An investigation of stair mode in optical phased arrays

An investigation of stair mode in optical phased arrays using tiled apertures Mark F. Spencera and Milo W. Hyde IVb Department of Engineering Physics, bDepartment of Electrical and Computer Engineering,

a

Air Force Institute of Technology, 2950 Hobson Way, Wright-Patterson AFB, OH 45433-7765 ABSTRACT With an optical phased array, the individual phases of a multi-fiber laser source can be manipulated by exploiting highbandwidth phase loops to correct for aero-optical flow over the turret and free-stream atmospheric effects along the line of sight; however, rough surface scatter through laser-target interaction adds the additional constraints of speckle and depolarizing effects. In particular, speckle phenomena can cause unobservable modes to arise in the beam control system of optical phased arrays. One such unobservable mode is termed stair mode and is appropriately identified by a stair-step pattern of piston phase across the individual subapertures that comprise a tiled aperture. This paper investigates the effects of stair mode using wave-optics simulations. To represent different array fill factors in the source plane, both seven and 19 element hexagonal close-packed tiled apertures are used in the simulations along with both Gaussian and flat-top outgoing beamlets. Peak Strehl ratio and power in the bucket are calculated in the target plane for all simulation setups and are then averaged for multiple random realizations of stair mode step sizes. In addition, the stair mode target irradiance patterns are imaged with cameras which have decreasing aperture stop diameters. Initial results show that low resolution imaging conditions, i.e. an aperture stop on the order of a subaperture diameter, makes it difficult to distinguish between different realizations of stair mode using a separate camera sensor. Keywords: stair mode, array tilt, optical phased arrays, unobservable modes, fiber lasers, adaptive optics

1. INTRODUCTION In a recent publication, Tyler identified unobservable modes as being part of the beam control system of an optical phased array performing target-based phasing1. We define one such unobservable mode as stair mode. Past research efforts may refer to stair mode as array tilt; however, because this particular unobservable mode resembles a stair-step pattern of piston phase across a tiled aperture, we appropriately identify it as stair mode. When performing target-based phasing, the beamlets from an optical phased array actively illuminate an extended target. As such, three classes of aberrations result and in some cases need to be sensed and corrected for all along the propagation paths from source to target and back. These aberrations include contributions from individual telescopes that comprise the tiled aperture2,3,4, atmospheric effects such as turbulence and thermal blooming5,6,7, and rough surface scatter through laser target interaction which results in speckle phenomena8,9,10. A practical beam control system for an optical phased array corrects for the aberrations resulting from telescope and atmospheric disturbances. In turn, aberrations resulting from speckle phenomena need to be separated from the control signals, as they only serve as a noise component. Since unobservable modes result from target-based phasing, this separation is unfortunately not a possibility. The unobservable modes cannot be measured; thus, their effects cannot be corrected for. As a result, all unobservable modes need to be projected out of the estimated piston commands while performing target based phasing. Herein lays the limitations found in current phasing approaches which fail in the presence of an extended target11,12,13,14,15,16,17,18.



[email protected]; phone (937) 255-3636 x6155; fax (937) 656-6000; www.afit.edu/en/ Unconventional Imaging and Wavefront Sensing 2012, edited by Jean J. Dolne, Thomas J. Karr, Victor L. Gamiz, Proc. of SPIE Vol. 8520, 852006 · © 2012 SPIE CCC code: 0277-786/12/$18 · doi: 10.1117/12.942634 Proc. of SPIE Vol. 8520 852006-1

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Stair mode ultimately manifests itself as a global aberration from the contributions of atmospheric disturbances and speckle phenomena1. The resulting stair-step pattern of piston phase across a tiled aperture is a relative 2 phase ambiguity that causes energy to leak into the side lobes of the far-field irradiance pattern. This is an undesired effect for tactical engagement scenarios where engineering constraints add up quickly in an error-budget analysis3,19,20. In future optical phased array designs, a separate sensor will need to be employed to properly identify the effects of stair mode. A stair mode imager might serve such a purpose; however, the primary aperture diameter of the stair mode imager needs to be on the order of a single subaperture diameter if it is going to be used within a conformal beam director2,5,15,17,18. In this paper, we present results from a series of wave-optics experiments. Throughout the analysis, we assume that the beamlets have perfectly aligned polarizations and that stair mode is the only phase error present in the phase-locked loop. We make these simplifications so that we may properly investigate the effects of stair mode in optical phased arrays using tiled apertures. Section two of this paper includes analysis for the modeling of stair mode, while section three provides results for a performance study and a stair mode imager study. We conclude this paper in section four with a discussion on future work.

2. MODEL SETUP, EXPLORATION, AND VALIDATION This section discusses the setup, exploration, and validation needed for a series of wave-optics simulations which identify the effects of stair mode in an optical phased array comprised of a tiled aperture. Throughout the analysis, we use the principles taught in a recent publication by Schmidt21. 2.1 Model setup As shown in Fig. 1, we wish to model optical phased arrays comprised of tiled apertures in hexagonal close-packed (HCP) configurations with the effects of stair mode. For this purpose, we establish the optical disturbance in the source plane, U  x, y, z  0   U 0 ( x, y ) , using the following relationship21:  k  U 0  x, y   t A  x, y  P  x, y  exp  i x2  y 2  ,   2Z 

(1)

where t A  x, y  is the aperture transmittance function, and P  x, y  is the real-valued pupil function which accounts for apodization. We include the quadratic phase factor with angular wavenumber k , in Eq. (1), so that the beamlets within the tiled aperture individually focus to the same spot in the target and or image plane at z  Z . 4S x

y

3S x 2S x

W0

3

12 13

2

5 15

 3

9

0

4 14

3W0

10

11

6

9,1,17 2, 6

10, 0,16 3,5  S x 11, 4,15 2 S x

7

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x

18

17

D0

D0

8

1

16

Sx

7

8,18

d0

11,10,9 2 S y 12,3, 2,8 S y

D0 S y 2 S y

13, 4, 0,1, 7 14,5, 6,18

y

15,16,17

Figure 1. The geometry used to describe tiled apertures in hexagonal close-packed configurations with the effects of stair mode. Note that we label each subaperture with a beamlet number which we then match with the corresponding amount of stair mode piston phase in the x and y directions, respectively.

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Based on the geometry shown in Fig. 1, we determine the aperture transmittance function t A  x, y  for a seven element, N  7 , HCP tiled aperture as   x2  y 2 t A  x, y    A  x, y  circ   d0   

N 1            x, y      x  W0 cos     1  , y  W0 sin     1   .  3  3    1    

(2)

In Eq. (2), A  x, y  is the amplitude function of the outgoing beamlets, which have an initial diameter d 0 because of multiplication with the two-dimensional circle function, circ



x2  y2 d 0



21

;  is the notation used to represent two-

dimensional convolution in the x and y directions, respectively; and   x  x0 , y  y0  is the two-dimensional Dirac delta function centered at the points  x0 , y0  21, which are dependent on the subaperture spacing W0 . If we include the effects of stair mode, integer multiples of piston phase, with step-sizes S x in the x direction and S y in the y direction (measured in waves), must be included in the model at the appropriate locations. Thus, expanding Eq. (2) and referencing Fig. 1, we obtain   x2  y 2   t A  x, y    A  x, y  circ    d0       1 3  W0  exp  i 2  S x  S y      x  W0 , y  exp  i 4 S x     x  W0 , y  2 2      1 3  1 3  W0  exp  i 2  S x  S y      x, y     x  W0 , y  W0  exp i 2  S x  S y     x  W0 , y  2 2 2 2       1 3  W0  exp i 2  S x  S y      x  W0 , y  exp  i 4 S x     x  W0 , y  2 2   

. (3)

Equation (3) describes the aperture transmittance function t A  x, y  for a seven element HCP tiled aperture with the effects of stair mode. Using the geometry in Fig 1, we can derive a similar expression for the 19 element case. We can compute the optical disturbance in the target and or image plane, U  u , v, z  Z   U Z (u , v) , by substituting Eq. (1) into the Fresnel diffraction integral, so that21 U Z  u, v  

1  k exp i  u 2  v 2   t A  x, y  P  x, y  fx  u , f y  v , i Z  2Z Z Z

(4)

where   is the notation used to represent a two-dimensional Fourier transform with respect to x and y. In the analysis to follow, we assume that each beamlet has a flat-top beam amplitude, A  x, y   4 P0  d 02  , with initial beamlet power P0 and no apodization, P  x, y   1 . As such, if we substitute Eq. (3) into Eq. (4), we obtain the following relationship: U Z  u, v  



where jinc c x 2  y 2

 P0 d 0 d   k  u 2  v 2   jinc  0 u 2  v 2  Q  u , v  , exp i  i 2 Z  2Z   Z 

 is the two-dimensional sombrero function

(5)

, and Q  u, v  is the interference function given by

21


 W  Q  u, v   1  2 cos  2  0 u  2S x      Z .   W0   W0   3W0 3W0 u v  S x  S y    2 cos  2  u v  Sx  S y  2 cos  2    2 Z 2 Z   2 Z   2 Z  

(6)

Note that we utilized the trigonometric identity, 2 cos  x   exp  ix   exp  ix  , in writing the form of the interference function Q  u, v  in Eq. (6). These results correspond to the seven element case; however, Eq. (5) is generic in the sense that it can also be used for the 19 element case provided we derive the appropriate interference function Q  u , v  . We can also compute the irradiance in the target and or image plane, I  u, v, z  Z   I Z  u , v  (measured in W m 2 ), by taking the square modulus of Eq. (5), so that 2

P  d  d  I Z  u , v   0  0  jinc2  0 u 2  v 2  Q 2  u , v  . 4  Z   Z 

(7)

Note that the terms in front of the interference function squared Q 2  u, v  , in Eq. (7), serve as an envelope for the coherent beam combination that occurs all along the propagation path. This envelope requires that all interference peaks fall within its diffraction-limited bucket diameter, DZ  2.44 Z d 0 , which we refer to here as the incoherent bucket diameter. What is more, in the presence of no stair mode, the on-axis irradiance is equal to the peak irradiance, I Z  u  0, v  0   I P , and is N 2 times the on-axis envelope magnitude: 2

IP  N 2

 P0  d 0  . 4   Z 

(8)

This peak irradiance is surrounded by a central lobe that falls within the diffraction-limited bucket diameter, d Z  2.44 Z D0 , which we refer to here as the coherent bucket diameter—a quantity dependent on the aperture diameter, D0  2nW0  d 0 , where n  0, 1, 2, ... is the number of active subaperture rings within the HCP tiled aperture. We now wish to explore the implications of our model setup. 2.2 Model exploration

Provided our model setup above, in Fig. 2 and Fig. 3 we plot normalized irradiance, I Z  u , v  I P , using Eqs. (7) and (8), for increasing stair mode step sizes, S x , y   0,1 . In Fig. 2, we set the subaperture spacing equal to the subaperture diameter, W0  d 0 , while in Fig. 3, W0  1.5d 0 . To create Fig. 2 and Fig. 3, we use N G  1024 grid points across with a grid spacing,  u ,v  2 DZ N G , in the target and or image plane. Both Fig. 2 and Fig. 3 identify the effects of stair mode as a relative 2

phase ambiguity—integer multiples of stair mode piston phase, 2 S x , y  2 m , where

m  0, 1, 2, ... , results in the presence of no stair mode. As such, the effects of stair mode should not be confused with array steering. At optical wavelengths, separate electro-optic devices, such as Risley Prisms, would be needed for largescale steering of the beamlets,    d 0 . However, if the effects of stair mode could be measured, high-bandwidth phase loops could then be exploited for fine-scale steering of the beamlets,    D0 11,12,13,14,15,16,17,18.


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Figure 2. Normalized target and or image plane irradiance patterns for increasing stair mode step sizes (a) |Sx| and (b) |Sy|. Note that we represent the incoherent bucket diameter DZ by dashed white circles and the coherent bucket diameter dZ by dash dot white circles. Here, the analysis corresponds to N = 7 subapertures with flat-top beamlet amplitudes, and the subaperture spacing is equal to the subaperture diameter, W0 = d0.


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Figure 3. Normalized target and or image plane irradiance patterns for increasing stair mode step sizes (a) |Sx| and (b) |Sy|. Note that we represent the incoherent bucket diameter DZ by dashed white circles and the coherent bucket diameter dZ by dash dot white circles. Here, the analysis corresponds to N = 7 subapertures with flat-top beamlet amplitudes, and the subaperture spacing is equal to 1.5 times the subaperture diameter, W0 = 1.5d0.


We now wish to quantify performance in the target plane. In order to do so, we will calculate two quantities. The first is peak Strehl ratio (PSR), S P , which we calculate using the following relationship: SP 

IM . IP

(9)

In Eq. (9), the quantity, I M  max  I Z  u, v   , is the maximum irradiance in the target plane in the presence of stair mode, while the quantity, I P , is again the peak irradiance in the target plane in the presence of no stair mode. Thus, PSR S P , as defined in Eq. (9), provides a normalized unit of measure for peak performance in the target plane. The second quantity of interest is power in the bucket (PIB), PB . We calculate this quantity by integrating the target-plane irradiance I Z  u , v  , so that

 u 2  v2  I u v , circ   Z    DB 

PB  





  dudv .  

(10)

Note that PIB PB , as defined in Eq. (10), gives a measure for the encircled power or average performance in the target plane. This encircled power falls within the bucket diameter DB because of multiplication with the two-dimensional circle function, circ



u 2  v 2 DB



21

, in Eq. (10).

As shown in Fig. 4, we illustrate how suabaperture spacing W0 affects PSR S P and PIB PB in the target plane. To create Fig. 4, we use the same generic setup as found in Fig. 2 and Fig. 3. For instance, in Fig. 4a and Fig. 4b, we plot PSR S P versus stair mode step size for both the x and y directions, S x , y  [0,1] . In Fig. 4a, we set the subaperture spacing equal to the supaperture diameter, W0  d 0 , while in Fig. 4b, W0  1.5d 0 . Furthermore, in Fig. 4c and Fig. 4d, we plot normalized PIB PB PT versus bucket diameter DB for multiple stair mode step sizes S x . Here, PT is the total power in the target plane, which we calculate using Eq. (10) by setting the bucket diameter equal to grid length, DB  2 DZ . Note again that in Fig. 4c, W0  d 0 , while in Fig. 4d, W0  1.5d 0 . Figure 4 supports two trends which we cannot overlook in the analysis. The first is the fact that as subaperture spacing W0 increases, PSR S P increases. We see this trend in Figs. 4a and 4b, where we also note the difference between x and y direction PSR S P values due to the asymmetry of the HCP tiled aperture in orthogonal axes. In general, as the subaperture spacing W0 increases, the side lobe spacing in the far-field irradiance pattern decreases. We see this behavior in Fig. 2 and Fig. 3 above. Under such circumstances, the side lobes become more distinguished due to modulation from the envelope found in Eq. (7). Any energy that leaks into the side lobes because of the effects of stair mode essentially leaks into more distinguished side lobes with increased subaperture spacing W0 . This explains why stair mode has less of an effect on peak performance in Fig. 4b when compared to Fig. 4a. It is also interesting to note that as the subaperture spacing W0 increases, the aperture diameter D0 increases and the coherent bucket diameter d Z decreases. These effects help in explaining the second trend found in Fig. 4. We see in Fig. 4c and Fig. 4d that as the subaperture spacing W0 increases, the PIB PB generally decreases. The solid horizontal and vertical magenta colored lines help to demonstrate this trend. In Fig. 4c, we place the vertical line at the location of the coherent bucket diameter, d Z DZ  0.33 . The corresponding horizontal line represents the fraction of encircled power, PB PT  0.80 . Similarly, in Fig. 4d, d Z DZ  0.25 and PB PT  0.38 . We note that the placement of these lines shift down and to the left when comparing Fig. 4c to Fig. 4d. This shift corresponds to less PIB PB . In Fig. 4d, we also note that the presence of stair mode for some bucket diameters DB has more PIB PB than in the presence of no stair mode, S x  0 . Thus, in some cases, stair mode can actually improve average performance in the target plane. We now use Eqs. (7) and (8) to validate our simulations.


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Figure 4. The effects of subaperture spacing W0 on peak Strehl ratio SP and normalized power in the bucket PB / PT in the target plane. Note that PT is the total power in the target plane, so that the bucket diameter is two times the incoherent bucket diameter, DB = 2DZ. In (a) and (c), W0 = d0 and in (b) and (d) W0 = 1.5d0, where d0 is the subaperture diameter. Also, in (c) and (d), we represent the location of the coherent bucket diameter dZ by the solid vertical magenta lines, while the solid horizontal magenta lines represent the corresponding fraction of encircled power. Here, the analysis corresponds to N = 7 subapertures with flat-top beamlet amplitudes.


2.3 Model validation

In the analysis to follow, we wish to numerically simulate the effects of stair mode. To do so, we must first validate that our numerical simulations are correct. Thus, in Fig. 5, we compare the analytical result for normalized irradiance, I Z  u , v  I P , obtained from Eqs. (7) and (8), to the numerical result obtained from using the DFT to compute Eq. (4) provided the aperture transmittance function t A  x, y  in Eq. (3) and the assumptions made above. Figure 5 shows the normalized irradiance cross-section results in the u and v directions, respectively, for arbitrary stair mode step sizes, S x , y  0.1 . Note that the numerical result matches the analytical result to two decimal places. To make Fig 5, we used the same generic setup as in Fig. 2 and Fig. 3 with N G  1024 grid points across and a grid spacing,  x , y  15D0 N G , in the source plane. Also note that we can relate the source-plane coordinates to the target-plane coordinates using the variable substitutions found in Eq. (4)21.

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Figure 5. Normalized target and or image plane irradiance cross-section results in the (a) u and (b) v directions, respectively, for arbitrary stair mode step sizes, Sx,y = -0.1. Here, the analysis corresponds to N = 7 subapertures with flattop beamlet amplitudes, and the subaperture diameter is equal to the subapture spacing, d0 = W0. Note that the envelope is N2 times the target and or image plane irradiance cross-section associated with one beamlet.

3. EXPERIMENTAL ANALYSIS AND DISCUSSION In this section, we present further analysis in support of the model setup, exploration, and validation given in the previous section. For instance, we conduct two studies. In the first study, we calculate average performance in the target plane for multiple realizations of stair mode and a variety of source-plane configurations. The second study investigates the use of a stair mode imager with varying aperture stop diameters to detect the effects of stair mode. 3.1 Average performance in the target plane

In the current study, we wish to simulate different array fill factors in the source plane using both seven and 19 element HCP tiled apertures along with both flat-top and Gaussian outgoing beamlet amplitudes. For this purpose, there are three definitions for array fill factor that we will use to help define the experimental parameter space. We refer to the


first definition of array fill factor f a as the area fill factor because we compute it as the ratio of the total active aperture area to the total composite aperture area, where3 fa 

N  d0 2 

2

  2n  1W0 2  

2

.

(11)

Here again, N is the number of active subapertures, and n  0, 1, 2, ... is the number of active subaperture rings within the HCP tiled aperture. Voronstov and Lachinova derived the second definition of array fill factor f s as the mode field diameter divided by the subaperture diameter2, so that fs 

2 w0 , d0

(12)

where w0 is the Gaussian beamlet waist. As a result, we refer to this array fill factor f s as the subaperture fill factor. It should be noted that performance metrics for a truncated Gaussian beamlet amplitude compare well with those of a flattop beamlet amplitude when f s  0.89 2,3. The third array fill factor is the product of the first two definitions of array fill factor, f e  f a f s . We refer to this array fill factor f e as the effective fill factor in accordance with the analysis of Motes and Berdine3. The values shown in Table 1 define the experimental parameter space for our current study provided our definitions for array fill factor above. Accordingly, the values in Table 1 represent a multitude of aperture diameters D0 . In the results to follow, we wish to quantify performance in the target plane for each source-plane configuration described in Table 1. Table 1. Values for the experimental parameter space.

Subap. diameter, d 0 , (cm)

Subap. spacing, W0 , (cm)

N 7 area fill factor, f a

N  19 area fill factor, f a

Subap. fill factor, f s

N 7 effective fill factor, fe

N  19 effective fill factor, fe

N 7 aperture diameter, D0 , (cm)

N  19 aperture diameter, D0 , (cm)

5

7.5

0.35

0.34

0.89

0.31

0.30

20

35

6

9.5

0.31

0.30

0.89

0.28

0.27

25

44

8

11

0.41

0.40

0.89

0.37

0.36

30

52

10

12.5

0.50

0.49

0.89

0.44

0.43

35

60

12

14

0.57

0.56

0.89

0.51

0.50

40

68

To generate results for this study, we assume uniformly distributed stair mode step sizes in the source plane from S x , y  1 to S x , y  1 . Consequently, we can then simulate 1000 random realizations of stair mode for each of the aperture diameters D0 listed in Table 1. The corresponding results for average PSR S P and average PIB PB are found in Table 2. In order to create the results found in Table 2, we use the following state of the art parameters:   1  m , P0  1 kW , and Z  5 km with the same grid setup and numerical procedure used to create Fig. 5. Note that we also used no apodization, P  x, y   1 , flat-top beamlet amplitudes, A  x, y   4 P0  d 02  , and Gaussian beamlet amplitudes, A  x, y   2 P0  w02  exp  x 2  y 2  w02  . Also note that we set the bucket diameter equal to the coherent bucket diameter, DB  d z , in order to compute average PIB PB using Eq. (10).


Table 2. Results for average performance in the target plane. These results correspond to the experimental parameter space defined in Table 1. Here, λ = 1 μm, P0 = 1 kW, and Z = 5 kM.

N  7 average performance metrics

Aperture diameter, D0  20 cm





Flat-top beamlet amplitude PSR, SP

0.83

0.84

0.80

0.76

0.74

Gaussian beamlet amplitude PSR, SP

0.86

0.87

0.83

0.80

0.78

Flat-top beamlet amplitude PIB, PB , (kW)

1.4

1.4

1.6

1.7

2.0

Gaussian beamlet amplitude PIB, PB , (kW)

1.2

1.1

1.4

1.5

1.7

N  19 average performance metrics






Flat-top beamlet amplitude PSR, SP

0.82

0.84

0.78

0.75

0.73

Gaussian beamlet amplitude PSR, SP

0.85

0.87

0.82

0.80

0.78

Flat-top beamlet amplitude PIB, PB , (kW)

1.4

1.3

1.4

1.8

2.0

Gaussian beamlet amplitude PIB, PB , (kW)

1.1

1.1

1.1

1.5

1.7

Based on the results in Table 2, there are three trends which cannot be overlooked in the analysis. The first and most important is the fact that stair mode effects system performance by 15-30%. This point can be seen in the recorded values for average PSR S P . Remember, that throughout the analysis, we assumed that stair mode is the only phase error present in the phase-locked loop; thus, we should note that auxiliary engineering constraints will only decrease system performance in addition to the effects of stair mode. The second trend reveals that as the array fill factor increases, average PIB PB increases. This occurs despite the presence of stair mode, which is in agreement with the analysis presented in Fig. 4. In general, as array fill factor increases, the side lobe spacing in the far field irradiance pattern decreases. The fact that this trend remains true despite the presence of stair mode is a nice reality. However, the third trend shows that average PSR S P is significantly affected as array fill factor increases, which is also in agreement with the analysis presented in Fig. 4. The results in Table 2 show that there is a distinct tradeoff between the values recorded for average PSR S P to those recorded for average PIB PB . These engineering tradeoffs may prove useful for future optical phased array designs.


3.2 Stair mode imager

In this study, we investigate the use of a simple imaging system or camera to detect the effects of stair mode. Remember that stair mode is an unobservable mode in the beam control system of optical phased arrays performing target-based phasing. Thus, we need a separate sensor to detect the effects of stair mode. The biggest constraint in using an imaging system to detect the effects of stair mode is the fact that the diameter of its primary aperture needs to be on the order of subaperture diameter within a tiled aperture if it’s going to be used as part of a conformal beam director. However, if the primary aperture diameter of the imaging system becomes too small, apodization occurs through diffractive effects. Apodization is an undesired effect for most detection and estimation schemes since it essentially decreases imaging resolution. To define the experimental parameter space for this study, we assume that the target to be illuminated is a perfectly flat and perfectly aligned mirror, as shown in Fig. 6. In so doing, we are able to use the same model setup developed above, so that the aperture transmittance function t A  x, y  for the stair mode imager is the same as that found in Eq. (3). We then account for different aperture stop diameters by letting the real-valued pupil function in Eq. (1) equal a circle function, P  x, y   circ



x 2  y 2 DS



21

. Here, DS is the diameter of the stair mode imager’s aperture stop. In making

this claim, we assume that the aperture stop diameter DS is the same size as the entrance pupil and exit pupil diameters for the simple imaging system, i.e. the stair mode imager has unit magnification after collimation from the target. HCP tiled aperture

r

Stair mode imager Focal plane array

10d Z

DS

Perfect mirror

Figure 6. Experimental setup for the stair mode imager study.

The results for the stair mode imager study are found in Fig. 7. Here, we simulate a single stair mode step size in the source plane, S x  0.5 , and vary the aperture stop diameter from the size of a single subaperture to the size of the active aperture, DS   d 0 , D0  . We do this with Gaussian beamlet amplitudes for both experimental cases in Table 1 with the

same aperture diameter, D0  35 cm . In Fig. 7a, N  7 subapertures with a subaperture diameter, d 0  10 cm , and a subaperture spacing, W0  12.5 cm , while in Fig. 7b, N  19 , d 0  5 cm , and W0  7.5 cm . To make Fig 7, we use the same grid setup, state of the art parameters, and numerical procedure used to create the results found in Fig. 5 and Table 2; however, here we interpolate the image plane results to correspond to a square focal plane array with N P  256 pixels across and a pixel spacing,  u ,v  10d Z N P , in the image plane. Note that both experimental cases have the same coherent bucket diameter, d Z  3.5 cm . It should also be noted that we plot normalized irradiance, I Z  u , v  I M , for each aperture stop diameter DS in Fig. 7. Remember that the quantity, I M  max  I Z  u, v   , is the maximum irradiance in the image plane in the presence of stair mode.


15

1

0.9

'0)

10

DS

5

0.8

/D o = 0.29

D$ /D

o

= 0.38

0.7

DS / Do = 0.46

0.6 N

0.5

0

0.4 DS/Do =0.64

DS/Do = 0.55

DS/Do = 0.73

0.3

0.2 - I () 0.1 DS / D0 = 0.82

DS / D0 = 0.91

-10

-15

DS / D0 = 1

-5

h/It

IO

5

O

15

/ Io

/

\

\/ DS/Do=0.14

/

-5

-I5

\_,/

)

0.8

\.. \_,/ \_,/

0.7

\

\ I\

D/D=036

DS/Da=0.25

//

i----

//

i- \ \

/

)

I

\

/

O

r. O

DS /Do =0.79

Ds/Do= 0.89

-IO

0 It/41

0.5

0.4

o

/B0

I

0.6

DS /D =0.68

= 0.57

D5

o

^

I0

\

DS /D = 0.46

/ -In

' \\

O.9

o

i

l

.

I J

\ DS /Do =1

0.3 0.2 0.1

0

IQ

(hi Figure 7. Normalized image-plane irradiance patterns for increasing aperture stop diameters DS, Gaussian beamlet amplitudes, and one stair mode step size, Sx = 0.5. In (a) N = 7 subapertures with a subaperture diameter, d0 = 10 cm, and a subaperture spacing, W0 = 12.5 cm, while in (b) N = 19, d0 = 5 cm, and W0 = 7.5 cm. Note that we represent the incoherent bucket diameter DZ by dashed white circles and the coherent bucket diameter dZ by dash dot white circles. Here, no apodization occurs when the aperture stop diameter is equal to the aperture diameter, DS / D0 = 1.


Based on the results in Fig. 7, there is one basic trend which cannot be overlooked in the analysis. As array fill factor decreases, the effects of apodization have less of an effect on the stair mode imager’s imaging resolution. We further emphasize this point in Fig. 8. Here, we plot normalized irradiance, I Z  u , v  I P , cross-sections in the v direction for multiple aperture stop diameters DS and a single stair mode step size in the source plane, S x  0.5 . The quantity, I P , is again the peak irradiance in the image plane in the presence of no stair mode. In Fig. 8a, N  7 subapertures with a subaperture diameter, d 0  10 cm , and a subaperture spacing, W0  12.5 cm , while in Fig. 8b, N  19 , d 0  5 cm , and W0  7.5 cm . These preliminary results show that imaging resolution is severely degraded in both cases for aperture stop diameters less than half the size of the aperture diameter, DS D0  0.5 . Auxiliary engineering constraints such as noise and atmospheric effects will only decrease imaging resolution further. We leave this analysis for future research.

---

Envelope DS

---

/D0 =1

Envelope

DS /D0 =1 _- .DS /DD =0.75

- _- .DS /D0 =0.75

Ds /Do =0.5

DS /D0 =0.5 1

0.8

Va 0.6 N

0.4

I

I

1

I

I

0.2

0-0 5

1

O.\

Il

1

/1

I

Ij

\ 0

1.

05

y /D (a) Figure 8. Normalized image-plane irradiance cross-section results in the v direction for increasing aperture stop diameters DS, Gaussian beamlet amplitudes, and one stair mode step size, Sx = 0.5. In (a) N = 7 subapertures with a subaperture diameter, d0 = 10 cm, and a subaperture spacing, W0 = 12.5 cm, while in (b) N = 19, d0 = 5 cm, and W0 = 7.5 cm. Here, the envelope is N2 times the image plane irradiance cross-section associated with one beamlet and no apodization, DS / D0 = 1.

4. CONCLUSION Future work in optical phased arrays will need to account for the effects of stair mode. As such, we propose two directions for future research efforts. The first research direction should better characterize the effects of stair mode on system performance. This needs to be done in addition to other sources of phase errors within the optical phased array such as telescope aberrations, platform jitter, turbulence, thermal blooming, target jitter, and rough surface scattering. An error budget analysis of this nature will better identify any engineering constraints associated with the effects of stair mode and it will also inform the directed energy community on ways to improve future optical phased array designs. The second research direction should develop a suitable sensor to detect the effects of stair mode. A stair mode imager is one approach; however, there are numerous engineering constraints which need to be identified in the road ahead. Apodization through diffractive effects is one such engineering constraint. Advanced image processing techniques might prove useful in combating the effects of apodization; however, we need more analysis to justify this claim.


ACKNOWLEDGEMENTS The authors would like to thank David Mann and Glenn Tyler from tOSC and Wesley Green and Dan Marker from AFRL/RD for all technical discussions relating to this work. The views expressed in this paper are those of the authors and do not necessarily reflect the official policy or position of the Air Force, the Department of Defense, or the U.S. Government.

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