Optical Microscopy

In many areas of natural sciences, optical microscopes play a major role. Their ability of magnification and therefore to allow for looking at very small things and seeing details that are not visible with the naked eye makes them an indispensable tool for scientists. They are used so often, that they became a symbol of scientific research [150]. Here, an overview on the basic operation principle and properties of optical microscopes is given.

The easiest way to achieve magnification is to use a simple magnifying glass. It increases the angle α under which light from different points of an object hits the eye, what forms a virtual enlarged image as shown in Figure 4.1 (a,b) [151]. In contrast to a magnifying glass (or more complicated variants thereof, which may involve more elements), a compound microscope consists of two distinct parts: an objective and an eyepiece. The objective is used to collect the light from a sample and form a real intermediate image inside the microscope, which is in turn imaged by the eyepiece. A sketch of this is shown in Figure 4.1 (c).

4.1.1. The Microscope Objective

In this thesis, mostly other variants of microscopes, like a confocal microscope (see Section 4.1.3), are used which have no eyepieces and direct the light they collect directly onto a detector. In any case, the most important component is the first lens system, the microscope objective. It directly determines which fraction of light is collected and how good the quality of an image can be. Modern microscopes

f f a

b

α

αm

c αm

f1 f1 f2 f2

Figure 4.1.: Magnifying glass and microscope. In (a) an observer at the right is looking at an object without additional optics. The angle α between rays from different point of the object is a measure for how big the object looks. In (b) a magnifying glass is applied. Light from the object is refracted what leads to a larger, magnified αm. In (c) the principle of a microscope is sketched. A first lens (objective) collects light from the object and forms a real image, which is then images by a second lens (ocular) forming an enlarged virtual image.

usually use so called infinity corrected systems in which the microscope objective, as introduced in Figure 4.1 (c), is split in two parts: One part collecting the light and sending out a collimated beam (i.e., focussed to infinity) which is still called objective and another part, the tube lens, which forms the intermediate image.

In this thesis, only this system is used due to the great advantage in flexibility it offers. A selection of the important properties of microscope objectives in this sense is reviewed in the following.

• Numerical aperture. The numerical aperture (NA) is given by [152]:

N A=nsinα, (4.1)

with n being the refractive index of the surrounding medium (n= 1 for air objectives andn >1 for immersion type objectives) andαbeing the maximum angle with the optical axis, that light stemming from the focal point can have while still being collected. An increased angle α leads to increased photon collection, therefore, to collect many photons, the NA for a fixed immersion medium should be as high as possible. Typically, for air objectives, NAs as high as 0.95 are commercially available. Using them, about one third of the full solid angle is covered.

• Magnification. For an infinity corrected objective, the quantity magnifica-tion only makes sense with the corresponding tube lens, since the objective

4.1. Optical Microscopy

alone does not form an image. What can lead to confusion is that different companies use different focal lengths for the tube lenses in their microscopes.

Hence, a more useful quantity is the objective’s focal length. It can be calcu-lated by treating the system as telescope. Using the equation for a telescope’s magnification [151] leads to:

f_objective=f_tube/M, (4.2)

with f_objective as the objectives focal length,f_tube the tube lens’ focal length as intended by the manufacturer and the M the nominal magnification of the objective.

• Point spread function. The point spread function (PSF) is the pattern, which is illuminated or observed by a lens (or system of lenses) at its fo-cal plane, provided there is an incident collimated beam or a point emitter, respectively [153]. This generally three-dimensional function is often approx-imated in two dimensions as an Airy function, the so called Airy disk. This is only a a valid approximation in the paraxial case where the NA is small (see [152] and [153] for details). For ideal imaging systems, the PSF only is dependent on the NA and the wavelength. If this is the case in good approx-imation, a system is called diffraction limited, since the limit in size is the diffraction taking place due to the finite NA. To achieve diffraction limited PSFs for high NA objectives at different wavelengths and for large fields of view, corrections for chromatic and spherical aberrations are applied, leading to a large variety of different objective types with different properties. To de-scribe them would go beyond the scope of this description of the parameters most important for this thesis.

• Back aperture size. A property of an objective, which is very important when light is coupled in or out of the objective, is the size of the microscopes back aperture. This aperture determines which diameter D an outcoming beam from a point source has or which diameter D an ingoing beam has to have ideally. It can be calculated as follows [154]:

D= 2·f·N A. (4.3)

This looks like what is expected for paraxial approximation only, but for most microscope objectives (so called aplantic objectives) it is also true for high NAs. The reason for this lies in the fact that in order to get a good image also for points not exactly on the optical axis, an optical system has to obey Abbe’s sine condition (see [151] for more information).

4.1.2. Resolution

Resolution is the ability of a microscope to resolve light stemming from adjacent points. Clearly, this is given by the size of the PSF. When the PSF gets larger, it will get harder to resolve distinct points. Traditionally, there are different criteria on how to judge if two points are considered as resolved. The most common criteria are the Rayleigh and Sparrow criteria. In the Rayleigh criterion, two equally bright points are resolved when the intensity maximum of one point coincides with the first intensity minimum of the other, while the Sparrow criterion states that two equally bright points are resolved when their combined intensity yields a local minimum or a flat top instead of a peak [155]. Assuming an Airy function for the lateral PSF, for the Rayleigh criterion this leads to [152]:

dlateral,R = 0.61 λ

N A, (4.4)

while it is [156]:

d_lateral,S = 0.51 λ

N A (4.5)

for the Sparrow criterion. Note that these criteria used for resolution are defined for practical reasons: If one would consider a completely noiseless case or has some additional information on the sample to be measured, even much closer objects could be measured, e.g., by deconvolving with the PSF. As an easy example, if it is known that there is just a single emitter, it is easily possible to localise it better than the resolution limit just by looking for the centre of the measured signal. Many advanced superresolution microscopy schemes employ a similar approach in order to localise many emitters [157].

4.1.3. Confocal Microscopy

In contrast to standard optical microscopy, in confocal microscopy only one point is illuminated at a time and also detection of the signal only takes place for this point.

The basic principle can be seen in Figure 4.2 (a). A light source, commonly a laser, is focused by an objective lens and detection happens by imaging the focal point on a pinhole. As Figure 4.2 (b) and (c) show, light that is not stemming from the illuminated point is suppressed by the detection pinhole. Even more, these points are also not illuminated. Mathematically this can be expressed by multiplying the PSF of the excitation light with the PSF of the detection pinhole. This is called the confocal PSF. For a suitable small pinhole size, the confocal PSF is the original PSF squared, what slightly increases the resolution of the microscope. Note that this is only the case for the Sparrow criterion, since the minima used in the Rayleigh criterion do not shift for a squared PSF. However, more important than