ESTUDIOS Y ANTECEDENTES DEL CONFLICTO

In its simplest form, an optical microscope consists of two positive lenses: an objective lens of short focal length that images the object and a magnifier that functions as an eyepiece. Most mod- ern microscopes use infinity-corrected optics, i.e., the objective forms the intermediate image at infinity (rays are parallel) and another lens to focus the intermediate image before the eyepiece. The advantage of this design is that additional elements (e.g. polarizers, prisms, dichroic flats, spatial filters, etc.) can be inserted into the optical train as needed since the light will remain parallel so long as the elements do not focus the rays.

A bright field microscope image is the consequence of the interference of direct light from the light source and transmitted light diffracted by the specimen. This concept was introduced by Ernst Abbe in 1870 and is illustrated in Figure 3.4. Light from a point source in the form of spherical wavefronts is collimated by a condenser and converted into plane waves which are sent into the specimen. Some of this light is diffracted by regions of varying index of refraction in the specimen while some pass through unaltered. Both the diffracted and undiffracted light are collected by the objective and focused in the Back Focal Plane (BFP) of the objective. The intermediate image is then formed by the interference of the light. Interference is the mechanism by which contrast is generated in the image of the specimen. Thus image contrast depends

4 um

(A) _(B)

Intermediate Image

Plane

Back Focal Plane

Objective

Specimen

Condenser

Light Source

Figure 3.4: (A) Schematic of image formation in a bright field microscope [62]. (B) Bright field

image of a 4µmdiameter Polystrene (n = 1.6) particle taken with a 20X water immersion NA =

0.7 objective.

on both the variations in index of refraction of the specimen and also on the coherence of the light source. A concrete example of interference-based image formation process can be found in a bright field image of a particle. As a consequence of interference, the intensity profile of the particle is not a circularly symmetric Gaussian, as is expected for an incoherently self- luminous particle in an aberration-free fluorescence imaging setup, but rather the particle’s image is an Airy disk consisting of central bright maximum, surrounded by alternating bright and dark circularly symmetric rings as shown in Figure 3.4B. The rings are the result of interference between the light diffracted from different regions of the particle and the undiffracted light.

The schematic of the simple microscope setup also motivates the concept of reciprocal and conjugate planes in microscopy. Two planes are called reciprocal planes when points in one plane are mapped onto the other via a lens and vice versa. In the setup of Figure 3.4A, the BFP is the reciprocal plane of the specimen image plane. Another way of looking at the relationship between reciprocal planes is as a spatial Fourier transform. For instance, the BFP is the spatial Fourier transform of the specimen image plane since the objective focuses the plane waves from the specimen plane to a point via conversion into spherical waves. By contrast, two planes which share common focus are called conjugate planes. In Figure 3.4A, the specimen and intermediate image planes are conjugate planes, as are the BFP and the light source plane. Modern micro- scopes typically contain two sets of conjugate planes, image and aperture planes. The images planes include field diaphragm, specimen plane, intermediate image plane, and retina. The aper- ture plane consists of the light source, condenser diaphragm, objective back focal plane, and pupil. These planes are illustrated in Figure 3.5. Most high-end microscopes, including ours, contain a removable lens known as a Bertrand lens that can be used toggle between the two con- jugate planes, permitting the user to observe the back focal plane of the objective. This is useful for aligning the phase ring for phase-contrast microscopy and for doing quick-and-dirty Bragg scattering measurements of e.g. colloidal crystals in a microscope.

Figure 3.5: Image beam path (left) and illumination beam path (right) in Kohler illumination design [4].

In order to achieve optimal image quality it is important to set up Kohler illumination in the microscope’s optical train. Kohler illumination is an alignment protocol that ensures every point in the specimen plane is evenly illuminated with parallel light rays emanating from the lamp filament, as shown in Figure 3.5. Essentially, this makes the illumination plane reciprocal to the image plane, eliminating contamination in the form of granularities from dirty surfaces which may be present in the aperture planes. For instance, this scheme has the effect of ensuring that the lamp filament is not imaged along with the specimen, a major problem in the early days of microscopy. A good step-by-step procedure for achieving Kohler illumination can be found in [82].

The most important optical parameter of a lensing element (e.g. objective or condenser) is the numerical aperture (NA) defined as

N A=nsinθ, (3.3)

where n is the index of refraction of the medium between the objective or condenser and the

coverslip andθis half-cone angle of light captured by the lensing element (Figure 3.6). Common

values of n are n = 1.00 (Air), n = 1.33 (water), and n = 1.5 (immersion oil). The system NA sets both the working distance and the lateral resolution of the of the lensing element, i.e., the minimum distance between two diffraction-limited objects that can be resolved in the image plane. For transmitted light (bright field) illumination, this distance is

For self-luminous objects, as in reflection fluorescence (epi-fluorescence) illumination where the objective focuses both excitation and emission, the resolution r is determined solely by the NA of the objective lens and is

r = 1.22λ0/2N Aobj. (3.5)

Eqns. 3.4 and 3.5 are statements of the Rayleigh criterion which dictates that two non- interfering Airy disks are barely resolvable when the first minimum of one and zeroth-order peak of the other are separated by a distance r. Higher resolution corresponds to a smaller value of r and is produced by increasing the NA. Conversely, lower resolution corresponds to a larger value of r and occurs when the NA is reduced. High-end microscopes contain irises in the condenser back focal plane which can be used to adjust the working NA by the modulating the

angleθ, as shown in Figure 3.6. Reducing the condenser iris has the dual effect of reducing the

N Acond and increasing the coherence of the illumination light since the light that is collected

then originates from a smaller region of the illuminating filament. This reduction of condenser iris diameter has the effect of increasing image contrast due to increased diffraction in the image from the enhanced coherence, but this gain comes at a cost of lower resolution.

Another important optical microscope imaging parameter controlled by the NA is the depth of field, d. The depth of field sets the longitudinal resolution of the optical system. The depth of field is the axial distance from the nearest object plane in focus to the farthest plane that also appears in focus. It is given by

d= 1.22 λ0n

N A2 +

n

Figure 3.6: Numerical apertures and paths of light rays in the condenser and objective lens. The

working numerical aperture of the condenser isN Acond =n′sinθ′and the working numerical

aperture of the objective isN Aobj = nsinθ. N Acond is proportional to r′, the radius of the

condenser iris opening [51].

where nand λ0 are defined as before, and the variable e is the smallest distance that can be

resolved by a detector placed in the image plane of the microscope objective whose lateral mag-

nification isM.