Glass hides a black net and not a white voile

A sheer’s privacy is the error it multiplies read a room’s reflectance back from the street through a curtain and found that every error arrives multiplied by the reading over the room’s own image. That is the reciprocal of the share of the view that is room, the same number a sheer hides whichever side is darker had computed as the curtain’s privacy. A white voile multiplies by sixty-four in daylight and a black net by two.

That account, like opacity is not cover before it, left the window out, and most curtains hang behind glass. A pane reflects the street like a weak mirror, and what it reflects reaches the passer-by’s eye along with everything the curtain sends. The obvious guess is that glass adds to the veil and so to the privacy, a little for every curtain. For a white voile straight on, the guess has the wrong sign. For a black net it is right, and by a large factor, far more than for any curtain chosen for privacy.

Two things a pane does, pulling opposite ways #

A pane does two separate things to the light a passer-by receives, and the bare-cloth account had neither.

It mirrors the street. Some fraction of the light arriving from across the road is reflected at the glass straight back into the passer-by’s eye. That light has never been near the curtain or the room. It is an image of the houses opposite, faint and reversed, laid over the window, in the way a float reflects into a line turns a lamp into a streak on a satin: a surface returning one direction of the surroundings rather than scattering all of them. In the curtain arithmetic it counts as veil: light in the view that carries no information about the room, added to the light that does.

And it dims the street’s light on the cloth. A curtain behind glass is lit through the glass, and the glass reflects away part of whatever tries to pass it — the same light that a cloth is more opaque than it is closed followed into the threads and back out. The cloth’s own veil, the threads lit by the street, is therefore fainter behind a pane than it would be in an open window. So the pane takes veil away as well as adding it.

Written in the units the curtain essays used, with the room’s illuminance as one, the reading becomes

$$\ell = T(\theta)\,\bigl[\,o\,\rho + (1-o)\,(R_t\,T_d\,R + \tau)\,\bigr] + P(\theta)\,\rho_s\,R,$$

with $P$ and $T$ the pane’s reflectance and transmittance at the passer-by’s angle $\theta$, $T_d$ its transmittance for the street’s diffuse light on the way in, and $\rho_s$ the reflectance of the street the pane mirrors. The rest is the bare account: $o$ the open area, $\rho$ the room, $R_t$ and $\tau$ the thread’s optics, $R$ the ratio of street light to room light. With no pane, $P$ is zero and $T$ and $T_d$ are one, and the expression is the old one to the last digit.

What Fresnel says a window reflects #

The pane’s two numbers are not assumptions. Glass is a dielectric, and how much of the light meeting its surface is reflected was worked out exactly by Fresnel in 1823 from its refractive index, about 1.52 for window glass, and the angle.

Straight on, a pane reflects 8.2 per cent: each face returns about 4.3 per cent, and light bouncing between the two faces adds a little more. The reflectance barely moves for the first forty degrees and then climbs steeply: 9.7 per cent at 45°, 15.7 at 60°, 38.5 at 75°, three quarters at 85°. A pane is a poor mirror when looked at squarely and a good one from along the pavement, which anybody who has tried to see into a shop window from beside it already knows.

The other number is fixed by the same equations. A street’s light does not arrive at the curtain from one direction. It comes from the whole half of the sky and the road in front of the window, much of it at a slant, and at a slant the glass reflects more. Averaged over that half-space with the weight an illuminance carries, the pane turns back 15.4 per cent of the street’s light before it reaches the cloth, nearly twice the straight-on loss.

Head-on, the white voile comes out ahead #

Put the two effects side by side for the white voile of the curtain essays, half open, its threads throwing back six tenths of what falls on them.

The dimming removes 15.4 per cent of the street’s contribution to the cloth’s veil. In daylight that contribution is almost the whole veil: 0.51 of the window is thread, and each unit of it returns 0.6 of a street thirty times brighter than the room. So the glass takes away $0.51 \times 0.6 \times 0.154 \approx 0.047$ of the street’s light, per unit of it, from the veil.

The mirror adds $P\rho_s/T$ per unit of the street’s light: 8.2 per cent of a street reflecting three tenths, divided by the 92 per cent the pane passes on the way out, which comes to 0.027. The pane takes off nearly twice the veil it adds. A white voile behind clean glass, looked at squarely, has a daylight multiplier of 59.8 against 64 without the glass. That is slightly less privacy, and a slightly better measurement.

The bars put the reason in one picture, and it is the picture two layers are the product on average and nowhere drew for a second curtain, with the difference that this second layer has no holes and no threads. Behind a white voile the mirror is a sliver at the end of a bar that is almost all cloth, because the voile’s own veil is already enormous. Taking a sixth off that veil and adding a thin mirror leaves the room’s share where it was, or slightly better: 1.6 per cent bare, 1.7 behind glass straight on. At 75°, where the pane mirrors 38 per cent of the street, the mirror is two fifths of the bar and the room falls to 1.1 per cent.

Past fifty-seven degrees it loses #

The mirror grows with the angle and the dimming does not, because the dimming belongs to the street’s light arriving from every direction, while the mirror belongs to the one direction the passer-by looks from. So there is an angle at which they balance. For the white voile it is 56.7°, and the hero figure marks it: steeper than that, the glass adds more veil than it removes and the voile becomes harder to read than it would be bare. At 60° the multiplier is 65.7, at 75° it is 92.7.

A passer-by on an ordinary pavement looking at a ground-floor window a couple of metres away, from a few metres along, is somewhere between thirty and seventy degrees off the glass’s normal — the same range over which a curtain is gathered so that it is seen edge-on found the cloth’s own holes closing. The white voile’s privacy behind glass is therefore within a few per cent of its privacy in an open window for most of the places a person can stand, and noticeably better only from far along the street, where the view is mostly mirror anyway.

The balance angle happens to fall within a hundredth of a degree of Brewster’s angle for glass, 56.7°. That is a coincidence of the numbers chosen for the voile, not of the physics: the balance is set by the cloth’s openness and its threads’ reflectance, and a voile three tenths open breaks even at 62°, one seven tenths open at 31°. Brewster’s angle does play a part in this story, but in a different place, further down.

The hour cancels out of the balance #

The balance angle has a property the curtain essays’ other thresholds did not. It does not depend on the light. The veil the glass removes is the cloth’s reflection of the street, proportional to the street’s light. The veil it adds is the street mirrored, also proportional to the street’s light. The ratio of street to room light multiplies both and falls out of the comparison:

$$\frac{P}{1-P} < \frac{(1-o)\,R_t\,(1-T_d)}{\rho_s}.$$

Nothing on either side is an hour. A curtain that is helped by its glass at noon is helped by the same glass at the same angle at dusk, by a smaller absolute amount because there is less street light to handle either way. The measuring hour that the read-back found, when image and veil become comparable, moves with the glass; the angle at which the glass helps or hurts does not.

The condition also says which cloths can be helped at all. The left side is smallest straight on, at $P/(1-P) = 0.089$, so a cloth is helped at some angle only if $(1-o)R_t(1-T_d)/\rho_s$ exceeds that. The cloth has to be bright and mostly closed: a white thread helps up to 71 per cent open, a thread of tone 0.7 only up to 55. Grey and black threads never qualify at any openness. They send so little of the street back that there is almost no veil for the glass to take off, and the mirror is pure addition.

The net that goes private #

That is the black net’s situation, and the numbers are large. A black net eight tenths open is barely a curtain at all: its threads throw back four hundredths of the light that meets them, and a sheer’s privacy put its daylight multiplier at 2.0 — half of what a passer-by sees is room.

Behind glass, looked at squarely, the mirror is 64 per cent of the view and the room 19 per cent. The multiplier rises to 5.2. At 60° it is 8.8; at 75°, where the mirror is 93 per cent of the view, it is 25. The glass does for the net what the voile’s white threads do for the voile, and does it with a veil that has no thread in it.

This explains an ordinary observation that the bare curtain arithmetic could not: a ground-floor room behind a dark, open net is surprisingly hard to see into from the pavement on a bright day, much harder than the net’s openness suggests, and the difficulty does not go away when the passer-by steps up and looks squarely, because straight on the pane still mirrors two thirds of what reaches the eye. The net is not doing the hiding. The glass is, and the net is doing just enough to keep the room darker than the reflected street. It is the arrangement a net in front gives the figure to the room described for a pair of curtains, with the pane as the outer layer. What a sheer hides is decided by the furniture found that a sheer’s privacy depends on the room behind it; behind glass it depends at least as much on the street across the road.

Cupping the hands defeats the glass, not the curtain #

The gesture everybody uses on a shop window takes out one term of the equation and leaves the others. A person who presses cupped hands to the glass round their eyes shuts out the part of the street the pane would mirror into them. The mirror term, $P\rho_s R$, falls to the reflection of a pair of dark palms, which is nearly nothing. Nothing else in the reading changes: the cloth is still lit by the street through the rest of the pane, and the room is still where it was.

So cupped hands convert a window behind glass into a bare curtain, exactly. For a black net that is the difference between a multiplier of 5 and one of 2, and the room appears. For a white voile it makes almost no difference, and anybody who has cupped their hands against a window hung with white voile will recall that it does not help. The gesture removes the glass’s veil and not the curtain’s, and the arithmetic says in advance which windows it will work on.

A polariser at Brewster’s angle keeps only the help #

The mirror has one more property the cloth’s veil does not share: it is polarised. Light reflected from glass is richer in the polarisation perpendicular to the plane of incidence, and at Brewster’s angle, $\arctan 1.52 = 56.7°$, the other polarisation is not reflected at all. The cloth’s veil is diffuse light scattered from fibres, and a passer-by sees it close to unpolarised.

A polarising filter turned to pass only the polarisation the glass does not reflect therefore removes the mirror entirely at Brewster’s angle, and most of it for twenty degrees either side. It also removes half of the room’s image and half of the cloth’s veil, but it removes the same half of both, so the multiplier does not notice.

At Brewster’s angle, through the filter, every sheer reads better behind glass than bare: the white voile at 54 against 64, the grey voile at 23 against 27, the black net at 1.86 against 2.02. What is left of the pane is its dimming of the street on the cloth, and that always helps. It is the same filter a photographer uses to see through a shop window or into water, and for the same reason, and it works best at exactly the angle at which glass reflects least of the other polarisation.

Dusk still comes, a little later or a little earlier #

The question the read-back left open was whether glass moves the measuring hour: whether, with a pane in front, there is still a light ratio at which a white voile can be read to within half the room’s reflectance.

There is, and it barely moves. Bare, the white voile reaches half its room’s reflectance as its total error when the street is 3.3 times as bright as the room. Behind glass looked at squarely it gets there at 3.7 to 1, a little earlier in the evening, because head-on the glass helps. Seen from 75° it gets there at 2.4 to 1, a little later. The mirror scales with the street’s light like everything else in the veil, so as the street darkens it goes with the rest; at equal light the voile’s multiplier behind glass at 75° is 4.6 against 3.6 bare, and at night the difference has shrunk to two per cent.

The black net’s hour moves much further. Bare, it is readable to half its room at twenty to one, which is overcast daylight; behind glass from 75° only below five to one. Glass pushes the net’s readable hour a long way toward nightfall and moves the voile’s hardly at all.

The pane is exact and still costs an error #

Fresnel’s equations are exact, and the pane’s reflectance at a known angle is not an uncertain input. Its reflectance at an unknown angle is, and so is the brightness of whatever it mirrors.

Straight on, the glass’s own terms are small. The mirrored street’s reflectance, known to three hundredths, puts ±0.16 on the room; the viewing angle puts nothing on it, because at the normal the reflectance curve is flat and a degree either way changes nothing. The total, ±1.20 on a room of 0.3, is actually below the bare voile’s ±1.37, because the dimming has shrunk the thread-reflectance term that dominated before.

From 75°, the mirrored street becomes the largest single input, ±1.15. The angle adds ±0.09 per degree of uncertainty. The total rises to ±1.82. The measurement that was already hopeless by day is now hopeless mainly because of the houses across the road.

For the black net the glass changes the character of the budget rather than its size. Straight on, its total is ±0.22 behind glass against ±0.23 bare: the mirror’s uncertainty replaces part of the cloth’s. From 75° it is ±0.90, three times the room, and nearly all of it is the street and the angle.

Privacy and precision part company #

The read-back found that a curtain’s privacy and a measurement’s precision were one number. Glass separates them, and the separation says something about what the identity was.

The identity was always about the reading’s own error: an uncertainty in the brightness of the window is multiplied by reading over image, which is the reciprocal of the room’s share of the view. That still holds behind glass, and the black net’s multiplier of 5.2 multiplies a photograph’s calibration error by 5.2. What no longer holds is that the multiplier dominates the budget. Straight on behind glass, the net’s total error is almost unchanged while its multiplier has risen two and a half times, because the extra veil is Fresnel’s, known exactly, and contributes only through the uncertainty in the street it reflects.

The comparison that survived the bare curtain’s daylight, two readings of one window with everything held, does follow the multiplier, because its error is the reading’s repeatability and nothing else. A change that the bare black net betrays at 0.004 of reflectance, the net behind glass betrays only at 0.011 straight on and 0.054 from 75°. The glass hides a moved chair from a camera on the pavement far better than it hides the room’s colour from a calibrated one.

What the account assumes #

The glass is ideal: one clean pane of index 1.52, no absorption, no coating. Clear float glass absorbs a per cent or two, which dims both the image and the cloth’s veil in equal proportion and so cancels from the multiplier. A low-emissivity coating raises the reflectance, and double glazing has four faces rather than two, roughly doubling it straight on. Every such pane lowers the curves in the break-even figure and raises the net’s multiplier further; none of them reverses a direction.

The mirrored scene is the street, reflecting three tenths of the same light that falls on the cloth. From a low viewpoint looking up at an upper window, the pane mirrors the sky, which can be several times brighter than any street. That multiplies the mirror term, moves every break-even angle down, and puts the white voile on the losing side at any angle at all once the sky is bright enough. The model has one number for the mirrored scene, and the sky’s brightness is the input that would change it most.

The cloth’s veil is taken as unpolarised, which is what multiple scattering in a fibre assembly produces. The fibre surfaces themselves reflect a little specularly, as a satin mirrors the top of the sky computes for a satin face and no cloth shines under a sky averages away under uniform light; a curtain’s crowns do so too, and that part of the veil would be partly polarised and partly removed by the filter. For a matt voile the share is small; for a lustrous one the polariser’s advantage would shrink.

Inter-reflection between the pane and the curtain is left out. Light the cloth sends toward the glass is partly reflected back onto the cloth, raising its brightness by a few per cent of a few per cent. It is second order in both the pane’s reflectance and the cloth’s, and it is the first term a more careful account would add.

The pane is required to reflect $2r/(1+r)$ straight on, to reflect none of the p polarisation at Brewster’s angle, and to become a mirror toward grazing. A pane of index one is required to reproduce the bare curtain’s multiplier and error budget to twelve figures at every sheer and light. The break-even angle is required to leave the multiplier unchanged at every light ratio from 0.3 to 300, a black net is required to be at least doubled at every angle, and a polariser at Brewster’s angle is required to leave every sheer better off than bare.

What the figures cannot show #

The mirror is an image, and the eye reads it as one. A figured sheer is a negative from one side found that the eye reads a curtain’s pattern against whatever is behind it; the reflection is a pattern too, and it lies in front. The account treats the reflected street as veil, a uniform addition to the view, and in the arithmetic it is. A passer-by does not see it as uniform. They see the houses opposite, their windows, a parked car, laid over the room, and a sharp reflected edge draws the eye away from the room’s faint image in a way a uniform haze does not. Whether that makes the glass hide more than its share of the light says, or less because the eye can learn to look through it, is a question about vision that the multiplier does not answer.

And the view from inside is left out. At night the room is the bright side, the pane mirrors the room back at its occupants through the curtain, and the same arithmetic run the other way says how much of the dark street they can see. It is the reason a lit room’s window looks black from inside, and it has the same two terms with the sides exchanged.

Who worked out which part #

Fresnel’s reflection coefficients are from 1823, and Brewster’s observation that reflected light is completely polarised at one angle is from 1815; both are textbook optics. The polarising filter as a way of seeing past a reflection has been in photographers’ kits since polarising sheet was made cheaply in the 1930s.

The image-and-veil account of a curtain, the read-back and its multiplier are from the curtain essays that came before this one. What is added here is the pane as a two-term correction to them, one term thread-less veil and one term dimming; the break-even condition, and the observation that the light ratio cancels from it; the reading that glass hides a dark open net and leaves a white voile nearly as it was; and the parting of the privacy multiplier from the measurement’s total error once part of the veil is known exactly.

Still open: a sky in the glass #

Every number here takes the pane to mirror a street as bright as the one lighting the cloth. From below an upper window it mirrors the sky, and a clear sky near the horizon can be ten times a street’s luminance, an overcast one three or four. A satin mirrors the top of the sky already carries a sky with a gradient in it, brightest overhead under cloud and at the horizon under a clear sun.

Folding that sky into the mirror term would say, window by window, whether a first-floor white voile is ever helped by its glass as seen from the street below, or whether the sky puts every upper-storey curtain on the losing side at every angle. The arithmetic is one substitution. The judgement it needs is where in the sky a pane at a given height mirrors for a passer-by at a given distance, and that is geometry this account has not yet drawn.