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DOE OSTI2024

Coherence in Chemistry: Foundations and Frontiers

Coherence refers to correlations in waves. Because matter has a wave-particle nature, it is unsurprising that coherence has deep connections with the most contemporary issues in chemistry research (e.g., energy harvesting, femtosecond spectroscopy, molecular qubits and more). But what does the word "coherence" really mean in the context of molecules and other quantum systems? We provide a review of key concepts, definitions, and methodologies, surrounding coherence phenomena in chemistry, and we describe how the terms "coherence" and "quantum coherence" refer to many different phenomena in chemistry. Moreover, we show how these notions are related to the concept of an interference pattern. Coherence phenomena are indeed complex, and ambiguous definitions may spawn confusion. By describing the many definitions and contexts for coherence in the molecular sciences, we aim to enhance understanding and communication in this broad and active area of chemistry.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1985

Coherent and incoherent lidar - An overview

Characteristics of coherent and incoherent lidars are described. The carrier-to-noise ratio (CNR), SNR relationship, coherence diameter, speckle averaging, multiple pulse averaging, and the possibility of single-frequency pulse generation from the laser transmitter for the two lidar types are compared. Diagrams of the incoherent lidar's direct detection system and the coherent lidar's heterodyne detection system are presented. The transverse coherence diameter is discussed and equations for backscatter from a distributed medium, transverse coherence length, and return path refractive index variations are provided. The importance of transverse coherence length in the coherence lidar system design is considered. The effectiveness of the lidar systems is evaluated in terms of CNR and SNR. The differences in generation of noise-equivalent power are studied and it is observed that the coherent lidar system achieves higher CNR values. The relation between signal fluctuation due to speckle effects and SNR is analyzed; equations for SNR in coherent and incoherent lidar systems are given.

Menzies, R. T.↗

NASA NTRS2010

Estimation of Signal Coherence Threshold and Concealed Spectral Lines Applied to Detection of Turbofan Engine Combustion Noise

Combustion noise from turbofan engines has become important, as the noise from sources like the fan and jet are reduced. An aligned and un-aligned coherence technique has been developed to determine a threshold level for the coherence and thereby help to separate the coherent combustion noise source from other noise sources measured with far-field microphones. This method is compared with a statistics based coherence threshold estimation method. In addition, the un-aligned coherence procedure at the same time also reveals periodicities, spectral lines, and undamped sinusoids hidden by broadband turbofan engine noise. In calculating the coherence threshold using a statistical method, one may use either the number of independent records or a larger number corresponding to the number of overlapped records used to create the average. Using data from a turbofan engine and a simulation this paper shows that applying the Fisher z-transform to the un-aligned coherence can aid in making the proper selection of samples and produce a reasonable statistics based coherence threshold. Examples are presented showing that the underlying tonal and coherent broad band structure which is buried under random broadband noise and jet noise can be determined. The method also shows the possible presence of indirect combustion noise. Copyright 2011 Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America.

Miles, Jeffrey Hilton↗

DOE OSTI2025

Correlated Lattice Fluctuations in CsPbBr 3 Quantum Dots Give Rise to Long-Lived Electronic Coherence

Electronic coherence is central to numerous areas of science, from quantum biology to quantum materials. In quantum materials, lead-halide perovskite (LHP) quantum dots (QDs) have been shown to support electronic coherence through observation of coherent single-photon emission and superfluorescence arising from spatial coherence at low temperatures. In contrast, direct measurement of temporal coherence between exciton states has been lacking. Here, we employ coherent multi-dimensional spectroscopy to observe an electronic coherence between exciton states in CsPbBr 3 QDs that is long-lived at room temperature, surviving nearly three times longer than the electronic dephasing time. This observation of a long-lived electronic coherence at room temperature points to nearly perfectly correlated lattice fluctuations for each excitonic state in the superposition. In conclusion, these experiments reveal that the properties of LHP QDs extend to lattice dynamics that give rise to correlated fluctuations in the basis exciton states, a process that may next be optimized by design.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Unraveling Electronic and Vibrational Coherences Following a Charge Transfer Process in a Photosystem II Reaction Center

A reaction center is a unique biological system that performs the initial charge separation within a Photosystem II (PSII) multiunit enzyme, which eventually drives the catalytic water-splitting in plants and algae. The possible role of quantum coherences coinciding with the energy and charge transfer processes in PSII reaction center is one of the active areas of research. Here, we study these quantum coherences by using a numerically exact method on an excitonic dimer model, including linear vibronic coupling and employing optimal parameters from experimental two-dimensional coherent spectroscopic measurements. This enables us to precisely capture the excitonic interaction between pigments and the dissipation of the energy from electronic and charge-transfer (CT) states to the protein environment. We employ the time nonlocal (TNL) quantum master equation to calculate the population dynamics, which yields numerically reliable results. The calculated results show that, due to the strong dissipation, the lifetime of electronic coherence is too short to have direct participation in the charge transfer processes. However, there are long-lived vibrational coherences present in the system at frequencies close to the excitionic energy gap. These are strongly coupled with the electronic coherences, which makes the detection of the electronic coherences with conventional techniques very challenging. Additionally, we unravel the strong excitonic interaction of radical pair (PD1 and PD2) in the reaction center, which results in a long-lived electronic coherence of >100 fs, even at room temperature. Our work provide important physical insight to the charge separation process in PSII reaction center, which may be helpful for better understanding of photophysical processes in other natural and artificial light-harvesting systems.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2022

Quantum coherent energy transport in the Fenna–Matthews–Olson complex at low temperature

In the primary step of natural light harvesting, the solar photon energy is captured in a photoexcited electron–hole pair, or an exciton, in chlorophyll. Its conversion to chemical potential occurs in the special pair reaction center, which is reached by downhill ultrafast excited-state energy transport through a network of chromophores. Being inherently quantum, transport could in principle occur via a matter wave, with vast implications for efficiency. How long a matter wave remains coherent is determined by the intensity by which the exciton is disturbed by the noisy biological environment. The stronger this is, the stronger the electronic coupling between chromophores must be to overcome the fluctuations and phase shifts. The current consensus is that under physiological conditions, quantum coherence vanishes on the 10-fs time scale, rendering it irrelevant for the observed picosecond transfer. Yet, at low-enough temperature, quantum coherence should in principle be present. Here, we reveal the onset of longer-lived electronic coherence at extremely low temperatures of ∼20 K. Using two-dimensional electronic spectroscopy, we determine the exciton coherence times in the Fenna–Matthew–Olson complex over an extensive temperature range. At 20 K, coherence persists out to 200 fs (close to the antenna) and marginally up to 500 fs at the reaction center. It decays markedly faster with modest increases in temperature to become irrelevant above 150 K. At low temperature, the fragile electronic coherence can be separated from the robust vibrational coherence, using a rigorous theoretical analysis. We believe that by this generic principle, light harvesting becomes robust against otherwise fragile quantum effects.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2024

Single Electron Self-coherence and Its Wave/Particle Duality in the Electron Microscope

Intensities in high-resolution phase-contrast images from electron microscopes build up discretely in time by detecting single electrons. A wave description of pulse-like coherent-inelastic interaction of an electron with matter implies a time-dependent coexistence of coherent partial waves. Their superposition forms a wave package by phase decoherence of 0.5 - 1 radian with Heisenbergs energy uncertainty ΔE H = $\hbar$/2 Δt -1 matching the energy loss ΔE of a coherent-inelastic interaction and sets the interaction time Δt. In these circumstances, the product of Planck's constant and the speed of light hc is given by the product of the expression for temporal coherence λ 2 /Δλ and the energy loss ΔE. Experimentally, the self-coherence length was measured by detecting the energy-dependent localization of scattered, plane matter waves in surface proximity exploiting the Goos–Hänchen shift. Chromatic-aberration Cc-corrected electron microscopy on boron nitride (BN) proves that the coherent crystal illumination and phase contrast are lost if the self-coherence length shrinks below the size of the crystal unit cell at ΔE > 200 eV. Finally, in perspective, the interaction time of any matter wave compares with the lifetime of a virtual particle of any elemental interaction, suggesting the present concept of coherent-inelastic interactions of matter waves might be generalizable.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Effect of sink strength on coherency loss of precipitates in dilute Cu-base alloys during in situ ion irradiation

In situ irradiations with 1 MeV Kr ions at 50~613 K up to a fluence of 6.25 × 10 14 ions/cm 2 (~1.25 displacements per atom, dpa) have been performed on pre-aged dilute Cu-0.9%Co, Cu-0.9%Fe and Cu-0.8%Cr alloys containing uniform matrix dispersions of coherent precipitates in order to study the effects of initial precipitate sink strength, damage dose and irradiation temperature on radiation-induced coherency loss of precipitates. Coherent precipitates with different point defect sink strengths (2πNd, where N and d are the precipitate density and diameter) were used in this work to examine potential differences in atomic relaxation during absorption of point defects. In all cases, irradiation to low doses (<~1 dpa) was very effective at inducing loss of precipitate coherency. At low sink strengths (~10 13 m -2 ), loss of precipitate coherency could be induced for doses ~0.01 dpa. This suggests there might be an efficient preferential medium-range strain-induced bias for absorption of interstitial defects due to tensile strains emanating from the undersized precipitates, which induces relatively rapid loss of coherency at low precipitate sink strengths. High precipitate sink strength (~10 14 m -2 ) conditions were relatively resistant to the radiation-induced loss of coherency (requiring higher doses approaching ~1 dpa) and this might be due to nearly equal numbers of interstitial and vacancy defects arriving at the precipitate interface for such high sink strength conditions. The precipitate coherency loss was observed to have a weak dependence on irradiation temperature. Molecular dynamics simulations confirm a strong effect of precipitate sink strength on the probability of interstitial absorption at precipitates.

36 MATERIALS SCIENCE↗

DOE OSTI2024

From Coherence to Function: Exploring the Connection in Chemical Systems

The role of quantum mechanical coherences or coherent superposition states in excited state processes has received considerable attention in the last two decades largely due to advancements in ultrafast laser spectroscopy. These coherence effects hold promise for enhancing the efficiency and robustness of functionally relevant processes, even when confronted with energy disorder and environmental fluctuations. Understanding coherence deeply drives us to unravel mechanisms and dynamics controlled by order and synchronization at a quantum mechanical level, envisioning optical control of coherence to enhance functions or create new ones in molecular and material systems. In this frontier, the interplay between electronic and vibrational dynamics, specifically the influence of vibrations in directing electronic dynamics, has emerged as the leading principle. Here, two energetically disparate quantum degrees of freedom work in-sync to dictate the trajectory of an excited state reaction. Moreover, with the vibrational degree being directly related to the structural composition of molecular or material systems, new molecular designs could be inspired by tailoring certain structural elements. In the realm of chemical kinetics, our understanding of the dynamics of chemical transformations is underpinned by fundamental theories, such as transition state theory, activated rate theory, and Marcus theory. These theories elucidate reaction rates by considering the energy barriers that must be overcome for reactants to transform into products. Those barriers are surmounted by the stochastic nature of energy gap fluctuations within reacting systems, emphasizing that the reaction coordinate, the pathway from reactants to products, is not rigidly defined by a specific vibrational motion but encompasses a diverse array of molecular motions. While less is known about the involvement of specific intramolecular vibrational modes, their significance in certain cases cannot be overlooked. In this Account, we summarize key experimental findings that offer deeper insights into the complex electronic–vibrational trajectories encompassing excited states afforded from state-of-the-art ultrafast laser spectroscopy in three exemplary processes: photoinduced electron transfer, singlet–triplet intersystem crossing, and intramolecular vibrational energy flow in molecular systems. We delve into the rapid decoherence, or loss of phase and amplitude correlations, of vibrational coherences along promoter vibrations during subpicosecond intersystem crossing dynamics in a series of binuclear platinum complexes. This rapid decoherence illustrates the vibration-driven reactive pathways from the Franck–Condon state to the curve crossing region. We also explore the generation of new vibrational coherences induced by impulsive reaction dynamics rather than by the laser pulse in these systems, which sheds light on specific energy dissipation pathways and thereby on the progression of the reaction trajectory in the vicinity of the curve crossing on the product side. Another property of vibrational coherences, amplitude, reveals how energy can flow from one vibration to another in the electronic excited state of a terpyridine–molybdenum complex hosting a nonreactive dinitrogen substrate. In conclusion, a slight change in vibrational energy triggers a quasi-resonant interaction, leading to constructive wavepacket interference and ultimately intramolecular vibrational redistribution from a Franck–Condon active terpyridine vibration to a dinitrogen stretching vibration, energizing the dinitrogen bond.

Electrical energy↗

DOE OSTI2025

Evaluation of a generalized least squares algorithm for infrasound beamforming with coherent background noise

Infrasonic signals of interest can occur during periods with persistent, coherent, background noise, which may be natural or anthropogenic. For high signal-to-noise (SNR) ratio transient signals, an ‘overprinting’ of the coherent background may occur, and the signal may still be detected. However, this approach fails for low SNR signals of interest, which may be obscured by coherent noise. An infrasound beamforming method based on generalized least squares (GLS) is investigated for detecting transient signals of interest in the presence of coherent and incoherent background noise. This approach relies on an estimate of the noise covariance, captured in a covariance matrix, to effectively null contributions to the array response from noisy directions of arrival. Synthetic array data is used to investigate the performance of the GLS beamformer compared to the Bartlett beamformer when coherent and incoherent backgrounds are present. Additionally, the effects of array element number and relative strength of the interfering signal on the GLS estimates is investigated. GLS empirical area under the curve estimates suggest that the beamformer can recover coherent power for a signal of interest lower in amplitude than the coherent background, but this effectiveness degrades more quickly with SNR for a four element array compared to a six or eight element infrasound array. Finally, infrasound from the Forensic Surface Experiment, a bolide signal observed at IMS array I37NO, and a volcanic signal recorded at the Alaska Volcano Observatory array ADKI are used to evaluate GLS performance on recorded data. A ten minute window was used to capture the background noise, and the coherent background signal was nulled in all three examples.

58 GEOSCIENCES↗

DOE OSTI2025

Tracing Long-Lived Atomic Coherences Generated via Molecular Conical Intersections

Accessing coherences is key to fully understand and control ultrafast dynamics of complex quantum systems like molecules. Most photochemical processes are mediated by conical intersections, which generate coherences between electronic states in molecules. We show with accurate calculations performed on gas-phase methyl iodide that electronic coherences of spin-orbit-split states persist in atomic iodine after dissociation. Our simulation predicts a maximum magnitude of vibronic coherence in the molecular regime of 0.75% of the initially photoexcited state population. Upon dissociation, one-third of this coherence magnitude is transferred to a long-lived atomic coherence where vibrational decoherence can no longer occur. To trace these dynamics, we propose a tabletop experimental approach—heterodyned attosecond four-wave-mixing spectroscopy. This technique can temporally resolve small electronic coherence magnitudes and reconstruct the full complex coherence function via phase cycling. Hence, heterodyned attosecond four-wave-mixing spectroscopy leads the way to a complete understanding and optimal control of spin-orbit-coupled electronic states in photochemistry.

Chemical Physics↗

NASA NTRS1976

Two-photon coherent states of the radiation field

The concept of a two-photon coherent state is introduced for applications in quantum optics. It is a simple generalization of the well-known minimum-uncertainty wave packets. The detailed properties of two-photon coherent states are developed and distinguished from ordinary coherent states. These two-photon coherent states are mathematically generated from coherent states through unitary operators associated with quadratic Hamiltonians. Physically they are the radiation states of ideal two-photon lasers operating far above threshold, according to the self-consistent-field approximation. The mean-square quantum noise behavior of these states, which is basically the same as those of minimum-uncertainty states, leads to applications not obtainable from coherent states or one-photon lasers. The essential behavior of two-photon coherent states is unchanged by small losses in the system. The counting rates or distributions these states generate in photocount experiments also reveal their difference from coherent states.

Yuen, H. P.↗

NASA NTRS1996

Optical amplifiers for coherent lidar

We examine application of optical amplification to coherent lidar for the case of a weak return signal (a number of quanta of the return optical field close to unity). We consider the option that has been explored to date, namely, incorporation of an optical amplifier operated in a linear manner located after reception of the signal and immediately prior to heterodyning and photodetection. We also consider alternative strategies where the coherent interaction, the nonlinear processes, and the amplification are not necessarily constrained to occur in the manner investigated to date. We include the complications that occur because of mechanisms that occur at the level of a few, or one, quantum excitation. Two factors combine in the work to date that limit the value of the approach. These are: (1) the weak signal tends to require operation of the amplifier in the linear regime where the important advantages of nonlinear optical processing are not accessed, (2) the linear optical amplifier has a -3dB noise figure (SN(out)/SN(in)) that necessarily degrades the signal. Some improvement is gained because the gain provided by the optical amplifier can be used to overcome losses in the heterodyned process and photodetection. The result, however, is that introduction of an optical amplifier in a well optimized coherent lidar system results in, at best, a modest improvement in signal to noise. Some improvement may also be realized on incorporating more optical components in a coherent lidar system for purely practical reasons. For example, more compact, lighter weight, components, more robust alignment, or more rapid processing may be gained. We further find that there remain a number of potentially valuable, but unexplored options offered both by the rapidly expanding base of optical technology and the recent investigation of novel nonlinear coherent interference phenomena occurring at the single quantum excitation level. Key findings are: (1) insertion of linear optical amplifiers in well optimized conventional lidar systems offers modest improvements, at best, (2) the practical advantages of optical amplifiers, especially fiber amplifiers, such as ease of alignment, compactness, efficiency, lightweight, etc., warrant further investigation for coherent lidar, (3) the possibility of more fully optical lidar systems should be explored, (4) advantages gained by use of coherent interference of optical fields at the level of one, or a few, signal quanta should be explored, (5) amplification without inversion, population trapping, and use of electromagnetic induced transparency warrant investigation in connection with coherent lidar, (6) these new findings are probably more applicable to earth related NASA work, although applications to deep space should not be excluded, and (7) our own work in the Ultrafast Laboratory at UAH along some of the above lines of investigation, may be useful.

Fork, Richard↗

NASA NTRS2011

Relationship Between Optimal Gain and Coherence Zone in Flight Simulation

In motion simulation the inertial information generated by the motion platform is most of the times different from the visual information in the simulator displays. This occurs due to the physical limits of the motion platform. However, for small motions that are within the physical limits of the motion platform, one-to-one motion, i.e. visual information equal to inertial information, is possible. It has been shown in previous studies that one-to-one motion is often judged as too strong, causing researchers to lower the inertial amplitude. When trying to measure the optimal inertial gain for a visual amplitude, we found a zone of optimal gains instead of a single value. Such result seems related with the coherence zones that have been measured in flight simulation studies. However, the optimal gain results were never directly related with the coherence zones. In this study we investigated whether the optimal gain measurements are the same as the coherence zone measurements. We also try to infer if the results obtained from the two measurements can be used to differentiate between simulators with different configurations. An experiment was conducted at the NASA Langley Research Center which used both the Cockpit Motion Facility and the Visual Motion Simulator. The results show that the inertial gains obtained with the optimal gain are different than the ones obtained with the coherence zone measurements. The optimal gain is within the coherence zone.The point of mean optimal gain was lower and further away from the one-to-one line than the point of mean coherence. The zone width obtained for the coherence zone measurements was dependent on the visual amplitude and frequency. For the optimal gain, the zone width remained constant when the visual amplitude and frequency were varied. We found no effect of the simulator configuration in both the coherence zone and optimal gain measurements.

Gracio, Bruno Jorge Correia↗

NASA NTRSDate not supplied

Comparison of Corcos-based and experimentally-derived coherence factors for BFFs estimation

In this paper, high-spatial-resolution unsteady Pressure Sensitive Paint (uPSP) data are utilized to compare two methods for panel Buffet Forcing Functions (BFF) estimation for the Space Launch System (SLS). Such methods are based on discrete pressure measurements within a panel but employ coherence factors to account for partially-correlated fluctuating pressures across the whole panel. In one method, coherence factors are derived based on the Corcos model, whereas the second method utilizes experimentally-derived coherence factors. To simulate discrete measurements using uPSP data, suitable subsets of the data are extracted. When full uPSP resolution is retained, uPSP data provide a benchmark to assess discrete-measurements-based methods. The analysis focuses on the peak SLS buffet environment located downstream of the Forward Attachment Hardware (FAH) between the core stage and solid rocket boosters. Trends of Corcos-based and experimentally-derived coherence factors are in reasonable agreement with the benchmark. However, at certain frequencies, experimentally-derived coherence factors are sensitive to the separation distance between pressure measurements utilized to compute coherence lengths. Such sensitivity originates from deviation of the experimental-based coherence function from an exponential decay assumption. On the other hand, the present implementation of the Corcos model fails to capture certain non-turbulent boundary layer related environments, such as a subharmonic of FAH vortex-shedding. For all methods presented in this paper, at near transonic conditions, increased pressure coherence and spatial nonuniformity lead to BFF overestimation and sensitivity to the pressure measurement location within the panel.

transonic buffet↗

NASA NTRSDate not supplied

Comparison of Corcos-Based and Experimentally-Derived Coherence Factors for Buffet Forcing Functions Estimation

In this paper, high-spatial-resolution unsteady Pressure Sensitive Paint (uPSP) data are utilized to compare two methods for panel Buffet Forcing Functions (BFF) estimation for the Space Launch System (SLS). Such methods are based on discrete pressure measurements within a panel but employ coherence factors to account for partially-correlated fluctuating pressures across the whole panel. In one method, coherence factors are derived based on the Corcos model, whereas the second method utilizes experimentally-derived coherence factors. To simulate discrete measurements using uPSP data, suitable subsets of the data are extracted. When full uPSP resolution is retained, uPSP data provide a benchmark to assess discrete-measurements-based methods. The analysis focuses on the peak SLS buffet environment located downstream of the Forward Attachment Hardware (FAH) between the core stage and solid rocket boosters. Trends of Corcos-based and experimentally-derived coherence factors are in reasonable agreement with the benchmark. However, at certain frequencies, experimentally-derived coherence factors are sensitive to the separation distance between pressure measurements utilized to compute coherence lengths. Such sensitivity originates from deviation of the experimental-based coherence function from an exponential decay assumption. On the other hand, the present implementation of the Corcos model fails to capture certain nonturbulent boundary layer related environments, such as a subharmonic of FAH vortex-shedding. For all methods presented in this paper, at near transonic conditions, increased pressure coherence and spatial nonuniformity lead to BFF overestimation and sensitivity to the pressure measurement location within the panel.

buffet↗

NASA NTRS2023

Comparison of Corcos-Based and Experimentally-Derived Coherence Factors for Buffet Forcing Function Estimation

In this paper, high-spatial-resolution unsteady Pressure Sensitive Paint (uPSP) data are utilized to compare two methods for panel Buffet Forcing Function (BFF) estimation for the Space Launch System (SLS). Such methods are based on discrete pressure measurements within a panel but employ coherence factors to account for partially-correlated fluctuating pressures across the whole panel. In one method, coherence factors are derived based on the Corcos model, whereas the second method utilizes experimentally-derived coherence factors. To simulate discrete measurements using uPSP data, suitable subsets of the data are extracted. When full uPSP resolution is retained, uPSP data provide a benchmark to assess discrete-measurement-based methods. The analysis focuses on the peak SLS buffet environment located downstream of the Forward Attachment Hardware (FAH) between the core stage and solid rocket boosters. Trends of Corcos-based and experimentally-derived coherence factors are in reasonable agreement with the benchmark. However, at certain frequencies, experimentally-derived coherence factors are sensitive to the separation distance between pressure measurements utilized to compute coherence lengths. Such sensitivity originates from deviation of the experimentally-based coherence function from an exponential decay assumption. On the other hand, the present implementation of the Corcos model fails to capture certain nonturbulent boundary layer related environments, such as a subharmonic of FAH vortex-shedding. For all methods presented in this paper, at near transonic conditions, increased pressure coherence and spatial nonuniformity lead to BFF overestimation and sensitivity to the pressure measurement location within the panel.

transonic buffet↗

NASA NTRSDate not supplied

Coherent Mode Decomposition for Kolmogorov Optical Turbulence in a Finite Aperture

An analysis of the coherent mode decomposition of an optical field after propagation through atmospheric turbulence is presented. The coherent modes represent an ideal basis by which to decompose the field for design of mode-limited optical systems. Using the rotational symmetry of the Fredholm integral operator for Kolmogorov optical turbulence, it is shown that the coherent modes exhibit separable solutions classified by radial and azimuthal quantum numbers. The study of the coherent modes is then reduced that of the radial functions determined by the aperture-coherence diameter ratio and obscuration ratio. Analysis of the spectrum of eigenvalues yields sharp bounds on the efficiency of receivers using incoherent or coherent combining with mode-limited photonic devices in the presence of Kolmogorov turbulence. The effective number of modes needed to represent Kolmogorov optical turbulence is studied via the von Neumann entropy, purity, and largest eigenvalue, and the differences in the different definitions is discussed. The similarity of the coherent modes to linearly polarized (LP) fiber modes is quantified yielding a precise characterization of the maximum gain that can be achieved in mode-limited systems via mode shaping techniques. As a final application, a mode sorting technique is presented for optimally splitting power from atmospherically degraded light into a finite number of modes simultaneously maximizing total coupling efficiency and minimizing the average and instantaneous power ratio between channels.

free-space optical communication↗