Appendix C: Fourier Transform in Optics - Exploring Optics: From Fundamentals to Advanced Applications

“The Fourier transform is a tool that breaks down a signal into its constituent frequencies. In optics, it reveals how light decomposes into its spatial frequency components.”

Joseph Fourier (paraphrased)

The Fourier transform stands as one of the most powerful mathematical tools in all of optics, providing a bridge between the spatial domain (where we observe intensity patterns) and the frequency domain (where we understand the wave components). Far from being merely computational convenience, Fourier analysis reveals deep physical insights about how light propagates, diffracts, and forms images. This appendix will guide you through the essential concepts, building from fundamental principles to sophisticated applications that illuminate the behavior of optical systems.

1C.1 What Is the Fourier Transform and Why Do We Need It?¶

1.1C.1.1 The Fundamental Concept¶

In temporal signal processing, we’re familiar with the idea that any complex waveform can be decomposed into sinusoidal components of different frequencies. The Fourier transform extends this concept to spatial dimensions: any spatial pattern can be decomposed into spatial sinusoidal components with different spatial frequencies.

Consider a simple example: a photograph contains both large-scale features (the overall shape of objects) and fine details (textures, edges). Fourier analysis separates these into:

Low spatial frequencies: Large-scale variations, overall structure
High spatial frequencies: Fine details, sharp edges, textures

In optics, this decomposition is crucial because optical systems often act differently on different spatial frequencies—a lens might blur fine details while preserving large-scale structure.

1.2C.1.2 Mathematical Definition¶

For a two-dimensional function $f(x,y)$ (such as an optical field or intensity distribution), the Fourier transform is:

F(k_x, k_y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) e^{-i(k_x x + k_y y)} dx dy

(1)

The inverse Fourier transform recovers the original function:

f(x,y) = \frac{1}{(2\pi)^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} F(k_x, k_y) e^{i(k_x x + k_y y)} dk_x dk_y

(2)

1.3C.1.3 Physical Interpretation in Optics¶

In optics, the Fourier transform has profound physical meaning:

Table 1:Fourier Transform Interpretations

Domain	Variable	Physical Meaning	Units
Spatial Domain	$f(x,y)$	Field amplitude or intensity at position	V/m or W/m²
Frequency Domain	$F(k_x, k_y)$	Amplitude of spatial frequency component	Complex amplitude
Wave Vector	$(k_x, k_y)$	Direction and spatial frequency of plane wave	rad/m

The key insight is that any optical field can be viewed as a superposition of plane waves, each traveling in a slightly different direction. The Fourier transform tells us the amplitude and phase of each plane wave component.

1.4C.1.4 Why Fourier Analysis Works So Well in Optics¶

Fourier methods are particularly powerful in optics because:

Linear systems: Most optical systems are linear, and Fourier transforms preserve linearity
Translation invariance: Many optical elements affect all spatial frequencies in the same way
Plane wave solutions: Maxwell’s equations have plane wave solutions, which are the building blocks of Fourier analysis
Convolution property: Optical propagation often involves convolutions, which become simple multiplication in Fourier space

2C.2 Essential Properties and Theorems¶

2.1C.2.1 Linearity¶

The Fourier transform is linear:

\mathcal{F}[af(x,y) + bg(x,y)] = a\mathcal{F}[f(x,y)] + b\mathcal{F}[g(x,y)]

(3)

Physical meaning: If light consists of multiple components, we can analyze each component separately and add the results.

2.2C.2.2 Shift Theorem¶

A spatial shift in the object domain becomes a phase factor in frequency domain:

\mathcal{F}[f(x-x_0, y-y_0)] = F(k_x, k_y) e^{-i(k_x x_0 + k_y y_0)}

(4)

2.3C.2.3 Scaling Theorem¶

Magnifying an object compresses its Fourier transform:

\mathcal{F}[f(ax, by)] = \frac{1}{|ab|} F\left(\frac{k_x}{a}, \frac{k_y}{b}\right)

(5)

Physical meaning: A telescope that magnifies an image spreads out the angular frequencies (making them finer), which is exactly what we observe—magnification reveals finer angular details.

2.4C.2.4 Convolution Theorem¶

This is perhaps the most important property for optics:

\mathcal{F}[f \star g] = \mathcal{F}[f] \cdot \mathcal{F}[g]

(6)

where $\star$ denotes convolution: $(f \star g)(x,y) = \int \int f(x',y') g(x-x', y-y') dx' dy'$

2.5C.2.5 Parseval’s Theorem¶

Energy is conserved between domains:

\int \int |f(x,y)|^2 dx dy = \frac{1}{(2\pi)^2} \int \int |F(k_x, k_y)|^2 dk_x dk_y

(7)

Physical meaning: The total optical power is the same whether we calculate it in the spatial domain or the frequency domain.

3C.3 Common Fourier Transform Pairs in Optics¶

3.1C.3.1 Fundamental Building Blocks¶

Table 2:Essential Fourier Pairs

Function f(x,y)	Fourier Transform F(k_x, k_y)	Optical Example
Delta function $\delta(x,y)$	1 (constant)	Point source → plane wave
Constant 1	$(2\pi)^2 \delta(k_x, k_y)$	Uniform illumination → single frequency
Plane wave $e^{i(k_0 x + k_1 y)}$	$(2\pi)^2 \delta(k_x - k_0, k_y - k_1)$	Single spatial frequency
Cosine grating $\cos(k_0 x)$	$\pi[\delta(k_x - k_0) + \delta(k_x + k_0)]\delta(k_y)$	Diffraction grating

3.2C.3.2 Aperture Functions¶

Rectangular aperture (width $a$ , height $b$ ):

\text{rect}\left(\frac{x}{a}\right)\text{rect}\left(\frac{y}{b}\right) \leftrightarrow ab \cdot \text{sinc}\left(\frac{k_x a}{2}\right)\text{sinc}\left(\frac{k_y b}{2}\right)

(8)

where $\text{sinc}(u) = \sin(u)/u$ .

Circular aperture (radius $R$ ):

\text{circ}\left(\frac{r}{R}\right) \leftrightarrow R^2 \frac{J_1(k_r R)}{k_r R/2}

(9)

where $J_1$ is the first-order Bessel function and $k_r = \sqrt{k_x^2 + k_y^2}$ .

Physical meaning: This gives the famous Airy pattern for circular aperture diffraction.

3.3C.3.3 Gaussian Functions¶

The Gaussian function is its own Fourier transform (up to scaling):

e^{-(x^2 + y^2)/2\sigma^2} \leftrightarrow 2\pi\sigma^2 e^{-\sigma^2(k_x^2 + k_y^2)/2}

(10)

4C.4 The Angular Spectrum Representation¶

4.1C.4.1 Plane Wave Decomposition¶

One of the most powerful concepts in Fourier optics is the angular spectrum representation. Any optical field $U(x,y,z)$ can be written as a superposition of plane waves:

U(x,y,z) = \int \int A(k_x, k_y) e^{i(k_x x + k_y y + k_z z)} dk_x dk_y

(11)

where:

$A(k_x, k_y)$ is the angular spectrum (Fourier transform of the field at z = 0)
$k_z = \sqrt{k^2 - k_x^2 - k_y^2}$ for propagating waves
$k = 2\pi/\lambda$ is the magnitude of the wave vector

4.2C.4.2 Propagation in Free Space¶

The beauty of the angular spectrum approach is that free space propagation becomes incredibly simple. If we know the field at plane z = 0, the field at any other plane z is:

U(x,y,z) = \mathcal{F}^{-1}[A(k_x, k_y) e^{ik_z z}]

(12)

4.3C.4.3 Fresnel and Fraunhofer Approximations¶

Fresnel approximation (near field): For small angles, $k_z \approx k - \frac{k_x^2 + k_y^2}{2k}$

U(x,y,z) \approx e^{ikz} \mathcal{F}^{-1}[A(k_x, k_y) e^{-i(k_x^2 + k_y^2)z/(2k)}]

(13)

Fraunhofer approximation (far field): For $z \gg$ aperture size squared/wavelength:

U(x,y,z) \approx \frac{e^{ikz}}{z} A\left(\frac{kx}{z}, \frac{ky}{z}\right)

(14)

5C.5 Applications in Optical Systems¶

5.1C.5.1 Lens as a Fourier Transformer¶

A converging lens performs a spatial Fourier transform under the right conditions. If we place an object at the front focal plane of a lens, the field in the back focal plane is proportional to the Fourier transform of the object:

U_{back}(x,y) \propto \mathcal{F}[U_{front}(x,y)]

(15)

5.2C.5.2 Transfer Functions¶

Every linear optical system can be characterized by its optical transfer function (OTF):

H(k_x, k_y) = \frac{\text{Output spectrum}}{\text{Input spectrum}}

(16)

For an aberration-free lens with circular aperture of radius $R$ :

H(k_x, k_y) = \begin{cases} 1 & \text{if } \sqrt{k_x^2 + k_y^2} \leq \frac{2\pi R}{\lambda z} \\ 0 & \text{otherwise} \end{cases}

(17)

5.3C.5.3 Coherent vs. Incoherent Imaging¶

Coherent imaging: The field amplitudes add, and the system transfer function operates on the field:

U_{image} = \mathcal{F}^{-1}[H_{coherent}(k_x, k_y) \cdot \mathcal{F}[U_{object}]]

(18)

Incoherent imaging: The intensities add, and the system operates on intensity:

I_{image} = \mathcal{F}^{-1}[H_{incoherent}(k_x, k_y) \cdot \mathcal{F}[I_{object}]]

(19)

The incoherent transfer function is related to the coherent one:

H_{incoherent}(k_x, k_y) = \int \int H_{coherent}(k_x', k_y') H_{coherent}^*(k_x' - k_x, k_y' - k_y) dk_x' dk_y'

(20)

6C.6 Diffraction Analysis Using Fourier Methods¶

6.1C.6.1 Fraunhofer Diffraction Patterns¶

For Fraunhofer diffraction, the far-field pattern is the Fourier transform of the aperture function:

Single slit (width $a$ ):

I(\theta) = I_0 \text{sinc}^2\left(\frac{ka\sin\theta}{2}\right)

(21)

Double slit (slit separation $d$ , slit width $a$ ):

I(\theta) = I_0 \text{sinc}^2\left(\frac{ka\sin\theta}{2}\right) \cos^2\left(\frac{kd\sin\theta}{2}\right)

(22)

Circular aperture (radius $R$ ):

I(\theta) = I_0 \left[\frac{2J_1(kR\sin\theta)}{kR\sin\theta}\right]^2

(23)

6.2C.6.2 Fresnel Diffraction¶

For Fresnel diffraction, we need to include the propagation phase:

U(x,y,z) = \frac{e^{ikz}}{i\lambda z} \int \int U_0(x',y') e^{ik[(x-x')^2 + (y-y')^2]/(2z)} dx' dy'

(26)

This can be evaluated using Fresnel integrals for specific geometries like straight edges and circular apertures.

6.3C.6.3 Babinet’s Principle¶

If apertures A and B are complementary (A + B = complete aperture), then:

U_A(\mathbf{r}) + U_B(\mathbf{r}) = U_{complete}(\mathbf{r})

(27)

In Fourier terms:

F_A(k_x, k_y) + F_B(k_x, k_y) = F_{complete}(k_x, k_y)

(28)

This principle allows us to calculate diffraction from complex apertures by relating them to simpler complementary shapes.

7C.7 Spatial Filtering and Optical Processing¶

7.1C.7.1 Spatial Frequency Filtering¶

In the Fourier plane of a 4f system, we can place physical masks to modify specific spatial frequencies:

Low-pass filter: Blocks high spatial frequencies (smooths image)

H_{LP}(k_x, k_y) = \begin{cases} 1 & \text{if } \sqrt{k_x^2 + k_y^2} < k_c \\ 0 & \text{otherwise} \end{cases}

(29)

High-pass filter: Blocks low spatial frequencies (enhances edges)

H_{HP}(k_x, k_y) = 1 - H_{LP}(k_x, k_y)

(30)

Band-pass filter: Selects a range of spatial frequencies

H_{BP}(k_x, k_y) = \begin{cases} 1 & \text{if } k_1 < \sqrt{k_x^2 + k_y^2} < k_2 \\ 0 & \text{otherwise} \end{cases}

(31)

7.2C.7.2 Phase Contrast and Differential Interference¶

Phase contrast: Converts phase variations to intensity variations by introducing a phase shift to the zero-order (DC) component.

Zernike phase contrast:

H_{PC}(k_x, k_y) = 1 + \Delta\phi \cdot \delta(k_x, k_y)

(32)

where $\Delta\phi$ is typically $\pi/2$ .

7.3C.7.3 Optical Correlation¶

Cross-correlation can be performed optically by placing a filter with the Fourier transform of a reference pattern:

g(x,y) = \mathcal{F}^{-1}[F(k_x, k_y) \cdot G^*(k_x, k_y)]

(33)

This technique is used for:

Pattern recognition
Target tracking
Image registration
Defect detection

8C.8 Digital Implementation and Computational Aspects¶

8.1C.8.1 Discrete Fourier Transform (DFT)¶

For computational implementation, we use the discrete Fourier transform:

F[m,n] = \sum_{j=0}^{N-1} \sum_{k=0}^{N-1} f[j,k] e^{-2\pi i(mj + nk)/N}

(34)

The Fast Fourier Transform (FFT) algorithm makes this computationally efficient, reducing complexity from $O(N^4)$ to $O(N^2 \log N)$ for an $N \times N$ image.

8.2C.8.2 Sampling and Aliasing¶

Nyquist sampling theorem: To avoid aliasing, the sampling frequency must be at least twice the highest spatial frequency:

\Delta x < \frac{\pi}{k_{max}}

(35)

Practical considerations:

Zero-padding: Improves interpolation in Fourier domain
Windowing: Reduces artifacts from finite sample size
Periodic boundaries: FFT assumes periodic boundary conditions

8.3C.8.3 Computational Propagation¶

Angular spectrum method for numerical propagation:

import numpy as np

def propagate_angular_spectrum(field, wavelength, distance, dx):
    """Propagate optical field using angular spectrum method"""

    # Fourier transform of input field
    Field_fft = np.fft.fft2(field)

    # Create spatial frequency grids
    N = field.shape[0]
    kx = np.fft.fftfreq(N, dx) * 2 * np.pi
    KX, KY = np.meshgrid(kx, kx)

    # Calculate propagation phase
    k = 2 * np.pi / wavelength
    kz = np.sqrt(k**2 - KX**2 - KY**2)

    # Handle evanescent waves
    kz = np.where(KX**2 + KY**2 < k**2, kz, 1j*np.sqrt(KX**2 + KY**2 - k**2))

    # Apply propagation operator
    propagator = np.exp(1j * kz * distance)
    Field_prop_fft = Field_fft * propagator

    # Inverse transform
    field_propagated = np.fft.ifft2(Field_prop_fft)

    return field_propagated

9C.9 Advanced Topics and Modern Applications¶

9.1C.9.1 Fractional Fourier Transform¶

The fractional Fourier transform interpolates between the spatial and frequency domains:

F_a(u) = \int_{-\infty}^{\infty} f(x) K_a(u,x) dx

(36)

where the kernel $K_a(u,x)$ depends on the fractional parameter $a$ .

Applications:

Beam propagation: Fresnel propagation can be viewed as fractional Fourier transformation
Lens design: Graded-index lenses perform fractional transforms
Signal processing: Time-frequency analysis

9.2C.9.2 Wavelet Analysis¶

While Fourier analysis provides frequency information, wavelets provide both frequency and spatial localization:

W(a,b) = \int_{-\infty}^{\infty} f(x) \psi^*\left(\frac{x-b}{a}\right) dx

(37)

Optical applications:

Edge detection: Better localization than Fourier methods
Texture analysis: Multi-scale feature extraction
Adaptive optics: Real-time wavefront analysis

9.3C.9.3 Holography and Fourier Optics¶

Digital holography combines Fourier analysis with interference:

Recording: Interference between object and reference beams
Reconstruction: Fourier analysis separates different diffraction orders
Numerical focusing: Propagate to different planes using Fourier methods

9.4C.9.4 Metamaterials and Fourier Optics¶

Subwavelength structures require careful Fourier analysis:

Effective medium theory: Homogenization using spatial averaging
Resonant structures: Localized modes affect Fourier spectrum
Negative index materials: Unusual propagation characteristics

10C.10 Practical Considerations and Common Pitfalls¶

10.1C.10.1 Experimental Implementation¶

10.2C.10.2 Numerical Implementation¶

10.3C.10.3 Physical Interpretation¶

11C.11 Worked Examples¶

11.1C.11.1 Design of a Spatial Filter¶

Problem: Noise Removal

An image contains high-frequency noise that you want to remove while preserving the main features. Design an optical spatial filter.

Solution:

Setup: Use a 4f optical processor with lenses of focal length f = 200 mm.

Input: Noisy image at front focal plane of first lens Fourier plane: At back focal plane of first lens (also front focal plane of second lens) Output: At back focal plane of second lens

Filter design: Place a circular aperture in the Fourier plane to block high spatial frequencies.

Cutoff frequency calculation: If we want to preserve features larger than 10 μm, the cutoff spatial frequency is:

k_c = \frac{2\pi}{10 \times 10^{-6}} = 6.28 \times 10^5~\text{rad/m}

(38)

Physical aperture size: In the Fourier plane, spatial frequency maps to position:

r = \frac{k_r \lambda f}{2\pi} = \frac{k_c \lambda f}{2\pi}

(39)

For λ = 632.8 nm (He-Ne laser):

r = \frac{6.28 \times 10^5 \times 632.8 \times 10^{-9} \times 0.2}{2\pi} = 12.7~\text{mm}

(40)

Filter: Circular aperture with radius 12.7 mm in the Fourier plane.

11.2C.11.2 Diffraction Grating Analysis¶

Problem: Grating Resolution

A diffraction grating has 600 lines/mm and is illuminated by λ = 500 nm light. Calculate the angular positions and widths of the diffraction orders.

Solution:

Grating period: $d = 1/(600 \times 10^3) = 1.67 \times 10^{-6}$ m

Grating equation: $\sin\theta_m = m\lambda/d$

Order positions:

m = 0: $\theta_0 = 0°$ (zero order)
m = ±1: $\sin\theta_1 = \pm 500 \times 10^{-9}/(1.67 \times 10^{-6}) = ±0.30$ , so $\theta_1 = ±17.5°$
m = ±2: $\sin\theta_2 = ±0.60$ , so $\theta_2 = ±36.9°$
m = ±3: $\sin\theta_3 = ±0.90$ , so $\theta_3 = ±64.2°$

Angular width of each order: If the grating has width W = 10 mm, the angular width of each order is approximately:

\Delta\theta \approx \frac{\lambda}{W\cos\theta_m}

(41)

For m = 1: $\Delta\theta_1 \approx \frac{500 \times 10^{-9}}{0.01 \times \cos(17.5°)} = 5.2 \times 10^{-5}$ rad = 0.003°

11.3C.11.3 Lens Resolution Calculation¶

Problem: Microscope Resolution

A microscope objective has numerical aperture NA = 1.4 and operates at λ = 400 nm. What is the theoretical resolution limit?

Solution:

Rayleigh criterion: Two point sources are just resolved when the peak of one Airy pattern coincides with the first zero of the other.

Resolution limit: For a circular aperture:

\Delta x = \frac{1.22\lambda}{2 \times NA} = \frac{1.22 \times 400 \times 10^{-9}}{2 \times 1.4} = 174~\text{nm}

(42)

Spatial frequency cutoff: The highest spatial frequency that can be transmitted is:

k_c = \frac{2\pi \times NA}{\lambda} = \frac{2\pi \times 1.4}{400 \times 10^{-9}} = 2.2 \times 10^7~\text{rad/m}

(43)

Transfer function: The optical transfer function is:

H(k_x, k_y) = \begin{cases} 1 & \text{if } \sqrt{k_x^2 + k_y^2} \leq k_c \\ 0 & \text{otherwise} \end{cases}

(44)

This circular cutoff in frequency space corresponds to the finite aperture of the objective lens.

12C.12 Summary and Physical Insights¶

12.1C.12.1 Fundamental Principles¶

The Fourier transform succeeds in optics because it captures the wave nature of light:

Table 3:Core Insights

Mathematical Property	Physical Principle	Optical Manifestation
Superposition	Waves add linearly	Interference and diffraction
Frequency decomposition	Plane wave solutions	Angular spectrum representation
Convolution theorem	System linearity	Transfer functions
Uncertainty principle	Wave packet localization	Resolution limits

12.2C.12.2 Connections to Other Physics¶

Quantum mechanics: The wavefunction in quantum mechanics follows the same Fourier relationships as optical fields—position and momentum are Fourier transform pairs.

Signal processing: Digital image processing techniques directly parallel optical spatial filtering.

Crystallography: X-ray diffraction patterns are Fourier transforms of crystal structures.

12.3C.12.3 Modern Applications¶

13C.13 Practice Problems¶

13.1C.13.1 Basic Fourier Analysis¶

13.2C.13.2 Diffraction Patterns¶

13.3C.13.3 Optical System Design¶

13.4C.13.4 Numerical Implementation¶

14C.14 Final Thoughts: The Unity of Wave Physics¶

The Fourier transform reveals a profound truth about wave phenomena: every localized disturbance is composed of extended waves, and every pure wave requires infinite space. This duality—localization versus frequency purity—appears throughout physics and represents a fundamental limitation of wave-based information processing.

“The Fourier transform is a tool that reveals the hidden periodicities in data. In optics, it shows us the plane wave components that combine to create complex optical fields.”

Ronald Bracewell, “The Fourier Transform and Its Applications”

In optics, this manifests as:

Heisenberg uncertainty: $\Delta x \Delta k \geq 1/2$
Resolution limits: Finite apertures limit spatial frequency bandwidth
Beam propagation: Gaussian beams balance spatial and spectral width

As you continue your journey through optics, remember that Fourier analysis isn’t just a mathematical technique—it’s a way of understanding how information propagates through optical systems. Whether designing a telescope, analyzing a laser beam, or processing an image, the Fourier transform provides the conceptual framework that connects the mathematics to the physics.

The Fourier transform stands at the heart of modern optics, connecting the wave nature of light to the practical demands of optical system design. Master this tool, and you’ll have gained access to the full power of spatial frequency analysis—one of the most elegant and useful concepts in all of physics.