Title: An Introduction to the TFI Etalon Doc #:
Date:
Rev:
JWST-STScI-002059, SM-12
July 15, 2010
-
Authors: A.R. Martel, A.
Fullerton
Phone: 410–
338-4888
Release Date: 10 May 2011
1 Abstract
We present the basic principles and equations underlying an ideal Fabry-Pérot etalon and the important parameters that describe its behavior: the Airy function, the free spectral range, the full-width at half-maximum, the finesse, the resolving power, and the contrast. The presentation is then generalized to the more complicated but realistic case of the etalon in the Tunable Filter Imager (TFI) by introducing wavelength-dependent reflectivities, phase shifts, absorption, and plate surface defects. A short discussion on the important concept of wavelength detuning is also given.
2 The Ideal Fabry-Pérot Etalon
2.1
The Basic Equation
In traditional astronomical applications, a Fabry-Pérot interferometer or etalon operates at a high spectral resolving power, large gaps, and high orders. Its output is a series of rings corresponding to different orders of interference and wavelengths. In contrast, an etalon used as a tunable filter operates at a low resolving power, small gaps, and low orders. It essentially “stretches” one of these rings radially into a large monochromatic region. The enabling component of the TFI is an etalon used in this latter fashion.
An ideal tunable filter consists of two perfectly plane-parallel plates whose inner surfaces are coated with films of high reflectivity and no absorption. Anti-reflective coatings cover the outer surfaces of the semi-transparent substrate, which is typically glass for optical applications or silicon for near-IR applications. The transmission function of an etalon arises from interference between the multiple reflections of light between the two inner surfaces. Constructive interference occurs if the transmitted beams are in phase, and this corresponds to a transmission peak. If the transmitted beams are out of phase, destructive interference occurs and this corresponds to a transmission minimum.
Operated by the Association of Universities for Research in Astronomy, Inc., for the National
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A simple diagram of an etalon is shown in Figure 1. The interior surfaces of the two plates are coated with thin dielectric films of reflectivities (reflection coefficients),
!
!
!
!
and
, while the exterior surfaces of the substrate have an anti-reflective (AR) coating. The gap between the plates is !
; the index of refraction of the medium between the plates is !
( µ = 1 for the TFI; note that the index of refraction outside the cavity is 1); and the angle of incidence for reflections within the cavity is !
, which is measured from the normal of the etalon plates. For an ideal etalon, we assume that the reflectivities are the same,
!
!
= !
!
, independent of wavelength.
Figure 1: Schematic illustration of an ideal etalon showing multiple reflections of an incident beam of radiation. The different quantities are defined in the text.
Consider a beam of monochromatic radiation of wavelength λ incident on the etalon at an angle !
and an adjacent beam created by reflection at the shiny inner surface. The geometric path difference (GPD) can be derived from Figure 1:
GPD = AB + BC =
!
cos !
1 + cos 2 !
since AB = !
cos !
and BC = AB cos 2 !
. Using the identity   cos 2 !
= 2 cos 2   !
− 1 ,
Equation (1) simplifies to
GPD =
!
cos !
1 + cos 2 !
= 2 !
cos !
(1)
(2) and consequently the optical path difference (OPD) is 2 !" cos !
.
The net phase difference between any pair of rays is a combination of the OPD expressed in terms of the wavelength of the incident light and the wavelength-dependent
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JWST-STScI-002059
SM-12 phase change per reflection, !
!
plate:
, which is assumed for simplicity to be the same for each
!
=
2
!
!
2 !" cos !
− 2 !
!
radians .
(3)
The rays interfere constructively when !
= 2 !" , where the order, m, is an integer, i.e.,:
2 !" =
4 !"# cos !
!
− 2 !
!
radians or
!" = 2 !" cos !
−
!
!
!
!
!
m   where the micron has been adopted as the unit of length. Further rearrangement gives
!
+
!
!
!
!
= !
eff
!
= 2 !" cos !
!
m   .
(4)
(5)
(6)
This definition of the effective order, !
eff
, recovers the simplest form of the interference equation, with the important caveat that in general !
wavelength. Note in particular that when the case for the TFI.
!
!
≈ !
,   !
eff eff
is a function of
≈ !
+ 1 .
This is approximately
The distinction between m and !
eff
(5) and (6) to obtain:
can be explored further by rearranging Equations
!
!
eff
= 1 −
!
!
!
1
!
eff
= 1 −
1
2
!
!
!
!"
!
cos !
(7)
Equation (7) shows that m approaches !
eff
when !
!
is large or when first condition does not apply to an etalon used as a tunable filter.
!
!
is small. The
Finally, note that differences in phase, Δ !
, can be converted to differences in wavelength, Δ !
,   via the simple relation:
!" = !"
!"
!"
The derivative that relates increments of wavelength to increments of phase is easily evaluated from Equation (3) and simplified by using Equation (6):
(8)
!"
!"
= − 2 !
1
!
!
+
!
!
!
+
1
!
!
!
!
!"
= −
2 !
!
!
eff
+
!
!
!
!
!
!"
(9)
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2.2
The Airy Function
The instrumental response function of an etalon is known as the Airy function. Since this is a key property, it is instructive to outline how it is derived from simple consideration of the amount of light transmitted after multiple reflections in the cavity defined by the etalon plates. More detailed derivations can be found in most textbooks on optics; see, e.g., Born & Wolf (2001).
For simplicity, let !
= !
etalon and !
= !
!
= !
!
!
= !
!
be the reflection coefficient of the inner surfaces of the
be the transmission coefficient through the layers of substrate.
Consider a ray of light entering the etalon cavity, e.g., from the upper left in Figure 1.
If !
!
is the incident amplitude of radiation, then a fraction !
!
At point “A”, a fraction to point “B”.
!
!
!
!
!
enters the cavity. emerges from the etalon, while a fraction !
!
!" is reflected
At “B”, the fraction of light reflected back through the cavity is !
!
emerges.
!
!
!
, of which !
!
!
!
!
!
Each subsequent passage of the reflected beam through the cavity reduces the amplitude of the light transmitted by a factor of !
!
. In addition, each passage through the cavity accrues a net phase shift !
, which is given by Equation (3). When expressed in complex exponential notation, it is readily apparent that the total amplitude of radiation emerging from the etalon, !
!
, is a geometric series with ratio !
an infinitely long etalon cavity, the total amplitude is:
!
!
!" < 1 . In the limit of
!
!
= !
!
!
!
+ !
!
!
!
!
!
!
!"   +   !
!
!
!
!
!
!
!
!
!
+ ⋯ =
!
!
!
!
1 − !
!
!
!"
.
(10)
Since the intensity of the transmitted beam, I
T
, is proportional to the squared modulus of this complex amplitude, it follows that
!
!
=
!
!
!
!
1 − !
!
!
!" !
=
!
!
!
!
1 − !
!
!" !
=
!
!
!
!
1 − !
!
!" 1 − !
!
!
!"
(11) where !
!
= !
!
!
, !
= !!
∗ = !
transmittance, and reflectance 1
!
,   and   !
= !
!
∗ = !
!
are the incident intensity,
, respectively. After expansion, the denominator becomes
1 − 2 !
cos !
+ !
!
= 1 − !
!
1 +
4 !
1 − !
!
sin 2
!
2
.
(12)
By substituting Equation (12) into Equation (11) and letting !
!
for the instrumental profile becomes
≡ !
!
1
This quantity is sometimes termed the power reflectivity or the intensity reflectivity.
!
!
, the expression
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!
!
=
!
!
1 − !
!
1 + where the Airy function, !
( !
; !
) , is
4 !
1 − !
!
sin 2
!
2
!
!
=
1
!
−
!
!
!
!
!
; !
(13)
!
!
; !
≡ 1 +
4 !
1 − !
!
sin 2
!
2
!
!
.
In the ideal case of a loss-less interface (i.e., no absorption), !
+ !
= 1 , so !
!
= !
!
;
(14)
!
Figure 2 illustrates the Airy function for an etalon similar to the one in the TFI. As expected from inspection of Equation (5), the Airy function for a specified gap exhibits a series of maxima that correspond to different orders, m . Larger orders occur at progressively smaller wavelengths. With an appropriate choice of blocking (i.e., order separating) filter in the light path, a specific order can be isolated and the resultant Airy function becomes the effective bandpass for a narrow-band filter.
For a given order, the Airy profile of an etalon can be tuned to transmit a specific central wavelength in one of three ways: (a) by varying the index of refraction, µ , in the etalon cavity; (b) by changing θ (“tilt scanning”); or (c) by changing the gap d (“gap scanning”). In recent years, gap scanning has become the method of choice because of the availability of highly accurate piezoelectric transducers (e.g., ceramics like leadzirconate-titanate [PZT]), which allow the etalon gap to be changed rapidly and controlled accurately via capacitance micrometry (Hicks et al. 1984).
.
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Figure 2: Top: The Airy function for an ideal etalon, computed with constant input quantities typical of the TFI. Top: !
= !
.
!" ( !
!
= !" ) , !
= !
.
!" , !
= !
.
!
!
m , !
= !
.
!
, !
!
= !"# ° , and !
= !
Successive orders are labeled, along with the free spectral range, FSR , and the full-width at half
° . maximum, !
!
!
.
Bottom: Same as above except for !
= !
.
!"   !
!
= !"# and !
= !
.
!" .
2.3
Etalon Performance Metrics
In this section, the following figures of merit that describe the performance of an etalon are defined and derived:
1) the free spectral range, FSR ;
2) the full-width at half maximum (FWHM) of a peak in the Airy profile, !
!
!
;
3) the reflective finesse, !
!
;
4) the resolving power, ℜ , which is sometimes loosely termed the “resolution; ” and
5) the contrast, !
.
2.3.1
Free Spectral Range
The free spectral range ( FSR ) is the separation in wavelength between adjacent orders for a specified gap (see Figure 2). It determines the properties of the blocking filters needed to exclude the transmission of adjacent orders.
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In terms of the angular phase, successive peaks are separated in the Airy function by
2 π radians, in the sense that higher orders are associated with larger phases.
Consequently, in terms of wavelength the FSR , Δλ , is given by
!" ≡ !
!
− !
!
!
!
= − 2 !
= !"
!"
!"
Making use of Equation (9), the FSR between order m and order m+1 is just
!" =
!
eff
!
+
!
!
!
!
!
!"
=
!
+
!
!
!
!
+
!
!
!
!
!
!"
.
(15)
(16)
For an ideal etalon, the phase change on reflection is taken to be a constant, and for simplicity most textbooks assume that it is negligible. In this case,
!" ≡ !
!
− !
!
!
!
=
!
!
!
!
!
.
(17)
Thus, as shown in Figure 2, the FSR decreases with increasing order, i.e., the peaks in the
Airy function become more closely spaced at shorter wavelength (higher orders). For the specific case illustrated in Figure 2, !
!
= 180° for all wavelengths, so Equation (16) shows that
!" =
!
!
!
!
!
+ 1
, which can be verified directly from the plot.
(18)
2.3.2
Full Width at Half Maximum
The sharpness of the transmission peak associated with a given order of the Airy profile is characterized by its FWHM, which can be calculated by setting
!
!
!
!
; !
= 0 .
5 = 1 + and solving for the half-width at half-maximum
4 !
1 − !
!
sin 2
!
!
!
2
!
!
(19)
!
!
!
= 2   sin !
!
1 −
4 !
!
!
≈ 2
1 − !
4 !
!
(20) where the final approximation is valid as long as the argument of the inverse sine function is small, i.e., as long as R is close to 1. With this approximation, the FWHM in radians, !
!
!
, !
, is
!
!
!
, !
= 2 !
!
!
= 4
1 − !
!
4 !
=
2 1 − !
!
, (21)
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SM-12 which depends only on the reflectance of the etalon coatings, R . Equation (9) can be used to convert this phase increment to wavelength:
!
!
!
= !
!
!
, !
×
For an ideal etalon with !
!
!"
!"
=
!
!
1 −
!
!
!
eff
+
!
!
!
!
!
!"
= 0 , this expression simplifies to
!
!
!
=
!
!"
1 − !
!
,
!
!
.
(22)
(23) which is the expression usually found in textbooks. Note in particular the proportionality with wavelength.
2.3.3
Reflective Finesse
The sharpness of a peak in the Airy profile may also be characterized by a dimensionless number know as the “finesse” of the etalon. The finesse is defined to be the ratio of distance between successive transmission peaks (i.e., the FSR ) expressed in units of the FWHM (i.e.,   !
!
!
). Since Equations (16) – (19) and (21) – (23) show that in the ideal case this ratio depends only on the reflectance of the etalon coatings, the instrumental broadening expressed in this way is termed the “reflective finesse” of the etalon, !
!
.
The expression for !
!
FWHM, respectively:
is easily obtained from Equations (16) and (22) for the FSR and
!
!
≡
Δ !
!
!
!
=
1
!
−
!
!
.
(24)
The reflective finesse is therefore completely described by the reflectance: a higher reflectance provides higher finesse, which implies sharper Airy profiles and better separated orders. These characteristics are evident in Figure 2, which compares !
!
(top panel) with !
!
= 105 (bottom panel).
= 20
Finally, note that the Airy function can be rewritten in terms of the reflective finesse:
!
!
; !
!
= 1 +
2 !
!
!
!
sin 2
!
2
!
!
.
(25)
2.3.4
Resolving Power
The resolving power of an etalon is defined conventionally for a given order of the
Airy function by
ℜ
!
≡
!
!
!
!
, (26)
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JWST-STScI-002059
SM-12 where the wavelength refers to the transmission peak. Equations (16) and (24) can be used to obtain:
ℜ
!
=   !
eff
+
!
!
!
!
!
!"
!
!
.
(27)
For an ideal etalon with !
!
= 0 , this expression simplifies to   ℜ =   !
!
!
. Hence, for a constant reflective finesse (or equivalently, a constant reflectance, R ), larger resolving powers are achieved by operating in higher orders. Conversely, for a given order, larger resolving powers require larger reflectances. However, since etalons used as tunable filters do not require the highest resolving power, low orders and modest reflectances are sufficient.
2.3.5
Contrast
The contrast of an etalon is defined to be the ratio between the peak transmission intensity and the minimum, inter-order intensity in the Airy profile. From Equation (14), this ratio for an idealized etalon is
!
=
!
!
= 2 !" ;   !
!
!
= 2 !
+ 1 !
;   !
=
1
1
+
−
!
!
!
(28) where !
= 0 , 1 , 2 , ⋯ . Thus, when the instrument broadening is dominated by the reflective finesse, the contrast of a transmission peak is independent of the order, and depends only on the reflectance of the etalon coating. The higher contrast associated with larger reflectance is visible in Figure 2, and can be attributed to the reduced interorder intensity in the far wings of a transmission peak.
3 Real Fabry-Pérot Etalons
The characteristics and performance of real etalons deviate in several respects from the ideal case presented in Section 2:
1) The reflection and transmission coefficients are wavelength-dependent. This adds an explicit wavelength dependence to all the equations presented in Section 2 where these terms appear.
2) There are wavelength-dependent phase shifts upon reflection at the surfaces of the plates, which are particularly significant when the etalon is used as a tunable filter because the gap is comparable to the thickness of the multilayer coatings. This wavelength dependence was carried throughout the derivations in Section 2, so the general form of the various equations is already available.
3) There is wavelength-dependent absorption from the coatings and substrates. This absorption is !
− !
!
. The net result is to decrease the maximum transmission associated with each peak in the Airy function from its previous value of !
!
!
1
!
−
=
!
1
!
−
!
!
!
[Equation (13)] to !
!
!
!
!
+ !
!
!
.
4) Real etalon plates typically exhibit a variety of defects, which include small deviations from parallelism, small amounts of bowing, and small amounts surface roughness. The treatment of these defects is discussed in the next section.
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3.1
Etalon Defects and the Concept of Effective Finesse
In Section 2.3.3, the concept of “finesse” was introduced as a figure of merit for the sharpness of a peak in Airy function compared with its inter-order spacing. In the ideal case, the angular width of a peak depends only on the quality of the reflective coating of the etalon cavity (i.e., the reflective finesse), as shown by Equation (25).
In reality, there are additional sources of instrumental broadening that must be considered. The concept of finesse is generalized to include the effect of each of these sources of additional broadening, !
!
, which are assumed to be independent and to add in quadrature
!
!
total
= !
!
!
!
+ !
!
!
(29) where the first term is the FWHM expressed in wavelength units [Equation (22)]. Since the physical mechanisms associated with these additional sources of instrumental broadening do not, on average, change the location of the peaks in the Airy function,
Equation (29) can be normalized by the FSR , Δ !
, which recasts it in terms of different contributions to the “effective finesse”, !
eff
:
!
!
!
1
!
!
eff
≡
!
total
Δ !
=
!
!
!
Δ !
+
!
!
Δ !
=
1
!
+
1
!
!
!
.
(30)
Although the subscript has been suppressed, all these contribution to the effective finesse are in general a function of wavelength.
Atherton et al. (1981) characterize three potential defects associated with real etalon plates that are potential contributors to the instrumental broadening function:
1) Bowing of the plates by an amount that is small compared with the gap (d) . A maximum departure from planarity of amplitude !"
!
microns produces a defect finesse of !
!
, !
=   !
( 2   !"
!
)   .
2) Departures from parallelism (“wedging”) between the plates that are small compared with the gap (d) . If the maximum wedge has amplitude !"
!
microns, the associated defect finesses is !
!
, !
=   !
( 3   !"
!
) .
3) Surface irregularities (roughness) on the plates, which have a small amplitude compared with the gap (d) , e.g., due to random polishing errors. If the surface irregularities follow a Gaussian distribution with dispersion (standard deviation) of !"
!
microns, the associated defect finesse is !
!
, !
=   !
( 4 .
7   !"
!
) .
The net “defect finesse”,   !
!
, associated with these defects in the etalon plates is
1
!
!
!
=
!
1
!
!
, !
+
!
1
!
!
, !
+
!
1
!
!
, !
=
4 !
!
!
!
+ 3 !
!
!
!
!
!
+ 22 !
!
!
!
.
(31)
In addition, Atherton et al. (1981) define an “aperture finesse”,   !
!
, which is due to the finite solid angle associated with radiation entering the etalon. Since not all incoming
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SM-12 rays traverse the etalon at the same angle, a peak in the Airy function is broadened slightly, essentially by the cos !
term in Equation (6). Atherton et al. (1981) express the aperture finesse as
!
!
=
2 !
!
Ω where m is the order and Ω is the solid angle of the incoming beam.
Thus, in general, the effective finesse of a real etalon is:
(32)
1
!
!
eff
=
1
!
!
!
+
1
!
!
!
+
1
!
!
!
.
(33)
3.2
The Airy Function Revisited
Equation (25) recast the Airy function for an ideal etalon in terms of the reflective finesse, !
!
, which provides the requisite scaling information to evaluate the broadening of a peak. For a real etalon, the reflective finesse must be replaced with the effective finesse, which yields the following more general expression for the Airy function
!
!
; !
eff , !
= 1 +
2 !
eff , !
!
!
sin 2
!
2
!
!
(34) where !
is given by Equation (3), as before.
With the inclusion of absorption, the transmission through a real etalon becomes
!
!
=
!
!
1 − !
!
!
!
!
; !
eff , !
=
!
!
!
!
+ !
!
!
!
!
; !
eff , !
(35)
4 The TFI Etalon
Dr. Craig Haley of COM DEV provided a detailed coating model for the proto-flight model (PFM) of the TFI etalon in 2009 September. This model consisted of tables listing the reflectance, transmittance, absorption, and phase shift of the etalon as a function of wavelength. Although this model had not been correlated in detail with the “as built” properties of the PFM etalon when we received it, it provided a sufficiently accurate representation of the device to permit the exploration of a variety of issues. Subsequent discussion to resolve minor discrepancies showed that COM DEV uses separate reflectivities for each etalon plate, while only a single value was provided to us. This makes little differences to the analysis presented here, which also assumes perfect AR coatings.
4.1
Basic Properties of the TFI Etalon
Figure 3 shows the variation of !
!
over the full operational range of the TFI, which extends from 1.5 to 5.0
!
m.
The reflectance is large (85 to 90%) and the absorption small ( ~ 0 .
001% ) over most of this region. However, the absorption becomes very
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SM-12 strong in the region between ~2.5 and ~3.2 !
m . This “non-functional” region 2 is unusable for scientific observations. It effectively divides the operational wavelength range of the TFI into two regions: a short wavelength region between 1.5 – 2.5 !
m , which is often referred to in a relative sense as the “blue” region and a long wavelength
(“red”) region between 3.2 and 5.0 !
m.
Within the functional region, the transmission is very high so that !
!
≈ 1 for peaks in the Airy function.
Figure 3 also shows the phase change upon reflection, which is not negligible and a significant function of wavelength. It varies from about 2.5
π to 3.5
π over the functional wavelength region, and is responsible for adding nearly one full order of interference to that provided by the optical path difference alone.
Figure 3: The variation of !
!
and !
!
for the PFM etalon over the operational wavelength range of the TFI. Significant absorption makes the region between 2.6 and 3.0 !
m unusable (shaded black).
The unusable region allowed by requirements is shaded in grey (2.5 to 3.2 !
m ).
2 The original design of the TFI contained two etalons, one each in short-wavelength and long-wavelength arms, which together provided complete coverage of the 1.5 – 5.0 !
m region. This design was descoped in
2005 to address problems with the ISIM mass budget. The unusual coating design of the PFM etalon was developed to enable scientific observations over as much of the original 1.5 – 5.0 !
m region as could be accommodated by a single device. The non-functional region is an artifact of these design choices.
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Figure 4: The relation between the etalon gap and the transmitted wavelength. The TFI will be routinely operated in the third order ( !
= !
; blue) on the short-wavelength side of the nonfunctional region and in the first order ( !
= !
;   red) on the long-wavelength side. Other orders can be accessed for engineering purposes.
Figure 4 shows the relation between the etalon gap, d , and the transmitted wavelength as computed from Equation (6). On the short-wavelength side of the non-functional region, the etalon is operated in the third order and on the long-wavelength side, in the first order. By “tuning” the separation between the etalon plates, the user will be able to choose an Airy bandpass centered on a wavelength of interest and the appropriate order will be isolated by one of eight blocking filters. Three examples that cover the full range of etalon gaps are shown in Figure 5. As expected from the discussion in Section 2.3, the
Airy profiles for a given order shift to longer wavelengths as the gap is increased and become significantly broader.
There are two wavelengths that are of particular operational importance to the TFI:
1) 2.054 µ m, which is the wavelength at which the “etalon check” procedure is performed; and
2) 4.600 µ m. which is the wavelength for which the coronagraphs and imaging with the non-redundant mask have been optimized.
With the current etalon coating model, these wavelengths correspond to etalon gaps of
4.126
µ m and 5.107
µ m in their respective operational orders.
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Figure 5a: Transmission profiles for the PFM etalon with a gap of 3.0 µ m. The reflective finesse was used to characterize the FWHM of the peaks. In flight, blocking filters will isolate the two operational orders, !
= !
on the short-wavelength side of the non-functional region and !
= !
on the long-wavelength side.
Figure 5b: Same as Figure 5a but for a gap of 4.2 µ m.
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Figure 5c: Same as Figure 5a but for a gap of 5.4 µ m.
4.2
The Effective Finesse
Equation (33) provides a general expression for the effective finesse of an etalon. In the specific case of the TFI, some further simplifications are possible if we assume that the etalon plates are not significantly bowed or wedged. In this case, the microroughness of the surface dominates the defect finesse. Furthermore, for an extended object that covers the entire field-of-view of the TFI (i.e., 2.2 arcmin
!
!
= 1 .
5 × 10 !
!
. Since the TFI operates in small orders, contributes negligibly to !
eff
.
!
!
2 ),
is very large and
Consequently, the effective finesses of the TFI can be approximated by:
!
1
!
eff, λ
≈
1
!
+
1
!
!
!
, !
=
1 − !
!
!
!
!
!
+
4 .
7
!
!
!
!
!
(36) where an explicit wavelength dependence has been indicated for the reflectance. This treatment characterizes the surface roughness under the assumption that the magnitude of the irregularities follows a Gaussian distribution. The irregularities can also be modeled with a fractal (i.e., power-law) distribution. M. Beaulieu (U de Montréal; private communication) reports that the behavior of the etalon with such an analysis is very similar to that of the standard Gaussian treatment.
The roughness is often expressed as the peak-to-valley (PTV) amplitude. For the TFI etalon, the PTV irregularities have a magnitude of approximately !
20 at the HeNe laser wavelength of 632.8 nm. This value corresponds to approximately !
63 (PTV) at 2 µ m.
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To convert from PTV to a standard deviation, i.e., the root-mean-square (RMS) deviation, we assume that the PTV amplitude corresponds to
3
of a Gaussian distribution, i.e., PTV = 6   !
!
!
. With this conversion, the RMS surface defects are approximately !
380 (or 5.3 nm) at 2 µ m, and even less at longer wavelengths.
The reflective finesse !
!
, !
is plotted in Figure 6 − it spans a range of approximately 17
!
380 at 2 µ m
(red curve), the value calculated above for the PFM etalon. We see that the effective finesse of the TFI etalon approaches closely its ideal reflective finesse provided that other defects of the plates are indeed negligible (as assumed in Equation [36]). To illustrate the sensitivity of the etalon performance to the plate roughness, the effective finesse is also plotted for !
!
!
=   !
100   and   !
200 , both of which are substantially worse than expected for the space-qualified TFI etalon and a small fraction (PTV) of the gap. In this and subsequent figures, the RMS surface defect is characterized as
!
!
=   !
100 ,   !
200 ,   and   !
380 for the fiducial wavelength of !
= 2   !
m . These
!
correspond to RMS dispersions of 20 nm, 10 nm, and 5.3 nm, respectively. Transmission profiles are plotted in Figure 8 for the same amplitudes of surface defects. As the defects become more important, the profiles become broader and their contrast decreases (see
Section 4.5). Surface irregularities of small amplitude can quickly degrade the effective finesse and the etalon bandpass.
Figure 6: The reflective finesse for the TFI etalon as a function of wavelength.
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Figure 7: The effective finesse for a range of surface roughness, !
!
!
specified for the fiducial wavelength of 2 µ m. The effective finesse (and performance) of the TFI etalon (red) approaches closely its reflective finesse (black; same as Figure 6). The etalon performance degrades substantially with increasing roughness of the etalon plates.
Figure 8: Transmission profiles for an etalon gap of 4.2 µ m are plotted for a range of surface irregularities specified for the fiducial wavelength of 2.0 µ m. The lower black curve is the profile for the reflective finesse and corresponds to Figure 5b.
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4.3
Free Spectral Range and Full-Width at Half-Maximum
Equations (16) and (22) provide general expressions for the (wavelength-dependent)
FSR and FWHM, respectively, which include the effects of the phase shift !
but neglect
!
broadening due to plate defects. The FSR and FWHM are plotted in Figs. 9 and 10, respectively, for several orders.
However, the more interesting parameterization of the FWHM of a peak in the Airy function is obtained from the effective finesse:
!
!
!
, !
=
Δ !
!
!
ef f , !
.
This FWHM is plotted in Figure 11 for an assumed range of surface defects: !
!
!
=
!
100 ,   !
200 ,   and   !
380 specified for the fiducial wavelength of 2.0 µ m. As the magnitude of the irregularities decreases (due, e.g., to better polishing), the FWHM approaches the ideal FWHM defined by the reflective finesse.
(37)
Figure 9: The free spectral range is plotted as a function of wavelength for different orders. The ripples on the short-wavelength side of the non-functional region are due to variations in the reflectance (Figure 3).
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Figure 10: The FWHM of the Airy profiles is plotted for several orders for the optimal case when the broadening is only due to the reflective finesse. Larger orders produce narrower peaks.
Figure 11: The FWHM of the Airy profiles in the operating orders of the TFI ( !
= !
on the short wavelength side of the non-functional region, !
= !
on the long-wavelength side) plotted for a range of surface defects. The lower black line is the FWHM for the reflective finesse (Figure 10).
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4.4
Resolving Power
Equation (27) provides the general expression for the wavelength-dependent resolving power of the TFI etalon, though the effective finesse should replace the reflective finesse:
ℜ
!
=   !
eff
+
!
!
!
!
!
!"
!
eff, !
.
(38)
This quantity is plotted in Figure 12 for the optimistic case   !
eff, !
= !
!
, !
. The average resolving power is about 100 over the operating orders of the TFI. The peak resolving power is 174 at a wavelength of 1.71 µ m on the blue side ( !
= 3 ) and 114 at 3.49 µ m on the red side ( !
= 1 ).
In Figure 13, the degradation of the resolving power when surface irregularities become significant is shown for the operating orders of the TFI. The resolving power is reduced by ~40% when the magnitude of the surface irregularities reaches !"
!
= !
100 at the fiducial wavelength of 2.0
µ m, which is much rougher than the TFI etalon plates.
Figure 12: The resolving power of the PFM etalon is plotted for several orders assuming that instrumental broadening is dominated by the reflective finesse.
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Figure 13: The resolving power of the TFI is shown for a range of surface defects and in the operating orders of the TFI, !
= !
on the short-wavelength side of the non-functional region and
!
= !
on the long-wavelength side. The upper black line is the resolving power for the reflective finesse (Figure 12). The actual performance of the PFM etalon is represented by the   !
/ !"# curve
(red).
4.5
Contrast
The etalon contrast is simply the ratio between the maximum and minimum transmission intensities. When Equation (34) is used, the generalized expression for the contrast is
!
!
= 1 +
2 !
eff, !
!
!
.
(39)
The contrast is plotted in Figure 14 for the optimistic case !
eff, !
= !
!
, !
. It spans a range of ~100 to ~600. As for the finesse, FWHM, and resolving power, surface irregularities will degrade the contrast. This is apparent in Figure 15, which shows the contrast for the same amplitude range of surface defects as before.
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Figure 14: The contrast of the PFM etalon is plotted assuming that the instrumental broadening is dominated by the reflective finesse.
Figure 15: The contrast of the PFM etalon for values of the effective finesse determined by a range of surface defect amplitudes. The upper black curve is the contrast for the reflective finesse (Figure
14).
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5 Detuning
When used as a tunable filter for narrow-band imaging, it is desirable for an etalon to produce a monochromatic image over the largest possible field-of-view 3 . However, as indicated by Equations (1) – (6), path differences within the etalon depend not only on the separation of the plates, d , but also on the angle of incidence, !
.
4 Consequently, variations in !
produce systematic shifts in the wavelength of the transmission peak across the field of view (FOV) of the TFI.
Equation (6) can be rearranged to emphasize the angular dependence of a transmission peak in the Airy function
!
!
, !
eff
, !
=
2 !
!
eff cos !
= !
!
cos !
≈ !
!
1 −
!
!
2
, (40) where !
!
≡ !
!
, !
eff
, !
= 0° = 2 !
!
eff
is the wavelength at the center of the field of view (i.e., !
= 0° , which is presumed to coincide with the optical axis of the instrument) and where the last approximation assumes that !
is small. Hence, when the etalon is
“tuned” to transmit wavelength !
!
at the center of the FOV, the wavelength of the transmission peak is systematically “detuned’ toward shorter wavelengths (i.e., blueshifted) with increasing radial distance from the optical axis. Detuning is an intrinsic property of an etalon, which depends only on the optical coating and the layout of the optical system.
The etalon of the TFI is very nearly at a pupil plane, and consequently the range of input angles over the FOV is quite small: 0° ≤ !
≤ 4 .
2° . The fractional change in wavelength from the center of the FOV to the edge is only !
!
2 = 0 .
3   % . This wavelength shift is small and can be ignored for many astronomical applications.
However, in a follow-up report, Martel & Fullerton (2010) present simulations of the effect of detuning for a variety of target spectra. Even though the fractional change in wavelength across the FOV is small, the effects of detuning on the observed count rate can be significant when steep gradients are present in the spectral energy distribution of a source. Detuning causes the Airy profile – which is itself a rapidly varying function near its peak – to “slide” along the rapidly varying spectral energy distribution of a target, which can result in large, position-dependent changes in the observed count rate of a given source or population of sources.
3 The monochromatic region produced by an etalon is often called the Jacquinot Spot.
4
The off-axis angle and !
tel collimator is !
!
= 5 .
94   ×   10 col
!
!
!
at the etalon can be related to the angle !
on the sky by !
=
are the focal length of the collimator and telescope, respectively. The focal length of the TFI
= 0 .
78   m and the effective focal length of the JWST OTE is !
tel
!
.
=
!
×   !
131 .
col
4   m
!
tel
where !
, so col
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6 Conclusion
This report is intended to serve as a general introduction to the optical properties of
Fabry-Pérot etalons, particularly when they are used as tunable filters. It also provides an initial assessment of the key properties of the etalon of the TFI.
Once the final design of the TFI etalon has been established and its basic properties have been characterized, such as the plate reflectivities, the derived parameters and associated figures in this report will be updated, including possible modifications resulting from the ground calibration campaign at cryogenic temperatures.
7 References
Atherton, P.D., Reay, N.K., Ring, J., & Hicks, T.R. 1981, “Tunable Fabry-Pérot Filters”,
Optical Engineering, Vol. 20, No. 6, pp. 806-814
Born, M., & Wolf, E. 2001, Principles of Optics, 7 th
University Press)
edition (Cambridge: Cambridge
Hicks, T. R., Reay, N. K., & Atherton, P. D. 1984, “The Application of Capacitance
Micrometry to the Control of Fabry-Pérot Etalons”, J. Phys. E. Sci. Instrum. Vol.
17, pp. 49--55
Martel, A.R. & Fullerton, A. 2010, Consequences of Etalon Detuning for TFI Science,
JWST-­‐STScI-­‐002190,  SM-­‐12
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