Introduction: From Fourier Optics to Image Formation
Building on the Fourier optics concepts you learned in previous lectures, we now explore how these mathematical tools explain how microscopes work. The microscope is one of the most important optical instruments in science and medicine—it reveals structures invisible to the naked eye by magnifying and enhancing contrast.
In this lecture, we focus on Brightfield and Darkfield microscopy, the two most fundamental imaging modalities. Both rest on a elegant theoretical foundation: Abbe’s theory of imaging, which interprets image formation as a double Fourier transform. This perspective unifies optical physics with practical microscopy and explains why some objects are visible while others remain invisible.
By the end of this lecture, you will understand: - Why image formation is fundamentally a Fourier-transform problem - How the objective lens acts as a Fourier transformer - Why transparent (phase) objects are invisible in brightfield - How darkfield unlocks contrast by Fourier filtering - The resolution limits imposed by wavelength and numerical aperture
1. Image Formation as a Double Fourier Transform
1.1 Abbe’s Theory of Imaging
The revolutionary insight of Ernst Abbe (1873) was recognizing that image formation involves two sequential Fourier transforms:
Object plane → Back focal plane of the objective: The objective lens takes the object and computes its Fourier transform in the back focal plane (BFP).
Back focal plane → Image plane: A second Fourier transform by the objective converts the Fourier spectrum back into an image.
Let us denote: - \(o(x, y)\) = object (amplitude or phase distribution) - \(O(u, v)\) = Fourier transform of the object (spatial frequency spectrum) - \(i(x', y')\) = image
The forward propagation is described by:
\[O(u, v) = \mathcal{F}[o(x, y)]\]
where \(u, v\) are spatial frequencies in the back focal plane. The objective has focal length \(f\), and the relationship between Fourier frequencies and physical positions in the BFP is:
\[u = \frac{x_{\text{BFP}}}{\lambda f}, \quad v = \frac{y_{\text{BFP}}}{\lambda f}\]
The image is then formed by a second Fourier transform:
\[i(x', y') = \mathcal{F}^{-1}[O(u, v)]\]
However, not all Fourier components reach the image plane. The objective lens has a finite aperture, so only spatial frequencies within a certain radius are transmitted. This frequency cutoff is the key to understanding resolution and contrast.
1.2 Numerical Aperture and the Frequency Cutoff
The numerical aperture (NA) is the most important parameter in microscopy:
\[\text{NA} = n \sin(\alpha)\]
where \(n\) is the refractive index of the medium (typically 1 for air, 1.515 for oil immersion) and \(\alpha\) is the half-angle of the cone of light accepted by the objective.
The highest spatial frequency transmitted is:
\[u_{\max} = \frac{\text{NA}}{\lambda}\]
All spatial frequency components with \(\sqrt{u^2 + v^2} > u_{\max}\) are blocked. This acts as a low-pass filter:
This fundamental low-pass filtering explains why high-frequency details are lost in microscopy and sets the ultimate resolution limit.
1.3 The Objective as a Fourier Transformer
The objective lens can be understood as implementing a Fourier transform operation. When the object is placed at the front focal plane of the objective, light propagates to the back focal plane, where the Fourier spectrum of the object appears.
The lens equation ensures that: - Object at distance \(s_1\) from front focal plane - Image at distance \(s_2\) from back focal plane - Conjugate relationship: \(1/s_1 + 1/s_2 = 1/f\)
In standard microscopy, the object is very close to the front focal plane (within one focal length), and the image is formed far from the back focal plane (at a standard tube length, typically 160 mm), achieving large magnification.
2. Köhler Illumination
2.1 Why Köhler Illumination Matters
Proper illumination is just as important as the objective lens. There are two classical approaches:
Critical Illumination (Abbe’s original setup): - Light source directly imaged onto the specimen - High intensity but very uneven illumination - Dust, imperfections on condenser lens appear in image - Poor for quantitative work
Köhler Illumination (modern standard): - Condenser lens forms an image of the light source in the back focal plane - Object is illuminated by an essentially parallel, uniform beam - Light source imperfections do not affect the specimen image - Allows proper use of the condenser aperture diaphragm
2.2 Köhler Illumination Setup
The key components are:
Field diaphragm (in object plane): Controls the illuminated area
Condenser aperture diaphragm (in back focal plane): Controls the cone of illumination (NA of the condenser)
Condenser lens: Forms an image of the light source in the back focal plane
For optimal Köhler illumination: - The condenser NA should match the objective NA - The field diaphragm is closed just outside the field of view - The condenser aperture is opened to approximately 80–90% of the objective aperture
The proper setting of the condenser aperture diaphragm is crucial for controlling coherence in the illumination, which affects contrast and image quality.
2.3 Illumination Coherence
Coherent illumination (plane wave, zero-coherence length): - All light arrives in phase - Produces sharp diffraction patterns - Poor for resolution of fine details - Causes speckle and diffraction artifacts
Incoherent (spatially incoherent) illumination (extended source): - Light arrives from different directions - Different spatial frequency components excited independently - Better resolution and more “real” images - Standard in modern microscopy
The degree of coherence is controlled by the coherence length:
where \(\Delta \theta\) is the angular size of the illumination source and \(\text{NA}_c\) is the condenser aperture.
For incoherent imaging (which is optimal), we want \(\ell_c \ll\) specimen thickness.
3. Coherent vs. Incoherent Imaging
3.1 Linear vs. Nonlinear Response
Coherent illumination (fully coherent plane wave): - Field amplitude at the image is a Fourier transform of the field at the object - Image intensity is the square of this transformed field - Contrast depends on the phase relationships between object and scattered light - Nonlinear system: doubling incident amplitude changes the image contrast
Incoherent illumination (spatially incoherent): - Different regions of the illumination source act independently - The detected intensity is a sum of images from each source point - Mathematical: intensity image is the convolution of the object intensity with the point-spread function (PSF) - Linear system for intensity: doubling incident intensity doubles the image intensity
3.2 Point-Spread Function (PSF)
The point-spread function is the image of a point source (infinitesimal bright spot):
where \(k = \pi \text{NA} / \lambda\) and \(J_1\) is the first-order Bessel function.
4. The Optical Transfer Function (OTF) and Coherent Transfer Function (CTF)
4.1 Optical Transfer Function (OTF) for Incoherent Imaging
The Optical Transfer Function (OTF) describes how spatial frequency components are attenuated and phase-shifted as they propagate through the imaging system. It is the autocorrelation of the pupil function:
where \(k = \sqrt{u^2 + v^2}\) is the radial spatial frequency and \(k_{\max} = \text{NA}/\lambda\) is the cutoff frequency.
Key properties of OTF: - Maximum at DC (\(k = 0\)): \(H_{\text{OTF}}(0) = 1\) (all low frequencies transmitted) - Cutoff at\(k_c = 2 \text{NA}/\lambda\) (twice the coherent cutoff!) - Monotonically decreasing: higher frequencies attenuated more - Smooth roll-off: no abrupt edges, unlike the pupil function itself
4.2 Coherent Transfer Function (CTF) for Coherent Imaging
For coherent illumination (plane wave), the imaging system is linear in field amplitude, not intensity. The transfer function is simply the pupil function itself:
The CTF has no phase information (for a uniform pupil), but it acts as a sharp low-pass filter at:
\[k_c^{\text{coherent}} = \text{NA}/\lambda\]
4.3 Comparing OTF and CTF: Incoherent vs. Coherent
Property
Incoherent (OTF)
Coherent (CTF)
Function
Autocorrelation of pupil
Pupil itself
Cutoff frequency
\(2 \text{NA}/\lambda\)
\(\text{NA}/\lambda\)
Shape
Smooth, monotonic decay
Sharp circular edge
Better resolution?
Yes, by factor of ~2
No, but sharp edges
Typical use
Modern microscopy
Holography, lensless imaging
The factor of 2 difference in cutoff frequency is fundamental: incoherent imaging can resolve finer details than coherent imaging with the same NA and wavelength. This is why modern microscopy uses incoherent (spatially incoherent) illumination.
4.4 The Modulation Transfer Function (MTF)
The Modulation Transfer Function is the magnitude of the OTF:
\[M(k) = |H_{\text{OTF}}(k)|\]
It represents the contrast preservation as a function of spatial frequency: - \(M(0) = 1\): perfect contrast at low frequencies - \(M(k_c) = 0\): no contrast beyond cutoff - In between: progressive loss of contrast for higher frequencies
The MTF is what limits the visibility of small structures: fine details become lower in contrast and eventually invisible when their spatial frequency exceeds \(k_c = 2 \text{NA}/\lambda\).
5. Brightfield Microscopy
5.1 Contrast in Brightfield
In brightfield (transmitted light) microscopy, the object is illuminated from below, and contrast arises from:
Amplitude objects (absorbing structures):
Material absorbs light → reduced transmitted intensity
Good contrast in cells with pigments, stains, or metal nanoparticles
No absorption, but refractive index differs from surroundings
Introduce phase shifts: \(\phi = 2\pi n \Delta n \cdot d / \lambda\)
Phase shifts do not affect intensity in brightfield!
Most biological structures (proteins, lipids, nuclei) are essentially transparent
Result: No contrast → invisible in brightfield without staining
This is a fundamental limitation: You cannot see unstained biological samples in brightfield microscopy.
5.2 Contrast in Terms of Fourier Components
In brightfield, the image intensity is proportional to \(|E_{\text{transmitted}}|^2\).
For a weak phase object, the transmitted field is: \[E(x, y) \approx E_0 e^{i \phi(x, y)} \approx E_0 [1 + i \phi(x, y)]\]
where \(\phi(x, y) = 2\pi n(\vec{r}) / \lambda\) is small.
The intensity is: \[I(x, y) = |E|^2 \approx E_0^2 [1 + 2 \text{Im}[\phi(x, y)]]\]
But for a purely real phase shift, \(\text{Im}[\phi] = 0\), so: \[\Delta I \approx 0\]
The phase information is encoded in the phase of the Fourier components in the back focal plane, but the intensity (what we measure) doesn’t directly reveal it. This is the fundamental problem of imaging phase objects.
5.3 Resolution in Brightfield
The minimum resolvable distance between two point sources is given by the Rayleigh criterion:
\[d = 0.61 \frac{\lambda}{\text{NA}}\]
For visible light (\(\lambda = 550\) nm) with a typical objective (NA = 1.4, oil immersion): \[d \approx 0.24 \, \mu\text{m} = 240 \, \text{nm}\]
This represents the fundamental diffraction limit. No microscope can beat this limit without exceeding the speed of light or using wavelengths shorter than visible light.
6. Fourier Filtering in the Back Focal Plane
5.1 The Pupil Plane as a Filtering Plane
The back focal plane (BFP) is where the Fourier spectrum of the object appears. By placing apertures, filters, or masks in the BFP, we can selectively control which spatial frequency components contribute to the final image.
Low-pass filtering (blocking high frequencies): - Objective aperture naturally blocks all frequencies beyond \(u_{\max} = \text{NA} / \lambda\) - Closing the aperture further increases the blur but removes noise and high-frequency artifacts - Useful for improving contrast in noisy images
High-pass filtering (blocking low frequencies): - Achieved by blocking the center of the Fourier spectrum - Removes large-scale variations, enhancing small features - Basis of darkfield microscopy (see next section)
Phase contrast (phase plate in BFP): - Introduces a phase shift of \(\pi/2\) to zero-order light - Converts phase objects into amplitude contrast - We’ll discuss this in detail in the next lecture
5.2 Quantitative Effect of Aperture Restriction
Closing the objective aperture diaphragm reduces the cutoff frequency: \[u'_{\max} = \frac{\text{NA}_{\text{eff}}}{\lambda} < u_{\max}\]
The resulting image is a convolution with a wider PSF: \[i_{\text{restricted}}(x, y) = \int \int i_{\text{unrestricted}}(s, t) \cdot \text{PSF}_{\text{wider}}(x - s, y - t) \, ds \, dt\]
This improves contrast at the cost of resolution.
7. Darkfield Microscopy
7.1 Principle: Blocking the Zero-Order
In darkfield microscopy, we block the unscattered, straight-through light (the zero-order component in the Fourier plane). Only scattered light from the object reaches the detector.
The principle is elegant: 1. Illuminate the specimen with light at an oblique angle (or ring-shaped illumination) 2. Place a central stop in the back focal plane to block the zero-order light 3. Only light scattered by the object can reach the objective lens 4. The detector sees dark background with bright scattered objects
7.2 Darkfield Setup
Ring-illumination darkfield (most common): - Light enters the objective at large angles from all sides - A circular stop blocks light traveling straight through the center - Only scattered light is collected
Oblique illumination (Schlieren darkfield): - Light approaches from one direction at an angle - A blade or edge stops the zero-order - More directional; useful for detecting edge features
The objective lens’ acceptance angle (NA) must be larger than the illumination cone, otherwise no light reaches the detector at all.
7.3 Contrast Mechanism in Darkfield
In darkfield, the image forms by scattered light interference:
where the integral is over the object surface, and the derivative represents scattering.
For an object with: - Phase changes (refractive index variations) → scatters light forward - Surface roughness → scatters light in all directions - Edges → strong diffraction and scattering
Result: Objects with phase variations or discontinuities become bright against a dark background.
7.4 Objects Visible in Darkfield
Darkfield is excellent for visualizing: - Living cells (no stain needed!) - Nanoparticles (gold, silver, latex) - Thin structures (fibers, filaments, axons) - Phase variations (organelles, membranes) - Refractive index boundaries
The contrast arises from scattering, not absorption or staining. This is why darkfield is invaluable in biology.
7.5 Darkfield as Spatial Filtering
Mathematically, darkfield corresponds to high-pass filtering in the Fourier domain. The darkfield transfer function is the brightfield transfer function minus the DC component:
where \(u_{\text{stop}}\) is the radius of the central stop and \(u_{\max} = \text{NA}/\lambda\) is the objective aperture cutoff.
Physical interpretation: - The darkfield filter removes the DC component (unscattered light, uniform background) - Only scattered light from high spatial frequencies contributes to the image - This enhances edges and fine structures while suppressing large-scale variations - Objects invisible in brightfield (transparent, weak phase contrast) become visible through scattering
This is a direct application of spatial filtering in the Fourier plane: by blocking the low frequencies, we highlight fine details that would otherwise be overwhelmed by the uniform transmitted light. The connection to the OTF (section 4) is that darkfield uses the high-frequency tail of the OTF while suppressing its peak at DC.
8. Oblique Illumination and Schlieren Methods
8.1 Oblique Illumination
When illumination comes from one side at a steep angle, we get oblique illumination. This is a hybrid between brightfield and darkfield:
The zero-order light still passes through (unlike pure darkfield)
But illumination is asymmetric
Shadowing effect: bright on one side, dark on the opposite edge
Enhances edge contrast even for phase objects
The resulting image has a characteristic 3D appearance, with apparent depth.
8.2 Schlieren Method
The Schlieren technique is a refined form of oblique darkfield: 1. Oblique illumination at a shallow angle 2. A Schlieren blade (knife edge) positioned to partially block the scattered light 3. Fine control over contrast by blade position
Schlieren is particularly useful for: - Visualizing phase gradients (refractive index changes) - Air flow visualization (optical path length changes) - Detecting mechanical waves and stress patterns - Biological structures with fine phase variations
9. Practical Considerations in Microscopy
9.1 Numerical Aperture and Immersion
Air objectives: - NA up to ~1.0 - Working distance: moderate (long) - No immersion liquid needed
Oil immersion objectives: - NA up to ~1.4 (occasionally 1.5 with special oil) - Working distance: very small (~0.1 mm) - Cedar oil (\(n = 1.515\)) increases light cone angle in the object space - Must match oil refractive index precisely; mismatch degrades image
Water immersion objectives: - NA up to ~1.2 - Used for living cells in aqueous environment - \(n_{\text{water}} = 1.33\)
The higher the NA, the better the resolution, but: - Smaller working distance → difficult to focus live cells - Optical aberrations increase (spherical, chromatic) - More expensive objective
# Create a simple object: parallel linessize =256x = np.linspace(-20, 20, size)y = np.linspace(-20, 20, size)X, Y = np.meshgrid(x, y)# Object: three vertical linesline_spacing =8object_img = (np.abs(X % line_spacing - line_spacing/2) <1.0).astype(float)object_img +=0.3* (np.abs(Y -5) <2).astype(float) # Add horizontal lineobject_img = np.clip(object_img, 0, 1)# Compute Fourier transformobject_ft = np.fft.fftshift(np.fft.fft2(object_img))object_spectrum = np.abs(object_ft)# Create circular aperture (NA filter)# NA = 0.4 corresponds to cutoff frequencycutoff =int(size *0.15) # NA relative to image sizeU, V = np.meshgrid(np.linspace(-1, 1, size), np.linspace(-1, 1, size))aperture = (U**2+ V**2<=0.15**2).astype(float)# Apply filterfiltered_spectrum = object_ft * aperture# Reconstruct image via inverse Fourier transformreconstructed_img = np.abs(np.fft.ifft2(np.fft.ifftshift(filtered_spectrum)))reconstructed_img /= reconstructed_img.max()# Create figure with subplotsfig, axes = plt.subplots(2, 2, figsize=get_size(14, 12))# Objectax = axes[0, 0]ax.imshow(object_img, cmap='gray', extent=[-20, 20, -20, 20], origin='lower')ax.set_xlabel('x (μm)')ax.set_ylabel('y (μm)')ax.set_aspect('equal')# Fourier spectrum (log scale for visibility)ax = axes[0, 1]spectrum_log = np.log1p(object_spectrum)im = ax.imshow(spectrum_log, cmap='hot', extent=[-1, 1, -1, 1], origin='lower')circle = plt.Circle((0, 0), 0.15, fill=False, edgecolor='cyan', linewidth=1.5, linestyle='--')ax.add_patch(circle)ax.set_xlabel('u (norm.)')ax.set_ylabel('v (norm.)')ax.set_aspect('equal')# Filtered spectrumax = axes[1, 0]filtered_spectrum_display = np.log1p(np.abs(filtered_spectrum))im = ax.imshow(filtered_spectrum_display, cmap='hot', extent=[-1, 1, -1, 1], origin='lower')circle = plt.Circle((0, 0), 0.15, fill=False, edgecolor='cyan', linewidth=1.5)ax.add_patch(circle)ax.set_xlabel('u (norm.)')ax.set_ylabel('v (norm.)')ax.set_aspect('equal')# Reconstructed imageax = axes[1, 1]ax.imshow(reconstructed_img, cmap='gray', extent=[-20, 20, -20, 20], origin='lower')ax.set_xlabel('x (μm)')ax.set_ylabel('y (μm)')ax.set_aspect('equal')fig.set_layout_engine('constrained')plt.savefig('img/abbe_imaging.png', dpi=150, bbox_inches='tight')plt.show()print("High-frequency details are lost due to the finite NA of the objective.")
Figure 1— Abbe’s double Fourier transform theory of imaging. (a) Object: a pattern of lines. (b) Fourier spectrum in back focal plane (hot colormap: log intensity). (c) Spectrum after low-pass filtering by the objective aperture (NA = 0.4, hot colormap: log intensity). (d) Reconstructed image via inverse Fourier transform.
High-frequency details are lost due to the finite NA of the objective.
Example 2: Effect of Numerical Aperture on Resolution
Code
# Create object: Ronchi grating with varying frequenciessize =256x = np.linspace(-10, 10, size)X, Y = np.meshgrid(x, x)# Multi-frequency gratingobject_grating = (np.sin(2* np.pi *3* X) +1) /2# 3 lines/10 μmobject_grating +=0.5* (np.sin(2* np.pi *6* X) +1) /2# 6 lines/10 μmobject_grating +=0.3* (np.sin(2* np.pi *12* X) +1) /2# 12 lines/10 μmobject_grating /=2.8object_grating = np.clip(object_grating, 0, 1)# Different NA valuesna_values = [0.2, 0.4, 0.6, 1.0, 1.4]images_na = []U, V = np.meshgrid(np.linspace(-1, 1, size), np.linspace(-1, 1, size))object_ft = np.fft.fftshift(np.fft.fft2(object_grating))for na in na_values: aperture = (U**2+ V**2<= (na *0.15)**2).astype(float) filtered_ft = object_ft * aperture img = np.abs(np.fft.ifft2(np.fft.ifftshift(filtered_ft))) images_na.append(img / img.max())# Create figurefig, axes = plt.subplots(2, 3, figsize=get_size(15, 10))# Top row: object and NA = 0.2ax = axes[0, 0]ax.imshow(object_grating, cmap='gray', extent=[-10, 10, -10, 10], origin='lower')ax.set_xlabel('x (μm)')ax.set_ylabel('y (μm)')ax.set_aspect('equal')for i, na inenumerate(na_values): row = (i +1) //3 col = (i +1) %3if row >=2:break ax = axes[row, col] ax.imshow(images_na[i], cmap='gray', extent=[-10, 10, -10, 10], origin='lower') ax.set_xlabel('x (μm)') ax.set_ylabel('y (μm)') ax.set_aspect('equal')# Bottom row: intensity profile at different NAsax = axes[1, 2]y_center = size //2for i, na inenumerate(na_values): intensity_line = images_na[i][y_center, :] ax.plot(x, intensity_line, label=f'NA = {na:.1f}', linewidth=1.5)ax.set_xlabel('x (μm)')ax.set_ylabel('Intensity')ax.legend(fontsize=8)ax.grid(True, alpha=0.3)ax.set_xlim([-5, 5])plt.tight_layout()plt.savefig('img/na_resolution.png', dpi=150, bbox_inches='tight')plt.show()print(f"Higher NA resolves finer details. Resolution: d = 0.61λ/NA")
Figure 2— Effect of numerical aperture on resolution. Top row: object (high-frequency grating). Middle rows: images at different NA values. Bottom: intensity profile showing how increased NA resolves finer details.
Higher NA resolves finer details. Resolution: d = 0.61λ/NA
/var/folders/t4/_9qps8wj56jc60nwkr3nrcr00000gn/T/ipykernel_11904/1430197535.py:149: UserWarning:
constrained_layout not applied because axes sizes collapsed to zero. Try making figure larger or Axes decorations smaller.
/Users/fci/python_default_env/.venv/lib/python3.13/site-packages/IPython/core/pylabtools.py:170: UserWarning:
constrained_layout not applied because axes sizes collapsed to zero. Try making figure larger or Axes decorations smaller.
Figure 4— Resolution limit and Airy patterns. (a) PSF cross-section at NA = 1.0 with Rayleigh distance marked. (b) 2D PSF (hot colormap: intensity). (c) NA vs. Rayleigh limit. (d) Two sources at the Rayleigh limit (hot colormap: intensity, cyan crosses mark source positions). (e) Two unresolved sources. (f) Intensity profiles (blue: at Rayleigh limit, orange: unresolved). (g) Rayleigh limit vs. wavelength.
Rayleigh criterion for NA = 1.0: d = 0.336 μm
Example 5: Optical Transfer Function (OTF) for Incoherent Imaging
Code
# Compute OTF for a circular aperturedef compute_otf_circular(k, na, wavelength):""" Optical Transfer Function for a circular aperture (incoherent imaging). OTF is the autocorrelation of the pupil function. For a circular uniform pupil: OTF(k) = (2/π) * [arccos(k/2k_max) - (k/2k_max)*sqrt(1 - (k/2k_max)²)] where k is spatial frequency, k_max = NA/λ """ k_max = na / wavelength k_normalized = k / (2* k_max)# Avoid division by zero and out-of-range arccos k_normalized = np.clip(k_normalized, 0, 1) otf = np.where(k <=2* k_max, (2/ np.pi) * (np.arccos(k_normalized) - k_normalized * np.sqrt(1- k_normalized**2)),0)return otf, k_max# Parameterswavelength =0.55# μm (green light)na_values = np.array([0.3, 0.6, 1.0, 1.4])# Spatial frequency rangek = np.linspace(0, 3, 300) / wavelength # in units of 1/μm# Compute OTF for each NAfig, axes = plt.subplots(1, 3, figsize=get_size(15, 5))# Plot 1: OTF profiles at different NAax = axes[0]for na in na_values: otf, k_max = compute_otf_circular(k, na, wavelength) ax.plot(k * wavelength, otf, linewidth=2, label=f'NA = {na:.1f}', marker='o', markersize=3, markevery=15)# Mark cutoff frequency k_c =2* na / wavelength ax.axvline(x=k_c, linestyle='--', alpha=0.4, linewidth=1)ax.set_xlabel('Spatial frequency $k$ (in units of $\lambda^{-1}$)')ax.set_ylabel('OTF($k$)')ax.set_xlim([0, 3])ax.set_ylim([0, 1.05])ax.legend(fontsize=9)ax.grid(True, alpha=0.3)# Plot 2: Coherent vs. Incoherent cutoffax = axes[1]na_range = np.linspace(0.2, 1.6, 50)k_c_coherent = na_range / wavelengthk_c_incoherent =2* na_range / wavelengthax.plot(na_range, k_c_incoherent * wavelength, 'b-', linewidth=2.5, label='Incoherent (OTF)')ax.plot(na_range, k_c_coherent * wavelength, 'r--', linewidth=2.5, label='Coherent (CTF)')ax.fill_between(na_range, k_c_coherent * wavelength, k_c_incoherent * wavelength, alpha=0.2, color='blue', label='2× resolution gain')ax.set_xlabel('Numerical Aperture')ax.set_ylabel('Cutoff frequency $k_c$ ($\lambda^{-1}$)')ax.legend(fontsize=9)ax.grid(True, alpha=0.3)# Plot 3: Modulation transfer function (contrast preservation)ax = axes[2]na_ref =1.0otf, k_max = compute_otf_circular(k, na_ref, wavelength)ax.fill_between(k * wavelength, 0, otf, alpha=0.3, color='steelblue', label='Passband')ax.plot(k * wavelength, otf, 'steelblue', linewidth=2.5)# Add annotationsk_c_ref =2* na_ref / wavelengthax.axvline(x=k_c_ref, color='red', linestyle='--', linewidth=2, label=f'Cutoff at NA={na_ref}')ax.text(k_c_ref +0.1, 0.5, f'$k_c = 2 \\times \\text{{NA}}/\\lambda$', fontsize=10)# Mark some key frequenciesfor frac in [0.25, 0.5, 0.75, 1.0]: k_test = frac * k_c_ref otf_val = compute_otf_circular(np.array([k_test]), na_ref, wavelength)[0][0] ax.plot(k_test, otf_val, 'ro', markersize=6) ax.text(k_test +0.05, otf_val +0.05, f'{frac:.0%}', fontsize=8)ax.set_xlabel('Spatial frequency $k$ ($\lambda^{-1}$)')ax.set_ylabel('$|H_{\\text{OTF}}(k)|$ (Modulation)')ax.set_xlim([0, 3])ax.set_ylim([0, 1.1])ax.legend(fontsize=9)ax.grid(True, alpha=0.3)plt.tight_layout()plt.savefig('img/otf.png', dpi=150, bbox_inches='tight')plt.show()# Print summary tableprint("="*70)print("OPTICAL TRANSFER FUNCTION (OTF) SUMMARY")print("="*70)print(f"\nFor wavelength λ = {wavelength} μm (green light):\n")print("NA | CTF Cutoff (NA/λ) | OTF Cutoff (2NA/λ) | Resolution Gain")print("-"*70)for na in na_values: ctf_cutoff = na / wavelength otf_cutoff =2* na / wavelengthprint(f"{na:.1f} | {ctf_cutoff:8.2f} μm⁻¹ | {otf_cutoff:9.2f} μm⁻¹ | 2× (incoherent wins)")print("\nKey insight: Incoherent imaging can resolve details at twice the")print("frequency compared to coherent imaging with the same NA!")print("="*70)
Figure 5— Optical Transfer Function (OTF) for incoherent imaging at different NA values. (a) OTF profiles showing how higher NA extends the passband to higher spatial frequencies. (b) Comparison of coherent (CTF) vs. incoherent (OTF) cutoff frequencies. (c) How OTF modulates contrast as a function of spatial frequency.
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OPTICAL TRANSFER FUNCTION (OTF) SUMMARY
======================================================================
For wavelength λ = 0.55 μm (green light):
NA | CTF Cutoff (NA/λ) | OTF Cutoff (2NA/λ) | Resolution Gain
----------------------------------------------------------------------
0.3 | 0.55 μm⁻¹ | 1.09 μm⁻¹ | 2× (incoherent wins)
0.6 | 1.09 μm⁻¹ | 2.18 μm⁻¹ | 2× (incoherent wins)
1.0 | 1.82 μm⁻¹ | 3.64 μm⁻¹ | 2× (incoherent wins)
1.4 | 2.55 μm⁻¹ | 5.09 μm⁻¹ | 2× (incoherent wins)
Key insight: Incoherent imaging can resolve details at twice the
frequency compared to coherent imaging with the same NA!
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Summary
In this lecture, we have developed a complete understanding of how microscopes form images, based on Abbe’s theory and Fourier optics:
Image formation is a double Fourier transform: Object → Fourier plane → Image
Numerical aperture sets the frequency cutoff and resolution limit
Brightfield microscopy suffers from poor contrast for transparent (phase) objects
Darkfield microscopy exploits Fourier filtering to reveal phase structure and scattering
Resolution is fundamentally limited by diffraction: \(d = 0.61 \lambda / \text{NA}\)
In the next lecture, we will explore Phase Contrast Microscopy, which solves the phase-object problem in brightfield by introducing a phase shift in the Fourier plane, converting invisible phase variations into visible intensity variations—an elegant application of Fourier filtering.
Acts as a low-pass filter: transmits only spatial frequencies with \(\sqrt{u^2+v^2} \leq \text{NA}/\lambda\)
Pupil function
The transfer function in field space (coherent): \(H_{\text{CTF}}(u,v) = \text{pupil}(u,v)\), sharp cutoff at \(\text{NA}/\lambda\)
Point-spread function (PSF)
Fourier transform of the pupil function: \(\text{PSF} = \|\mathcal{F}^{-1}[\text{pupil}]\|^2\)
Optical Transfer Function (OTF)
Autocorrelation of the pupil function (incoherent imaging): smooth modulation of spatial frequencies, cutoff at \(2\text{NA}/\lambda\)
Resolution limit
Set by the cutoff frequency: \(d = 0.61\lambda/\text{NA}\) (incoherent), or equivalently, \(d = \lambda/(2\text{NA})\) from the OTF cutoff \(2\text{NA}/\lambda\)
Brightfield microscopy
No high-pass component: DC (zero-order) dominates; phase objects are invisible because phase information is encoded in the imaginary part of Fourier components, but we only measure intensity
Brightfield phase problem
Phase information \(\phi(x,y)\) modulates the Fourier spectrum as \(e^{i\phi}\), but intensity detection loses this phase: \(\|\mathcal{F}[e^{i\phi}]\|^2\) loses the phase of the components
Darkfield microscopy
High-pass filter: removes DC component by blocking the zero-order light. Transfer function: \(H_{\text{DF}}(u,v) = H_{\text{BF}} - \delta(u,v)\). Only high-frequency scattered light contributes, revealing edges and phase variations
Fourier filtering
Placing masks in the back focal plane directly modulates which spatial frequencies reach the image plane: low-pass (blur), high-pass (edges), band-pass (fine detail only)
Coherent vs. incoherent
Coherent: CTF is the pupil itself (sharp edges); Incoherent: OTF is pupil autocorrelation (smooth decay). Incoherent wins because its cutoff is twice as high, allowing finer resolution with the same NA
Key takeaway: Every concept in microscopy can be understood as a filter in spatial frequency space. The ultimate goal is to design imaging systems that extend the passband to higher frequencies (higher NA) or selectively amplify contrast at specific frequencies (darkfield, phase contrast, structured illumination).
Experimental Connections
Image formation in microscopy is best understood by doing it:
Abbe diffraction experiment Place a fine grating (e.g., a diffraction grating or a piece of fine mesh) on the microscope stage. In the back focal plane of the objective, observe the discrete diffraction orders. Progressively block higher orders with a slit or iris — watch how the image loses fine detail. This is Abbe’s original experiment from 1873.
Measuring the point spread function Image fluorescent beads smaller than the diffraction limit (e.g., 100 nm beads with a 0.5 μm resolution objective). The bead images are direct measurements of the PSF. Fit with an Airy function to extract the effective NA. Compare objectives of different NA.
Darkfield microscopy in practice Block the central (DC) order at the back focal plane with a small opaque disk. Only scattered light reaches the image plane — transparent specimens that are invisible in brightfield suddenly glow against a dark background. Try with unstained onion skin cells or colloidal nanoparticles in water.
Resolution limit demonstration Image a USAF 1951 resolution test target with objectives of different NA. Identify the smallest resolved group/element. Compare to the theoretical Rayleigh limit \(d = 0.61\lambda/\text{NA}\). Discuss why the measured resolution is often worse than the theoretical limit (aberrations, vibrations, sample preparation).
Spatial filtering Place masks in the Fourier plane of a 4f system: low-pass (blur), high-pass (edge detection), directional (enhance features along one axis). This directly connects Fourier optics theory to practical image processing.
