APOD: Webb's First Deep Field (2022 Jul 13)

[VictorBorun]

I wonder why the dots in the spikes makes a sort of chess-board pattern.
Is it because the obstacles are many and not quite parallel?

[Chris Peterson]

I wonder why the dots in the spikes makes a sort of chess-board pattern.
Is it because the obstacles are many and not quite parallel?

It's because the pattern is being simulated using a monochromatic source, so you're seeing minima and maxima produced by interference.

[VictorBorun]

I wonder why the dots in the spikes makes a sort of chess-board pattern.
Is it because the obstacles are many and not quite parallel?

It's because the pattern is being simulated using a monochromatic source, so you're seeing minima and maxima produced by interference.

were the obstacle just one straight-line edge, the pattern of minima and maxima should be one thin dotted-line spike

were the obstacle just two opposite straight-line edges, the pattern of minima and maxima should be two thin dotted-line spikes in the opposite directions

But a chess-board of dots? How come?

[Chris Peterson]

I wonder why the dots in the spikes makes a sort of chess-board pattern.
Is it because the obstacles are many and not quite parallel?

It's because the pattern is being simulated using a monochromatic source, so you're seeing minima and maxima produced by interference.

were the obstacle just one straight-line edge, the pattern of minima and maxima should be one thin dotted-line spike

were the obstacle just two opposite straight-line edges, the pattern of minima and maxima should be two thin dotted-line spikes in the opposite directions

But a chess-board of dots? How come?

The hexagonal pupil changes the PSF from an ordinary (circular) Airy function to an "Airy hexagon" with six-fold symmetry. As you add the additional edges, you shift more energy out of the central spot and into the arms. The shape of the interference spots becomes more complex as you shift the location of the "rings" for different orientations. You can play with this by performing an FFT on different aperture shapes (a lot of imaging processing software offers this ability).
_

[VictorBorun]

It's because the pattern is being simulated using a monochromatic source, so you're seeing minima and maxima produced by interference.

were the obstacle just one straight-line edge, the pattern of minima and maxima should be one thin dotted-line spike

were the obstacle just two opposite straight-line edges, the pattern of minima and maxima should be two thin dotted-line spikes in the opposite directions

But a chess-board of dots? How come?

The hexagonal pupil changes the PSF from an ordinary (circular) Airy function to an "Airy hexagon" with six-fold symmetry. As you add the additional edges, you shift more energy out of the central spot and into the arms. The shape of the interference spots becomes more complex as you shift the location of the "rings" for different orientations. You can play with this by performing an FFT on different aperture shapes (a lot of imaging processing software offers this ability).
_
Screenshot 2022-07-17 100406.png

wow. So perfectly parallel multiple edges generate a chess board lattice. ok

[johnnydeep]

It's because the pattern is being simulated using a monochromatic source, so you're seeing minima and maxima produced by interference.

were the obstacle just one straight-line edge, the pattern of minima and maxima should be one thin dotted-line spike

were the obstacle just two opposite straight-line edges, the pattern of minima and maxima should be two thin dotted-line spikes in the opposite directions

But a chess-board of dots? How come?

The hexagonal pupil changes the PSF from an ordinary (circular) Airy function to an "Airy hexagon" with six-fold symmetry. As you add the additional edges, you shift more energy out of the central spot and into the arms. The shape of the interference spots becomes more complex as you shift the location of the "rings" for different orientations. You can play with this by performing an FFT on different aperture shapes (a lot of imaging processing software offers this ability).
_
Screenshot 2022-07-17 100406.png

So to make it painfully clear to me, if Webb had just one mirror segment (and no secondary mirror with support struts), would the spikes look like the third image from this pic from the JWST site?:

Are the "spikes" we see just the "max" portions of the max/min diffractions patterns, or not?

[Chris Peterson]

were the obstacle just one straight-line edge, the pattern of minima and maxima should be one thin dotted-line spike

were the obstacle just two opposite straight-line edges, the pattern of minima and maxima should be two thin dotted-line spikes in the opposite directions

But a chess-board of dots? How come?

The hexagonal pupil changes the PSF from an ordinary (circular) Airy function to an "Airy hexagon" with six-fold symmetry. As you add the additional edges, you shift more energy out of the central spot and into the arms. The shape of the interference spots becomes more complex as you shift the location of the "rings" for different orientations. You can play with this by performing an FFT on different aperture shapes (a lot of imaging processing software offers this ability).
_
Screenshot 2022-07-17 100406.png

So to make it painfully clear to me, if Webb had just one mirror segment (and no secondary mirror with support struts), would the spikes look like the third image from this pic from the JWST site?:

With one hexagonal segment, yes. It's just what a camera on a backyard telescope would see if the scope was fitted with a hexagonal mask at the aperture and a narrowband filter.

Are the "spikes" we see just the "max" portions of the max/min diffractions patterns, or not?

You could think of it that way (although even the min sections are brighter than zero, so appearance depends a lot on how the image is processed).

[johnnydeep]

The hexagonal pupil changes the PSF from an ordinary (circular) Airy function to an "Airy hexagon" with six-fold symmetry. As you add the additional edges, you shift more energy out of the central spot and into the arms. The shape of the interference spots becomes more complex as you shift the location of the "rings" for different orientations. You can play with this by performing an FFT on different aperture shapes (a lot of imaging processing software offers this ability).
_
Screenshot 2022-07-17 100406.png

So to make it painfully clear to me, if Webb had just one mirror segment (and no secondary mirror with support struts), would the spikes look like the third image from this pic from the JWST site?:

With one hexagonal segment, yes. It's just what a camera on a backyard telescope would see if the scope was fitted with a hexagonal mask at the aperture and a narrowband filter.

Are the "spikes" we see just the "max" portions of the max/min diffractions patterns, or not?

You could think of it that way (although even the min sections are brighter than zero, so appearance depends a lot on how the image is processed).

Good. And if the ACTUAL spikes we see with the Webb 18 segment mirror are due to the overlapping - and also mutually interfering? - patterns created by all the different separate patterns from the 18 segments (plus those created by the 3 support struts), then perhaps I can finally claim I am understanding it the right way?

[Chris Peterson]

So to make it painfully clear to me, if Webb had just one mirror segment (and no secondary mirror with support struts), would the spikes look like the third image from this pic from the JWST site?:

With one hexagonal segment, yes. It's just what a camera on a backyard telescope would see if the scope was fitted with a hexagonal mask at the aperture and a narrowband filter.

Are the "spikes" we see just the "max" portions of the max/min diffractions patterns, or not?

You could think of it that way (although even the min sections are brighter than zero, so appearance depends a lot on how the image is processed).

Good. And if the ACTUAL spikes we see with the Webb 18 segment mirror are due to the overlapping - and also mutually interfering? - patterns created by all the different separate patterns from the 18 segments (plus those created by the 3 support struts), then perhaps I can finally claim I am understanding it the right way?

That's a reasonable way to understand the broad picture. Just keep in mind that there are two related but different things going on. You have diffraction, which results in light that ideally would create a perfect point being distributed outside that point by how it interacts with edges and obstructions, and you have interference, which creates additional regions of constructive and destructive combination... light and dark zones within the diffraction structures.

[alter-ego]

With one hexagonal segment, yes. It's just what a camera on a backyard telescope would see if the scope was fitted with a hexagonal mask at the aperture and a narrowband filter.


You could think of it that way (although even the min sections are brighter than zero, so appearance depends a lot on how the image is processed).

Good. And if the ACTUAL spikes we see with the Webb 18 segment mirror are due to the overlapping - and also mutually interfering? - patterns created by all the different separate patterns from the 18 segments (plus those created by the 3 support struts), then perhaps I can finally claim I am understanding it the right way?

That's a reasonable way to understand the broad picture. Just keep in mind that there are two related but different things going on. You have diffraction, which results in light that ideally would create a perfect point being distributed outside that point by how it interacts with edges and obstructions, and you have interference, which creates additional regions of constructive and destructive combination... light and dark zones within the diffraction structures.

I guess you're associated particle and wave duality of light to diffraction and interference. I don't think you can have diffraction without interference. Classical, Hugens - Fresnel diffraction is built entirely on wave theory, and quantum diffraction necessitates wavefunctions to describe it, hence the particle-wave duality emerges. I consider diffraction like a vector with particle and wave dynamics as orthogonal components. Each projection describes a particular diffraction behavior in that context, but point-to-point deviation is not the sole definition of diffraction or diffraction structures. The "diffraction pattern" intrinsically includes interference. If you include measuring energy quanta, you'll then have the point-to-point aspect of diffraction.

[Chris Peterson]

Good. And if the ACTUAL spikes we see with the Webb 18 segment mirror are due to the overlapping - and also mutually interfering? - patterns created by all the different separate patterns from the 18 segments (plus those created by the 3 support struts), then perhaps I can finally claim I am understanding it the right way?

That's a reasonable way to understand the broad picture. Just keep in mind that there are two related but different things going on. You have diffraction, which results in light that ideally would create a perfect point being distributed outside that point by how it interacts with edges and obstructions, and you have interference, which creates additional regions of constructive and destructive combination... light and dark zones within the diffraction structures.

I guess you're associated particle and wave duality of light to diffraction and interference. I don't think you can have diffraction without interference. Classical, Hugens - Fresnel diffraction is built entirely on wave theory, and quantum diffraction necessitates wavefunctions to describe it, hence the particle-wave duality emerges. I consider diffraction like a vector with particle and wave dynamics as orthogonal components. Each projection describes a particular diffraction behavior in that context, but point-to-point deviation is not the sole definition of diffraction or diffraction structures. The "diffraction pattern" intrinsically includes interference. If you include measuring energy quanta, you'll then have the point-to-point aspect of diffraction.

Indeed. That's what I meant by different but related. Just trying to keep things simple to give a sense of what's going on without getting too technical.

[alter-ego]

That's a reasonable way to understand the broad picture. Just keep in mind that there are two related but different things going on. You have diffraction, which results in light that ideally would create a perfect point being distributed outside that point by how it interacts with edges and obstructions, and you have interference, which creates additional regions of constructive and destructive combination... light and dark zones within the diffraction structures.

I guess you're associated particle and wave duality of light to diffraction and interference. I don't think you can have diffraction without interference. Classical, Hugens - Fresnel diffraction is built entirely on wave theory, and quantum diffraction necessitates wavefunctions to describe it, hence the particle-wave duality emerges. I consider diffraction like a vector with particle and wave dynamics as orthogonal components. Each projection describes a particular diffraction behavior in that context, but point-to-point deviation is not the sole definition of diffraction or diffraction structures. The "diffraction pattern" intrinsically includes interference. If you include measuring energy quanta, you'll then have the point-to-point aspect of diffraction.

Indeed. That's what I meant by different but related. Just trying to keep things simple to give a sense of what's going on without getting too technical.

I'm technical and can be intensely literal - not always a good combination
Thanks for the clarification.

[VictorBorun]

Each projection describes a particular diffraction behavior in that context, but point-to-point deviation is not the sole definition of diffraction or diffraction structures. The "diffraction pattern" intrinsically includes interference. If you include measuring energy quanta, you'll then have the point-to-point aspect of diffraction.

IMHO quanta do not matter here at all. It's not important if photons are rare and come one by one; even one photon does its own interference pattern like a wave, statistically: that one photon has some chances to slightly bend its way around the obstacle and, after a distant star's photon is focused to a point in the image plane, that photon still has some chances to get caught by a sensor pixel slightly aside from the focus point for that star

[johnnydeep]

With one hexagonal segment, yes. It's just what a camera on a backyard telescope would see if the scope was fitted with a hexagonal mask at the aperture and a narrowband filter.


You could think of it that way (although even the min sections are brighter than zero, so appearance depends a lot on how the image is processed).

Good. And if the ACTUAL spikes we see with the Webb 18 segment mirror are due to the overlapping - and also mutually interfering? - patterns created by all the different separate patterns from the 18 segments (plus those created by the 3 support struts), then perhaps I can finally claim I am understanding it the right way?

That's a reasonable way to understand the broad picture. Just keep in mind that there are two related but different things going on. You have diffraction, which results in light that ideally would create a perfect point being distributed outside that point by how it interacts with edges and obstructions, and you have interference, which creates additional regions of constructive and destructive combination... light and dark zones within the diffraction structures.

Ok, thanks. But I'm right back to not understanding the difference. Oh well, I've taken too much of your time already.

[Chris Peterson]

Good. And if the ACTUAL spikes we see with the Webb 18 segment mirror are due to the overlapping - and also mutually interfering? - patterns created by all the different separate patterns from the 18 segments (plus those created by the 3 support struts), then perhaps I can finally claim I am understanding it the right way?

That's a reasonable way to understand the broad picture. Just keep in mind that there are two related but different things going on. You have diffraction, which results in light that ideally would create a perfect point being distributed outside that point by how it interacts with edges and obstructions, and you have interference, which creates additional regions of constructive and destructive combination... light and dark zones within the diffraction structures.

Ok, thanks. But I'm right back to not understanding the difference. Oh well, I've taken too much of your time already.

Diffraction changes the direction of the light when it encounters an obstacle. Interference occurs when light interacts with itself constructively or destructively to either get brighter or dimmer.

[johnnydeep]

That's a reasonable way to understand the broad picture. Just keep in mind that there are two related but different things going on. You have diffraction, which results in light that ideally would create a perfect point being distributed outside that point by how it interacts with edges and obstructions, and you have interference, which creates additional regions of constructive and destructive combination... light and dark zones within the diffraction structures.

Ok, thanks. But I'm right back to not understanding the difference. Oh well, I've taken too much of your time already.

Diffraction changes the direction of the light when it encounters an obstacle. Interference occurs when light interacts with itself constructively or destructively to either get brighter or dimmer.

Well, I'm pretty sure I understand those words, but I can't understand how the different effects of diffraction and interference contribute to the Webb spikes. They are diffraction spikes after all, not interference spikes!

I thought it was the diffraction around the edges of the mirrors and support struts that is causing the interference that results in the min/max patterns we see. In the image of the single hexagonal mirror showing the diffraction pattern above, is what we see due to diffraction, interference, or both?

And just how much of the Webb spikes we see are due to diffraction versus interference, and can they be separated out? The answer may well be buried in the previous posts from you or others above, but if so I guess it went over my head.

[Chris Peterson]

Ok, thanks. But I'm right back to not understanding the difference. Oh well, I've taken too much of your time already.

Diffraction changes the direction of the light when it encounters an obstacle. Interference occurs when light interacts with itself constructively or destructively to either get brighter or dimmer.

Well, I'm pretty sure I understand those words, but I can't understand how the different effects of diffraction and interference contribute to the Webb spikes. They are diffraction spikes after all, not interference spikes!

I thought it was the diffraction around the edges of the mirrors and support struts that is causing the interference that results in the min/max patterns we see. In the image of the single hexagonal mirror showing the diffraction pattern above, is what we see due to diffraction, interference, or both?

And just how much of the Webb spikes we see are due to diffraction versus interference, and can they be separated out? The answer may well be buried in the previous posts from you or others above, but if so I guess it went over my head.

Both effects are present. Diffraction is why we have light outside of a central point. But if there were no interference, those spikes would all be smooth gradients. It is interference that creates the rings and other structure in the PSF. In fact, we can pretty much see the effects of diffraction without interference by using a white light source, because the structure is different for different wavelengths, and it all overlaps and washes out when you have white light.

[VictorBorun]

JWST User Documentation wrote:
JWST NIRCam offers 29 bandpass filters in the short wavelength (0.6–2.3 μm) and long wavelength (2.4–5.0 μm) channels.

The JWST NIRCam is the camera detecting the shortest infrared wavelengths. There is another camera too, which I think is called MIRI (Mid Infrared Instrument) (but you'll have to google, because I'm too lazy), which detects longer, mid infrared wavelengths.
So anyway. If I understand things correctly, and 0.6 μm is the same thing as 600 nm, then the shortest wavelength that JWST can detect corresponds to this visible color: ███
So this is the shortest wavelength that JWST can detect, and any emission at this wavelength will be mapped as blue ███. (The color sample I just showed you corresponds to a wavelength of 441 nm, which probably qualifies as the height of "blue-ness" that the human eye can detect.) The difference in wavelength between 600 nm and 441 nm is not so large, only some "160 steps on the wavelength meter" (or how do you put that in words?).
All right. But please note that 2.3 μm, 2300 nm, also counts as a short wavelength to JWST, and this wavelength will, perhaps, be mapped as this visible color: ███ (This color sample corresponds to 550 nm, which is a wavelength that is sometimes used by Hubble as a green filter.)
Let's assume that 2300 nm is mapped as "green", or 550 nm, by JWST. The difference between 550 nm and 2300 nm corresponds to "1750 steps on the wavelength meter". That's a lot.
2300 nm is way, way off the optical scale of wavelengths that the human eye can detect. After all, when Hubble takes pictures in the infrared, it is usually no more infrared than - if I remember correctly - 814 nm. 2300 nm is almost three times longer than the longest infrared wavelength than Hubble can see.

Ann

How do we compress a waveband uniformly if it is wide? And even wider than Hubble + Webb NIR, if you look at today's APOD: — which combine those with Hubble UF and Webb MIR.

It's as simple as this: we use logarithmic scale. Like tones in a melody, like decibels in a loudness and like pH in an acid.
You take your target band where we humans are sensitive to the wavelength: from violet 435 nm to red 625 nm.
You ignore the margins that we humans can see but can not tell the colour difference, < 435 nm or > 625 nm.
You take your source band which is from UV 200 nm to MIR 26000 nm..
And here is your mapping: present any source wavelength λ_source with base 10 logarithms as
10^(log(435) + (log(625) - log(435))(log(λ_source) - log(200))/(log(26000) - log(200))).

PS Coding in Wolfram's Mathematica, made it base e logarithms, to the same effect:

[Chris Peterson]

JWST User Documentation wrote:
JWST NIRCam offers 29 bandpass filters in the short wavelength (0.6–2.3 μm) and long wavelength (2.4–5.0 μm) channels.

The JWST NIRCam is the camera detecting the shortest infrared wavelengths. There is another camera too, which I think is called MIRI (Mid Infrared Instrument) (but you'll have to google, because I'm too lazy), which detects longer, mid infrared wavelengths.
So anyway. If I understand things correctly, and 0.6 μm is the same thing as 600 nm, then the shortest wavelength that JWST can detect corresponds to this visible color: ███
So this is the shortest wavelength that JWST can detect, and any emission at this wavelength will be mapped as blue ███. (The color sample I just showed you corresponds to a wavelength of 441 nm, which probably qualifies as the height of "blue-ness" that the human eye can detect.) The difference in wavelength between 600 nm and 441 nm is not so large, only some "160 steps on the wavelength meter" (or how do you put that in words?).
All right. But please note that 2.3 μm, 2300 nm, also counts as a short wavelength to JWST, and this wavelength will, perhaps, be mapped as this visible color: ███ (This color sample corresponds to 550 nm, which is a wavelength that is sometimes used by Hubble as a green filter.)
Let's assume that 2300 nm is mapped as "green", or 550 nm, by JWST. The difference between 550 nm and 2300 nm corresponds to "1750 steps on the wavelength meter". That's a lot.
2300 nm is way, way off the optical scale of wavelengths that the human eye can detect. After all, when Hubble takes pictures in the infrared, it is usually no more infrared than - if I remember correctly - 814 nm. 2300 nm is almost three times longer than the longest infrared wavelength than Hubble can see.

Ann

How do we compress a waveband uniformly if it is wide? And even wider than Hubble + Webb NIR, if you look at today's APOD: — which combine those with Hubble UF and Webb MIR.

It's as simple as this: we use logarithmic scale. Like tones in a melody, like decibels in a loudness and like pH in an acid.
You take your target band where we humans are sensitive to the wavelength: from violet 435 nm to red 625 nm.
You ignore the margins that we humans can see but can not tell the colour difference, < 435 nm or > 625 nm.
You take your source band which is from UV 200 nm to MIR 26000 nm..
And here is your mapping: present any source wavelength λ_source with base 10 logarithms as
10^(log(435) + (log(625) - log(435))(log(λ_source) - log(200))/(log(26000) - log(200))).

Which results in a palette that essentially tells us nothing more than any other palette.

The rational strategy in constructing a color image from data like this is to choose a palette that emphasizes what we most want to show. Which likely will not involve a monotonic relationship between source and display.

[ddorn777]

This is a Very good point! Why keep extraneous "corners" or cut off additional data, just to fit an archaic format?

Nice. What surprised me most is that the James Webb's images from its hexagonal mirrors still produces rectangular images with perfect 90 degree corners. Unlike my binoculars that has round lenses and it produces a round image. The explanation was for the old round camera lenses producing square images was because that is how the photo negative film was made. And of course science text books were rectangular and demanded rectangular photos. But nowadays when paper books are on the way out and most images are processed by computer there is no reason to still cut the round or hexagonal images into squares.

[ddorn777]

Okay, I responded before reading the next page...

Nice. What surprised me most is that the James Webb's images from its hexagonal mirrors still produces rectangular images with perfect 90 degree corners. Unlike my binoculars that has round lenses and it produces a round image. The explanation was for the old round camera lenses producing square images was because that is how the photo negative film was made. And of course science text books were rectangular and demanded rectangular photos. But nowadays when paper books are on the way out and most images are processed by computer there is no reason to still cut the round or hexagonal images into squares.

Like essentially all optical systems, the image produced by the telescope is round. The images as we see them are rectangular because the sensor is rectangular (like film in most cases), clipping out a section of the full image. The shape of the mirror or its segments doesn't matter.

[VictorBorun]

The rational strategy in constructing a color image from data like this is to choose a palette that emphasizes what we most want to show. Which likely will not involve a monotonic relationship between source and display.

Suppose we just want to identify possible multiple lensed and mirrored images of the same galaxy.
Let's overlay Webb's NIR + MIR, re-encoding NIR's RGB presentation as cyan to blue and MIR's RGB presentation as red to yellow:

Voila! We can easily see a slightly curved line of symmetry through the brightest midground galaxies of SMACS and distorted shapes but distinct colours of 4 pairs of images of background galaxies

"b" is the famous background galaxy with dots around
"a" is the galaxy with its spectrum taken, of both images:

[alter-ego]

Each projection describes a particular diffraction behavior in that context, but point-to-point deviation is not the sole definition of diffraction or diffraction structures. The "diffraction pattern" intrinsically includes interference. If you include measuring energy quanta, you'll then have the point-to-point aspect of diffraction.

IMHO quanta do not matter here at all. It's not important if photons are rare and come one by one; even one photon does its own interference pattern like a wave, statistically: that one photon has some chances to slightly bend its way around the obstacle and, after a distant star's photon is focused to a point in the image plane, that photon still has some chances to get caught by a sensor pixel slightly aside from the focus point for that star

A more complete understanding of diffraction does require particles (quanta).
In terms of detecting a single photon deviating from a straight line does not reveal the interference component, i.e. that measurement reveals just the particle component behavior of diffraction . The interference diffraction component cannot be revealed with a single photon. The probability of single photon detection at a specific detector location is result of the QM wavefunction for the photon. Only after a (very) large number of photons are detected will the interference component of diffraction will become evident. Quantum mechanics successfully describes the particle-wave duality of diffraction.
 

Click to play embedded YouTube video.

Interference pattern after thousands of photons collected.

In similar experiments conducted in the mid 1920s, the new field of quantum mechanics was used in conjunction with one Davisson-Germer experiment to ultimately confirm the matter-wave, de Broglie hypothesis. The experiment, using a low-energy beam of electrons incident on platinum, revealed similar photon-like, wave interference as a component of diffraction. Similar to photons, it took very many electron detections to reveal the interference part of diffraction.

[VictorBorun]

Only after a (very) large number of photons are detected will the interference component of diffraction will become evident. Quantum mechanics successfully describes the particle-wave duality of diffraction.

Click to play embedded YouTube video.

Two Slit Diffraction - Single Photons.jpg
In similar experiments conducted in the mid 1920s, the new field of quantum mechanics was used in conjunction with one Davisson-Germer experiment to ultimately confirm the matter-wave, de Broglie hypothesis. The experiment, using a low-energy beam of electrons incident on platinum, revealed similar photon-like, wave interference as a component of diffraction. Similar to photons, it took very many electron detections to reveal the interference part of diffraction.

oh the times when Monte Carlo runs were done physically rather than digitally!
Yes, you have to expose for many a photon to see all the pixels of a star's 8 spikes in a JWST pic
I was trying to say that those photons do no have to interference one another; each one itself is a full-blown wave and so the number of particles affect rather the clarity of the spikes than their size, shape and inner patterns.

[VictorBorun]

Note how one of the lensed elongated orange background galaxies, at center right, is surrounded by at least ten distinct and quite bright orange dots! Amazing! What are those things?
Ann

To my eye it's similar to SDSS J1226+2152 (Hubble) :


This, too, is a lensed galaxy with stellar globular clusters of the same colour and with another galaxy not far away in the frame which is more elongated by lensing