Dynamic Range

We measured the illuminator’s dynamic range in situ in the display of (Lincoln, Blate, et al. 2017). Measurements were made at two locations in the optical path: immediately after the projection lens and in the eye box (exit pupil) of the display (see Figure 7).

Measurements were made using an APDS-9250 digital sensor (Avago 2015). This is the same sensor used in the position-sensitive light sensor component reported in (Lincoln, Blate, et al. 2017); this

49

enabled automation of the measurements by using our existing hardware and software interfaces and APIs. This sensor was used in ambient light sensor mode, which produces readings in lux (adjusted for the wavelength-dependent perceptual sensitivity of the human eye).

The measurements made at the projection lens were used to calibrate the illuminator – i.e., to map DAC codes to binary-weighted intensities, perform color calibration, etc. (see also section 4.4.1 and Table 4.2 of (P. C. Lincoln 2017)).

Theoretical Dynamic Range

If we assume that the LEDs’ current-to-luminosity transfer functions are perfectly linear and that the current is programmed with a perfect 16-bit DAC, then we would expect the illuminator’s dynamic range to be 96.32 dB per color channel. However, we know that the current-to-luminosity transfer functions for the LEDs are not exactly linear. Additionally, the presence of parallel resistor RADJ (discussed in section 3.3.5) creates a non-trivial non-linearity on the low end, as explained below.

The non-linearity on the low end can be understood as follows. Referring to Figure 13, for a given programmed current, we can calculate the effective resistance of the LED: (VOUT-VSENSE)/ILED (Ohm’s law). At small programmed currents, the LED looks like a high-value resistor (≫100 kΩ); this resistance is in parallel with RADJ, which is on the order of 2-15 kΩ. As the programmed current

increases, the LED’s effective resistance drops exponentially, until it’s effective resistance is on the order of a few Ohms50. We thus move non-linearly from a condition where most of the programmed current is flowing through RADJ to a condition where most of the programmed current is flowing through the LED. The effect of a 1 LSB change in programmed current to the output luminosity is thus smaller for lower DAC codes (<28) than it is for larger DAC codes (≫28); equivalently, the slope of the luminosity versus programmed current curve starts out at essentially zero, grows non-linearly, and eventually becomes constant above some programmed current.

50 _{For example, at full-scale current (1A), the voltage drop across the blue LED is about 3 V, so the LED looks like a}

The reduction in the luminosity of DAC code 0x0001 (one LSB), which is our reference level, combined with the non-linearity described above, which essentially gives us higher resolution at lower currents, effectively extends the dynamic range on the low end. We therefore expect to measure a somewhat larger dynamic range than would be projected solely from knowledge of the current source.

Caveats

We note that, for these measurements, the illuminator was set up with RS=2.5 Ω for the green

channel and RS=5 Ω for the red and blue channels; this means that the absolute maximum currents (and

thus intensities) for red and blue were artificially limited. This configuration was later changed, such that all three channels use RS=2.5 Ω, but the data reported here is from the earlier configuration. The reduction

in the maximum intensities in red and blue resulted in a reduction in their respective measured dynamic ranges, as we will see below.

Table 2: Illuminator dynamic range

Channel Exit Optics (dB) Eye box (dB)* _{Exit Optics (bits per pixel) Eye box (bits per pixel)}*

Blue 90.3 49.2 15.0 8.2 Green 114.4 64.9 19.0 10.8 Red 91.2 61.0 15.2 10.1 Total 115.5 70.0 49.1 29.1

* _{The sensor’s resolution is 1 lx, so low luminosities could not be sensed. The images were visible to} the naked eye—light was being emitted but it was below the sensor’s sensitivity. Thus, the

measurements above at the eye box are lower than would be expected because of the sensor’s limited low-end range. Please refer to the text for more details.

Note that the sensors themselves read directly in lux and thus the light readings upon which the data above are based have already been adjusted for wavelength-dependent perceptual sensitivity. Calculation of total dynamic range in Decibels

Recall that, in the present context dynamic range is the ratio of the maximum luminosity (fMAX) to a

reference luminosity (fREF) – typically the luminosity of 1 LSB. The dynamic range, r, in field-scale

Decibels is given by:

𝑟 = 20 ∙ log₁₀(𝑓𝑀𝐴𝑋

𝑓_𝑅𝐸𝐹) (1)

The totals in decibels are calculated via logarithmic addition (Lothaire 2005). Given a set of dynamic ranges R={ri}in field-scale Decibels, the total dynamic range is given by:

𝑇𝑜𝑡𝑎𝑙(𝑅) = 20 ∙ log10( ∑ 10( 𝑟 𝑖 20) 𝑟𝑖∈𝑅

) (2)

That is, we convert each ri from Decibels back to a ratio (the term inside the summation), add them up,

Results

The results of our measurements are summarized in Table 2. The table includes the per-channel dynamic ranges in Decibels into “equivalent bits” for reference51. Equivalent bits totals are the algebraic sums of the per-channel equivalent bits figures52. The formula used to calculate dynamic range totals in Decibels is given in Equation (2), below Table 2.

Dynamic Range at Projection Lens

Measured immediately after the projection lens, the overall dynamic range for the blue and red channels is about 6 dB53 lower than theoretical (96.2 dB, as discussed above). This follows from the aforementioned caveat, i.e., that these channels were operating at half their maximum current. The green channel’s dynamic range, which was not artificially-limited, measured about 18 dB54

higher than theoretical. The overall dynamic range of the illuminator, calculated using Equation (2), is 115.5 dB, about 9.6 dB above theoretical (three 96.3 dB channels=105.8 dB). If the red and blue channels were run at full current and had dynamic ranges similar to the green channel55, then the overall dynamic range of the illuminator would be around 120 dB. Recall that we estimated typical scene dynamic range to be in the range of 100-120 dB. Our implementation falls comfortably within this range.

51 _{Equivalent bits is calculated as the dynamic range in dB divided by 6.02.}

52 _{As an example, if each of the RGB channels had 8 bits of dynamic range, then the total “equivalent bits” would be}

24 bpp.

53 _{6 dB ≅ a multiplicative factor of 2} 54 _{18 dB ≅ a multiplicative factor of 8} 55

The blue and green LED segments use the same InGaN process and have nearly identical current-luminosity characteristic curves; for this reason, we expect that the blue channel’s dynamic range would be similar to that of the green channel. The red LED segment uses a different process and has a somewhat different characteristic curve; that said, doubling the forward current will increase luminous output, though perhaps by a factor somewhat less than two.

Dynamic Range at Eye Box

Measured at the display’s eye box (see Figure 7), the overall dynamic range of the display was considerably smaller than theoretical – 70 dB versus 96.3 dB. Initially, this result was surprising to us, however, there are several factors that explain these results:

 The distance from the projection lens to the eye box is about 150 mm and luminance falls off with the square of distance. Equivalently, the light at the eye box is spread out over a much larger area than immediately after the projection lens. The resulting incident flux per unit area, then, is proportionally smaller.

 The optical path includes a light combiner. The light combiner has an insertion loss of about -1.93 dB, equivalent to a ~20% neutral density filter.

 The signal at the eye box is thus smaller (on the order of -30 to -42 dB) than at the projector’s exit optics. The light sensor’s resolution (LSB) is 1 lx, placing the low-order 5-7 bits (i.e., DAC codes 1 through 16-64, depending on the channel) below the sensor’s noise floor, i.e., the sensor could not resolve the intensities corresponding to the low-order bits and/or distinguish them from noise.

Due to the factors above, we were unable to measure about 30 to 42 dB (5-7 bits) of the

illuminator’s dynamic range at the eye box. Unfortunately, at the time of the writing of this dissertation, the display we used in (Lincoln, Blate, et al. 2017) is not available to us to redo the experiments with a more sensitive sensor and equal values of RS on all color channels. Note, however, because luminosities

that were clearly visible with the naked eye registered below the sensor’s noise floor, we estimate that the dynamic range at the eye box is on the order of 100 to 112 dB, referenced to a lower luminosity (i.e., a reference level on the order of 30-42 dB below that of the measurements at the projection lens). We have no reason to believe that the attenuation in luminance due to distance and the optical combiner would affect the ratio of the maximum and minimum intensities at the eye box.

Conclusion

We estimated the dynamic range of a typical real-life scene at 100-120 dB. Our measurements confirm that the illuminator’s output is within this range. We discuss methods for extending the dynamic range of our illuminator and DBI in general in section 3.5.1.

Figure 16: Illuminator full-scale pulse (control signal and analog response).

A zoomed-in view of the first 2 μs, the interval denoted by the red bracket, is shown in the next figure. The horizontal axis (time base) is 1 μs per division and the vertical scale for channels 1 and 2 is 500 mV per division. The 0 V is at the first vertical division and 2.5 V is at the sixth vertical division. The rising edge of chip select (CS, oscilloscope channel 4, red) corresponds to the COMMIT event. In response, DAC output VSET (oscilloscope channel 1, blue) and current sense VSENSE (oscilloscope channel 2, green) begin to rise and eventually settle at the set value. The short (low) pulse in CS, indicated on the far right, corresponds to the CLEAR event. In response, VSET and VSENSE begin to fall, settling back to zero. The full falling edge is off-screen.

CLEAR

2.5V

Figure 17: Illuminator full-scale transition: zoomed-in view of rising edges.

The red brace at the bottom of the figure corresponds to the similarly-denoted interval in the previous figure. Note that the time base (horizontal scale) is 200 ns per division. The vertical scale for channels 1 and 2 is 500 mV per division. The 0 V is at the first vertical division and 2.5 V is at the sixth vertical division. Observe the latency between the rising edge of CS (oscilloscope channel 4, dark red) and VSET (oscilloscope channel 1, blue), 174 ns (“C4 C1 DL” on the upper right), and between CS and VSENSE (oscilloscope channel 2, green), 360 ns (“C4C2 DL”). (These measurements are rise-to-rise delays between two channels, i.e., the timer starts when the first channel rises above ~50% of its peak-to-peak value and the timer stops when the second channel rises above ~50% of its peak-to-peak value. The red “T” superimposed on oscilloscope channel 4 is the location of the sweep trigger and is not related to these measurements.) Looking carefully, we note that the time from the rising edge of CS (COMMIT) to VSENSE reaching the sixth vertical division (the set-point of 2.5 V) is about 2 to 2.5 horizontal divisions or about 400-500 ns (indicated by the orange arrow).

2.5V