## Abstract

In operational electro-optical systems, infrared focal plane arrays (IR FPA) are integrated in cryocoolers which induce vibrations that may strongly affect their modulation transfer function (MTF). In this paper, we present the MTF measurement of an IR FPA sealed in its cryocooler. The method we use to measure the MTF decorrelates operational constraints and the technological limitations of the IR FPA. The bench is based on the diffraction properties of a continuously self imaging grating (CSIG). The 26 µm pixel size extracted from the MTF measurement is in good agreement with the expected value.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The modulation transfer function (MTF) of a detector is a very sought after figure of merit (FoM) by technologists and system designers alike. It provides the former valuable feedbacks on the quality of the process (etching steepness or depth for instance) and intrinsic properties of the material such as electronic diffusion length; the latter will harness the MTF to modelize pixel filtering and compare it to the optical filtering, in order to optimize the system overall performance.

There are quite many challenges when attempting to measure a detector MTF in the infrared domain (IR). First, the trend of smaller pixels means the diffraction limit and deconvolution become major concerns. As a result, constraints are harsher on any projection optics used in measurement benches. They have to reconcile low chromatism and high aperture while operating within a relatively large spectral band. Such attributes are difficult to achieve at the same time and likely come at a prohibitive cost. Spectral filters and reduced aperture can be used to lower the requirements on optics in the measurement bench but the Signal to Noise Ratio (SNR) is lowered as well. Second, the fact that most high performance IR technologies are cooled limits the physical access to the focal plane array (FPA). Alignment and focalization are then more delicate in the setup. Third, another issue to address is that operational IR focal plane arrays are often packaged as Integrated Detector Dewar Cooler Assembly (IDDCA), so the f-number is set by the cold shield. Moreover, IDDCA configuration implies the use of a cryocooler that introduces vibrations on the FPA. It translates into motion blur that affects the detector MTF one wants to measure. These vibrations have to be dampened or taken into account when attempting to measure the detector MTF. Our solution to address the three aforementioned issues is a MTF measurement bench exploiting the self-imaging panchromatic property of achromatic gratings to project a high resolution periodic pattern on the FPA. It requires neither high quality projection optics nor demanding alignments for the detector MTF to be measured.

The type-II superlattice (T2SL) FPA has been chosen to illustrate our MTF measurement technique since no such characterization has been made so far on the T2SL technology integrated in a cryocooler, that is to say in an operational packaging. Indeed T2SL have shown promising prospects thus far [1–8]. Thus, it is a suitable candidate for high performance infrared imaging applications in which the MTF measurement is demanded.

In this paper we present the MTF measurements of a 30 µm pitch 320×256 MWIR type-II superlattice infrared Focal Plane Array sealed in a cryocooler. We first present the bench principle and how it is designed. The second part of this paper is dedicated to the vibration study, as they can strongly affect the MTF. The use of accelerometers combined with low integration time measurements allows to distinguish the vibration effect from other contributions on the measured MTF.

## 2. Experimental setup

#### 2.1. Principle of measurement

The keystone of the bench presented in this paper is the Continuously Self Imaging Grating (CSIG). The CSIG diffracts a set number N of orders (usually CSIGs are engineered to diffract 12, 24 or 48 orders). The fact that these orders are designed to be localized on the Montgomery circle [9] grants the achromatic and propagation-invariant property to the projected pattern [10–12]. It completely deletes the need of projection optics (and their constraints along) and neither alignment nor focusing distance are as critical anymore which is quite convenient when measuring on an IDDCA. The resulting MTF measurement is sparse, as we tailored which frequencies are generated by the CSIG so we know which frequencies to look for in the image acquired by the IRFPA. This is a global method which assumes identical pixels across the FPA like other existing MTF measurement methods relying on interference, random pattern and speckle [13–16]. However, this hypothesis is acceptable because T2SL are very promising in term of uniformity. The average MTF and spatial response can be obtained in a single measurement. As a bonus, the pixel size can be directly extracted from the measurements. This key parameter is essential for system designers as well as technologists.

The bench is made of a blackbody source, a pinhole, a collimator and the CSIG as described in Fig. 1. To project the intended frequencies on the FPA, the CSIG has to be illuminated in parallel beams [12], hence the need of a collimator. The size of the projected image of the blackbody on the CSIG has a filtering impact so a pinhole is there to ensure that the spatial frequencies have not been filtered out before even reaching the CSIG. The projected pattern of the CSIG is known by design [17] by specifying its period *a*_{0} and its scaling factor *η*^{2} which defines the projected frequencies. The raw image obtained by the FPA will be searched for amplitudes corresponding to the spatial frequencies contained in the projected pattern, which allows a 2D global MTF measurement as explained in details in [10].

The total measured MTF can be written as the product of the MTFs assigned for each contributor as illustrated on Fig. 2. We can first split the global MTF into 3 main contributors: the MTF of the CSIG, the MTF bound to the bench itself, the MTF due to the full IDDCA, with *ν _{x},ν_{y}* being the spatial frequencies along x and y axes of the detector.

The CSIG generates a sparse MTF. Here it is made of 288 frequencies, which can be written as a sum of Dirac functions $\delta ({v}_{{x}_{i}},{v}_{{y}_{i}})$ assigned with a weight *c _{i}*:

The MTF* _{bench}* features a MTF

*as the pinhole acts as a low pass filter defined by its diameter*

_{pinhole}*ϕ*and modelized by an Airy disk function A, and a MTF

_{pinhole}*as well (whose focal length f will also define another cutoff frequency).*

_{collimator}The MTF* _{IDDCA}* includes the MTF

*we want to measure. Unfortunately, the MTF*

_{detector}*is also part of the MTF*

_{vibrations}*: it is modelized by a 2D gaussian function G with 2 characteristic lengths*

_{IDDCA}*δ*and

_{x}*δ*.

_{y}Furthermore, we can see MTF* _{detector}* as a product of MTFs. First, the MTF

*(modelized by a 2D sinc and 2 characteristic lengths ${a}_{x}^{pix}$ and ${a}_{y}^{pix}$) to describe the perfect behavior of an ideal rectangular pixel. The pixel is supposed to be rectangular, which allows the associated MTF as a 2D sinc function which cutoff frequency depends on ${a}_{x}^{pix}$ and ${a}_{y}^{pix}$ respectively. This assumption is critical and the reasons will be explained whenever this hypothesis is needed throughout the paper. Second, a MTF*

_{idealpixel}*(modelized by a 2D gaussian and parameters*

_{tech}*d*and

_{x}*d*), that describes effects such as electronic diffusion that alter the MTF

_{y}*. The MTF*

_{detector}*assumes that every effect altering the MTF*

_{tech}*can be encompassed and described by a single MTF with independent parameters*

_{idealpixel}*d*and

_{x}*d*.

_{y}An accelerometer is put on the cryocooler compressor as we can assume it is the source of the vibrations and two more are put on the bench and on the edge of the cooler assembly (see Fig. 1). Accelerometer measurements will be commented in part 3.

#### 2.2. Design of the bench

Measurements were made using a Smaract mechanical stage, a CI Systems collimator (f=762mm) and a 1473K blackbody. The IDDCA tested here is a MWIR 320×256 T2SL FPA with 30µm pitch, made by IRnova. The FPA is a InAs/GaSb MWIR type-II superlattice operating at 80K. Integration time *t _{i}* has been set to 6ms to ensure the measured pattern is well contrasted. The number of images has been set to 300 (with a framerate of 60Hz) in order to have enough images to improve the signal to noise ratio while maintaining a short measurement time.

The whole purpose of the bench designing phase is to ensure that MTF* _{detector}* is far more significant compared to MTF

*, MTF*

_{pinhole}*and, if possible, MTF*

_{collimator}*(respectively labeled as 1, 3, 4 and 2 in Fig. 2). As such, the collimator has been chosen so that*

_{vibration}*MTF*(

_{collimator}*ν*) is close to 1.

Three points demand special attention to make sure the MTF* _{detector}* is the main contributor of the MTF we will measure.

- - First, the choice of the CSIG itself is important as it defines by its period a
_{0}and scaling factor*η*^{2}the maximum frequency excited by the grating:The CSIG we chose is an amplitude grating, without antireflection coating. It has a

*η*^{2}of 650 (24 orders) and a period of 1mm, which gives*ν*= 51_{max}*mm*^{−1}. - - Second, we have to mitigate spatial filtering effect of the pinhole of diameter
*ϕ*by having its cutoff frequency as far as possible. This cutoff frequency is defined by the diameter of its image_{pinhole}*ϕ*′ on the FPA: where: with f the focal length of the collimator and d the CSIG-FPA distance. As the diameter of the pinhole diminishes, the cutoff frequency increases, but this also results in a reduced incoming flux on the CSIG, which does not favor the signal-to-noise ratio. We took a 500µm pinhole and ensured that the FPA still could detect the pattern. Assuming that*d*< 30*mm*and knowing that f=762mm, we have*v*>62_{pinhole}*mm*^{−1}. The pinhole is guaranteed not to be the limiting factor in this setup. - Last, the cryocooler creates and transmits vibrations to the FPA. Unfortunately, the signature of such vibrations is similar to technological effects and will have to be evaluated a posteriori (see Results and discussion).

Caution is needed on another point. On the one hand, the CSIG has to excite and project on the FPA frequencies beyond the cutoff frequency of the pixel. We chose a CSIG whose max frequency is as high as *ν _{max}* =51

*mm*

^{−1}, guaranteeing to be able to measure a pixel size greater than ${a}_{min}^{pix}=\frac{1}{{v}_{max}}=20\text{\mu m}$. This minimum size hypothesis is enabled precisely because we assume that the pixel is rectangular (indeed, the cutoff frequency is not guaranteed to be equal to the pixel size outside this hypothesis). On the other hand, the FPA has to properly sample the projected pattern and the FPA cannot sample properly frequencies beyond its Nyquist frequency of 16, 67

*mm*

^{−1}$\left(=\frac{1}{2pitc{h}^{pix}}\right)$ under normal conditions. To meet both requirements, we chose to oversample the image by microscanning, in order to push the Nyquist frequency of the equivalent pitch beyond

*ν*. A 5× oversampling with a 6µm step has been chosen, increasing the image resolution thus extending the Nyquist frequency to 83,3

_{max}*mm*

^{−1}.

Different microscanning patterns have been tested. The spiral pattern as shown on the left hand side of Fig. 3 was most faithfully reproduced by the mechanical stage, as probably this movement pattern helps mitigate the hysteris observed on other patterns.

#### 2.3. Results

The measured MTF is displayed in Fig. 4. The MTF is bi-dimensional. However, because we chose to represent the MTF as a function of $v=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}$, dispersion is expected. Indeed the diagonal length will also act as a cutoff frequency. That is to say the border length (corresponding to a cutoff frequency) will be different based on whether we look along x (or y) direction or the diagonal (x+y) direction (since we assumed that the pixel is rectangular). For this reason, models along x,y and x+y direction should be drawn. Since MTF* _{vibration}* and MTF

*have the same gaussian signature and MTF*

_{diffusion}*(*

_{collimator}*ν*) being close to 1, comparison is made with the following model simplified from Eq. (1) with G being the gaussian function which comprises here vibration and technological effects (see Fig. 2) :

*δ*and

_{x}*δ*are 2 adjustable parameters. Best agreement with the data is found for

_{y}*δ*=

_{x}*δ*= 10µm. Gaussian functions do not have zeros, so only two contributions may cancel the MTF: the pixel MTF and the pinhole MTF. The pinhole diameter has been chosen so that the cutoff is much further than the pixel cutoff so that the first cancellation of the measured MTF is due to the pixel MTF. It is important to note that the first zero of the pixel MTF is at $v=\frac{1}{{a}^{pix}}$ (again, the formula is valid because the pixel is supposed to be rectangular). It also happens to be $v=\frac{1}{pitc{h}^{pix}}$ only when the fill factor (which we define as the ratio between the actual pixel size and the pitch) of the FPA is equal to 1. It is worth noting that the pixel size is already extractable regardless of the other contributions of the global MTF. However, it may be difficult to estimate the first zero of the experimental measurement of the modulus of the MTF. It can be more reliable to look for a change of sign than just a zero to determine the pixel size. Therefore the argument of the IDDCA MTF is displayed on Figure 4 and we estimate

_{y}*ν*= 39

_{pixel}*mm*

^{−1}that is to say ${a}_{x}^{pix}={a}_{y}^{pix}=26\text{\mu m}$ for a 30 µm pitch. The 26µm square pixel size matches with the expected value. It means that the bench can be used as a non destructive technique to estimate the pixel size of an operational FPA.

## 3. Discriminating technological effects and vibrations

Up to now, we measured MTF* _{IDDC}A*. We still need to distinguish MTF

*from MTF*

_{vibration}*. Vibrations whose frequency is lower than $v=\frac{1}{2{t}_{i}}=83\phantom{\rule{0.2em}{0ex}}\text{Hz}$ can be seen as translations along x and y axes from an image to another. Because the frequencies excited by the CSIG are known, it is possible to view these same vibrations as phase shifts in the Fourier domain. Namely, be*

_{tech}*Img*a given image,

*TF*(

*Img*) its Fourier decomposition,

*x*

_{0}and

*y*

_{0}the translation in x and y axes, we have:

By making use of the Fourier decomposition on each of the 300 images and fitting the exponential in Eq. (12), it is possible to extract a (*x*_{0},*y*_{0}) translation for each position and estimate the typical vibrations amplitudes on x and y axes.

As shown in fig. 5, we estimate the vibration amplitude as the Full Width Half Maximum (FWHM) using Kernel Density estimations. An average amplitude *δ _{x}* = 1.2 µm is found in the image on the horizontal axis and 1.2 µm amplitude

*δ*on the vertical axis. The same pattern is found on each position, which is coherent with the fact that vibrations affects the whole FPA.

_{y}#### 3.1. Vibrations frequency range and their effects

Vibrations induce a spread on the position occupied by the FPA. Therefore they will affect the MTF measurement depending on the frequencies they contain. We will assume for clarity purpose that the vibration is a 1D sinusoid as depicted in Fig. 6 but what is explained still remains true in 2D. Four cases can be highlighted depending on the vibration frequency *f _{vib}*:

- - Case 0 is the ideal case where vibrations are non existent. Every single image is sharp and there is little dispersion in the returned position from an image to another.
- - Case 1: ${f}_{vib}\xab\frac{1}{2{t}_{i}}$. FPA barely moves during
*t*so single images are sharp but shifts will appear as averaging requires multiple images. Measuring a faithful MTF is achievable provided data processing (Shift-and-adding is a remedy in this case)._{i} - - Case 2: ${f}_{vib}\xbb\frac{1}{2{t}_{i}}$, the FPA on contrary describes the full periodic vibration pattern during the integration time. Single images are blurry but the dispersion in the returned position of each blurry image will be low, misleading into thinking that there is less vibrations than there actually is. The MTF is possibly distorted. Additional efforts are required to determine if the amplitude of the vibration is high enough to strongly distort the MTF.
- - Case 3 is when
*f*can compare to $\frac{1}{2{t}_{i}}$. The drawbacks of the two previous cases overlap. Some images are sharp, some are not. One can consider sorting out and using only the sharpest images (a.k.a. Lucky imaging) to return in case 1 but in our case, we will look for an upper bound of the impact on the MTF._{vib}

For each case, the periodic vibration is compared to *t _{i}* and

*t*on the left; on the right, the resulting effect on a single image and upon acquiring multiple images is displayed.

_{frame}#### 3.2. Accelerometer measurements

Given the importance of which vibrations are present in our analysis, we decided to implement accelerometer measurements using 3 crossbow CXL04GP3 accelerometers (200Hz bandwidth).

The first one is put on the compressor of the cryocooler as we assume it is the source of vibration. Besides, the second one is put on the measurement bench (on the breadboard), in order to verify that the lab environment does not transmit other vibrations. The last one is put near the edge of the IDDCA, (geographically) close to the FPA position. We ensured the bench does not receive any vibration from the environment before starting measurements by comparing the data acquired from the accelerometer bound to the cryocooler and the one on the bench. On the breadboard data, we can see a 50Hz peak that does not appear on the compressor data. This 50Hz on z axis stems from the fan air cooling system imbedded in the power supplies operating nearby the bench. The fans induced a 3D 50Hz vibration that has been mitigated in x and y axes because of the mass of the breadboard. Moreover, data from the edge of the IDDCA proves that there is no propagation in the outer mechanical structure of the IDDCA (see Fig. 7).

The data shows that there is no low frequency vibration but there is a significant peak at 31Hz. This frequency is neither far higher nor far lower than $\frac{1}{2{t}_{i}}$: we are typically in case 3. This peak shifts as we change the cryocooler operating temperature, meaning that this particular 31Hz frequency is due to this cryocooler operating at 80K.

### Evaluation of the 31Hz frequency vibration on the MTF

We need to evaluate in our configuration the impact of the vibration at 31Hz on the MTF data. For the sake of clarity, the FPA position can be modelized as a 1D sinusoid function of time pos(t), but this remains true in 2D (with 2 separate 1D sinusoids).

where t is the time,*f*the vibration frequency, A the half amplitude of the sinusoid. Maximum displacement

_{vib}*d*is then given by finding the maximum of pos(t) derivative and multiplying by

_{max}*t*:

_{i}*d*represents the upper boundary of the shift that can occur during

_{max}*t*. As it represents the worst case scenario in term of blur within an image, we will use Eq. (14) to evaluate the worst impact on the MTF this 31Hz vibration can have. While the amplitude of those vibrations on the FPA cannot be retrieved with the sole accelerometer measurement, it is still possible to have a good estimation of the upper boundary of the blur. As we noted in the right panel of Fig. 5, we estimated

_{i}*δ*,

_{x}*δ*=1.2 µm amplitude for the vibration, so this is why we can safely assume that 1.2 µm represents a good estimation of A. Applied with A=1.2 µm,

_{y}*t*=6ms and

_{i}*f*=31Hz we find

_{vib}*d*=1.4 µm on both axes here.

_{max}Let’s suppose that the FPA can move up to 1.4 µm during *t _{i}* within a single image (case 3). The impact of the shift within the elementary image can be seen as convoluting the pixel with a 1.4 µm wide window (or multiplying by a sinc in Fourier space). Considering

*MTF*is the MTF of the vibrating pixel,

_{vibpix}*MTF*the MTF due to the pixel only and

_{pixel}*MTF*the MTF due to vibration, we have in Fourier space:

_{vib}At Nyquist frequency $\left({v}_{Nyq}=\frac{1}{2pitc{h}^{pix}}\right)$, with *pitch ^{pix}* =30 μm we have MTF

*(*

_{vib}*v*) = 0:997, which is very close to 1.

_{Nyq}Let’s assume the other extreme scenario that consecutive images can shift by 2.4 µm (2 × *A*) (case 1). This 2.4 µm shift is totally similar to a line of sight stabilization residual in an electro-optical system. This can be modelized as :

*σ*=2.4μm.

_{stab}Numerical application at *v _{Nyq}* yields

*MTF*(

_{stab}*v*) = 0.97. Compared to the pixel filtering effect at Nyquist frequency (

_{x}*sinc*(

*v*)=0.72), we see that such a shift will not have sizeable influence on the MTF.

_{Nyq}We can conclude that the 31Hz vibration has been evidenced as non harmful on our MTF measurement. Only low and medium frequency vibrations have been studied thus far (cases 1 and 3 in Fig. 6). The high frequency vibrations (case 2) are integrated during *t _{i}* and may have significant effect on the MTF. Since the accelerometer is not perfectly adapted to detect the high frequencies due to its bandwidth (200Hz), this case 2 warrants dedicated studies and measurements that are presented hereafter.

#### 3.3. Low integration time complementary measurements

So far, we know there is a 31Hz vibration which does not strongly affect the MTF. We need now to ensure that no high frequency vibration has been integrated during *t _{i}*. Our approach is to estimate the

*δ*,

_{x}*δ*parameters as a function of

_{y}*t*. We can see

_{i}*δ*,

_{x}*δ*as the standard deviation of the position on the FPA during

_{y}*t*. It is important to remember that the SNR also decreases as

_{i}*t*is shorter. As demonstrated in [18] for a backgroundless scene (but can be extended to our case as adding a constant background does not alter the result),

_{i}*δ*and

_{x}*δ*vary as $\frac{1}{\sqrt{{t}_{i}}}$ when limited by photon noise. Our bench is limited by photon noise, as the background signal (the blackbody) is predominant and has to be subtracted before extracting the MTF. Should there be any high frequency vibrations, they should affect

_{y}*δ*and

_{x}*δ*estimations, thus no longer making them as function of $\frac{1}{\sqrt{{t}_{i}}}$ alone.

_{y}For that purpose, we estimated the vibration amplitude *δ _{x}* and

*δ*while changing the integration time, ranging from 16ms to 0.2ms, thus increasing the bandwidth of our measurement. We did not make any modification on the MTF measurement bench, meaning that the setup has not changed in terms of vibrations. Here, there is no microscanning as we are not interested in measuring the MTF but only in how well the positions are evaluated. This is equivalent to only measuring 1 of the 25 positions in the previous paragraph, and this time we take 900 images instead of 300. The vibration amplitudes

_{y}*δ*and

_{x}*δ*are now a function of

_{y}*t*and results are found in Fig 8. Both have been fitted with a function proportional to $F(t)=\frac{1}{\sqrt{{t}_{i}}}$.

_{i}Since there is good agreement with the experimental data we can tell that there are no high frequency vibrations impacting the MTF bench at the time of measurement. By combining accelerometer measurements and low *t _{i}* measurements we showed that the reported measurements are void of vibrations effects whose frequencies are up to $\frac{1}{2\times 0.2ms}=2.5\phantom{\rule{0.2em}{0ex}}\text{kHz}$

## 4. Conclusion

In this work, we report on a MTF measurement bench using the diffraction properties of a Continuously Self Imaging Grating (CSIG). This bench is used for the MTF measurement of a type-II superlattice infrared detector in operational packaging. The bench does not require high-end optics to project the pattern with known spatial frequencies. It also significantly lowers alignment and focusing requirements, a convenience especially when the detector needs to be cooled. At last, this study allows to tell vibrations effects and technologial contribution apart, making the bench a strong contender for MTF measurements on infrared FPA. The IDDCA configuration of the characterized 30µm pitch type-II InAs/GaSb superlattice FPA required to lead dedicated additional vibrations studies along the MTF measurement. Finally, the CSIG bench allows to determine the pixel size in the same process as well. In the particular case of the T2SL FPA, a 26µm pixel size has been measured and is relevant with the expected value, giving access to a key detector parameter.

## Funding

Labex FOCUS (ANR-11-LABX-0013).

## Acknowledgments

We would like to acknowledge the fruitful discussions with IRnova. We also thank the Labex Focus for its financial support.

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