Microscope Numerical Aperture Examples: A Comprehensive Guide for Physics Students

Microscope Numerical Aperture (NA) is a crucial factor that determines the light-gathering ability and resolution of a microscope objective or condenser. Understanding the principles of numerical aperture and its practical applications is essential for physics students working with microscopy techniques. This comprehensive guide will delve into the technical details, formulas, and examples of microscope numerical aperture to provide a valuable resource for your studies.

Understanding Numerical Aperture

The numerical aperture (NA) of a microscope objective or condenser is defined as the sine of half the angle of acceptance of the objective or condenser, multiplied by the refractive index of the medium between the objective or condenser and the specimen. The formula for calculating the numerical aperture is:

NA = n * sin(α/2)

where n is the refractive index of the medium, and α is the angle of acceptance.

The numerical aperture is a dimensionless quantity and is usually expressed as a decimal value. The higher the numerical aperture, the greater the light-gathering ability and resolution of the objective or condenser.

Factors Affecting Numerical Aperture

Several factors can influence the numerical aperture of a microscope system:

1. Objective Lens Design: The design of the objective lens, including the curvature of the lens surfaces and the number of lens elements, can affect the numerical aperture. Objectives with more complex designs and larger lens diameters generally have higher numerical apertures.

2. Refractive Index of the Medium: The refractive index of the medium between the objective lens and the specimen plays a crucial role in determining the numerical aperture. Immersion media, such as oil, water, or glycerin, can increase the refractive index and, consequently, the numerical aperture.

3. Angle of Acceptance: The angle of acceptance, or the maximum angle at which light can enter the objective lens, is another factor that affects the numerical aperture. Objectives with a larger angle of acceptance have a higher numerical aperture.

4. Condenser Numerical Aperture: The numerical aperture of the condenser lens also plays a significant role in the overall performance of the microscope. The condenser numerical aperture should match or exceed the objective numerical aperture for optimal resolution and contrast.

Numerical Aperture and Resolution

The numerical aperture of a microscope objective is directly related to the resolution of the microscope. The higher the numerical aperture, the greater the resolving power of the microscope. The relationship between numerical aperture and resolution is described by the Rayleigh criterion, which states that the minimum resolvable distance (d) between two points is given by:

d = 0.61 * λ / NA

where λ is the wavelength of the illuminating light.

For example, a typical objective with a numerical aperture of 0.45 can resolve details down to about 0.5 µm, while an objective with a numerical aperture of 1.40 can resolve details down to about 0.15 µm in the lateral dimension and 0.4 µm in the axial dimension.

Numerical Aperture and Depth of Field

The numerical aperture of a microscope objective also affects the depth of field, which is the distance over which the specimen remains in focus. A high numerical aperture objective has a shallow depth of field, while a low numerical aperture objective has a deeper depth of field.

The depth of field (DOF) can be calculated using the following formula:

DOF = n * λ / (NA^2)

where n is the refractive index of the medium, λ is the wavelength of the illuminating light, and NA is the numerical aperture of the objective.

For instance, a 100x objective with a numerical aperture of 1.40 will have a much shallower depth of field compared to a 10x objective with a numerical aperture of 0.25.

Immersion Objectives and Numerical Aperture

Immersion objectives are designed to be used with a specific immersion medium, such as oil, water, or glycerin, to increase the numerical aperture of the objective. The immersion medium has a higher refractive index than air, which allows for the collection of more light and the resolution of finer details.

For example, a 100x objective with a numerical aperture of 1.25 can be used with immersion oil to increase the numerical aperture to 1.45, resulting in improved resolution and contrast.

Numerical Aperture Examples

Let’s explore some practical examples of numerical aperture in microscopy:

1. Brightfield Microscope: A typical brightfield microscope objective with a numerical aperture of 0.45 can resolve details down to about 0.5 µm.

2. Phase Contrast Microscope: A phase contrast microscope objective with a numerical aperture of 0.65 can resolve details down to about 0.4 µm.

3. Confocal Microscope: A confocal microscope objective with a numerical aperture of 1.40 can resolve details down to about 0.15 µm in the lateral dimension and 0.4 µm in the axial dimension.

4. Fluorescence Microscope: A fluorescence microscope objective with a numerical aperture of 1.30 can resolve details down to about 0.17 µm in the lateral dimension and 0.45 µm in the axial dimension.

5. Electron Microscope: A transmission electron microscope (TEM) with an accelerating voltage of 200 kV and a numerical aperture of 0.006 can resolve details down to about 0.1 nm.

These examples illustrate the importance of numerical aperture in achieving high-resolution imaging in various microscopy techniques.

Numerical Aperture Calculations and Exercises

To further solidify your understanding of numerical aperture, let’s work through some calculations and exercises:

1. Calculating Numerical Aperture: Suppose you have a microscope objective with a refractive index of 1.52 and an angle of acceptance of 60 degrees. Calculate the numerical aperture of the objective.

“`
Given:
Refractive index (n) = 1.52
Angle of acceptance (α) = 60 degrees

Numerical Aperture (NA) = n * sin(α/2)
NA = 1.52 * sin(60/2)
NA = 1.52 * 0.5
NA = 0.76
“`

1. Resolving Power Calculation: A microscope objective has a numerical aperture of 1.40 and is used with a light source with a wavelength of 550 nm. Calculate the minimum resolvable distance between two points.

“`
Given:
Numerical Aperture (NA) = 1.40
Wavelength (λ) = 550 nm

Minimum resolvable distance (d) = 0.61 * λ / NA
d = 0.61 * 550 nm / 1.40
d = 240 nm
“`

1. Depth of Field Calculation: A 100x objective with a numerical aperture of 1.40 is used in a microscope. Calculate the depth of field of the objective, assuming the refractive index of the medium is 1.52 and the wavelength of the illuminating light is 500 nm.

“`
Given:
Numerical Aperture (NA) = 1.40
Refractive index (n) = 1.52
Wavelength (λ) = 500 nm

Depth of Field (DOF) = n * λ / (NA^2)
DOF = 1.52 * 500 nm / (1.40^2)
DOF = 0.61 µm
“`

These examples demonstrate how to apply the formulas and principles of numerical aperture to solve practical problems in microscopy.

Conclusion

Numerical aperture is a critical parameter in microscopy that determines the light-gathering ability and resolution of a microscope objective or condenser. By understanding the factors that influence numerical aperture, the relationship between numerical aperture and resolution, and the practical applications of numerical aperture in various microscopy techniques, physics students can gain a deeper understanding of the fundamental principles of microscopy and enhance their experimental capabilities.

References:
– Numerical Aperture – an overview | ScienceDirect Topics
– Microscopy Basics | Numerical Aperture and Resolution
– Numerical Aperture and Resolution – Navitar
– Numerical Aperture; NA, optical fiber, lens, objective … – RP Photonics
– Numerical Aperture – an overview | ScienceDirect Topics