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Partial Coherence : From Classical Optics to Quantum Optics in Optical Microlithography

Edvin Lundberg, Joel Grenehed

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

Over the past century the concept of optical coherence has evolved with the development of quantumoptics, alongside quantum mechanics. The use of lasers in microlithographic patterning is makingaccurate modeling and computation of projection systems using partial coherence increasingly im-portant in the modern semiconductor industry. This report aims to compile and compare theories ofpartial coherence, from mutual coherence functions to the density operator approach and the Lind-blad master equation. In classical optics, coherence is a measure of the correlation of intensities, andpartial coherence is viewed as a statistical degree of field correlation. Quantum optical coherence isdescribed in terms of photon statistics, with partial coherence originating from for example thermalfluctuations in the optical field or environmental decoherence. Mathematically the models are simi-lar, with the main difference being whether the field is described at a macroscopic or quantum scale.

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Over the past century the concept of optical coherence has evolved with the development of quantumoptics, alongside quantum mechanics. The use of lasers in microlithographic patterning is makingaccurate modeling and computation of projection systems using partial coherence increasingly im-portant in the modern semiconductor industry. This report aims to compile and compare theories ofpartial coherence, from mutual coherence functions to the density operator approach and the Lind-blad master equation. In classical optics, coherence is a measure of the correlation of intensities, andpartial coherence is viewed as a statistical degree of field correlation. Quantum optical coherence isdescribed in terms of photon statistics, with partial coherence originating from for example thermalfluctuations in the optical field or environmental decoherence. Mathematically the models are simi-lar, with the main difference being whether the field is described at a macroscopic or quantum scale.
Introduction In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point. The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons. In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.
Optical coherence is a well-established branch of physical optics in which the statistical properties of fluctuating optical fields are described in terms of correlation functions over continuous spatial and temporal degrees of freedom (DoFs). Nevertheless, in any practical setting, only discrete DoFs are ever accessible. In these scenarios, the modes are fixed, stable, and deterministic, and partial coherence arises from random relative complex amplitudes, a configuration we refer to as 'structured coherence'. Advances in structured coherence have recently unveiled new conceptual ground in optical communications and information processing in which partial coherence may be preferable to full coherence, which we call a 'coherence advantage'. We formulate structured coherence in terms of coherence matrices to investigate these recent theoretical and experimental breakthroughs. We first review optical fields characterized by a binary DoF via 2x2 Hermitian coherence matrices and introduce key concepts that take on new significance for larger-dimensional modal sets. Next, we examine the structured coherence of two binary DoFs, which can be described by 4x4 coherence matrices. We introduce the concept of coherence rank, entropy swapping, and optical cross-purity. In the perspective outlined here, coherence is viewed as a 'resource', which can be exchanged between DoFs, concentrated into a DoF or into particular modes, or spread over the DoFs. We then examine larger-dimensional modal sets, which allow for more versatile applications in optical information processing. The formulation presented here lends itself particularly to the manipulation of partial optical coherence in integrated photonic platforms, thereby opening myriad avenues for novel fundamental investigations of structured coherence and exploiting the coherence advantage in optical communications and information processing.
The statistical description of optical fields in coherence theory is based on underlying fluctuations originating at the source or during propagation through a random medium. We show that uncertainty (partial coherence) also stems from the act of ignoring a degree of freedom of a beam when observing another degree of freedom classically entangled with it. Specifically, we demonstrate that Bells measure, commonly used in tests of quantum non-locality, is as a quantitative tool that delineates intrinsic incoherence associated with statistical fluctuations from classical entanglement-based incoherence. This demonstrates, more generally, the applicability of concepts from quantum information processing to classical optical coherence theory and optical signal processing.
Optical spatial coherence, characterizing correlations between optical fields at distinct points, plays a key role in the understanding of classical and quantum phenomena. Recent decades’ technological advances, such as powerful lasers and fast, efficient photon counters enabled the study of nonclassical optical fields. Here, we describe single photon partial coherence, providing a theoretical model from quantum coherence theory and experimental observation. In particular, we make clear the role of quantum birth-zones for SPDC sources. These birth-zones are the quantum analog of classical emitters; they characterize the heralded single photon’s coherence properties. We make the first observation of the predicted inversion of the single photon degree of coherence.
The quantum theory of the electromagnetic field uncovered that classical forms of light were indeed produced by distinct superpositions of nonclassical multiphoton wavepackets. Specifically, partially coherent light represents the most common kind of classical light. Here, for the first time, we demonstrate the extraction of the constituent multiphoton quantum systems of a partially coherent light field. We shift from the realm of classical optics to the domain of quantum optics via a quantum representation of partially coherent light using its complex-Gaussian statistical properties. Our formulation of the quantum Gaussian-Schell model unveils the possibility of performing photon-number-resolving detection to isolate the constituent quantum multiphoton wavepackets of a classical light field. We experimentally verified the coherence properties of isolated vacuum systems and wavepackets with up to sixteen photons. Our findings not only demonstrate the possibility of observing quantum properties of classical macroscopic objects, but also establish a fundamental bridge between the classical and quantum worlds.