9750

Research Article

Vol. 58, No. 36 / 20 December 2019 / Applied Optics

Alignment precision of polarization components
Nathan Hagen,1, *Prathan Buranasiri,2 AND Yukitoshi Otani1

1

Utsunomiya University, Department of Optical Engineering, Center for Optical Research and Engineering (CORE), 7-1-2 Yoto, Utsunomiya,Tochigi 321-0954, Japan2
Department of Physics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand*Corresponding author: nh@hagenlab.org
Received 8 October 2019; revised 13 November 2019; accepted 14 November 2019; posted 15 November 2019 (Doc. ID 379897);
published 11 December 2019

Recent research publications in the polarization literature have discussed methods of correcting for azimuthalalignment errors of optical elements in postprocessing. However, we show that high angular precision is not difficult to achieve during system alignment, so that postprocessing correction should be unnecessary. We estimate thealignment precision achievable for linear polarizers and waveplates in polarization systems. This shows that using
an optical signal model for alignment allows a precision limited by the quality of the optics and detectors rather
than the quality of the mechanics, rendering millidegree alignment precision possible with ordinary rotational
mounts. © 2019 Optical Society of America
https://doi.org/10.1364/AO.58.009750

1. INTRODUCTION
Polarizers and waveplates are the fundamental components ofpolarization instruments, and using the optical extinction properties of crossed polarizers is by far the most common meansof aligning the azimuth angles of polarization components in asystem. Several recent researchers [1–6] have analyzed how todeal with alignment errors in postprocessing, which begs thequestion of how precisely we can expect polarization systems
to be aligned. Some of these authors discuss alignment errors
on the order of 0.5◦ . Such large errors seem outstandingly bad,but without a quantitative discussion of precision it is difficult
to say. In addition, some systems may require achieving errorsof < 0.01◦ , and this can be difficult to achieve with standard
mounts and methods. In order to address these issues, we estimate the precision that can be achieved during alignment oflinear polarizers and waveplates, finding that the techniques
for achieving high-precision alignment are not at all difficult.
Using the all-optical method that we discuss, one should be
able to assemble the components of a polarization system togreater precision than the alignment errors can be detected in
postprocessing. Achieving this precision is a matter of using
appropriate methods and of careful experimental procedures
rather than expensive equipment.
The standard procedure for aligning components in a polarization system is to first set one polarizer (Pref , the reference) at
horizontal (0◦ ) to define the orientation axis. All of the remaining components are then aligned to this reference axis. To align asecond polarizer P2 , we rotate its azimuth angle until we find thepoint at which the transmitted light through the polarizer pair is
a minimum (P2 at 90◦ ) or a maximum (P2 at 0◦ ). These two are
1559-128X/19/369750-07 Journal © 2019 Optical Society of America

the “crossed condition” and “parallel condition.” If the desired
azimuth angle differs from one of these two, then the usual
procedure is to use the indicators on the mechanical mount toadjust the angle to the intended azimuth, so that the precisionof the mount will limit the alignment precision. While a typicalmanual rotation mount often has a vernier precision of about
0.1◦ , mounts without a vernier are typically limited to precisionsof 0.25◦ ∼ 1◦ .
Another common procedure requires the use of an electronic
stepper-motor rotational stage with a position encoder. As the
stepper rotates P2 to a sequence of angular positions, the detector records the transmitted signal at each position. Once a fullrotation is completed, we can use the entire sinusoidal signal
to calibrate the encoder azimuth. This method relies on themechanical precision of the stepper motor, so that a mechanicalalignment precision between 0.1◦ and 0.003◦ is possible, assuming a good optical signal and high performance motors [7]. This
technique, however, requires having an electronically controlled
rotation stage for every component to be accurately aligned in
the system.
Finally, we describe a fourth procedure which depends
entirely on the optical properties of Malus’s law for alignment.
In this procedure, we measure the light level at the crossed
and parallel conditions; from these we can use Malus’s law toestimate the intensity at the desired azimuth angle of the component. The component is then rotated to the angle at which
the target intensity is reached. As this procedure does not rely
on mechanical encoding at all, it can achieve a precision limited
only by the quality of the optical components. Thus even acoarse mechanical mount can achieve high precision alignment.

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For aligning components to angles other than 0◦ and 90◦ , wecan summarize the four common alignment procedures:
Procedure 1, crossed condition (⊥): rotate P2 to the angle
where the minimum light level is reached and set that as the90◦ azimuth (error = 1θbot ). If a different angle than 90◦ isneeded, then use the mechanical accuracy of the mount to turnthe polarizer from the 90◦ position to the desired azimuth.
Procedure 2, parallel condition (k): rotate the polarizer
to the position where maximum light level is reached and set
that as the 0◦ azimuth (error = 1θtop ). If a different angle isneeded, then use the mechanical accuracy of the mount to turnthe polarizer from the 90◦ position to the desired azimuth.
Procedure 3, sinusoid fitting (“sin”): using a mount with anelectronic encoder, rotate the mount at a series of angles and record the light level at each angular position. After a full rotation,
calibrate the azimuth angles of the encoder from the full set.
Procedure 4, optical method (“opt”): rotate the polarizer to the position of minimum light, and record that signal
as Ibot (error = 1Ibot ). Next rotate the polarizer to the position of maximum light level and record that signal as Itop(error = 1Itop ). Calculate the desired position’s light level via
Ibot and Itop and the desired azimuth angle, and then rotate thepolarizer to achieve that intensity.
Note that the alignment procedures should ideally place thereference polarizer directly in front of the detector and not havethe component be aligned between the reference polarizer anddetector. The latter setup exposes the detector to changes in thepolarization state, so that the detector’s polarization dependence(typically on the order of 0.1–1%) introduces an unwanted
additional parameter that must be calibrated.

Fig. 1. Simulated sinusoidal intensity curve generated by rotatinga polarizer in front of a fixed polarizer, showing the noise-free curve(black line), noisy measurement (blue line), and the standard deviation
about the noise-free curve (gray region). The parameters assumed are
Ib = 10, σd = 3, and I0 = 500. Inset: the optical alignment layout.

any constant terms and find that the value of 1I declines
quadratically as θ moves away from 0◦ :1
[1I ]top = − I0 (1θ )2 .
2

2. ALIGNING A LINEAR POLARIZER

1
I (θ ) = Ib + I0 [1 + cos(2θ )] + n,4(1)

where Ib is the bias signal, I0 is the incident light intensity, θ isthe azimuth angle, and n is the measurement noise. Here noisemean hni = 0 and noise variance var(n) = hI i + vd , so that n
is a mix of Poisson (shot) noise and Gaussian (detector) noise,
where vd is the detector noise variance. Thus, n and I (θ ) arestochastic variables and have units of photoelectrons, while theremainder are deterministic parameters.
When the polarizers are aligned in parallel, we get
I = Ib + 12 I0 + n 0 , and when crossed we get I = Ib + n 90 .
First we consider the noise-free case.
Approximating the cosine curve of Eq. (1) about the peakwith a power series gives



1
1
Itop = Ib + I0 1 + 1 − (2θ )2 + ...
4
2≈ Ib +1
I0 (1 − θ 2 ).
2

For estimating alignment precision, we are interested in analyzing how small changes in intensity (1I ) depend on smallchanges in the polarizer angle (1θ ). Thus, we can ignore(2)

From this result, we can see that an angle error of 0.25◦ createsa fractional intensity change of 1I /I0 = (0.25π /180)2 =9.50 × 10−6 .
Performing the same analysis at the bottom of the sine curve(θ = 90◦ ), we obtain
1
I0 (1θ )2
2(3)

1
I0 (1θ )2 .
2

(4)

Ibot = Ib +Rotating one polarizer in front of another produces a measured
intensity given by Malus’s law (see Fig. 1) as

9751

and
1Ibot =

Finally, if we want to align the polarizer to the 45◦ position(the middle of the curve), then a Taylor expansion of aboutθ = π/4 leads toImid = Ib +1
I0 (1 − 21θ )
4

(5)

and
1
1Imid = − I0 1θ .
2

(6)

Thus, we find a linear dependence instead of quadratic.
For each of these three cases [Eqs. (2), (4), and (6)], we caninvert the equations to write the angular sensitivity in terms ofchanges in intensity:p1θbot = 21I /I0 ,(7)−21I /I0 ,(8)

1θmid = −21I /I0 .

(9)1θtop =

p

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A. Incorporating Measurement NoiseAt the crossed-polarizer position, the noise variance is given byvar(Ibot ) = Ib + vd ,(10)

a sum of the Poisson-distributed shot noise due to the detector
dark current (“dark shot noise”) and the Gaussian-distributed
detector noise (variance vd ). This assumes that any electronic
bias has been subtracted from the measurements prior toapplying any variance formulas.
At the parallel (maximum transmission) condition, the measurement intensity is larger, so that the corresponding shot noiseis also larger:
var(Itop ) = Ib +1
I 0 + vd .
2

(11)

If we want to estimate the intensity at the θ = 45◦ point of thetransmission curve, then we can average the maximum and minimum measured intensity values, such that1
Imid = (Ibot + Itop ),2(12)with the resulting variance11
var(Imid ) = var(Ibot ) + var(Itop )22= Ib +

1
I 0 + vd .
4

(13)

Thus, averaging the peak and valley intensities bears the same
variance as an intensity measurement taken at the middle
position itself.
Since the crossed and parallel alignment procedures require
only locating the minimum or maximum intensities Ibot or Itop ,we can simply equate the variance of the procedure to that of thesingle measurement: var(I⊥ ) = var(Ibot ), var(Ik ) = var(Itop ).
The optical procedure, however, requires first estimating
Imid and then using the noisy measurements obtained while
searching for the middle position. This doubles the variance tovar(Iopt ) = 2Imid = 2Ib +1
I0 + 2vd .
2(14)

We can now use Eqs. (7)–(9) to calculate the angular standard
deviation (for which we use  in place of 1θ ) as a function of
measurement noise by substituting the measurement standard
deviation σ for the intensity change 1I :


 √
1/2
2σbot 1/2
4 I b + vd
⊥ =
=
,
(15)
I0
I0
k =
opt =



2σtop
I0

1/2
≈

√ −1/42I0 ,√2σmid 2 8Ib + 2I0 + 8vd
=
,
I0
I0

(16)

(17)

where the approximation in Eq. (16) assumes that the shot noise
predominates over detector noise. From these results, we see thatwhile the precision of the parallel condition alignment method(Proc. 2) improves only slowly with increasing SNR, the crossed
condition (Proc. 1) and optical methods (Proc. 4) both improve
rapidly.
For angles other than 0◦ , 45◦ , and 90◦ , we can use Malus’s
law in the same way to estimate the target intensity value, andthen rotate the component to reach it. While angles between 90◦
and 45◦ can achieve high precision using the crossed and opticalmethods, as one approaches the 0◦ position (parallel condition)
the achievable alignment precision steadily worsens. Section 5
below provides numerical estimates of the precision values wecan expect.
In order to help keep track of the quantities discussed here,
we review the various labels here. Subscripts “top,” “bot,” and“mid” refer to the intensities measured while the sinusoidal
curve of intensity versus azimuth angle is at its brightest, darkest,or middle point. Subscripts ⊥, k, “sin,” and “opt” refer to thealignment procedure used—the crossed, parallel, sinusoid
fitting, or optical method.
B. Alignment Using a Mechanical Encoder

The previous analysis considers the case of manual alignment,with the user manipulating a rotation stage manually to achieve
the correct position. A rotational stage that has an electronic
encoder or stepper motor, however, allows the user to align thecomponent algorithmically. While this can be done in the same
way as for manual alignment, using a binary search procedure
to adjust the stage to achieve the desired intensity, a naturalprocedure for this setup is to collect the sinusoid intensity signal
from a full rotation and estimate the azimuth angle corresponding to the motor position. We can expect computer-controlled
motion to achieve a precision limited by either the measurement
noise (for low signal levels) or by the precision of the rotational
stage.
While many researchers have worked on the problem of
estimating sinusoid signals in the presence of noise, most of thework deals with independent identically distributed Gaussian(IG) noise rather than Poisson noise. For IG noise, the varianceof the angular estimate has been established as [8]var(θ̂ ) =

2vG
,
I02 N

(18)

where vG is the variance of the Gaussian noise and N the number of measurements. For Poisson-noise corrupted signals,
however, this result is useful only when the visibility of thesinusoid signal becomes small, in which case the shot noise
can be approximated as uniformly distributed Gaussian noise.
Reference [9] develops a more general noise model, with mixed
Poisson and Gaussian noise, but does not obtain a closed-form
result and focuses on estimating the period of the sinusoid rather
than its phase. Although an IG noise model underweighs the
low-intensity measurements and overweighs the high-intensity
ones, it nevertheless typically provides a reasonable approximation to the Poisson-noise model. Thus, we can give the angularstandard deviation for the stepper-motor-based procedure as
2vG 1/2
sin =
.
(19)
I02 N

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9753

C. Aligning Imperfect Polarizers

The measurement model we have constructed so far [Eq. (1)]assumes that we work with ideal polarizers that transmit
100% of light in the polarizing channel and block 100% of
the cross-channel. Real-life polarizers, however, are imperfectdiattenuators, which we can model with parameters q and r forthe transmission of two orthogonal polarization channels, anideal polarizer having q = 1 and r = 0. This gives two degrees
of freedom, but we will assume q = 1, so that only r determines the extinction ratio X of the polarizer via X = q /r . If weassume the two polarizers used for alignment are identical, themeasurement model becomesI (θ ) = Ib +1
I0 [(1 + r )2 + (1 − r )2 cos(2θ )].
4(20)

Almost all polarizer elements have better extinction ratios than10, so we can safely regard r as small, allowing us to approximate
the equation above as
I (θ ) ≈ Ib + 41 I0 [1 + 2r + (1 − 2r ) cos(2θ )].

(21)

The relation of small changes in intensity to angle therefore
becomes1I ≈

1
I0 (1 − 2r ) cos(21θ ),4(22)

so that all of the angle-sensitivity formulas Eqs. (7)–(9)acquire an extra factor of (1 − 2r ). Moreover, for small r ,1/(1 − 2r ) ≈ 1 + 2r , so that the 1I in Eqs. (15)–(17) can be
replaced with (1 + 2r )1I . This indicates that it would requirea poor polarizer indeed to have a substantial impact on the precision. For example, using a polarizer pair with extinction ratiosof 30 reduces the angular sensitivity
1θ (worsens the alignment√
precision) by a factor of 1 + 2r ≈ 1.07 for the “bottom”
and “top” configurations, and a factor of 1.14 for the “middle”
configuration.
3. ALIGNING A LINEAR RETARDER
So far we have analyzed the alignment of a linear polarizer to areference polarizer. Linear waveplates are also common elements
in polarimetry systems, and we should know their alignment
precision as well. We start with the most common type of
waveplate—a linear quarter-wave retarder.
For a rotating quarter-wave retarder at angle θ placed between
two fixed crossed polarizers (i.e., P2 at 0◦ and Pref at 90◦ ), we
obtain a transmitted intensity of (Fig. 2)
I (θ ) = Ib +1
I0 sin2 (2θ ) + n.
4

Fig. 2. Simulated sinusoidal intensity curve generated by rotating
a linear quarter-wave plate R between two crossed polarizers P1 andP2 , showing the noise-free curve (black line), noisy measurement(red line), and the standard deviation about the noise-free curve (greyregion). The parameters assumed are the same as those used in Fig. 1 :Ib = 10, σd = 3, and I0 = 500. Inset: the optical alignment layout.

(23)

Taking the same approach as in Section 2, we approximate the equation for small angular deviations about threepositions—θbot = 0◦ , θtop = 45◦ , and θmid = 22.5◦ —giving theresults

1Imid ≈ I0 (1θ ),(26)

which converted into small changes in angle give
p
1θbot = 1I /(2I0 ),(27)

1θtop =

(28)p

−1I /(2I0 ),1θmid = 1I /I0 .

(29)

These 1θ values for retarder alignment are all smaller than thecorresponding values for polarizer alignment [Eqs. (7)–(9)],so that waveplate alignment should be able to achieve a higher
precision than polarizer alignment can.
Next we incorporate noise. The measurement noise variances
are the same as in the polarizer case but with the Poisson contribution halved due to the smaller signal level. Thus,
var(I⊥ ) = Ib + vd ,1
I 0 + vd ,
2

(31)


1
(Ibot + Itop ) + vd ,2

(32)

var(Ik ) = Ib +var(Iopt ) = 2



(30)

= 2Ib + I0 + 2vd .

(33)

Following the same procedure as for Eqs. (15)–(17), we calculate
the waveplate angular error as a function of measurement noise:⊥ = (Ib + vd )1/2 (2I0 )−1/2 ,(34)

1Ibot ≈ 2I0 (1θ )2 ,(24)

k ≈ (2I0 )−1/4 ,(35)

1Itop ≈ −2I0 (1θ )2 ,(25)

opt ≈ (2I0 )−1/2 .

(36)

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These waveplate angle precision values are smaller (i.e.,
better) than the precision values for polarizer alignment[Eqs. (15)–(17)].
So far we have considered the case of a rotating quarter-wave
plate between crossed polarizers. We can also consider the case ofaligning the retarder between parallel polarizers. In this case, theintensity varies with waveplate rotation angle as
I (θ ) = Ib +1
I0 [1 + cos2 (2θ )] + n.
2(37)This is the same equation as Eq. (23) but with an added baseline
signal of 14 I0 . This has no effect on the sensitivity but increases
the noise so that the alignment precision will be worse.
When aligning a waveplate, searching for an intensity value
gives the appropriate angle to modulus 45◦ , since any angle
θ ± nπ/4 produces the same transmitted intensity as the angle
θ , for integer n. Coarse angular markings on the optical mounts
are sufficient to tell a user which azimuth positions correspondto which values for n.
In addition to quarter-wave plates, half-wave plates are also
common elements. Between crossed polarizers, a half-wave plateproduces the following signal as a function of its azimuth angle:
I (θ ) = Ib +1
I0 sin2 (2θ ) + n.
2(38)This is the same equation as Eq. (23) but with twice the factor in
front of I0 . Thus, a half-wave plate allows better alignment precision than a quarter-wave plate does.
An additional source of error to keep in mind is that whenaligning a linear retarder between a pair of crossed polarizers,
any alignment error in the polarizers will naturally bias thealignment in the retarder as well. A quarter-wave plate placed
between crossed polarizers generates an intensity at the detectorofI (θ ) = 1 − cos(21) cos2 (2θ ) − sin(21) sin(2θ ) cos(2θ )≈ 1 − cos2 (2θ )[1 − 21 sin(2θ )](39)

Fig. 3. Simulated incident spectrum transmitted by a 0.5 mm
thick quartz retarder placed between crossed polarizers, across the
350–1050 nm spectral range. Each of the curves indicates a different
azimuth angle for the retarder, while the dashed black curve indicatesthe envelope of the intensity spectrum.

its maximum, so that the integrated intensity across the spectrum is also a maximum. Above 45◦ , the modulation contrast
begins to decrease once again, in the same pattern as in Fig. 2.
Thus, if we integrate the signal across the spectrum, we can treat
the high-order retarder alignment in exactly the same way that
we treat the quarter-wave plate’s alignment.
To show this, we can write the expression for the intensity transmitted by the high-order retarder between crossed
polarizers,11
I0 − I0 [cos2 (2θ ) + cos(δ(λ))sin2 (2θ )]+n,44(40)where retardance δ(λ) is wavelength-dependent. The integral ofthis spectrum gives
Z λmax1
I (θ )s =I (δ(λ), θ )dλλmax − λmin λmin
I (δ, θ ) = Ib +≈ Ib +

1 0
I [1 − cos2 (2θ )] + n,
4 0

(41)

for retarder azimuth θ and (small) polarizer azimuth error 1.
Thus, a polarizer that is misaligned by 1◦ from its nominal
crossed-polarization position will in turn bias the alignmentof the linear waveplate by approximately twice the sameangle—2◦ —for quarter-wave plate azimuth angles near 0◦ .

which is equivalent to drawing a curve halfway between theupper (black-dashed curve) and lower (solid black line) envelopes shown in Fig. 3, producing an expression equivalent to
Eq. (23).

4. ALIGNING HIGH-ORDER LINEAR RETARDERS

5. QUANTITATIVE ALIGNMENT PRECISIONESTIMATES

High-order linear retarders are used in channeled spectropolarimetry [10,11] for detecting spectrally resolved polarization.
These systems use spectra and not just single intensity values,
so that the transmission of this setup varies with wavelength.
Figure 3 shows the amplitude modulation resulting from placing a high-order waveplate between crossed polarizers, forfast-axis orientation angle θ from 0◦ to 45◦ and for a Gaussianshaped spectrum. At 0◦ , there is no modulation in the spectrum,
so that the integrated intensity across the spectrum is a minimum, just as for the quarter-wave plate. As the fast axis azimuth
increases towards π/4, the modulation amplitude approaches

Providing experimental validation of the expressions ofSections 2 and 3 is difficult, since it requires equipment thathas a higher precision than the optical method does. This can bequite hard to come by. However, our experience has been thatusing the optical approach (Procedure 4) in place of Procedure 1(i.e., the crossed condition followed by using rotation mountindicators) has noticeably improved our measurements.
To make the results more concrete, we can calculate theexpected angular precision that can be achieved if we use a typical modern scientific detector array to align a polarizer using
optical-only methods to positions of 0◦ , 90◦ , and 45◦ . To start,

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we assume detector pixels with a full well capacity of 50,000
electrons (typical for 6 µm pixels), a read noise σd of 15 electrons, and a mean dark current signal of 10 electrons. We will
ignore the dark current itself by assuming that the frames arebackground-subtracted but keep the dark current shot noise.
We assume that the integration time of the detector is set toachieve a maximum signal at 90% of full well capacity, i.e.,
I0 = 45,000 electrons. Calculating the angular precision for our
three alignment approaches, we use Eqs. (15)–(17) to obtain⊥ = 2(15 + 10)2

1/4(45,000)−1/2= 0.037 rad = 2.1 ,9755

about 0.61 A/W, and at the HeNe laser wavelength of 633 nm,
the responsivity drops further to about 0.49 A/W. If we apply afrequency-domain filter to the output equivalent to integrating
the signal over a half second of time, then the equivalent bandwidth of the output will be 1 Hz, for an NEP of 1.2 × 10−14 W.
For the source, we consider Thorlabs’ HNL020RB HeNe laser,which has an optical output of 2 mW. As a result, the parameters
for our formulas are
I0 = (2 × 10−3 W)(0.49 A/W)(0.5 s)(6.25 × 1018 e− /s)◦k = 21/2 (45,000)−1/4 = 0.097 rad= 5.6◦ ,opt ≈ 23/2 (45,000)−1/2 = 0.13 rad= 0.76◦ .
These initial estimates, however, are unrealistic in that theyignore our ability to average (or sum) together many measurements. For the case of Poisson noise, this increases the detectedsignal faster than it does the noise. In the lab, we are also usually free to increase the intensity of the light source, by a factor
approaching the extinction ratio of the polarizer pair—106 forgood polarizers. This is the real strength of the crossed-polarizer
condition. The parallel and midpoint conditions are not free toincrease the light level, since doing so leads to detector saturation. Finally, in the above model we have also assumed a single
6 µm pixel, and most systems provide a much larger detection
area than that.
For a more reasonable model, we modify the setup to allow
temporal summing over 100 measurements, spatial summingover a 100 × 100 array of pixels, and a 104 × boost in theillumination level for the crossed condition; then, we obtain⊥ = 2[(1.5 × 107 )2 + 1.0 × 107 ]1/4 (4.5 × 1014 )−1/2= 0.37 mrad= 0.021◦ ,k = 21/2 (4.5 × 1010 )−1/4 = 3.1 mrad= 0.18◦ ,opt ≈ 23/2 (4.5 × 1010 )−1/2 = 0.013 rad= 0.00076◦ .
Not only does this quantify the conventional wisdom thatcrossed-polarizer alignment achieves a much better precision
than the parallel condition can, but we also see that alignment at
45◦ using the all-optical method can achieve a higher precisionthat the cross-condition can. Moreover, with careful measurements, there is still considerable room to improve on even thesemillidegree precision values. Precision commercial mechanicalstepper motor stages, on the other hand, are capable of achieving
absolute accuracy down to only about 3 mdeg, even though theirstep sizes are smaller (0.2 ∼ 0.3 mdeg) [7].
For a second example alignment setup, we consider usinga photodiode detector with a HeNe laser light source to aligna quarter-wave plate to azimuth angles of 0◦ , 22.5◦ , and 45◦ .
These correspond to the crossed, mid, and parallel condition
respectively, modeled in Eqs. (34)–(36). The photodiode we
consider is a Thorlabs FDS100, with a stated noise-equivalent
power (NEP) of 1.2 × 10−14 W · Hz1/2 at λ = 900 nm, a
responsivity of 0.65 A/W at λ = 980 nm, and a dark currentof 1 nA. For operating at 900 nm, the responsivity drops to= 3.06 × 1015 photoelectrons,
Ib = (1 nA)(0.5 s)(6.25 × 1018 e− /s) = 3.13 × 109 electrons,
vd = (1.2 × 10−14 W·Hz1/2 )(1 Hz)(0.61 A/W)× (6.25 × 1018 e− /s) = 4.58 × 104 electrons.
Since the detector noise vd is negligible in comparison to thephoton shot noise and dark current shot noise, Eqs. (34)–(36)give the estimated alignment precisions as⊥ ≈ (3.13 × 109 )1/2 (6.25 × 1018 )−1/2 = 2.23 × 10−5 mrad
= 1.28 mdeg,
k = 2(6.25 × 1018 )−1/4 = 4.00 × 10−5 mrad = 2.29 mdeg,
opt ≈ 2(6.25 × 1018 )−1/2 = 7.98 × 10−10 rad = 45.7 ndeg.
The precision estimated for the optical alignment method
when setting the waveplate to the 22.5◦ position is given in nanodegrees. Clearly, the alignment precision here will be limited
entirely by the mount and by deviations from the measurement
model, such as waveplate and polarizer nonuniformities, rather
than by noise. For the crossed and parallel alignment methods
for setting the waveplate to positions of 0◦ and 45◦ , we see that
there is surprisingly only a factor of 2 difference between them
due to the high dark current that this system suffers. Thus, thisphotodiode is well-suited to bright illumination applications.
6. CONCLUSIONS
Although the alignment of polarization component azimuth
angles is a straightforward process, the recent research literature has shown that it is not generally known what alignment
precision we should expect. A number of researchers haveshown how to correct for polarization component alignment
errors in postprocessing, but our precision analysis indicatesthat researchers’ efforts are better directed at getting the alignment precise in the first place, during system assembly. Taking
advantage of the quality of polarization optics and using theoptical signal as a guide, a patient technician should have nogreat difficulty in achieving millidegree-order alignment errors,so that expensive mechanical mounts and precision electronic
encoders are unnecessary. In addition, we have shown that theconventional crossed-polarized method does not achieve thehighest precision possible when the source light level is fixed.
Aligning a polarizer to the 45◦ position, or a waveplate to the9756

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22.5◦ position, allows for greater precision for fixed source light.
If one has the freedom to increase the source intensity as desired,
then the crossed-polarization condition may be more accurate.
While alignment precision at the millidegree level therefore
appears to be relatively easy to achieve in principle, precisionat this level is also difficult to assess. Higher-order effects suchas polarizer and retarder spatial nonuniformity will begin toplay a significant role. Higher-order mechanical problems inthe mounts also become problematic, such as eccentricity error(radial runout) in the rotational stage and wobble in the stage’s
rotational axis.
The discussion above has focused on linear polarizers andlinear waveplates, but more complex polarization elements
certainly exist. One commonly found such element is a “biplate”
retarder, typically constructed by cementing two retarders with
their azimuth angles rotated by 90◦ from one another [12]. Eachof the two constituent waveplate components is high-orderlinear retarders and thus can be aligned using the methods
suggested in Section 4. If the azimuth angles of the two plates
are not quite aligned to one another, however, the result is anelement that has a fast axis whose azimuth angle varies withwavelength [13]. Aligning composite elements such as thisrequire building a measurement model describing how theelement azimuth angle affects the detected intensity.
Funding. King Mongkut’s Institute of Technology
Ladkrabang (Academic Melting Pot Fellowship 2019).
Disclosures. The authors declare no conflicts of interest.
REFERENCES AND NOTES
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Research Article
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