm,n (a) that is carried by an unfocused Um,n beam by
within the area delimited by a circle of radius a. Note that in obtaining the above expression we have used Eq. (1). Equation 13 is plotted as a function of the normalized circle radius a/w 0 in Fig. 8.
This figure reveals that in fact for a/w 0<0.7, corresponding to F=0.7, U 1,3 and U 1,5 carry fractionally more energy than U 1,1. However, any larger F values the U 1,1 beam carries significantly more energy than either the U 1,3 or U 1,5 beams. A perhaps surprising result in Fig. 8 is the fact that the FWHM values of the focused U 1,3, U 1,5, U 2,4 and U 2,6 beams can become smaller than that corresponding to the U 1,1 beam. At the same time it appears that at those fill factors when this occurs the encircled energy is higher that that corresponding to the U 1,1 beam. However, for fill factors larger than unity secondary lobes of the Gauss-Laguerre beam are no longer cut by the aperture. When this happens they form small secondary peaks, as seen in the inset of Fig. 8, close to the optic axis after focusing. Their intensity, however, is so small that it is negligible when compared to the main peak. Nevertheless, when our program calculated the FWHM curves it automatically found these small peaks. We can also confirm the presence of these lobes of the unfocused beams in the encircled energy diagram; the sag on any given curve corresponds to a secondary peak appearing within the aperture This in turn corresponds to the radius of maximum intensity which scales with root m as pointed out before by Padgett and Allen .
Consequently, we may deduce from the figure that a fill factor of F<0.75 seems to achieve a good FWHM value with high power efficiency. Of course, as pointed out above, the FWHM values can be improved if, e.g. F=0.85 is used with the U 1,5 beam. However, the peak intensity due to the secondary lobe will be reduced by a factor of 8.3× when compared to that due to the main peak of the same beam. Ultimately, the light budget of a given experimental setup will determine whether this is a permissible loss of laser power.
We now move to discuss the effect of spherical aberration on the STED generation volume. Of course, when a water immersion lens is used and the refractive index of the specimen is very close to that of water significant spherical aberration will not occur. However, in cases when the sample is sandwiched between the microscope slide and a coverslip spherical aberration does occur due to the mismatched refractive indices of the different layers. Since spherical aberration affects both the excitation and the STED beams it is no longer possible to only analyze the latter one. For this reason, we introduce a metric that is a possible way of modelling the STED volume. Let the STED generation volume metric (I SV) be given by:
Here, the maximum cross section for stimulated emission σmax is a dimensionless quantity in contrast to its usual m2 dimension. This is so because α n is normalised to a value between 0 and 1. The above expression is a general STED generation volume metric: it permits us to calculate the STED generation volume when n STED beams are used. For the case when the STED beam does not saturate σmax=1. Because it would be difficult to carry out a meaningful analysis valid for all cases we initially assume that the STED beam does not saturate, that is σmax=1. The consequences of this assumption, and the behaviour of the Gauss-Laguerre STED beam under saturation will be discussed at the end of this section.
We first use the STED generation volume metric to analyse a typical STED beam aberrated by a glass (n=1.55) to water (n=1.33) interface. Such a beam, defined for example in , is created using a phase function of the form ΨSTED(θ,ϕ)=sign(sin(θ-θ 0))π/2 with θ 0=arcsin(sinα/(2)) where ϕ and θ assume the same roles as in Eq. (1) and Eq. (8) respectively. We employ an excitation beam linearly polarised in the x direction and an objective with NA of 1.4.
In the following simulations the position of the interface is progressively varied from the unaberrated Gaussian focus to a position closer to the objective lens. We refer to the displacement of the interface relative to the unaberrated Gaussian focus as the depth of the water layer or focusing depth. The interface is depicted in each animation with a partially transparent blue surface. Figure 9 shows an animation which demonstrates how the STED generation volume varies as the depth of water is increased from 0 to 15µm. It reveals that the STED generation volume becomes larger and begins to break up as the depth of water increases. This occurs because the phase function used to produce the STED beam is severely perturbed by the aberration introduced by the glass-water interface.
Consider now using a circularly polarised, m=1, n=1 Gauss-Laguerre beam as a STED beam. The plots in Fig. 6 reveal that using F=0.5 and NA of 1.4 will provide an acceptable, although not as bright as possible, FWHM and peak intensity in the STED beam. Figure 10 shows an animation which demonstrates how the STED generation volume for the Gauss-Laguerre STED beam varies with the depth of water. In this case, the shape of the STED generation volume remains reasonably constant and stable as the depth of the water is increases. It is noticeable, however, that the STED generation volume is quite long in the axial direction.
The STED generation volume may be further reduced by employing a second STED beam directed along the y-axis of the image space. A circularly polarised m=1, n=1 Gauss-Laguerre beam is once again employed however this time a convergence angle of α=17° is used in order to effectively reduce the axial length of the STED generation volume. If the NA of the second beam is too high then the multiple peaks in the aberrated excitation beam will cause the STED generation volume to break up. For this simulation the second STED beam is assumed to be focused in to 10µm of water and is shifted along the z-axis of the image space so as to be aligned with the aberration induced offset of the other beams. The results of this simulation are shown in Fig. 11. This animation demonstrates how the STED generation volume may be substantially reduced by employing a second rotated illumination objective. Note that this configuration should not be confused with multiple objective observation reported in the literature [16, 17].
The animations in Figs. 10 and 11 show how a Gauss-Laguerre beam may be employed to produce a STED generation volume which is robust to aberration. This is so because Gauss-Laguerre beams have a singularity of phase at the centre of the beam. This singularity is not destroyed by the aberration of the interface. It is thus the presence of the phase singularity that makes Gauss-Laguerre STED beams in terms of spherical aberration more robust, especially in the lateral direction, than conventional STED beams.
Finally we move to examine the effect of saturation on the Gauss-Laguerre beam. With reference to Fig. 12 it is clearly visible that the unaberrated conventional STED beam (dash-dotted line) exhibits a zero intensity in focus. When the focusing depth is increased to 2µm the conventional STED beam (continuous line) is shifted and the intensity in focus is no longer zero (indicated by an arrow). Conversely, the intensity of the Gauss-Laguerre STED beam (dashed line) along the optic axis stays always zero no matter how much aberration is present.
The consequence of this behaviour is severe. When the conventional STED beam is saturated, which is the normal operating mode of this type of microscopy, as long as there are no aberrations present the STED generation volume will form as usual. When, however, aberrations are present the intensity in the original minimum is no longer zero. When the conventional STED beam is saturated this non-zero intensity level will elevate which in turn results in the STED generation volume to change location or in severe cases to disappear completely. On the other hand, because the Gauss-Laguerre STED beam always remains zero along the optic axis, saturation will not elevate the intensity here. Consequently, the Gauss-Laguerre STED beam performs in an ideal manner especially under realistic experimental conditions, when the STED beam saturates.
We have presented an analysis of Gauss-Laguerre beams focused by high numerical aperture lenses and their applications in STED microscopy. Because these beams are extremely aberration-resilient they can be used to generate STED generation volumes that might be better confined under extreme aberration conditions than their conventional counterpart. We have established that the first order, circularly polarized Gauss-Laguerre beams are the most beneficial for STED purposes. We have shown that the Gauss-Laguerre STED beam has a superior performance under realistic experimental conditions.
Part of this work was supported by the European Union within the framework of the Future and Emerging Technologies-SLAM programme.
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