Optical Transfer Function (OTF)

The Optical Transfer Function (OTF) describes how an optical system transmits spatial frequencies from object to image. It is the Fourier transform of the point spread function:

$$ \text{OTF}(f_x, f_y) = \mathcal{F}\{\text{PSF}(x, y)\} $$

Definition

For a linear shift-invariant imaging system:

$$ \tilde{G}_{\text{image}}(f_x, f_y) = \text{OTF}(f_x, f_y) \cdot \tilde{G}_{\text{object}}(f_x, f_y) $$

The OTF is the transfer function of the imaging system in the spatial frequency domain.

Components

The OTF is generally complex-valued and can be decomposed into:

$$ \text{OTF}(f_x, f_y) = \text{MTF}(f_x, f_y) \cdot e^{j \, \text{PTF}(f_x, f_y)} $$

• MTF (Modulation Transfer Function): The magnitude $|\text{OTF}|$. Describes contrast reduction at each spatial frequency.
• PTF (Phase Transfer Function): The phase $\arg(\text{OTF})$. Describes spatial phase shifts (lateral displacement of features).

Relationship to Pupil Function

For a diffraction-limited system with pupil function $P(x, y)$, the OTF is the autocorrelation of the pupil:

$$ \text{OTF}(f_x, f_y) = \frac{\iint P(\xi, \eta) \, P^*(\xi - \lambda z_i f_x, \eta - \lambda z_i f_y) \, d\xi \, d\eta}{\iint |P(\xi, \eta)|^2 \, d\xi \, d\eta} $$

This is normalized so that $\text{OTF}(0, 0) = 1$.

Cutoff Frequency

For a circular aperture of diameter $D$ at f-number $F = z_i / D$:

$$ f_{\text{cutoff}} = \frac{1}{\lambda F} = \frac{D}{\lambda z_i} $$

The OTF is zero for all frequencies above this cutoff — the optical system acts as a low-pass filter.

Key Properties

Property	Description
$\text{OTF}(0,0) = 1$	Normalized to unity at zero frequency
$\text{OTF}(f) = \text{OTF}^*(-f)$	Hermitian symmetry (for real PSF)
$\|\text{OTF}\| \leq 1$	Contrast can only decrease, never increase
Cutoff at $1/\lambda F$	No transmission above diffraction limit

Physical Interpretation

• Low frequencies (large features): Transmitted with high contrast
• High frequencies (fine details): Attenuated, with cutoff at diffraction limit
• Aberrations reduce MTF below the diffraction-limited ideal
• Defocus causes the MTF to oscillate and can produce contrast reversal

Related Concepts

• Point spread function — spatial domain equivalent
• Airy disk — PSF of a circular aperture
• Fraunhofer diffraction — far-field limit where OTF analysis applies
• Transfer function — general concept in linear systems