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Insert the slit-limited spectral resolving power $R=\lambda q/p$ into the invariant expression:Here $q=|dx/d\lambda |$ is a magnitude; a signed linear spectral dispersion would require the corresponding absolute value. Since $q$ has units of photodetector length per unit wavelength, $L$ has units of length.

The optical telescope focuses the sky onto an optical slit; the collimator turns the selected light into a beam incident on the reflection diffraction grating; the optical camera images each diffracted direction onto the photodetector. The drawing also shows the monochromatic optical slit image for uniform optical slit illumination in the geometric, slit-limited approximation.

Near the chosen wavelength $\lambda$, a small wavelength increment moves the image by $q\Delta \lambda$, where $q$ is the magnitude of the linear spectral dispersion. An optical slit image of physical width $p$ therefore has an apparent spectral width $p/q$. With one optical slit width as the adopted separation criterion,Uniform illumination gives a top-hat monochromatic profile of width $p$; an unresolved stellar image need not illuminate the optical slit uniformly, so its profile follows the actual illumination convolved with the instrument response. Negligible optical slit diffraction and optical camera optical aberrations do not remove the finite diffraction grating diffraction width. The formula is the slit-limited result when that width is small compared with $p$, rather than a universal exact line-profile criterion.

Differentiate the grating equation with incidence fixed:The second formula is local to a spectrum centered on the optical camera optical axis, where $dx=f_{\mathrm{cam}}d\beta$ to first order. Since the illuminated surface width $W$ projects to optical camera beam width $A_{\mathrm{cam}}=W\cos \beta$,The expression takes positive diffraction order and width magnitudes. Off-axis, a photodetector mapping $x=f_{\mathrm{cam}}\tan (\beta -{\beta }_0)$ also has a ${\sec }^2(\beta -{\beta }_0)$ factor; the exam's compact formula uses the local paraxial mapping.

The fundamental angular floor is the diffraction limit of a telescope, of order $\lambda /D$. The ideal diffraction grating has $R_g=mW/d$, whereas its slit-limited resolving power has $R\theta D=mW\lambda /d$. Requiring $R\le R_g$ givesThe coefficient depends on the adopted line-resolution and aperture conventions. Below this scale, a narrower spaxel oversamples the same spatial mode rather than creating a new independently resolved element. Equivalently a spatial optical mode occupies etendue of order ${\lambda }^2$.

There is also an engineering floor before arbitrary shrinkage: the optical camera focal length, photodetector size and support clearances do not shrink, while the number of photodetectors increases as ${\theta }^{-2}$. Thus the shrinking-beam model's constant combined optical camera volume is eventually overtaken by fixed per-camera overheads. For fixed brightness, smaller spaxels receive fewer photons, so read noise and photon shot noise can constrain useful sampling before the formal optical floor. Diffraction limits independent spatial information; fixed photodetector and mechanical dimensions limit practical volume reduction.

The number of spaxels is $M=(\beta /\theta {)}^2$. To obtain definite scaling exponents, keep the spectral resolving power, wavelength interval, diffraction grating angles and groove spacing, input focal ratio, and photodetector sampling of a spectral resolution element fixed. A fixed photodetector then accommodates a fixed number $M_0$ of spectra, so $N\simeq M/M_0$ (rounded up in an actual instrument). The corresponding etendue per spaxel scales as ${\theta }^2$. Angular ratios are independent of whether both angles are in arcseconds; optical invariant equations use radians.

Write $b$ for the collimated beam diameter. The diffraction grating result $R\theta D=W(\sin \alpha +\sin \beta )$ gives $b\propto W\propto \theta$. A fixed collimator focal ratio gives $f_{\mathrm{coll}}\propto b\propto \theta$. With a fixed image width $p$ in photodetector detector pixels, the invariant $\theta D=p{A}_{\mathrm{cam}}/f_{\mathrm{cam}}$ and $A_{\mathrm{cam}}\propto \theta$ instead imply $f_{\mathrm{cam}}$ is constant: the optical camera focal ratio increases as $1/\theta$.

In a simple on-axis beam-envelope model, each collimator and collimated disperser space has area proportional to $b^2$ and length proportional to $b$, whereas the optical camera cone has area proportional to $b^2$ and fixed length. HenceThe constants include $1/M_0$ and fixed design parameters. At fixed field size, the combined collimator and diffraction grating volumes scale as $M{\theta }^3\propto \theta \propto M^{-1/2}$, while the combined optical camera beam-cone volume scales as $M{\theta }^2\propto 1$.

These are integral-field spectrograph volume scaling laws for the shrinking beam, not a claim that the whole apparatus can shrink without a floor. A photodetector of fixed transverse size $d_0$ needs space for its field: an optical camera envelope interpolating between optical pupil width $b$ and photodetector width $d_0$ has volume proportional to $f_{\mathrm{cam}}(b^2+b{d}_0+d_0^2)$, rather than just $f_{\mathrm{cam}}{b}^2$. Replication then adds terms scaling as $M\theta$ and $M$, which can grow as the spaxels shrink. Other field clearances and mechanical margins also change the asymptote. Without the fixed spectral/design assumptions above, the information in the question does not determine unique physical-volume exponents. The role of fixed photodetector size and beam spread is discussed in Allington-Smith's instrument scaling model.

Let ${\beta }_0$ be the direction of the optical camera optical axis. The grating equation on that axis isThe central wavelengths in adjacent diffraction orders are therefore ${\lambda }_m=K/m$ and ${\lambda }_{m+1}=K/(m+1)$, with separation$K$ is the fixed order-times-wavelength product directed onto the optical camera axis, or the optical path difference between adjacent grooves in that direction. In a Littrow configuration, $K=2d\sin \alpha$. This adjacent-central-wavelength spacing is a conventional free spectral range of an echelle grating; at high order it is approximately ${\lambda }_m/m$.

There is a qualification in the wording of the printed definition. The exact interval for which order $m$ lies closest to the $y$ axis is not, in general, identical to the spacing of two adjacent order centres. For a concrete counterexample take ${\beta }_0=0$, $0<\alpha <\pi /2$, and a photodetector mapping $x=f_{\mathrm{cam}}\tan \beta$. Equality of the distances from the axis for orders $m,m+1$ means ${\beta }_m=-{\beta }_{m+1}$, so their common boundary is $\lambda =K/(m+1/2)$. The other boundary with order $m-1$ is $K/(m-1/2)$. Consequently the literal closest-order interval has widthfor $m\ge 2$ and accessible non-grazing orders. This differs from the printed $K/[m(m+1)]$. For a nonzero ${\beta }_0$, the exact boundary is determined by ${\beta }_m+{\beta }_{m+1}=2{\beta }_0$ and need not even have the simple half-order form. The two definitions agree to leading order $K/m^2$ when $m$ is large. Thus the printed expression is correct as an adjacent-centre convention or a high-order coverage estimate, rather than an exact consequence of the literal nearest-axis definition. Gap-free coverage should be checked using the actual photodetector mapping and neighbouring order endpoints.

Choose photodetector $x$ along increasing main grating dispersion, and $y$ along increasing wavelength for the cross-disperser. The echellogram shows two order traces and their intersections with the $y$ axis, which passes through the optical camera optical axis.

For fixed incidence, $d\beta /d\lambda =m/(d\cos \beta )>0$, so increasing wavelength goes to the right along each order. The cross-disperser also sends increasing wavelength upward in the chosen orientation; the traces therefore slope upward. At $x=0$, ${\lambda }_m=K/m>{\lambda }_{m+1}=K/(m+1)$, so the trace labeled $m$ lies above the trace labeled $m+1$. Reflecting either plotting axis would reverse its arrow but leave the optical content unchanged. The cross-disperser separates orders that would overlap in the main dispersion direction.

An optical pupil ripple of spatial period $\Lambda$ diffracts starlight into two sidebands displaced by $\theta \simeq \lambda /\Lambda$ along the ripple's spatial frequency vector. Substituting the axial Nyquist period gives the deformable-mirror control radiusThe square actuator lattice has component cutoffs $|u_x|,|u_y|\le N/2$, hence an ideal square corrected region with this half-width in each photodetector direction. Its diagonal reaches $\sqrt{2}$ times the axial radius; an isotropic conservative restriction is the inscribed disk. Using $\lambda /D$ as the scale of a spatial resolution element, the square is about $N$ elements on a side and $N^2$ in area; the disk has about $\pi N^2/4$ elements. These are angular area counts, not the quadrant coefficient count in part (ii).

A circular illuminated optical pupil has fewer active actuator degrees of freedom than the complete square array, and influence functions alter the usable boundary. Moreover one real wave phase deformable mirror produces conjugately related corrections at opposite speckles: arbitrary complex-field correction of both independent wave amplitude and wave phase errors generally needs additional control, or a restricted half-plane. The square therefore describes ideal spatial-frequency access, rather than a guarantee that every intensity element inside it can be independently set to zero. The Fourier-domain square control region is derived in Bordé and Traub's speckle-nulling analysis.

Let $k=m/2$. After offset subtraction, each summed image has mean $N$ analogue-to-digital units per detector pixel. For a constant detector conversion gain $g$ and independent Poisson distribution detected counts, $\mathrm{Var}(A)=\mathrm{Var}(B)=N/g$: summing $k$ independent frames adds both their means and variances. Here $N$ is the mean of the sum, not one frame.

Write $A=N+a$, $B=N+b$. At high counts, the delta method gives $A/B\simeq 1+a/N-b/N$. Independence therefore givesThus the formula is a high-signal gain estimation from flat-field ratios, not an exact identity for a ratio of random variables. A Poisson distribution denominator can even be zero at low counts. The exposures must be linear and unsaturated, offsets removed, and read noise negligible. Being close to saturation increases the signal but does not establish linearity; clipping or charge-induced correlations biases a variance estimate. A large ensemble of suitable detector pixels supplies the measured standard deviation. In normal photodetector terminology $g$ is collected electrons per ADU, consistent with detected photons here only under the stated unit-yield convention; see the instrument gain definition.

If temporal variation is spatially uniform, summing all frames in each group still gives independent Poisson distribution totals, but their means may now be $N_A,N_B$ with ratio $c=N_A/N_B\ne 1$. At high counts and negligible read noise, direct propagation givesAfter dividing the ratio image by its measured mean $c$, the appropriate estimator isThe original formula is recovered only for $N_A=N_B=N$. Uniform brightness drift does not automatically create a spatial pattern or extra ratio scatter beyond these changed noise levels; its leading problem is unequal group normalization. Interleave the groups in time or pair exposures of comparable flux, measure their separate means, and propagate shot and read noise when rescaling. If the spatial pattern or photodetector sensitivity also evolves, the ratio develops systematic structure that must be modeled or those frames rejected. A normalization inferred from finitely many detector pixels introduces a small shared correlation, which vanishes in the large-pixel limit and can be included in a precise uncertainty analysis.

Two applications are ultraviolet photodetection and ultraviolet/near-band-edge light emission. The bulk edge corresponds approximately toAbove-gap absorption creates carriers that can be collected in a photodetector. Measure dark current and wavelength-dependent response to separate photocarriers from heating or leakage. An optically or electrically excited emitter instead uses electron-hole recombination. quantum confinement in sufficiently small dots can shift absorption and emission above the bulk band gap. Surface states can produce broad defect luminescence or nonradiative loss, so a direct bulk gap does not guarantee an efficient, narrow-band emitter.

One application is a semiconducting nanotube transistor array. Removing metallic tubes prevents conducting bridges that would bypass gate control; a narrow chirality distribution also narrows the spread of band gaps and switching behaviour. High charge carrier mobility and thin flexible channels are attractive, but contact resistance, tube alignment, residual metallic contamination, dielectric traps and the cost/yield of sorting still limit uniform large-area circuits.

A second is a wavelength-selective near-infrared photodetector or emitter. A selected chirality concentrates absorption and emission around its characteristic nanotube optical transitions, avoiding the broad mixed-tube response. Optical selectivity can be tailored by choosing the species. Limitations include low absolute emission yield, dark exciton states, contact-related recombination, environmental spectral shifts and quenching by bundles or metallic contaminants. Optical and electrical characterization after device processing is needed; the sorting label alone does not guarantee device performance.

The optical telescope supplies collecting area and spatial selection, the spectral selector supplies a passband, and the photodetector converts received radiant flux into a measured signal. For an extended source, a useful schematic photon count iswhere $A\Omega$ is the etendue, $\tau$ the integration time, $\eta$ the net transmission and quantum efficiency, and $L_{\lambda }$ the spectral radiance. An illustrative independent-noise model givesHere $N_b$ and $N_d$ denote background and dark counts, and ${\sigma }_{\mathrm{read}}$ is readout standard deviation. This makes the roles of the three components explicit.

A larger optical telescope collects more photons at fixed angular pixel size and improves the diffraction-limited resolution. Its limits include optical aberrations, alignment, launch size, stray light and pointing. Conservation of etendue constrains simultaneous collecting area and angular acceptance. Increasing spatial resolution by narrowing a pixel's solid angle reduces its photons.

A narrower spectral passband or higher spectral resolving power separates diagnostic features and rejects unwanted background, but usually reduces photons per channel and hence signal-to-noise ratio. Selector transmission, leakage, slit width, detector format and the required time cadence limit the trade. A wider band helps only insofar as it adds useful signal rather than mostly background.

The photodetector's quantum efficiency, dark current, read noise, dynamic range and response time determine conversion losses and added noise. Cooling can suppress thermal background and dark current, but cannot remove photon shot noise. Longer integration improves a stationary signal's signal-to-noise ratio until motion, saturation, drift or required cadence becomes limiting.

A Michelson interferometer uses a beam splitter to divide collimated input light between a fixed mirror and a moving mirror. Returning beams recombine at the beam splitter; one output reaches a photodetector. A compensator plate can equalize the glass traversed in the two arms. Moving one mirror by $x$ changes the optical path difference by $2x$.

The usual nominal Fourier transform spectroscopy specification is $\delta \nu \simeq 1/d_{\max }$, where $d_{\max }$ is the maximum optical path difference measured from equal arms. It givesThis is a nominal spectral resolution element convention.^[ref] A symmetric rectangular scan from $-d_{\max }$ to $d_{\max }$ instead has a sinc instrumental line shape with first-zero offset $1/(2d_{\max })$. Quoting that first-zero criterion would give $d_{\max }=1$ cm; interferogram apodization and other linewidth criteria change the numerical factor. An exact mathematical minimum requires such a criterion.

In the measurement chain, the Michelson interferometer is the spectral analyzer or modulator, between the collection optics and the photodetector. It encodes the incoming radiation spectrum into an interferogram; detector electronics digitize that signal and a Fourier transform reconstructs the radiation spectrum. It does not itself turn radiant power into an electrical signal.

Photon counting registers discrete detector events and their times. A beam splitter sends the selected light to two photodetectors; a coincidence circuit or time-recording system compares the outputs. With a vacuum input at the unused optical port, splitter fractions $R,T$ and independent quantum efficiencies ${\eta }_1,{\eta }_2$, a selected input photon reaches the two recorded channels with probabilities $p_1=R{\eta }_1$ and $p_2=T{\eta }_2$. Conditional on input occupation $N$, each pair of distinct photons contributes $p_1{p}_2$ to the cross-channel pair moment. ThereforeTheir normalized ratio recovers $g^{(2)}(0)$. This assumes number-resolving counts, or sufficiently dilute gates for threshold detectors to approximate them. One photon cannot create both channel counts in this ideal model.

A point spread function is an imaging system's response to an optical point source. Here it means the distribution of image light intensity: an ideal point object becomes a finite spot, often with faint rings, because of diffraction. An unresolved star and a sufficiently small bead emitting fluorescence are practical approximations to an optical point source. Their image tells us how nearby object details will overlap, and reveals blurring from optical aberrations, defocus, and, for an optical telescope, astronomical seeing.

A useful convention separates shape from throughput. At a fixed image plane, let $p(\mathbf{x})\ge 0$ be the point spread function with ${\int }_{{\mathbb{R}}^2}p(\mathbf{x})d^2\mathbf{x}=1$. Its units are inverse area. A point delivering detected radiant flux $F$ at image position ${\mathbf{x}}_0$ produces light intensity $Fp(\mathbf{x}-{\mathbf{x}}_0)$; this point spread function normalization makes the integral over the image equal to $F$. A profile divided by its own peak is convenient for comparing widths, but is a different point spread function normalization and does not by itself compare total radiant flux.

In incoherent imaging, different object points have negligible mutual spatial optical coherence over the measurement, so their image light intensities add. If the response also obeys translation invariance over the chosen field of view, the expected continuous image is the convolutionHere $O$ is the object brightness distribution expressed in image-scaled coordinates, with magnification already included, and $\tau$ is constant transmission. For $O(\mathbf{u})=F_0\delta (\mathbf{u}-{\mathbf{u}}_0)$, where $\delta$ is the two-dimensional Dirac delta function, this gives $I=\tau F_{0}p(\mathbf{x}-{\mathbf{u}}_0)$. A changing response across the field of view instead requires a position-dependent integral kernel $p(\mathbf{x};\mathbf{u})$; one measured point spread function is then only a local calibration.

In coherent imaging, the superposition principle applies to the complex wave amplitudes. With amplitude spread function $h$, the corresponding ideal model isThus $|h\ast E{|}^2$ generally differs from $|h{|}^2\ast |E{|}^2$: coherent contributions produce wave interference terms that depend on their relative wave phases. Their time average vanishes for mutually incoherent emitters, which explains the intensity convolution. Partial optical coherence requires the correlations between object points as well.^[ref]

The figure is an original diffraction model, not measured data. All panels have the same total radiant flux: each member of a pair contributes half the single-source radiant flux. Dashed curves show the separate light intensity contributions; the solid curve is their sum. The point spread functions have tails, so even sources satisfying the Rayleigh criterion overlap. As separation decreases, the image changes continuously; there is no abrupt disappearance of all information at one separation.

A motivating measured example is Measurement of point-spread function (PSF) for confocal fluorescence microscopy by InCheon Song and others (2005). A $543.8\mathrm{nm}$ laser, delivered through an optical fiber, excited fluorescent beads on a slide through a $100\times$, numerical aperture $1.3$ oil-immersion objective. A moving stage scanned the bead; emitted light returned through the objective, an emission filter and a pinhole to a photodiode. The readout was fluorescence light intensity against scanned position. Under the same setup, nominal bead diameters $0.17$, $0.5$, and $1\mu \mathrm{m}$ produced progressively broader measured spots. The size comparison controls for the finite object: the measured response is the convolution of bead emission with the effective point spread function, rather than a pure point response. Inverting this convolution model gave a response compared with the theoretical one. The smaller bead gave a weaker signal relative to noise; uncertainty in actual bead diameter also affected the restored width. These are limits of the calibration. In confocal microscopy, excitation and pinhole detection jointly determine the effective point spread function; this experiment does not measure the ideal widefield Airy disk plotted above.^[ref]

For a new calibration, useful additional controls are to measure detector background, avoid saturating the photodetector, repeat the scan to check stability, and compare isolated beads at several positions in the field of view. Record wavelength, numerical aperture, pinhole setting, sampling interval and optical focus. These are practical recommendations, not extra measurements claimed for the published experiment.

Polarimetry measures optical rotation with a polarimeter. The vertical Vernier Go Direct instrument uses a $589\mathrm{nm}$ light-emitting diode source, fixed polarizer, sample cell, motor-driven polarimeter analyzer, photodetector and optical rotary encoder. Calibrate using the same solvent at $10\mathrm{cm}$ fill height; rinse with sample, fill to $10.0\mathrm{cm}$, and read the meniscus to $0.1\mathrm{cm}$. The readout plots illumination against analyzer angle; a maximum near zero or a cosine-squared fit locates $\alpha$.^[ref]

An actual teaching experiment in the Truman State University-hosted Optical Activity Laboratory handout, revised July 2013, uses sucrose chemical solutions. Weigh the sucrose, dissolve completely without heating, and measure final solution volume for concentration. Use one polarimeter throughout; measure the water blank's extinction reading, then each sample's extinction reading and actual liquid path length near $10\mathrm{cm}$. Subtract the blank and plot corrected optical rotation against concentration. The attached apparatus description specifies a vertical light path, an incandescent or tungsten source with yellow filter, sheet polarizers and a rotating eyepiece polarimeter analyzer. Its coarse demonstration permits temperature control to the nearest $5^{\circ }\mathrm{C}$, path length to the nearest millimeter and approximately $1^{\circ }$ angular precision. These are design tolerances, not measured results.^[ref]

For a quantitative comparison, equilibrate blank, reference and sample at the same recorded temperature; use a thermostatted cell when needed. This is a recommended control, not a claim that the teaching instruments provide a thermostat. A yellow filter also passes a band of wavelengths, so its result need not match a narrow-source reference exactly. The laboratory's concentration series tests the linear model rather than establishing it in advance; a second series varying only path length tests the other factor.

For our original ideal model, Malus's law giveswhere $I_0$ is light intensity immediately before the ideal polarimeter analyzer, after sample transmission. It is not the intensity of the original unpolarized source. The angle difference is evaluated in consistent units; the plotted degree values are converted to radians before computing the cosine. Assume transmitted light retains linear polarization, with negligible scattering and no polarization-dependent absorption. The pure positive model, $75:25$ mixture and racemate shift the maxima to $8^{\circ }$, $4^{\circ }$ and $0^{\circ }$, respectively. Minima shift by the same amount, to $98^{\circ }$, $94^{\circ }$ and $90^{\circ }$. Thus an extinction measurement and a maximum measurement recover the same shift after their respective blanks are subtracted.

Repeat blanks and use a reference to check drift and scale. Bubbles, particles, stressed windows and sample absorption can spoil the ideal curve. Vernier specifies $\pm 1^{\circ }$ accuracy; the manual's $\pm 0.1^{\circ }$ fit reproducibility does not establish that accuracy. It warns against interpreting its illumination readout quantitatively and requires a transparent homogeneous sample.^[ref] Neither a zero reading nor a matching specific rotation uniquely identifies a molecule.

A 2018 photonic experiment makes the distinction between populations and quantum coherence controllable. A 400-nm laser pumped a crystal to generate paired photons, filtered near 800 nm. One photon triggered collection; the other entered the apparatus. A half-wave plate and polarizing beam splitter selected horizontal photon polarization; another half-wave plate prepared balanced $H,V$ quantum superpositions. Separated polarization paths and diffraction gratings let a spatial light modulator control their frequency-dependent quantum phases and spectral weights. Recombined paths passed through quartz plates whose adjustable thickness controlled polarization–frequency coupling. Quarter- and half-wave plates plus a polarizing beam splitter selected tomography bases; jointly registered photodetector events supplied the readout. Quantum state tomography inferred polarization density matrices at each thickness. The dephasing model preserves diagonal populations while changing off-diagonal quantum coherence. Here the unobserved photon frequency acts as the environment for the polarization qubit. Finite modulator resolution and imperfect spectral shaping limited agreement with the intended dynamics; this experiment does not establish exact $I/2$ preparation or the ideal probabilities above. ^[ref]

A polarizing beam splitter sends two orthogonal linear polarizations to different output paths. Acting coherently on $\alpha |H⟩+\beta |V⟩$, it correlates path with photon polarization instead of measuring photon polarization unless a path is actually detected or discarded.

For an ideal polarizing beam splitter with one input, choose orthogonal unit electromagnetic polarization vectors ${\mathbf{e}}_H,{\mathbf{e}}_V$ and an incident power-normalized field $a_H{\mathbf{e}}_H+a_V{\mathbf{e}}_V$. If $H$ is transmitted and $V$ reflected, the output powers are $P_{\mathrm{transmitted}}=|a_H{|}^2$ and $P_{\mathrm{reflected}}=|a_V{|}^2$. The associated complex amplitudes can acquire different wave phases; the routing is not an equal-power split for every incident electromagnetic polarization. For linear polarization at angle $\theta$ to $H$, input power $P$ gives $P{\cos }^2\theta$ and $P{\sin }^2\theta$, respectively. At $\theta =30^{\circ }$, a stipulated $4\mathrm{mW}$ input gives $3\mathrm{mW}$ transmitted and $1\mathrm{mW}$ reflected. A $45^{\circ }$ input gives equal powers even though an $H$ input goes entirely into the transmitted output.^[ref]

Real polarizing beam splitters have leakage into the unwanted output and losses. Each linear polarization has its own field transformation and power coefficients. To make an ordinary power wave interference term between the two output contributions, first bring them into the same path with overlapping transverse profiles and a common electromagnetic polarization, for example using a polarizer. Merely placing two orthogonal linear polarizations on a polarization-insensitive photodetector adds their powers. Coherent routing can be reversed by suitable optics before detection; discarding a path loses access to that part of the field but supplies no recorded measurement outcome by itself.

// Target: optics.bigb

Fix a detected position and polarization of waves. Write the electric field operator as $\hat{E}(t)={\hat{E}}^{(+)}(t)+{\hat{E}}^{(-)}(t)$, where the positive-frequency part contains annihilation operators and ${\hat{E}}^{(-)}=({\hat{E}}^{(+)}{)}^{†}$. An ideal absorbing photodetector responds towith density matrix $\rho$. The diagonal gives the single-count rate up to the detector coupling. Multiple absorptions require higher, normal-ordered field products. Roy J. Glauber used these correlations to develop the quantum account of optical coherence.^[ref]

The ordering follows from a detection amplitude. If absorption takes an initial pure state $|i⟩$ to an unobserved final state $|f⟩$, its field factor is $⟨f|{\hat{E}}^{(+)}|i⟩$. Squaring its complex modulus and summing over a complete orthonormal basis of final states gives $⟨i|{\hat{E}}^{(-)}{\hat{E}}^{(+)}|i⟩$. Averaging over initial preparations replaces this expression by a trace. Thus a detector that absorbs cannot count the Fock vacuum, although the vacuum has quantum fluctuations.

For one selected normal mode, ${\hat{E}}^{(+)}(t)=\mathcal{E}u(t)a$. All Fock states with nonzero occupation, coherent states, and thermal light in this same mode havewhere both denominators are nonzero. Its magnitude is one even when the expected value of the field vanishes. Perfect first-order optical coherence therefore does not identify a coherent state. Pair detection supplies additional information.