The physical determinacy of signal-to-noise ratio by camera sensor quantum efficiency (QE) and full well capacity (FWC)

1. Definition of the two ultimate parameters of semiconductor photoelectric conversion

Each pixel of a digital camera is essentially a miniature photocapacitor (Photodiode):

• Quantum Efficiency (QE) - light capture rate:
• Definition: The physical percentage of the total number of photons irradiated on the photosensitive surface of the pixel that can be absorbed by the silicon wafer and successfully excite free photoelectrons (Photoelectrons);
• The QE of traditional front-illuminated (FSI) CCD/CMOS sensors is only about 35% to 45% (more than half of the photons are reflected or blocked by surface metal wiring and absorbed losses);
• Modern advanced back-illuminated (BSI) and stacked sensors adopt inverted architecture and anti-reflective microlens coating, and their peak QE even exceeds 75% to 85%, almost approaching the theoretical physical limit of single photon detection!
• Full Well Capacity (FWC) - electronic reservoir volume:
• Definition: The maximum number of free electrons that a single pixel photodiode can collect before voltage saturation overflow (Clipping / Overexposure) occurs (unit: (e^-));
• Small size pixels (such as 0.8(\mu\text{m}) pixels on mobile phones): FWC often only has a pitiful 5,000 to 8,000 electrons;
• Advanced full-frame bodies (such as Sony A7S3 12 million large pixels, pixel pitch 8.4(\mu\text{m})): FWC up to an astonishing 90,000 to 120,000 electrons!

2. Derivation of strict physical formulas for signal-to-noise ratio and dynamic range

Light itself is not a continuous fluid, but a discrete rain of photons. According to the Poisson distribution of quantum optics, pure light naturally contains Photon Shot Noise:

• Maximum signal-to-noise ratio formula (Max SNR):
  In the strong light highlight area, the readout noise is negligible, and the ultimate signal-to-noise ratio of the picture is completely determined by the total number of electrons collected (N):
  [ \text{SNR}_{\max} = \frac{N}{\sqrt{N}} = \sqrt{N} ]
• If FWC is 10,000 (e^-), the maximum signal-to-noise ratio is (\sqrt{10000} = 100) (about 40 dB);
• If the FWC is 100,000 (e^-), the maximum signal-to-noise ratio jumps to (\sqrt{100000} \approx 316) (about 50 dB)! The signal-to-noise ratio has been improved by more than 3 times, and the color purity and grayscale fineness have achieved a qualitative leap!
• Theoretical limit dynamic range formula of the sensor (Dynamic Range, DR):
  [ \text{DR} = 20 \log_{10} \left( \frac{\text{FWC}}{\text{Read Noise}} \right) \quad \text{Or expressed in gears:} \quad \text{DR (EV)} = \log_2 \left( \frac{\text{FWC}}{\text{Read Noise}} \right) ]
  The larger the full well capacity, the higher the upper limit of the highlight information that the sensor can accommodate. When exposing to the right, overexposure will never occur, and the upper limit of the dynamic range is directly locked by the physical lock!

3. Engineering Dialectics of Large Pixels (A7S Series) vs. High Pixels (A7R/Z8 Series)

Many photographers are confused: Should they buy 61 million high pixels or 12 million large pixels?

• Single pixel perspective (Pixel-Level):
  The large pixels have unparalleled single-point FWC and ultimate high-sensitivity purity, and the noise is extremely slight under 100% enlargement of screenshots;
• Whole-frame equal-frame perspective (Print-Level/Image-Level):
• Although the 61-megapixel body has a small single-pixel well capacity (about 25,000 (e^-), the signal-to-noise ratio of a single pixel is low);
• But in the same photo, it has a full 5 times as many pixels co-sampling the same scene!
• When you downsample a 61 million photo to a size of 12 million, the discrete photon shot noise of five adjacent pixels will cancel each other out in mathematical averaging ((\sqrt{5}) noise reduction factor). The overall dynamic range and total light signal-to-noise ratio of the entire picture are almost equal to those of large-pixel bodies, while retaining several times the ultimate spatial resolution!
• Conclusion:
• Static extreme scenery, commercial commercial printing: firmly choose the high pixel body, and enjoy the signal-to-noise ratio bonus and terrifying analysis after thumbnails;
• Pure low-light video recording, extremely high frame rate slow motion (no thumbnails required): Choose the large pixel body to enjoy ultra-low readout latency and natively large FWC.

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