## Column and Hadamard scan in DMD spectrometers

Ibsen’s PEAK spectrometers are using the Texas Instruments DMD micro-mirror technology to select and detect the intensity of wavelengths from a broadband source. This technology can not only be used for a simple sweep through the wavelength range using the so-called column scan mode but can also enable advanced detection schemes like the Hadamard method. On this page, you can read more about how the column scan and Hadamard scan methods work.

The micro-mirrors used in a DMD based spectrometer are arranged in a two-dimensional array as shown in the picture below. One axis (rows) of the DMD corresponds to wavelengths and the perpendicular axis (columns) corresponds to the spatial distribution of energy along the tall input slit.

If you want to read more about how DMD spectroscopy works in general you can visit our page about this topic here.

## Column scan

In column scan mode the wavelengths are simply scanned one by one by turning a complete column of mirrors to the ON-state one by one. See a simple example of a column scan on the small animation below.

## Hadamard scan

In Hadamard scan mode the mirrors of the DMD are switched ON and OFF in a sequence of specifc patterns. The overall benefit of using this mode is to improve the signal-to-noise ratio (SNR) of your measurements. If the number of pixels in the rows of the DMD is denoted n, then the Hadamard method will improve the SNR by √(n)/2 compared to the simple column scan method.

See a simple example of a Hadamard scan on the small animation below.

In order to understand how the Hadamard scan method works, we can consider a simple sequence of scans based on a 7 pixel wide DMD. Since n=7, we will improve our SNR by √(7)/2 = 1.3 times over the simple column scan mode. In a real situation, n is more likely going to be several hundred thereby providing 10 times improvement of SNR or more.

Each column of the DMD will be illuminated by an intensity I equal to the intensity of the input light at the wavelength hitting that specific column. So, the intensity at wavelength 1 hitting column 1 is I_{1}, the intensity at wavelength 2 hitting column 2 is I_{2}, and so on. The purpose of the spectrometer is to find the intensities I_{1}, I_{2}, … , I_{7}. The total intensity measured by the detector for a given Hadamard pattern is called R_{i}.

In the picture below we show pattern number 1. As can be seen, the mirrors in columns 1, 3, 5, and 7 are ON meaning that the intensity R_{1 }measured at the detector becomes I_{1} + I_{3} + I_{5} + I_{7}.

Below is the 7 different patterns that we are using illustrated along with the respective detector intensities R_{i}.

When we write the 7 equations for the intensity R, they become:

This can conveniently be combined into a matrix equation:

Or

Where ** H** is the Hadamard matrix. In order to solve this for the intensities

**, we need to invert**

__I__**like this:**

__H__Which can be done by standard numerical methods in software. In Ibsen’s PEAK evaluation software the generation and numerical solution of the Hadamard equation decribed above are all done automatically, so you don’t have to worry about that.

## The Hadamard Matrix

In order for a matrix to be a Hadamard matrix it must fulfill the following rule:

- In every two adjacent rows and columns half of the elements must be the same and half must be different.

From the example with 7 columns above we will require that 3 elements are the same and 4 different when comparing adjacent rows and columns. See the examples below for the first two rows and first column. You can check the rest of the matrix yourself to ensure that it is really a Hadamard matrix.