Photonics Essentials: Detector Operating Modes and Camera Pixels
================================================================

Source
------

Thomas P. Pearsall, *Photonics Essentials: An Introduction with Experiments*
(McGraw-Hill, 2003), Chapter 3, Section 3.3, ``Photodiode Operation: The
Photocurrent Mode and the Photovoltage Mode``, printed pages 47--48.

The two modes in Section 3.3
----------------------------

Illumination shifts the diode current by the photocurrent:

.. math::

   I(V)=I_s
   \left[
      \exp\left(\frac{qV}{nk_BT}\right)-1
   \right]
   -I_{\mathrm{ph}}.

The external circuit selects the operating point on this current-voltage
curve.

.. list-table::
   :header-rows: 1
   :widths: 19 24 24 33

   * - Operation
     - Electrical condition
     - Measured signal
     - Flux response
   * - Photocurrent mode
     - Usually reverse bias; voltage is controlled
     - :math:`I_{\mathrm{ph}}`
     - Approximately linear
   * - Photovoltage mode
     - Open circuit; :math:`I=0`
     - :math:`V_{\mathrm{OC}}`
     - Logarithmic and therefore nonlinear
   * - Camera integration
     - Reset, finite exposure, then readout
     - Stored charge followed by voltage
     - Approximately linear until saturation

In reverse bias, the ideal diode term is nearly constant:

.. math::

   I\approx-I_s-I_{\mathrm{ph}}.

The small current :math:`-I_s` that remains without illumination is the
ideal **dark current**.  Real dark current is larger because thermal
generation, defects, surfaces, and leakage paths also contribute.  The
Chapter 3 curve explorer shows this in **Photodiode I--V family
(Figs. 3.2--3.4)** with four-quadrant axes.

Why photovoltage is nonlinear
-----------------------------

At open circuit, set :math:`I=0` and solve for voltage:

.. math::

   V_{\mathrm{OC}}
   =
   \frac{nk_BT}{q}
   \ln\left(1+\frac{I_{\mathrm{ph}}}{I_s}\right).

The logarithm compresses large changes in photon flux into smaller voltage
changes.  That is useful when circuit simplicity or wide dynamic range is
more important than a faithful measurement of flux.  It can distort a
spectral line shape because relative optical intensities are not preserved.

What mode does a camera pixel use?
----------------------------------

A conventional CMOS or CCD camera pixel is closest to **photocurrent mode**,
but a more precise name is **photocurrent integration** or **charge
accumulation**.  It does not normally wait for the steady-state open-circuit
photovoltage described above.

During an exposure of duration :math:`T_{\mathrm{exp}}`, the photodiode
stores the charge generated by its photocurrent:

.. math::

   Q_{\mathrm{sig}}
   =
   \int_0^{T_{\mathrm{exp}}}I_{\mathrm{ph}}(t)\,dt.

For constant illumination,

.. math::

   Q_{\mathrm{sig}}
   =
   I_{\mathrm{ph}}T_{\mathrm{exp}}
   =
   \eta qN_\gamma,

where :math:`N_\gamma` is the number of incident photons that contribute to
the signal and :math:`\eta` is the quantum efficiency.  Readout electronics
convert charge to a voltage on a sensing capacitance:

.. math::

   \left|\Delta V\right|
   =
   \frac{Q_{\mathrm{sig}}}{C_{\mathrm{sense}}}.

The voltage sign depends on the pixel circuit's node convention.  The
important result is

.. math::

   \boxed{
   \text{raw pixel signal}
   \mathrel{\propto}
   \text{photon flux}
   \times
   \text{exposure time}
   }.

CMOS and CCD readout
--------------------

A common four-transistor CMOS pixel follows this sequence:

#. Reset the sensing node and pinned photodiode.
#. Accumulate photoelectrons in the pinned photodiode during the exposure.
#. Open the transfer gate and move the stored charge to a floating diffusion.
#. Measure the resulting voltage with the pixel source follower and column
   electronics.

Correlated double sampling subtracts a reset measurement from the signal
measurement.  A CCD also integrates a charge packet, but shifts that packet
through the sensor to an output amplifier instead of placing an amplifier in
each pixel.

Where camera response becomes nonlinear
---------------------------------------

The sensor RAW value is designed to remain approximately linear with
collected photons, but not under every condition:

* At **full well**, the pixel cannot store more charge and saturates.
* Junction and sensing capacitances can vary with voltage.
* Charge transfer, source-follower, column amplifier, and ADC errors can add
  nonlinearity.
* Dark current adds charge even without light.
* Gamma correction, tone mapping, and JPEG processing are deliberately
  nonlinear, even when the underlying RAW measurement is nearly linear.

There are also special-purpose exceptions.  Logarithmic high-dynamic-range
pixels deliberately use a response closer to photovoltage operation.  SPAD
pixels use Geiger-mode avalanche events and count photons rather than
integrating ordinary photodiode current.

The concise classification is therefore

.. math::

   \boxed{
   \text{ordinary camera pixel}
   =
   \text{photocurrent generation}
   +
   \text{finite-time charge integration}
   }.

Continue interactively
----------------------

Use :doc:`ch03_interactive_lab` to compare the linear I--V photocurrent
family, logarithmic open-circuit photovoltage, and the Chapter 3 absorption
models.