Optics

Optics is the science of light. Due to wave-particle duality, light can be described as both an electromagnetic wave and a stream of particles (photons). Usually, one of these two properties is more prominent when light interacts with matter. Key types of interaction include the bending of light (refraction) and the conversion of light energy into heat (absorption), which attenuates the light's intensity. Refraction is particularly useful in optical devices like lenses (e.g., in glasses) and light microscopes to magnify objects.

Light is perceived in different colors. White light consists of all wavelengths in the visible spectrum and is perceived as colorless. When a beam of white light is separated into its component wavelengths, a process known as dispersion, beams of colored light are produced.

Wave-particle duality

Light is a form of electromagnetic radiation that propagates in waves but also consists of discrete particles called photons (or light quanta). Depending on the phenomenon being observed, either its wave or particle properties will be more apparent.

• Electromagnetic waves are transverse waves, consisting of electric and magnetic fields oscillating perpendicular to each other and to the direction of energy propagation.
• Light waves are mathematically described by their frequency and wavelength.

Speed of light

In a vacuum, photons (and all electromagnetic waves) travel at a constant speed, the speed of light (c).

• Wave equation: c = λ × f
  • Unit: m/s = m × 1/s
  • c = speed of light (m/s), λ = wavelength (m), f = frequency (Hz = 1/s)
  • Since c is constant in a vacuum, the product of λ and f is also constant: c = constant → λ × f = constant
    • The wavelength and frequency of light are inversely proportional to each other: As one increases, the other decreases.
    • Accordingly, each frequency has a specific wavelength and vice versa.
    • The entirety of all wavelength-frequency pairs is referred to as the electromagnetic spectrum.
• Speed of light: c = 3 x 10⁸ m/s = 300,000 km/s (in a vacuum)

Electromagnetic spectrum

The electromagnetic spectrum is a continuum of radiation with varying wavelengths (and thus frequencies).

• Electromagnetic spectrum: sorted from low-frequency/long-wavelength to high-frequency/short-wavelength
  • Microwave radiation: 0.1–30 cm
  • Infrared radiation (thermal radiation): 750 nm to 0.1 cm (corresponds to a frequency of 10¹² to 10¹⁴ Hz)
  • Visible range: approximately 400–750 nm
  • Ultraviolet light (UV light): 10–400 nm

ROY G BIV (Red, Orange, Yellow, Green, Blue, Indigo, Violet): red light has the longest wavelength and lowest energy.

Color

The color of light depends on its wavelength: red ≈ 750 nm, yellow ≈ 550 nm, green ≈ 500 nm, blue ≈ 460 nm, and violet ≈ 400 nm. Photon energy increases as wavelength decreases. Sunlight is white light, which contains all colors that combine through additive color mixing to be perceived as white.

• Additive color mixing: The perception of the human eye and the subsequent processing in the brain cause light that contains the entire visible spectrum to be seen as colorless or white.
  • Blue + red = magenta
  • Green + blue = cyan
  • Green + red = yellow
• Monochromatic light (monochromatic = single-colored): Light with only one wavelength is perceived as having a specific color.

Energy

The energy of a single photon is the product of Planck's constant (h) and the frequency of the radiation (f).

• Photon energy: E = h × f
  • Unit: J (Joule)
  • E = energy (J]), h = Planck's constant ≈ 6.626 × 10^-34 J·s, f = frequency (1/s = Hz)

Light can interact with matter in various ways:

Reflection of light

• Definition: the bouncing of light rays off a surface
• Law of reflection: angle of incidence (θ₁) = angle of reflection (θ₂)
• Spherical mirrors: mirrors with a spherical reflective surface, classified as concave or convex
  • Properties
    • Center of curvature (C): the point in the centre of the mirror surface that passes through the curve of the mirror and has the same tangent and curvature at that point
    • Radius of curvature (R): the distance from the mirror's surface to C; R = 2f
    • Focal point (F): the point where parallel light rays converge (or appear to diverge from) after reflecting
    • Focal length (f): the distance from the mirror's surface to F.
      • For spherical mirrors, the focal length is equal to half of the radius of curvature of the spherical mirror: f = R/2
        • f = focal length of the spherical mirror, R = radius of curvature of the spherical mirror
  • The mirror equation: relates the object distance (d_o), the image distance (d_i), and the focal length (f).
    • f1 = d_o1+d_i1
  • Magnification (m): the image's orientation and size relative to the object.
    • m = − d_i/d_o
  • Mirror types
    • Concave (converging) mirrors: the reflective surface curves inward
      • Can form both real and virtual images depending on the object's position
        • Object farther than focal point: image real and inverted (e.g., telescope)
        • Object closer than focal point: image virtual, upright, magnified (e.g., makeup mirror)
        • Object at C: image real, inverted and same size
    • Convex (diverging) mirrors: the reflective surface curves outward
      • Always form virtual, upright, and reduced images (e.g., security mirror in a store, car's side-view mirror)
  • Image types
    • Real image: formed where light rays actually converge; can be projected onto a screen
      • By convention, real images have a positive image distance (+d_i).
    • Virtual image: formed where light rays appear to diverge from; cannot be projected onto a screen
      • By convention, virtual images have a negative image distance (-d_i).

A concave lens or mirror "caves in."

A conVEX mirror gives a VURy small view (Virtual, Upright, Reduced). The objects in the mirror are closer than they appear!

Refraction of light (bending of light)

• Definition: the change in a light beam's direction that occurs at the interface between two materials with different refractive indices (n) because of a change in the light's speed
• Speed of light in a vacuum (c): c ≈ 3 x 10⁸ m/s
• Speed of light in a medium (v)
  • Light travels slower than c in any other medium (e.g., water or glass) due to interactions with the medium's molecules.
  • The speed is always constant for a given medium.
• Refractive index (n): a dimensionless number that quantifies how much a medium slows down light
  • Formula: n = c/v
    • Unit: dimensionless
    • n = refractive index, c = speed of light in a vacuum (≈ 3.0 × 10⁸ m/s), v = speed of light in the medium
  • For a vacuum, n = 1. For all other media, n > 1.
• Snell's law of refraction: quantifies the change in a light ray's direction due to refraction
  • Formula: n₁sin(θ₁) = n₂sin(θ₂)
    • n = refractive index, θ₁ = angle of incidence, θ₂ = angle of refraction
• Refraction of light at the interface between two materials:
  • If n₂ > n₁ (e.g., air to water) = fast to slow:
    • Light slows down (and its wavelength becomes shorter, as frequency remains constant).
    • The light ray bends toward the normal.
    • Angle of refraction < angle of incidence
  • If n₁ > n₂ (e.g., water to air) = slow to fast:
    • Light speeds up.
    • The light ray bends away from the normal.
    • Angle of refraction > angle of incidence
• Total internal reflection: a phenomenon where a light ray is completely reflected back into its original medium instead of being refracted into the second medium
  • Conditions
    • Light must be traveling from a medium with a higher refractive index to a medium with a lower refractive index (n₁ > n₂).
    • The angle of incidence must be greater than the critical angle (θc).
  • Critical angle (θc): the specific angle of incidence for which the angle of refraction is 90°
    • Formula: sin(θc) = n₂/n₁
  • Example: fiber optics

FST = Fast to Slow, light ray bends Towards Normal, and SFA = Slow to Fast, light ray bends Away from Normal.

A higher refractive index (n) means the medium is more "optically dense" and light travels slower.

When light enters a new medium, its frequency remains constant, while its speed and wavelength change.

Sample calculation

Light traveling through air (n = 1) hits a water surface (n = 1.33) at an angle of 36°. Calculate the angle of refraction.

• Required: θ_water
• Given: angle of incidence θ_air, refractive index n_air, refractive index n_water
  • n_airsin(θ_air) = n_watersin(θ_water)
  • sin(θ_water) = (n_air/n_water) × sin(θ_air) = (1/1.33) × sin(36°) ≈ (1/1.33) × 0.588 ≈ 0.442
  • θ_water = sin^-1(0.442) ≈ 26.2°
  • The angle of refraction is approximately 26°.

Dispersion of light

The effect of additive color mixing can be reversed by separating a beam of light into its different wavelengths.

• Dispersion: the splitting of a polychromatic light beam into its individual constituent wavelengths
  • Cause: The speed of light in a medium is wavelength-dependent, causing different colors to be refracted at slightly different angles at a surface, thereby separating them into monochromatic rays.
  • Examples: a rainbow, which results from the dispersion of white sunlight by small water droplets in the atmosphere; a prism creating a rainbow

Polarization of light

• Definition: the process of filtering randomly oscillating (unpolarized) light, such as sunlight, so that its oscillations are confined to a single plane (polarized light)
• Types of polarized light
  • Linearly (plane) polarized light: oscillates in a single plane
    • Example: A polarization filter used in photography allows only light waves oscillating in a specific direction to pass through, reducing glare and enhancing image quality.
  • Circularly polarized light: results from the vector sum of the electric field oscillations of two linearly polarized transverse waves with equal amplitude, traveling along the same axis but with a phase difference of 90 degrees
    • Example: Circularly polarized light is used in circular dichroism (CD), a technique that helps identify the secondary structure of proteins by measuring the differential absorption of left-handed and right-handed circularly polarized light.

Interference and diffraction of light

Interference of light

• Definition: two or more, distinct waves interfere with each other and can either amplify (constructive interference) or cancel each other out (destructive interference)
• Young's double-slit experiment: demonstrates the wave nature of light
  • Setup: monochromatic light (light of a single wavelength, λ) is directed through two narrow, parallel slits, separated by a distance,d
  • Observation
    • Instead of two bright lines on a screen behind the slits, a pattern of alternating bright and dark bands (called fringes) appears. This pattern is caused by the light waves from each slit interfering with each other.
    • Bright fringes (constructive interference)
      • Occur when light waves arrive at the screen in phase (e.g., crest meets crest), amplifying each other
      • This happens when the path length difference from the slits to the screen is an integer multiple of the wavelength
      • Formula for bright fringes (maxima): d sin(θ) = mλ
        • d = distance between the slits, θ = angle of the fringe from the central maximum, m = the order of the fringe (m = 0 is the central bright fringe, m = 1 is the first bright fringe, etc.), λ = wavelength of the light
      • Formula for bright fringes for small angles: x ≈ mλL/d
        • x = distance from the central maximum on the screen, m = integer representing the order of the fringe, λ = wavelength of the light, L = distance from the slits to the screen, d = distance between slits
    • Dark fringes (destructive interference), minima
      • Occur where waves arrive out of phase (crest meets trough), canceling each other out.
      • This happens when the path length difference is a half-integer multiple of the wavelength
      • Formula for minima: d sin(θ) = (m + 1/2)λ
        • d = distance between the slits, θ = angle of the fringe from the central maximum, m = the order of the fringe (m = 0 is the central bright fringe, m = 1 is the first bright fringe, etc.), λ = wavelength of the light

The iridescence of an oil film is an example of thin-film interference, where light waves reflecting from the top and bottom surfaces of the film interfere constructively and destructively to produce a rainbow of colors.

Diffraction of light

• Definition: occurs when a wave bends as it passes through a narrow opening or around an obstacle
• Single-slit diffraction
  • Setup: monochromatic light passes through one narrow slit
  • Observation: creates a diffraction pattern with a very bright and very wide central maximum and smaller, dimmer maxima on either side
  • Formula to locate dark fringes (minima): a sin(θ) = mλ
    • a = the width of the single slit (replaces d), θ = from the central axis to the location of the dark fringe, m = the order of the dark fringe (m = 1, 2, 3... but not 0), λ = the wavelength of the light
• Diffraction grating: an optical component with many equally spaced slits (e.g., thousands per centimeter)
  • Function: produces sharper and more intense interference patterns than a double slit, making it useful in high-resolution spectroscopy for separating light into its constituent wavelengths
    • Formula for maxima: d sin(θ) = mλ
      • d = distance between adjacent slits, θ = angle from the center line to the fringe, m = fringe order (0, 1, 2, …), λ = the wavelength of the light
• X-ray diffraction: a technique used to determine the three-dimensional structure of crystalline solids, including proteins and nucleic acids
  • Mechanism: Because X-rays have wavelengths similar to the spacing between atoms in a crystal, they diffract when passing through a crystallized molecule. By analyzing the resulting diffraction pattern, scientists can determine the 3D structure of molecules like proteins and DNA.

“Wide slits, tight fringes. Narrow slits, wide pattern”: Making the slit(s) wider (increasing d or a) causes the fringes to become tighter (closer together). Making the slit(s) narrower (decreasing d or a) causes the pattern to spread out.

Diffraction explains the lack of perfectly defined edges in shadows. As light waves bend into the areas designated as shadow, they create regions of reduced brightness.

Longer wavelengths (e.g., red light) increase the distance between fringes.

Be careful! The formula a sin(θ) = mλ gives you minima for a single slit but the very similar formula d sin(θ) = mλ gives you maxima for a double slit.

Absorption of light

• Absorption: the attenuation of a light beam's intensity as its energy is converted into other forms, like heat, by the medium it passes through
  • Example: an absorption spectrometer
• Transmittance (T): the fraction of incident light that passes through a substance
  • Formula: T = I/I₀
    • Unit: a dimensionless quantity, often expressed as a decimal (e.g., 0.8) or a percentage (e.g., 80%)
    • T = transmittance, I = intensity after interaction, I₀ = intensity before interaction
    • T_total = T₁ × T₂
• Absorbance of light (A): a measure of how much light is absorbed by a sample; logarithmic and related to concentration.
  • Formula: A = log (I₀/I)
    • A = absorbance, I = intensity after interaction, I₀ = intensity before interaction
    • A_total = A₁ + A₂+…
• Beer-Lambert law: describes the reduction in light intensity due to absorption
  • Formulas
    • A = εcl
      • A = absorbance (no units), ε = molar absorptivity L·mol^-1·cm^-1 (L/(mol·cm)) , c = concentration of the absorbing substance (mol/L^-1), l or b = the path length of the light beam through the solution (cm)
    • A = log (I₀/I)
      • A = absorbance (no units), I₀ = initial light intensity, I = transmitted light intensity
  • Use case: calculate an unknown concentration after measuring its absorbance in a spectrophotometer (c = A/εl)
• Light intensity (irradiance): the power delivered per unit area
  • Formula: I = P/A
    • Unit: W/m²
    • I = irradiance (W/m²), P = power (W), A = area (m²)

An absorbance of 1 means 90% of the light was absorbed. An absorbance of 2 means 99% was absorbed.

Remember: absorbances add, while transmittances multiply.

If you double the concentration (c) or the path length (l), you double the absorbance (A).

Sample calculation

In a photometric experiment, the concentration of a dissolved substance is measured. Monochromatic light is passed through a cuvette with a length of 1 cm. The measured absorbance is 1.147. The substance's molar absorptivity is 20 L·mol^-1·cm^-1L/(mol·cm). Calculate the concentration of the solution.

• Required: concentration c (mol/L = M)
• Given: absorbance A (dimensionless), molar absorptivity ε (L·mol^-1·cm^-1), path length l or b
  • Beer-Lambert law: A = εcl
  • c = A/εl
  • c = 1.147 / (20 L/(mol·cm) × 1 cm) = 0.057 mol/L = 5.7 x 10^-2 M

Spectroscopy techniques analyze the interaction between electromagnetic radiation and matter to determine molecular structure. Different regions of the electromagnetic spectrum probe different molecular properties.

Ultraviolet-visible spectroscopy

• Principle: UV-Vis spectrometer measures the absorption of UV or visible light, which causes electronic transitions (e.g., promoting π-electrons to higher energy orbitals)
• Applications
  • Conjugated systems
    • Primarily used to study molecules containing conjugated π systems (alternating single and double bonds)
    • The extent of conjugation correlates with the wavelength of maximum absorbance (λmax); more conjugation results in absorption at longer wavelengths (a redshift).
  • Color perception: The color of a substance is the complement of the color of light it absorbs.

Infrared spectroscopy

• Principle: IR spectrometer measures the absorption of infrared radiation, which causes intramolecular vibrations (stretching, bending, twisting) in covalent bonds
• Application: used to identify the functional groups present in a molecule
• IR spectrum: a plot of transmittance vs. wavenumber (cm^-1); absorption peaks correspond to specific bond types
  • Characteristic IR peaks:
    • O-H stretch (alcohols, carboxylic acids): broad peak around 3300 cm^-1
    • N-H stretch (amines): peak around 3300 cm⁻¹, sharper than O-H
    • C=O stretch (carbonyls): sharp, strong peak around 1700 cm^-1
    • Fingerprint region: The region from 1500–400 cm^-1 contains complex vibrations unique to a molecule, allowing for identification by comparison to a known spectrum.

Nuclear magnetic resonance spectroscopy

• Principle: The nuclei of certain atoms (e.g., ¹H, ¹³C) have a magnetic spin. When placed in a strong external magnetic field and exposed to radio waves, these nuclei can absorb energy and flip their spin states. The exact frequency of absorption provides information about the local electronic environment of the nucleus.
• Application: provides detailed information about the carbon-hydrogen framework of a molecule
• Features of a ¹H NMR spectrum
  • Chemical shift (δ): indicates the location of a signal on the spectrum, measured in parts per million (ppm)
    • The local electronic environment influences a proton's resonance frequency. Electron-withdrawing groups "deshield" a proton, shifting its signal downfield (to a higher δ value).
  • Equivalent protons: Protons in identical electronic environments are chemically equivalent and produce a single signal.
  • Integration: the area under a signal, which is proportional to the number of equivalent protons that the signal represents
  • Splitting (multiplicity): The division of a signal into multiple smaller peaks due to the influence of neighboring, non-equivalent protons. The number of peaks follows the n+1 rule, where n is the number of adjacent protons.

Refraction of light at lenses

Because the speed of light differs in air and glass, light rays are refracted when passing through lenses. The exact path of a light ray depends on the shape of the lens, which in turn determines the type, location, and size of the resulting image. The properties of a lens can be demonstrated with a simple optical setup, which consists of:

• Object: the item being viewed through a lens
  • Object height: h_o; the height of the object
  • Object distance: d_o; the distance from the object to the center of the lens. By convention, d_o is always positive for a single lens.
• Image: the replica of the object formed by the lens
  • Image height: h_i; the height of the image produced by the lens
  • Image distance: d_i; the distance of the image from the center of the lens
  • Types of images
    • Real image: formed by the actual convergence of light rays; always inverted and can be projected onto a screen
    • Virtual image: formed where light rays appear to originate from; always upright and cannot be projected onto a screen
• Lens: a transparent component, usually made of glass, that refracts light
  • Properties
    • Focal point: the point at which light rays initially parallel to the optical axis intersect after passing through the lens
    • Focal length (f); the distance from the center of the lens to the focal point, where parallel rays converge (or appear to diverge from)
  • Types
    • Converging lens (convex): a lens that is thicker in the middle and converges parallel light rays to a real focal point
      • Ray path:
      • “Plus lens“: has a positive focal length (f > 0) and refractive power
      • Image type: can be real or virtual, depending on the object's location
        • Object beyond the focal length (d_o > f): the image is real and inverted (e.g., eye/camera lens)
        • Object within the focal length (d_o < f) : the image is virtual, upright, and enlarged (e.g., magnifying glass)
        • Object at the focal point (d_o = f): no image formed (rays are parallel)
      • Example: glasses for farsightedness (hyperopia), magnifying glasses
    • Diverging lens (concave): a lens that is thinner in the middle and causes parallel rays to diverge
      • Ray path:
      • “Minus lens“: has a negative focal length (f < 0) and refractive power
      • Image type: always a virtual, upright, and reduced image, regardless of the object's location
      • Example: glasses for nearsightedness (myopia)
  • Magnification equation: relates the image height (h_i) and object height (h_o) to the distances
    • Formula: M = h_i/h_o = - d_i/d_o
      • M = magnification, h_i = image height, h_o = object height, d_i = image distance, d_o = object distance
    • A positive M means the image is enlarged and upright (same orientation as the object).
    • A negative M means the image is reduced and inverted (upside down).
    • M = 1: image same size
  • Thin lens equation: relates the object distance (d_o), image distance (d_i), and focal length (f); the reciprocal of the focal length is the refractive power (P) in diopters
    • Formula: P = 1/f = 1/d_o + 1/d_i
      • Unit: diopters [dpt = 1/m]
      • P = refractive power, f = focal length, d_o = object distance, d_i = image distance
    • Sign conventions
      • Focal length (f): positive for converging (convex) lenses, negative for diverging (concave) lenses
      • Object distance (d_o): positive if the object is on the same side as the incoming light (typical case)
      • Image distance (d_i)
        • Positive (+) for real images (formed on the opposite side of the lens from the object)
        • Negative (‑) for virtual images (formed on the same side of the lens as the object)
  • Lens combinations (e.g., glasses on an eye): the total refractive power of two thin lenses placed close together is the sum of their individual powers
    • Formula: P_total = P₁ + P₂ = 1/f₁ + 1/f₂
      • Unit: diopters [dpt = 1/m]
      • _Ptotal = total refractive power, P₁ = refractive power of the first lens, P₂ = refractive power of the second lens

A helpful mnemonic for image types is "UV-IR": Upright images are always Virtual. Inverted images are always Real. This applies to all single-lens and single-mirror systems.

Think "Real is Positive, Virtual is Negative" for the image distance d_i.

The power (P) of a lens in diopters is 1/f.

Sample calculation

A lens magnifies an object 10-fold when the object is 5 cm away. What is the refractive power of this lens?

• Required: refractive power (P) of the lens
• Given: magnification (M) = +10 , object distance d_o = + 5 cm = + 0.05 m
  • The lens is a magnifying glass, which is a converging lens used with d_o < f. This means we expect a positive power (P > 0).
  • Calculate the image distance (d_i)
    • Using the magnification formula: M = - d_i/d_o
    • 10 = - d_i /5 cm
    • d_i = 10 × (- 5 cm) = - 50 cm = - 0.5 m
  • Calculate the refractive power (P) using the thin lens equation
    • P = 1/d_o + 1/d_i (using distances in meters)
    • P = 1/(0.05 m) + 1/(- 0.5 m) = 20 dpt - 2 dpt = 18 dpt
    • The power of lens is +18 diopters.

Optical aberrations

Optical aberrations are deviations from the ideal ray path that cause imperfections in the image. They can be chromatic or monochromatic.

• Chromatic aberration: occurs because a lens refracts different wavelengths (colors) of light by slightly different amounts (e.g., blue light bends more than red light)
• Monochromatic aberrations, e.g.
  • Spherical aberration
    • Occurs because rays hitting the edge of a spherical lens are refracted more strongly than rays hitting the center
    • The human eye can compensate for spherical aberration by constricting the pupil, which blocks rays that would otherwise hit the edge of the lens.
  • Astigmatism, see: “Refractive errors”

The human eye as an optical instrument

• Optical components
  • Cornea: the transparent outer layer that performs most of the initial refraction of light entering the eye
  • Lens: a converging lens located behind the pupil that fine-tunes the focus of light onto the retina
• Focusing mechanism
  • Accommodation: the process by which the eye changes its optical power to maintain a clear image of objects at varying distances
    • Mechanism: The ciliary muscles contract or relax, changing the curvature and focal length of the lens.
    • For distant objects, the ciliary muscles relax, flattening the lens and increasing its focal length.
    • For near objects, the muscles contract, making the lens more convex and decreasing its focal length.
  • Retina: the light-sensitive tissue at the back of the eye where the real, inverted image is formed
• Refractive errors
  • Myopia (nearsightedness): The eye converges light too strongly, so the image forms in front of the retina. Corrected with a diverging (concave) lens.
  • Hyperopia (farsightedness): The eye is too weak at converging light, so the image forms behind the retina. Corrected with a converging (convex) lens.

My Nearsightedness Needs a Negative lens.

Physical principles of light microscopy

To magnify small structures, the optical system must improve resolution, which is the ability to distinguish between two closely spaced points. A light microscope accomplishes this by passing light through a series of lenses to produce a magnified image.

• Structure: A compound microscope uses multiple converging lenses to magnify an image in stages.
  • Eyepiece: the lens closest to the eye, which acts like a magnifying glass
    • Magnification: M = s₀/f
    • M = magnification, s₀ = near point vision (at 25cm), f = focal length
  • Objective: the lens closest to the object; its magnification depends on the tube length
    • Magnification: M = t/f
    • M = magnification, t = tube length, f = focal length
• Total magnification of the microscope
  • Formula: M_total = M_eyepiece × M_objective → M_total = (s₀/f_eyepiece) × (t/f_objective)
    • M = magnification, s₀ = near point vision, f = focal length, t = optical tube length
• Ray path: the path of light through a microscope
• Resolution: the smallest distance between two points at which they can still be distinguished as separate entities
  • Formula: d = λ / (n × sinθ)
    • Unit: nm
    • d = smallest resolvable distance, n×sinθ = numerical aperture (NA), n = refractive index of the medium between the objective and the specimen, θ = half the opening angle of the objective lens, λ = wavelength of light
  • Improving resolution
    • Using a shorter wavelength (λ) of light results in a smaller distance d (and thus higher resolving power).
    • Increasing the numerical aperture, e.g., by using an immersion medium with a higher refractive index (n), results in a smaller distance d (and thus higher resolving power).

The refractive index of a microscopic setup can be optimized by using an immersion medium, which has a higher refractive index than air!

Sample calculation 1

• Required: Total magnification M_total
• Given: M_eyepiece = 10x, t = 160 mm, f_objective = 4 mm (example values substituted for clarity)
  • First, calculate the objective magnification: M_objective = t/f_objective
  • M_objective = 160 mm / 4 mm = 40x
  • Next, calculate total magnification: M_total = M_eyepiece × M_objective
  • M_total= 10 × 40 = 400x

Sample calculation 2

• Required: Resolution d
• Given: Numerical aperture of a microscope NA = 0.12; λ = 550 nm
  • d = λ/NA
  • d = 550 nm / 0.12 ≈ 4583 nm
  • d ≈ 4.58 μm