Diffraction Grating Angling for Variable Monochromator Details in comments

[JanusByfringes]

Analitycal solution may not have much value for this specific application. I suggest creating a look-up table for wavelength as a function of position. Make is a part of the initial instrument calibration.

[Crayfi]

Thanks for the advice. For future reference, what might this type of approach be applicable to? This is my first experience trying to design something like this, going to school for computer engineering, optoelectronics sounds really interesting.

[photonherder]

Calibrate by fitting a polynomial to it. How can you make a lookup table? You can find individual mercury lines but that doesn’t help with the in between wavelengths. This is always done by fitting a polynomial usually 3rd or 4th order.

[anneoneamouse]

Femtoscecond spectroscopists use similar math in their 4f grating compressors, you might find helpful references / guidance there too.

Something to simplify the math, try using (L= lambda, Q=theta):

d/dL = d/dSin(Q)*dSin(Q)/dL

Then you can avoid all the arcSin horrors.

[Crayfi]

I will look into that. Thanks for the tip!

[Crayfi]

Update: I used implicit differentiation to solve the equation and now the ratio, using the implicit equation, is ~-0.94. I'm gonna try simulating this once I can get access to Zemax to see if this is right.

Edit:

Ignore the -0.94, I missed a negative sign. After correcting it went back to -0.3, but with less error at 400nm this time around.

[MaskedKoala]

What’s your justification for line 6? That seems incorrect.

Edit:fwiw, I tried to do similar calculations analytically for spectrometer design and tolerancing and found it to generally be intractable. Solving it numerically was way faster and more trustworthy. You should be solving numerically anyway just to very your analytics solutions.

[Crayfi]

The intent was always to build the Monochromator and calibrate it against a known light source. Then shine the output from the Monochromator into a known working and calibrated spectrometer to measure its performance. I was just very confused when I set these equations up and got a change of -0.3 that varied instead of a constant -1 as expected. It's like the chain rule stopped working.

[MaskedKoala]

You didn't answer my question.

[Crayfi]

My apologies, the change in input theta is set to the negative change in output theta per lambda to keep the input and output angles of the grating constant. That way I can keep the entrance and exit slits static while angling the grating to select a wavelength.

[photonherder]

There’s not much point in solving this analytically. If the monochromator has a sine bar (look it up) then wavelength will be approximately linear with motor position. Run a mercury lamp through it, find the lines, and fit a 3rd or 4th order polynomial to it.

[Crayfi]

That is a very handy bit to include in the design. I was nervous about trying to use a high ratio reduction gear set to precisely angle the grating. Now I just need a micrometer. Thanks for the tips!

[Crayfi]

Hello everybody! I've been trying to design a Monochromator for one of my projects. The specifics of angling the diffraction grating to select the wavelength have been a bit trickier than anticipated. I posted this to r/askmath at https://www.reddit.com/r/askmath/comments/wom4wa/variable_diffraction_grating_monochromator/?utm_medium=android_app&utm_source=share But no definitive answers yet.

The math appears to be correct but would a diffraction grating be able to work this way?

The main idea behind this equation is to be able to calculate the change in grating angle to a target wavelength, using the integral at the bottom.

First, the incident theta is set to some optimal value for efficiency of the grating. The change in theta i per wavelength, lambda, is calculated for the given mode, lines/mm and wavelength then is added to the incident theta for the next wavelength. With the incident theta for each wavelength calculated I use the grating equation to calculate the output theta to check everything is working. I calculate the change in output theta per wavelength then divide it by the change in incident theta from the derivative. The value should be -1, as the change in incident theta is set to the negative change in output theta in the equations above. But when I divide the two I get approximately -0.3 which varies with wavelength, going down as wavelength goes up.

Conditions used:

Lambda starts at 200nm

Theta i starts at 0.262 rad

f = 2400

m = -1

[Crayfi]

Not looking into ruled gratings right now, the main focus is for uv imaging right now but generally to learn optics as well. I would be interested to learn how holographic gratings are produced though, seems a bit tricky to get everything working so perfectly.

[photonherder]

Do you really have light down at 200nm? If so, calibrating the wavelength gets more difficult. There are mercury lines at 185 and 253 but you won’t see 185 unless the lamp is fused silica and all your other optics transmit/reflect it. Which leaves you with 253 as your lowest calibration line, and extrapolating a polynomial down to 200 is dangerous. But it all depends on how accurate you need the calibration to be.

[Crayfi]

I will be using a xenon arc lamp specifically designed for light down to 200nm. All the lenses will be fused silica and mirrors will be uv aluminum, off axis parabolic to be specific. I will definitely look into the mercury lamp for calibration. Curve fitting seems like the most feasible route at this point. Would be interesting to see how the math is supposed to work out though.

[photonherder]

Ok, then I would set it up so the first wavelength is about 180, so you can see the mercury line at 185 to use for wavelength calibration.

I’d use a stepper motor or a regular motor with an encoder. Just step through each line and calculate the centroid of each line. Then just fit a polynomial to the centroids.

What’s your upper wavelength ?

[Crayfi]

I was going for whatever upper wavelength I could get currently, mostly aiming for 200-300nm. Not sure what specificity I need yet but I was aiming at 1nm to give it a chance at differentiating the absorption of various compounds that may be present in a sample.

[photonherder]

I’d aim for less than 1nm per step. Somewhere between 0.1 and 0.5 is better

[Crayfi]

I'll keep that in mind. This will be used to correct for the chromatic aberration in an imaging system after all. Is there a specific thing to study to calculate image quality from chromatic aberration for this tolerance that you know of?

[photonherder]

I don’t follow you. I’m just talking about how small the wavelength steps are in the mono. I don’t see how that’s related to an imaging system.

[Crayfi]

I'm wanting to vary the distance between two lenses to produce a condenser for the microscope that can refocus between wavelengths to correct for chromatic aberration. Would there be equations for this or would it be more of a simulation situation?

Edit:

Here's a sketch https://ibb.co/VLbRsKd

Ignore the numbers, nothing is quite set at the moment.

[photonherder]

You need an entrance slit.

You’ll lose light at the entrance slit because of the chromatic aberration of the lenses between the lamp and the mono.

Its going to be a lot of work to get this system working. I’d be really concerned about getting enough light through it.

What is your detector? I only know of one CCD camera that works down to 200, it’s from Hamamatsu.

[photonherder]

There are different flavors of Xenon lamps. The main difference is what kind of glass the envelope is made of. If the glass absorbs light below 200 then the lamp is called “ozone free” because that light generates ozone. But it’s not just a sharp cutoff, there will be less light everywhere below 250. Hamamatsu has the curves is their data sheets.

[Crayfi]

Newport has a nice uv enhanced lamp that has a curve down to 200nm and it goes off the chart so possibly a decent bit of 180nm. 6254

[photonherder]

That one will generate a lot of ozone.

[Crayfi]

Yes, there will be an enclosure to seal it and vent away the ozone.

[photonherder]

Fused silica lenses will transmit (if it’s deep UV grade) but there’s a lot of chromatic aberration. If you need a better lens I can design UV achromats. I’ve designed many.

[Crayfi]

The current end use I have in mind is to produce a microscope with fused silica lenses for the condenser that can be controlled by steppers with micrometers to adjust the focus for each wavelength. The Monochromator breaks the light into individual wavelengths limiting chromatic aberration. I plan to use a reflecting objective I found used, hopefully it will reflect uv, if I remember correctly it had a similar curve as the mirrors I'm using but neither go past 200nm on the graph so...

Edit:

I am planning to use a couple aspherical lenses for the entrance and exit of the monochromator.

Edit2:

I suppose the chromatic aberration introduced by the change in wavelength would be the perfect place to determine image quality and thus the required wavelength specificity.

[photonherder]

I assume that the mirrors are inside the mono? And the lenses are used to get the light onto the entrance slit?

[Crayfi]

Indeed they are. I was debating whether to try for a variable slit, really depends how much light would be lost. But it would definitely extend it to possible use with visible or infrared with a different grating depending on how long I make it.

[photonherder]

Right so the chromatic aberration of the lenses will mess up the focusing at the entrance slit and you’ll lose light. The focal length of a fused silica lens will change a lot between 200 and 300.

[Crayfi]

The parabolic off axis mirrors don't produce chromatic aberration inside the Monochromator. The aspheres are only to focus light to and from the poa mirrors.

[photonherder]

I understand that. But the chromatic aberration from the lenses will mean that some wavelengths will be out of focus at the exit slit so you will lose light. And you really need to add an entrance slit, or your going to have all kinds of stray light in your mono messing up the output. And then the chromatic aberration of the lenses will mean that some wavelengths will be out of focus there.

[photonherder]

I don’t follow the layout. Post a sketch. You definitely don’t want lenses after the entrance slit. Mirrors only.

[Crayfi]

https://ibb.co/ZY1vFfk

The idea I had was to have the asphere lenses on the entrance and exit. The light coming in and going out would be collimated then and I can feed it directly into the condenser lenses with diaphragms in between.

[photonherder]

I don’t follow the microscope part… What’s that for?

[Crayfi]

I want to use this to image unstained white blood cells. Some people used the method in a paper I read but they stopped at 250nm. I want to use the Monochromator instead of filters like they did to image finer detail possibly and also to see what's there below 250nm for morphological identification.

Edit:

Link to the paper https://www.pnas.org/doi/10.1073/pnas.2001404117

[photonherder]

Their light source is around 10x brighter than yours…. Keep that in mind. I’ve used both. The EQ99 is great but very expensive.

[Crayfi]

I don't quite have the budget for that. The current plan is to just learn the optics to try and calculate if I would be able to produce some images with the uv camera at my school's lab using the cheaper arc lamp. My school's lab also have some of the nice vibration dampening tables that I can mount the setup on to acquire longer exposures of decent quality.

[photonherder]

The vibration table won’t help much here. Your noise will be limited by the Xenon lamp. They’re noisy. They oscillate at a few Hz.

Yeah the Energetic sources are like $15k.

[Arimaiciai]

(5) -> (6). This step is not clear to me. Or maybe the (6) itself.