Diffraction Effects and Artifacts in Telescopes like the JWST

[Devin-M]

Isn’t the maximum diffraction angle of a photon from a dim star the same as the maximum diffraction angle of a photon from a bright star for the reasons stated in @collinsmark previous post?

But identical to the double slit experiment, photons from a given light source have the same interference pattern regardless of the photon rate. It doesn't matter if many photons pass through in quick succession or pass through slowly, one at a time. The pattern on the sensor is the same.

 

[andrew s 1905]

Firstly an apology the FWHM is indeed not a function of star magnitude. I was mistaken it is a measure of "shape".

Yes the theoretical diffraction patter of a point source convoluted with the telescopes instrumental profile is infinite in extent but this is misleading.

The image is also the result of sampling by the detector and boxed both by its physical size and the length of the exposure. At some point as the intensity of the point source drops no photons will be captured in fainter parts of the point source image even with a noiseless detector. The image will either be finite and within the area of the detector and can be measurable using standard methods or be undefined if it extends to the detector edge.

This is true of both the core of the star image and the diffraction spikes. You can measure from the centroid to a point where the intensity of the object of interest falls to zero or often some defined fraction above the noise floor. This is routinely done in setting photometric apertures.

Regards Andrew

 

[andrew s 1905]

Whatever the captured bit depth it would be normal to do the calibration in 32 bit arithmetic and save the result in 32 bit files.
Regards Andrew

 

[Devin-M]

“the maximum angle for diffraction maxima is 90°”

https://physics.stackexchange.com/questions/508635/diffraction-grating-angle

 

[sophiecentaur]

Yes the theoretical diffraction pattern of a point source convoluted with the telescopes instrumental profile is infinite in extent but this is misleading.

It's very misleading because, in a real image there is not a point source and also there are a number of other sources in the vicinity of the low level parts of a diffraction spike. this constitutes a 'floor' which can be significantly above the least significant step in the ADC.

 

[collinsmark]

It's very misleading because, in a real image there is not a point source and also there are a number of other sources in the vicinity of the low level parts of a diffraction spike. this constitutes a 'floor' which can be significantly above the least significant step in the ADC.

If a pixel lies along the path of diffraction spike, and is dim enough such that expected value for that pixel is less than 1 analog-to-digital converter units (ADU) after compensating for noise (dark frame subtraction), there is still a finite probability that the pixel will register 1 or more ADU. For example, for a given exposure time, in a dim section of a diffraction spike, a pixel might only have a 50% chance of registering a single ADU or more. On even dimmer section of the spike, the probability drops to 25%. This is due to probabilistic nature of shot noise, which is inherently part and parcel of the signal. In other words, if you look along a diffraction spike in the vicinity of 50% probability, half of the pixels in that region will register at least 1 ADU above the noise (i.e., ~1 ADU after dark frame subtraction).

The point is that even if the expected value of a pixel that lies in a diffraction spike is less than the least significant step in the ADC (i.e., 1 ADU), it does not guarantee that the pixel will not register a signal. The signal still has an effect on the pixel registration in a probabilistic manner.

[Edit: and if a particular pixel is along the intersection of diffraction spikes/artifacts, say from two or three or more different stars, the probabilistic contributions add together linearly, even if the expectation value of anyone of the spikes/artifacts, or all of them, is less than 1 ADU in that region.]

And you don't need a point source for this. As I've essentially stated in post #404 the interference pattern applies to all photons that pass through the optics and reach the detector, whether those photons originate from stars, nebulosity, accretion disks, anything. Any photon that passes through the telescope's optics is subject to an interference pattern before that photon reaches the sensor, if it reaches the sensor at all. It matters not what the source of the photons are for this. All photons from any distant source that reach the sensor are subject an interference pattern before reaching the sensor.

There's nothing misleading about this. Diffraction patterns and interference patterns and the probabilistic nature of quantum mechanics are not "misleading." It's just how the universe works.

 

[Devin-M]

I decided to put a piece of window screen in front of my objective lens for a diffraction test…

I took a single 5 minute exposure of Polaris at 600mm focal f/9, 100iso with the window screen in front of the lens…

When I adjust the RAW conversion settings of half the 14 bit image (with the identical exposure / image data), the brighter Polaris diffraction pattern appears the same shape and size as the dimmer star in the upper right…

 

[andrew s 1905]

There's nothing misleading about this. Diffraction patterns and interference patterns and the probabilistic nature of quantum mechanics are not "misleading." It's just how the universe works.

Yes true but there comes a point where even with photon statistics etc. that the probability of detecting a photon is FAPP zero within the length of the exposure. No amount of signal processing can pull the signal from the noise.

If this were not the case there would be no lower limit to the faintness of stars we could detect and no need for larger telescopes.

Regards Andrew

 

[andrew s 1905]

When I adjust the RAW conversion settings of half the 14 bit image (with the identical exposure / image data), the brighter Polaris diffraction pattern appears the same shape and size as the dimmer star in the upper right…

Not sure what you are trying to show other than by signal processing you can manipulate how an image looks.

Can you tell me how you measured the size of the images to be the same as your earlier claim was they are not finite but extend off the image?

Regards Andrew

 

[collinsmark]

I'm afraid there are too many points in your post to be addressed.

OK, I already accepted that the sensor itself is linear over a wide range and that stacking will allow a substantial increase in effective bit depth by averaging out random noise. I also know that the maths of diffraction goes on and on, as far down as you like. Real life is not like that. We always run out of range because noise and interference are present.

Do you have a reference about 32bit ADCs on the JWST? Wherever I have found a bit depth of the sensor arrays mentioned, it's been 16bits. Stacking can be achieved in many ways and they are all based on non linear processing to reject spurious data so I have to assume that, in fact, the linearity / purity of the images is restricted to 16 bits. JWST doesn't need defending and 16 bit data has been quite adequate as a source of fantastic and revealing images.

I'm pretty sure the bit depth of the sensor itself is 16 bit, since the full-well value of any of the sensors in NIRCam's sensor array is less than 2^{16} = 65536. (See https://jwst-docs.stsci.edu/jwst-ne...detector-overview/nircam-detector-performance)

But each pixel value is stored in 32 bit floating point format before or during the steps where calibration is applied and subframes are stacked. This allows for sub-ADU resolution of each pixel. As a matter of fact, as described below, it's possible to achieve resolutions not just below that of an ADU, but even a single photon, if sufficient stacking is performed.

One obvious reason for stacking is to identify cosmic rays. It's not difficult to identify because a subframe pixel affected by a cosmic ray will be a statistical outlier compared to the corresponding pixels in the other subframes.

And, as you mentioned, you can increase the signal to noise ratio above that of any single subframe by stacking multiple subframes. One can use the central limit theorem to show that (given a few assumptions about the noise being uncorrelated) the signal to noise ratio increases by a factor of \sqrt{N} over a single subframe, where N is the number of subframes stacked.

Stacking also increases the bit-depth in another way due to the probabilistic nature of photon arrival. Even if some sublte detail in a target is less than 1 ADU, it still affects the pixels in a probabilistic manner. For example if a star's diffraction spike over a particular pixel is only 1/5 of an ADU, you would expect 2 out of 10 subframes to have an additional ADU above the background for that pixel. And if you stack 10 subframes (averaging them), you can get that extra 0.2 ADU detail in the result. If you stack enough subframes you can gain resolutions better than even a single photon.

I don't know how much stacking is typically done in JWST images, but there's definitely some stacking done. There are gaps between the sensors within the sensor array, so there needs to be at least some stacking overlap for that at least. (See https://jwst-docs.stsci.edu/jwst-near-infrared-camera)

NIRCam modules field of view

For what it's worth, here's an image from the recent Pillars Of Creation redo, specifically showing the stacking overlap of the individual sensor cores. I downloaded this particular image from Mast, then I used PixInsight to apply a quick-and-dirty stretch to it (otherwise it would look nearly all black), resized it for PF, and saved it as a JPEG. This image was acquired using the F090W filter.

 

[collinsmark]

Yes true but there comes a point where even with photon statistics etc. that the probability of detecting a photon is FAPP zero within the length of the exposure. No amount of signal processing can pull the signal from the noise.

Of course there are limitations with the sensor. I've never argued against that. What I object to are incorrect claims such as

• diffraction spikes are caused by nonlinearities (false)
• diffraction patterns themselves are inherently nonlinear (false)
• dim stars never produce diffraction spikes; diffraction spikes are only caused by bright stars (false)
• diffraction patterns are a nonlinear function of the star's brightness (false)
• for a given exposure it's impossible to ever gain more detail below 1 ADU (false. You can gain better resolution than 1 ADU by stacking multiple subframes and exploit the probabilistic nature of photon arrival).

I've never claimed that for a given exposure you can gain more detail than a single ADU by "signal processing." Of course not. But you can take that single exposure and stack it together with many other similar, single exposures, and get that detail back. Or, if saturation isn't an issue, just take longer exposures.

If what you said is true ("the probability of detecting a photon is FAPP zero within the length of the exposure") then the act of stacking multiple sub-exposures of the same length would also have "FAPP zero" probability of detecting a photon. But it's not. You can get that detail* back by stacking. The point being that the information of that subtle diffraction spike of that dim star is still there, albeit in a probabilistic manner (i.e., it takes more than one exposure, but it can be gotten).

*[Edit: here "detail" refers to small variations in intensity, not detail in terms of angular resolution.]

Applying all that to this discussion: Diffraction patterns/interference patterns are not the result of the exposure time or the result of sensor limitations. Diffraction patterns/interference patterns are ultimately a function of the telescope's optics.

If this were not the case there would be no lower limit to the faintness of stars we could detect and no need for larger telescopes.

Regards Andrew

There are several reasons for larger telescopes. Two in particular:

1. For the same angular resolution (or for a given focal length), a bigger telescope gathers more light and allow the image to be acquired in less time, all else being roughly equal.
2. And more importantly, the image produced by a given telescope is essentially a convolution of the diffraction pattern/interference pattern which we are discussing here. It's not possible to achieve more angular detail in the image than the angular detail in the diffraction pattern/interference pattern. Bigger telescopes have smaller/more detailed diffraction/interference patterns. So if you want more angular detail in the resulting image, you need a bigger scope.

 

[sophiecentaur]

I'm pretty sure the bit depth of the sensor itself is 16 bit,

So we agree on that. The only way to increase the effective number of bits is by using multiple images and you say there's no stacking. So the linearity range cannot go below one bit 1:1/(2¹⁶). That's not a big relative magnitude and can account for the 'smaller' / shorter spikes for the dimmer stars. I'm not sure why you took exception to this.

 

[sophiecentaur]

• diffraction spikes are caused by nonlinearities (false)
• diffraction patterns themselves are inherently nonlinear (false)
• dim stars never produce diffraction spikes; diffraction spikes are only caused by bright stars (false)
• diffraction patterns are a nonlinear function of the star's brightness (false)
• for a given exposure it's impossible to ever gain more detail below 1 ADU (false. You can gain better resolution than 1 ADU by stacking multiple subframes and exploit the probabilistic nature of photon arrival).

Where did you get this list from? You must have mis-read a lot of what I wrote or, at least, have confused the concept of "recorded image of a pattern" with the pattern itself. The linearity failure at low levels can destroy recorded spikes and 16 bits is where linearity fails. Some spikes are never there in 16 bit images.

 

[collinsmark]

So we agree on that. The only way to increase the effective number of bits is by using multiple images and you say there's no stacking. So the linearity range cannot go below one bit 1:1/(2¹⁶). That's not a big relative magnitude and can account for the 'smaller' / shorter spikes for the dimmer stars. I'm not sure why you took exception to this.

No, there is stacking. There's always at least some stacking, even with JWST's pristine sensors. I'm just not sure how much is typically done with JWST.

I don't take exception to the acknowledgment that there are practical limitations. Of course there are limitations such as finite amount of integration time for practical reasons. Of course.

What I take objection to is claims implying that it is impossible to detect small details such as the diffraction spikes of dimmer stars, even in principle. It's not impossible; the physics that cause the diffraction spikes of brigher stars is also equally present for dimmer stars. It just may take more integration time (either longer exposures or stacking of shorter exposures) to bring those spikes above the floor.

 

[collinsmark]

Where did you get this list from? You must have mis-read a lot of what I wrote or, [...]

I wasn't replying to you in-particular on that one.

 

[andrew s 1905]

I wasn't replying to you in-particular on that one.

Well I never made any of those claims so who were you replying to?
Regards Andrew

 

[andrew s 1905]

what you said is true ("the probability of detecting a photon is FAPP zero within the length of the exposure") then the act of stacking multiple sub-exposures of the same length would also have "FAPP zero" probability of detecting a photon. But it's not. You can get that detail back by stacking. The point being that the information of that subtle diffraction spike of that dim star is still there, albeit in a prob

This is not true. Whatever the exposure time (single or multiple images) there will be an intensity where FAPP there will be zero photons detected. Yes by increasing the exposue you can record fainter details but even here there is a limit due to non zero sky brightness and othe noise sources.

Regards Andrew

 

[collinsmark]

This is not true. Whatever the exposure time (single or multiple images) there will be an intensity where FAPP there will be zero photons detected. Yes by increasing the exposue you can record fainter details but even here there is a limit due to non zero sky brightness and othe noise sources.

Regards Andrew

If there are no photons at all, then of course there will be no photons detected.

But if there's even a dim source, The Central Limit Theorem disagrees with you.

Just like rolling a die (as in "dice") that has an ever so slightly greater chance of landing on a particular number compared to any other number, the discrepancy can be determined with enough rolls. Even if the imperfection is smaller, it can be determined with a greater number of rolls.

In the case of a dim object viewed from a telescope, if the source's photon flux is greater than its surrounding background, and the photon's wavelengths are within the bandwidth of the receiver (within the filter's/sensor's bandwidth), and if the statistics of the system are stationary (i.e., we're not talking about a dynamical system such as a one-off flash, or something changing its behavior in an aperiodic fashion), then the photons can be detected with sufficient integration.

For a given exposure time of subframes, the pixel value of interest can be treated as a random variable with a mean (i.e, "average" value) and standard deviation. The standard deviation of the pixel value is the result of all the noise sources combined.

We can estimate the true mean by summing together the pixel values of multiple subframes, and then dividing by N, the number of subrames in the ensemble. (in other words, take the average value of the pixel).

What does that do to the standard deviation, you might ask? That is, the standard deviation caused by the combination of all noise sources after summing multiple subframes together?

The Central Limit Theorem shows that standard deviation of the averaged ensemble tends toward zero as N increases. Specifically by a factor of \frac{1}{\sqrt{N}}.

Similarly, if instead of stacking, you wanted to take a longer exposure (and are not at risk of saturation), with exposure time T, the time averaged noise (per unit signal) decreases by a factor of \frac{1}{\sqrt{T}} for all noise sources except the read noise, and then \frac{\mathrm{read \ noise}}{T} is added on as final step. [Edit: I'm admittedly kind of sloppy here. The units of time here are not seconds, but rather the fraction of some fixed time interval such as that used for individual subframes described above.]

The implication here is that by increasing total integration time, the estimated mean approaches the true mean with arbitrarily close precision, as total integration time increases.

Of course there may be practical limitations in any real world system. Of course. But saying that it's not possible, even in principle, is incorrect.

 

[Devin-M]

The pertinent equations suggest a different color point source will have a different diffraction shape/size, not a different brightness.

 

[collinsmark]

The pertinent equations suggest a different color point source will have a different diffraction shape/size, not a different brightness.

Yes, the diffraction pattern is wavelength dependent. That's true.

 

[Devin-M]

Can you tell me how you measured the size of the images to be the same as your earlier claim was they are not finite but extend off the image?

https://www.cyberphysics.co.uk/topics/light/A_level/difraction.htm

 

[Devin-M]

Suppose we have 2 equal intensity (photon per second) point sources with different wavelengths. The detectable diffraction spikes of the redder point source will be larger in pixel radius. Or with long enough integration time and low enough noise, both will extend outside the image frame.

 

[Devin-M]

 

[andrew s 1905]

If there are no photons at all, then of course there will be no photons detected.

But if there's even a dim source, The Central Limit Theorem disagrees with you.

Just like rolling a die (as in "dice") that has an ever so slightly greater chance of landing on a particular number compared to any other number, the discrepancy can be determined with enough rolls. Even if the imperfection is smaller, it can be determined with a greater number of rolls.

In the case of a dim object viewed from a telescope, if the source's photon flux is greater than its surrounding background, and the photon's wavelengths are within the bandwidth of the receiver (within the filter's/sensor's bandwidth), and if the statistics of the system are stationary (i.e., we're not talking about a dynamical system such as a one-off flash, or something changing its behavior in an aperiodic fashion), then the photons can be detected with sufficient integration.

For a given exposure time of subframes, the pixel value of interest can be treated as a random variable with a mean (i.e, "average" value) and standard deviation. The standard deviation of the pixel value is the result of all the noise sources combined.

We can estimate the true mean by summing together the pixel values of multiple subframes, and then dividing by N, the number of subrames in the ensemble. (in other words, take the average value of the pixel).

What does that do to the standard deviation, you might ask? That is, the standard deviation caused by the combination of all noise sources after summing multiple subframes together?

The Central Limit Theorem shows that standard deviation of the averaged ensemble tends toward zero as N increases. Specifically by a factor of \frac{1}{\sqrt{N}}.

Similarly, if instead of stacking, you wanted to take a longer exposure (and are not at risk of saturation), with exposure time T, the time averaged noise (per unit signal) decreases by a factor of \frac{1}{\sqrt{T}} for all noise sources except the read noise, and then \frac{\mathrm{read \ noise}}{T} is added on as final step. [Edit: I'm admittedly kind of sloppy here. The units of time here are not seconds, but rather the fraction of some fixed time interval such as that used for individual subframes described above.]

The implication here is that by increasing total integration time, the estimated mean approaches the true mean with arbitrarily close precision, as total integration time increases.

Of course there may be practical limitations in any real world system. Of course. But saying that it's not possible, even in principle, is incorrect.

I am saying it is For All Practical Purposes not possible. I don't recall saying it was impossible in principle but given the finite life of stars I am inclined to think it is.

In your calculation there will come a point along the diffraction spike where the diffraction pattern will fall below the sky general background so your assumption is invalid.

We shall just have to disagree on this.

Regards Andrew

 

[Devin-M]

Forget stars, suppose we have 2 satellites within the field of view, each with a monochrome laser point source. The red laser has photon count per second of 10k and the blue laser 10k+1. Which has the larger diffraction spike— the brighter blue satellite or the dimmer red?

 

[sophiecentaur]

In your calculation there will come a point along the diffraction spike where the diffraction pattern will fall below the sky general background so your assumption is invalid.

This is the numb of the problem . Even the JWST is working in marginal conditions, even if the margins have been changed a lot. If they were operating the scope within those margins then they would be wasting many billions of dollars.
@collinsmark seems to be insisting that the limited model of his maths is all that needs to be considered but, in the limited situation of a 16 bit sensor and the presence of many other interfering low level sources there is a very real limit to how far the maths will follow reality. This is not a problem but it's what limits what we can see.

 

[collinsmark]

I am saying it is For All Practical Purposes not possible. I don't recall saying it was impossible in principle

Oh. "For All Practical Purposes" is what you meant by "FAPP" when you said:

the probability of detecting a photon is FAPP zero

I see now that "FAPP" is an acronym. That's the first time I've ever heard that word being used as an acronym. I thought you were shouting an expletive to emphasize your point. (To pound home your point, so to speak.)

I'm sorry. My bad. I think we are in agreement then. For practical reasons, yes, there are limitations. Of course.

By the way, if you didn't know, "fapp" has a much more commonly used colloquial meaning in the contemporary English language (i.e., slang) that I won't repeat here. 'Figure I should point that out so you know.

In your calculation there will come a point along the diffraction spike where the diffraction pattern will fall below the sky general background so your assumption is invalid.

Not necessarily, because sky background alone can be treated as a form of noise. The variation (and standard deviation) of a patch of sky background in an otherwise boring patch of sky tends toward zero with increased total integration time. That means that even an arbitrarily small blip above the background is detectable. Sure it might not have been detected in any one, given image. But such blips are detectable with sufficient total integration time.

 

[BillTre]

I see now that "FAPP" is an acronym. That's the first time I've ever heard that word being used as an acronym. I thought you were shouting an expletive to emphasize your point.

I certainly prefer it when a not obvious acronym is used that it is clearly spelled out in its first usage. It if only used once, lose the acronym, and do the work to make your communication clear.

 

[sophiecentaur]

The variation (and standard deviation) of a patch of sky background in an otherwise boring patch of sky tends toward zero with increased total integration time.

But the stars are stationary and will have their own images after a long exposure - not a grey background.

 

[collinsmark]

This is the numb of the problem . Even the JWST is working in marginal conditions, even if the margins have been changed a lot. If they were operating the scope within those margins then they would be wasting many billions of dollars.
@collinsmark seems to be insisting that the limited model of his maths is all that needs to be considered but, in the limited situation of a 16 bit sensor and the presence of many other interfering low level sources there is a very real limit to how far the maths will follow reality. This is not a problem but it's what limits what we can see.

I've said this before, but I'll try to say it again in different words. The bit depth of the sensor hardware is not the end-all-be-all of the overall bit depth that the camera is capable of achieving. You can increase the effective bit depth to some degree (sometimes to a very significant degree) though the process of stacking multiple subframes.

I think it's best now that I explain with examples. I'll start with some real-world examples and finish with a hypothetical, yet extreme example.

Here's an image of Mars that took with my backyard telescope, a couple of years ago:

The color resolution (bit depth) seems very smooth, does it not? However, the image was taken with an 8 bit sensor! The sensor only had 256 levels. Yet look at the final image. The final image is way, way higher level resolution than 8 bits. How did I do that? Integration. The final image was composed by integrating thousands of individual subframes.

Check out this Hubble Ultra Deep Field image (I didn't take this one; Hubble Space Telescope [HST] did):

HST pointed to very, very unpopulated (unpopulated by nearby stars, nebula, etc.) patch of sky, taking many individual, 20 minute subframes and stacking them to achieve this final image.

One might say, "that's impossible, all those galaxies would just blend into the background sky glow." Well, no, it's not impossible. As HST shows us here, it is possible. As I've said in a previous post, the standard deviation of the background sky glow tends toward zero with increased total integration time, and thus the background sky glow can be subtracted out.

One might also say, "that's impossible. The detail in objects so dim would be less than 1 ADU of the bit-depth of HST's sensor." Sure, some of the detail was less than 1 ADU of the sensor for a single, 20 minute exposure, but HST gained more bit depth by integrating many individual sub-exposures.

The image was taken with 4 different filters. All subexposures were approximately 20 minutes each. For the two shorter wavelength filters, 112 individual subframes were stacked for each filter. For the higher wavelength filters, 288 individual subframes were stacked for each filter.

That makes for a total integration time of over 11 days.

And no, integrating a whole 11 days worth of subframes to produce the "Hubble Ultra Deep Field" image, instead of settling on a single, 20 minute exposure, is not a waste of many billions of dollars.

I'm not just pulling math out of my butt. This is how real-world science is done. Right here.

------------

Now for a hypothetical, extreme example. Consider a 1 bit camera. Each pixel can represent either an on or off.

For the purposes of this example, assume that the camera has a high quantum efficiency and the camera is operating near unity gain, ~1 \mathrm{e^-}/ADU. Also, for this hypothetical example, assume the sensor's read noise is small.

Now, put the camera on a tripod and point it at your favorite sleeping kitten, where there are both bright and dark regions (maybe the cat is sleeping in a ray of sunlight from the window). Adjust the exposure time such that some pixels are consistenly black over many different subframes, some pixlels are consistently white over many different subfames, and the rest of the pixels randomly alternate between black and white from one exposure to the next, to some varying extent from pixel location to pixel location.

Now take and record 255 separate subexposures. You'll find that when analyzing the data, in the really dark regions, some pixels are black in all 255 subframes. But a few pixels are white in 1 of the 255 frames. Moving to a slightly brighter region, there are pixel locations that are white in 2 of the 255 frames. On the really bright regions, some pixels are white in all frames. But some pixels nearby are white in only 254 of the 255 frames. Others nearby are white in only 253 of the frames. In the regions that are neutral brightness, the number of white pixels seem to be consistently around 46 out of the 255 subframes.

Now sum (or average, if you store your data in floating point format) each pixel location over all 255 subframes. Blam! you've got yourself an image with a bit depth of 8 bits. You started with a camera with only 1 bit, and now you have an 8 bit image. There's black, there's white, and 256 levels of grey, total (0 to 255).

Sure, this particular image suffers quite a bit from shot noise, but you can reduce the shot noise by integrating further, and producing an image with a bit depth greater than 8 bits as a byproduct. You'll even find that by doing so, you can eke out more detail in the shadows that were previously, consistently all black.

Isn't math neat?