From 22c9d5db007dabdba14d5a0a02d03421b6bfd1f9 Mon Sep 17 00:00:00 2001
From: eegerferenc <44524311+eegerferenc@users.noreply.github.com>
Date: Fri, 18 Dec 2020 02:13:11 +0100
Subject: [PATCH] Add elaboration on optical integration

---
 partial_coherence.ods | Bin 0 -> 11069 bytes
 references            |   5 ++++
 uLED_optics           |  55 ++++++++++++++++++++++++++++++++++++++++++
 3 files changed, 60 insertions(+)
 create mode 100644 partial_coherence.ods
 create mode 100644 references
 create mode 100644 uLED_optics

diff --git a/partial_coherence.ods b/partial_coherence.ods
new file mode 100644
index 0000000000000000000000000000000000000000..00ad68d7b2098238bf72d849ae9f7954d676623f
GIT binary patch
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diff --git a/references b/references
new file mode 100644
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--- /dev/null
+++ b/references
@@ -0,0 +1,5 @@
+Photolithography Written by: Scotten W. Jones https://pdfslide.net/download/link/photolithography-chapter-by-scotten-w-jones
+Fundamental Principles of Optical Lithography: The Science of Microfabrication.  Chris Mack https://sci-hub.do/downloads/2019-11-19/2e/mack2007.pdf
+https://www.asml.com/-/media/asml/files/products/refurbished-products/pas-5500-100d.pdf
+http://www.smfl.rit.edu/pdf/manual/Manual_ASML_Stepper.pdf
+
diff --git a/uLED_optics b/uLED_optics
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+Considerations for optical integration of uLED-based Self-Emissive Programmable Photomasks into lithography steppers
+
+0. Foreword and assumptions
+
+It has to be noted at first, that the author of this note is not an expert in optics, therefore, all conclusions have to be taken with proper caution.
+
+For all assumptions and calculations behind this document, as usual, the ASML PAS5500 stepper is used as a "vet's horse". Its specs can be found in the indicated references. The display is a hypothetical display with specs based on existing JBD models.
+Center wavelength: 405nm
+Spectral FWHM: 15nm
+Pixel pitch: 2.5um
+
+1. Spectral characteristics
+
+The optical performance of the projection lens is highly dependent on the wavelength, and is designed for monochromatic operation. The design wavelength of the stepper is 365nm, while the center wavelength of the LED array is 405nm. However, InGaN/GaN LEDs can be "tuned" down to 365nm by changing the In/Ga ratio, with conventional LED units in this range are readily available for banknote checking and disinfection lamps.
+
+As per [2] 3.1.6, i-line sources typically have an FWHM of 5-10nm, while the LED array has 15nm, that is higher, but not too much. In addition, i-line steppers typically have chromatically corrected lens, which is assumed here for PAS5500 also. Therefore, it is not predicted that the FWHM of the LED array will cause significant chromatic aberration. However, this is an assumption, it needs to be verified in reality.
+
+2. Partial coherence, annular/off-axis illumination
+
+The most important consequence of using a LED matrix instead of a conventional reticle in an existing stepper is that instead of modulating the light generated by the stepper's illumination system, the "reticle" has to emulate all the functionality of the light source itself, including the provision of coherence and off-axis illumination. The stepper's illuminator will have no effect on the process (except for giving a trigger signal for the LED matrix "shutter"). In the illumination system, there is a relatively complex optics with many adjustable parts (e.g. changeable pupils for annular/two-pole/quadrupole, reticle masking), and there is one of it. In contrast, the LED matrix has simpler micro-optics at each LED consisting at most a microreflector and a microlens and possibly a shadow mask between them, that is not adjustable on the field. Consequentially, the flexibility of illumination will be sacrificed for the flexibility of pattern.
+
+Partial coherence in this environment can be controlled by adjusting the "ray-optics" half-angle of the emitters (that is, the half-angle coming from emitter geometry, not considering diffraction that will occour due to small size of the emitter itself). Providing a sigma value of 0.85 (that is still sufficient for imaging at 350nm, see [4]) is feasible with a half-angle of 5,8°. However, this needs to be verified in practice. (For comparison, standard 3mm LEDs have a half-angle in the range of 5-7°.)
+
+As for off-axis illumination, it is very hard to impossible to integrate the necessary pupil masking and imaging function into the micro-optics due to size limitations. In addition, as we will see later, due to the different behavior of the LED array compared to a conventional reticle, off-axis illumination would not provide the usual benefits it is normally used for.
+
+3. Effect of multiple-source arrangement
+
+When projection is performed using a single source and an intensity-modulated or phase-modulated photomask, the image forming process is governed by the following well-known principle: each opening on the mask behaves as a group of point sources (Huygens principle), and emits light that is "phase-locked" to the central source, with of course some constant delay between them in the function of position. These individual emissions then add up in the pupil plane by amplitude and by phase, forming a diffraction pattern. This diffraction pattern is the Fourier-transform of the original pattern, with components belonging to higher spatial frequencies are placed farther from the optical axis. Then, the finite pupil-size objective lens refocuses these spatial frequency components to the wafer plane, while rejecting components outside of the pupil (determined by the numerical aperture), acting as a low-pass filter. Finally, the refocused components interfere on the wafer plane, creating a low-pass filtered version of the mask image. When the pattern to be imaged is periodic, this process gives rise to discrete diffraction orders, out of which the zero-order and at least the first order must be transmitted to form an image.
+
+The LED matrix case differs from this in two major ways: first, the image is made up of discrete "pixels" instead of continuous patterns. Second, every pixel is illuminated by its own individual LED. This can be imagined as a mask with a pattern of small openings in a rectangular array, with an independent point source (or whatever other kind of source) behind each opening. In this case, each single opening will produce a diffraction pattern similar to a Bessel-function or a sin(x)/x function (or in general: a bell-shaped, ringing amplitude profile), with its width is dictated by the width of the opening. However, since the emissions from individual openings are not phase-locked to each other, these "bells" will not interfere, but will be added by intensity, resulting in either a single bell, or a bell spreaded-out by the size of the LED array (see the note on Köhler), with the amplitude distribution is more or less independent of the actual pattern being displayed, and information on the displayed pattern is primarily conveyed by the phase distribution. This resultant bell will be then low-pass-filtered by the pupil and refocused onto the wafer plane. Here, the components of the individual bells will interfere in themselves to re-create the low-pass-filtered image of the pixel created them, and these pixel images will reproduce the pattern being displayed. This results in many interesting consequences:
+
+- In case of a periodic pattern is being imaged, there will be no discrete diffraction orders, even if the pattern of the LEDs and pattern they are switched on is periodic.
+- Low-pass filtering by finite NA will primarily act on the images of the individual pixels, not on the image they form. If the image of the individual pixel is slightly degraded, it is beneficial, as it will "fill up" the space between pixels, preventing the resultant image of wires, trenches, etc. in the resist from falling apart into an array of unconnected spots. However, spreading of the individual pixel's image beyond the neighboring pixels will degrade resolution.
+- Images of neighboring pixels at the wafer plane will add by intensity, so no interference fringes will be created between them. This is important, as together with the linearity of the objective lens, it implies that if all pixels are imaged correctly, then the pattern displayed by them is also imaged correctly, and if one pixel is imaged correctly, then all pixels are imaged correctly (the latter is a strong statement that is not necessarily true, see the note on Köhler).
+- Since the pupil plane diffraction pattern is dependent only on the individual pixel and there are no discrete diffraction orders, resolution-enhancing techniques based on shifting the 1st order into the pupil (off-axis and annular illumination) will remain ineffective.  
+
+Of course, the lack of discrete diffraction orders does not mean that the resolution of the objective can be violated: as the size of a single pixel decreases below the NA-limited resolution, the pupil-plane Bessel pattern produced by it will spread out of the pupil, and the size of the pupil (therefore the NA) will dominate the transmission. On the wafer plane, the size of the single pixel's image will decrease until it reaches the size of the PSF. From that point on, further reduction in pixel size will result not in the reduction of the pixel image size, but the decrease of its intensity only. Therefore, the minimum size of a pixel's image on the wafer plane is still dominated by the resolution of the optics. However, the distance between images of neighboring pixels will continue to shrink with the LED array's pixel pitch, even beyond the resolution limit, causing neighboring pixel images to be merged. In other words, the lower limit of the MFS that can be imaged is the maximum of the optical (diffraction-limited) resolution and the pixel pitch of the LED array. It shall be noted here that in practice, the resolution of the LED array shall be at least 2-3 times finer than what is required for the given MFS, to prevent excessive ripple along the edges of the projected features. Also, it is recommended to form the micro-reflectors of the LED array to be squares rather than round, for the same reason.
+
+4. Köhler illumination and why it is important
+
+The definition of Köhler illumination can be found in [2] 2.3.7. The problem it is intended to solve (diffraction patterns of features on the mask are displaced on the pupil plane depending on their position on the mask, and therefore falling outside the aperture) is also relevant for the LED matrix case. Generally, in the previous sections, when it is stated that "if one pixel is imaged correctly, then all pixels are imaged correctly", it was assumed that there is aperture in front of all pixels, so imaging does not depend on the spatial position of the given pixel. Actually, this sentence shall be "If the centermost and the four outermost pixels are imaged correctly, then all pixels are imaged correctly". Out of these two conditions, the first one is straightforward: a standalone pixel in the center is the analog of a single contact hole on a conventional photomask. Therefore, if the size of that pixel is larger than the resolution of the objective, imaging shall be fine.
+
+However, the latter condition is more complicated. Let's assume that the micro-optics at all LEDs is the same, and all LEDs are emitting forward, perpendicular to the LED matrix. In this case, the diffraction patterns of the individual pixels at the pupil plane are laterally shifted with the pixel's position. If we consider that the pupil of the objective lens has finite diameter, it can be seen that the farther a pixel is from the center, the less its diffraction "bell" is overlapping with the pupil. Two problems can be seen immediately:
+- The farther a pixel is from the center, the lesser light energy is transmitted to the wafer, the lesser energy is available for image formation at the wafer plane, so the image of that pixel has lower intensity.
+- The farther a pixel is from the center, the less of the pixel's diffraction pattern is transmitted to the wafer, the lesser phase information is available for image reconstruction at the wafer plane, so the image of that pixel is more blurred.
+
+In case of a conventional photomask, this problem is solved by designing the condenser lens in the illumination system in such way to focus the image of the source at the pupil plane of the objective lens, and placing the mask at the pupil plane of the condenser lens, so the illumination at all points of the mask is tilted in such way to shift the diffraction pattern of the given point of the mask to the center of the objective pupil plane. In case of a LED matrix, there is no single condenser lens, as there is no single source. Instead, the function of a condenser can be implemented by making every LED in the array to emit towards a single point, in this case, towards the center of the pupil plane. One way to do so is reducing the pitch of the microlens layer relative to the pixel pitch, thus creating a displacement between the microreflector and the microlens in every pixel that is proportional to the distance from the center of the matrix and is pointing towards the center. By assuming that the sine of the angle of the deflection in emission, thus the displacement of the resulting diffraction pattern at the pupil plane, is proportional to the reflector-lens displacement, this will cause diffraction patterns from all pixels to be centered onto each other. However, designing such solution requires the knowledge of the distance between the reticle and the pupil plane of the objective (which is not necessarily at the physical entrance of the objective).
+
+4. References
+
+[1] Photolithography Written by: Scotten W. Jones https://pdfslide.net/download/link/photolithography-chapter-by-scotten-w-jones
+[2] Fundamental Principles of Optical Lithography: The Science of Microfabrication.  Chris Mack
+[3] https://www.asml.com/-/media/asml/files/products/refurbished-products/pas-5500-100d.pdf
+[4] http://www.smfl.rit.edu/pdf/manual/Manual_ASML_Stepper.pdf
+
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