Method of fabricating anisotropic optical interference filter

Abstract

In a method of manufacturing a one-dimensionally varying optical filter, a substrate is coated to form a stack of layers of two or more different types. The coating may, for example, employ sputtering, electron-beam evaporation, or thermal evaporation. During the coating, the time-averaged deposition rate is varied along an optical gradient direction by generating reciprocation between a shadow mask and the substrate in a reciprocation direction that is transverse to the optical gradient direction. In some approaches, the shadow mask is periodic with a mask period defined along the direction of reciprocation, and the generated reciprocation has a stroke equal to or greater than the mask period along the direction of reciprocation. The substrate and the shadow mask may also be rotated together as a unit during the coating. Also disclosed are one-dimensionally varying optical filters, such as linear variable filters, made by such methods.

Description

BACKGROUND

The following relates to the optical arts, optical filter arts, and related arts.

Optical filters with high spectral selectivity can be manufactured using a stack of layers with alternating layers of two (or more) constituent materials having different refractive index values. Such filters are sometimes called interference filters, and can be designed and variously combined to provide a designed pass-band, stop-band, high-pass, or low-pass output. A notch filter can be designed by constructing the stack of layers to form a Bragg reflector blocking a stop-band. The layer stack is deposited on a substrate that is optically transmissive for the wavelength or wavelength range to be transmitted, and may for example be a glass plate for an optical filter in the visible spectrum.

A linear variable filter is similar, but includes a linear gradient in the thickness(es) of one or more layers of the interference filter. The linear gradient produces a corresponding linear gradient in the wavelength (e.g. passband), though the correspondence is not necessarily linear in nature.

Some improvements are disclosed herein.

BRIEF SUMMARY

In one illustrative aspect, a method of manufacturing a one-dimensionally varying optical filter is disclosed. The method comprises: providing a shadow mask having an opening fraction along a direction of reciprocation that varies along a direction transverse to the direction of reciprocation; coating a substrate through the shadow mask to form a stack of layers of two or more different types; and during the coating, generating reciprocation between the shadow mask and the substrate in the direction of reciprocation. Also disclosed is a one-dimensionally varying optical filter manufactured by the foregoing method.

In another illustrative aspect, a method of manufacturing a one-dimensionally varying optical filter is disclosed. The method comprises: coating a substrate through a shadow mask to form a stack of layers of two or more different types; and, during the coating, generating reciprocation between the shadow mask and the substrate. The shadow mask has openings configured to, in combination with the generated reciprocation, define a coating duty cycle that varies along a direction transverse to the direction of reciprocation. Also disclosed is a linear variable filter manufactured by the foregoing method.

In another illustrative aspect, a method of manufacturing a one-dimensionally varying optical filter is disclosed. The method comprises coating a substrate to form a stack of layers of two or more different types and, during the coating, varying the time-averaged deposition rate along an optical gradient direction by generating reciprocation between a shadow mask and the substrate in a reciprocation direction that is transverse to the optical gradient direction. In some embodiments the shadow mask is periodic with a mask period defined along the direction of reciprocation, and the generated reciprocation has a stroke equal to or greater than the mask period along the direction of reciprocation.

In another illustrative aspect, a method of manufacturing a one-dimensionally varying optical filter is disclosed. The method comprises: providing a shadow mask having openings; coating a substrate through the shadow mask to form a stack of layers of two or more different types; and during the coating, generating reciprocation between the shadow mask and the substrate in the direction of reciprocation. The duty cycle of the reciprocating is varied during the coating in order to produce a multiplicity of filters having similar profiles but differing wavelength characteristics via a single deposition. In some embodiments, the openings of the shadow mask do not have an opening fraction along a direction of reciprocation that varies along a direction transverse to the direction of reciprocation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 diagrammatically represents a linear variable filter deposition system as disclosed herein.

FIG. 2 diagrammatically shows a plan view of a shadow mask suitably used in the system of FIG. 1 in generating one linear variable filter.

FIG. 3 (right side) diagrammatically shows a portion of the shadow mask of FIG. 2, while FIG. 3 (left side) plots the time-averaged percent coating time provided by the shadow mask operating in the deposition system of FIG. 1.

FIG. 4 diagrammatically shows a plan view of a shadow mask wafer suitably used in the system of FIG. 1, in which the shadow mask wafer includes three shadow masks of the type shown in FIG. 2 usable to simultaneously deposit three linear variable filter elements using the system of FIG. 1.

FIG. 5 diagrammatically shows a plan view of a wafer with three linear variable filter elements deposited using the system of FIG. 1 and the shadow mask wafer of FIG. 4.

FIGS. 6, 7, and 8 diagrammatically shows plan views of additional shadow masks that may be suitably used in the system of FIG. 1 in generating linear variable filters with different linear gradient profiles.

DETAILED DESCRIPTION

With reference to FIG. 1, a one-dimensionally varying filter manufacturing system deposits a one-dimensionally varying filter on a substrate 10 includes an illustrative two coating material sources 12, 14 controlled by respective shutters 16, 18. At the illustrative point in time shown in FIG. 1, shutter 16 is open allowing a deposition beam 20 from the coating material source 12 to impinge upon the substrate 10 through a shadow mask wafer 22, while the shutter 18 is closed thus blocking deposition from the coating material source 14. It will be appreciated that deposition by coating material source 14 can be performed by closing shutter 16 and opening shutter 18 (configuration shown in phantom in FIG. 1), or that both shutters 16, 18 could be opened to co-deposit using both coating material sources 12, 14. The illustrative coating sources 12, 14 can be thermal evaporation sources heated by a heat source such as a resistive heater or an electron beam, or sputtering sources in which energetic particles ejects material from a sputtering target, various combinations thereof, or so forth. While two illustrative coating sources 12, 14 are shown, a multiplicity of sources could be employed to overcome various limitations inherent in the individual sources or in the fabrication of layers with complex compositions such as ternary alloys, which in some embodiments may include continuously varying compositions (i.e. compositional grading) over the thickness of a layer.

With continuing reference to FIG. 1 and with further reference to FIG. 2, the shadow mask wafer 22 includes a shadow mask 30 comprising a one-dimensional array of wedge-shaped openings 32 along a direction of reciprocation (d_R), with each wedge extending from a narrow end to a wide end along a direction (d_G) transverse to the direction of reciprocation d_R. The shadow mask wafer 22 is arranged to reciprocate respective to the substrate 10 in the direction of reciprocation d_R during the coating of the layers making up the interference filter. In diagrammatic FIG. 1, the substrate 10 is mounted on a substrate holder 40, and a reciprocation mechanism includes an actuator arm 42 extending from the substrate holder 40 to connect with and reciprocate the shadow mask wafer 22. A support arm 44 supports the end of the shadow mask wafer 22 distal from its connection with the actuator arm 42. Note that the features 40, 42, 44 are shown in phantom (using dashed lines) to reveal otherwise-occluded features.

In the system of illustrative FIG. 1, the substrate 10 and shadow mask wafer 22 are rotated together as a unit by rotating the substrate holder 40 about a shaft 46. The rotation is optional, but if provided can enhance uniformity of deposition, especially if the coating material sources 12, 14 are located off-center. If rotation is provided, the substrate and shadow mask should rotate together as a unit. It is further contemplated that this entire system may rotate about a second axis in a so-called planetary configuration wherein the device as shown in FIG. 1 constitutes one of the planets.

It is to be appreciated that the illustrative substrate holder and reciprocation mechanism 40, 42, 44 is a diagrammatic representation, and various mechanisms can be employed to generate the reciprocation between the shadow mask 22 and the substrate 10 in the direction of reciprocation d_R. In the illustrative embodiment the shadow mask 22 reciprocates while the substrate 10 does not reciprocate; however, in other contemplated embodiments the reciprocation is generated by reciprocating the substrate while the shadow mask does not reciprocate. In embodiments that include rotation of the substrate/shadow mask unit, it is contemplated to use suitable gearing to convert rotary motion to generate the reciprocation of the shadow mask or the substrate.

It will also be appreciated that the deposition system of FIG. 1 may be disposed in a vacuum chamber or other chamber providing controlled ambient composition and pressure. Various other conventional features may be included which are not shown in diagrammatic FIG. 1, such as a substrate heater to control substrate temperature during deposition, a deposition monitoring sensor to measure deposition rate optically or by another mechanism, beam flux sensors to measure the flux of the deposition beam 20, and so forth.

With continuing reference to FIGS. 1 and 2 and with further reference to FIG. 3, the effect of reciprocation between the shadow mask 22 and the substrate 10 in the reciprocation direction d_R is to vary the time-averaged deposition rate along the optical gradient direction d_G, so that the deposited layers exhibit a thickness gradient along the optical gradient direction d_G. As already mentioned, the shadow mask 30 comprises a one-dimensional array of wedge-shaped openings 32 along the direction of reciprocation d_R, with each wedge extending from a narrow end to a wide end along the (optical gradient) direction d_G transverse to the direction of reciprocation d_R. The one-dimensional array of wedge-shaped openings 32 has a period T_M (also referred to herein as the “mask period” T_M) along the direction of reciprocation d_R. If the stroke of the generated reciprocation is greater than this mask period T_M then, as graphically depicted in FIG. 3, the coating experienced by any point on the substrate 10 is a time-averaged combination of a coating time corresponding to when the point on the substrate is exposed to the deposition beam 20 through one of the wedge-shaped openings 32 and the “blocked” time corresponding to when the point on the substrate is blocked from exposure to the deposition beam 20 by the shadow mask 30. The wedge-shaped openings 32 provide a greater time-averaged exposure for the wide end of the wedge as compared with the narrow end of the wedge. Accordingly, the time-averaged exposure varies gradually from a highest exposure (and hence largest thickness for the deposited layer) at the wide end of the wedge to a lowest exposure (and hence lowest thickness for the deposited layer) at the narrow end of the wedge. This is plotted on the left-hand side of FIG. 3, where in this illustrative example the percent coating (time averaged) ranges from 80% at the wide end of the wedge to 30% at the narrow end of the wedge. This gradient in time-averaged deposition exposure time translates to a corresponding gradient in the deposited layer thickness along the direction d_G transverse to the direction of reciprocation d_R—this direction d_G is therefore referred to as the optical gradient direction d_G, because in an interference filter the wavelength of the passband (or of the notch in the case of a notch filter) is controlled by the thicknesses of the layers and their respective refractive indices so that the passband (or notch) varies along the optical gradient direction d_G in accord with the varying of layer thicknesses.

Another way of describing this effect is that, during the coating, the reciprocation between the shadow mask and the substrate produces a coating “duty cycle” that varies along the (optical gradient) direction do oriented transverse to the direction of reciprocation d_R. In FIG. 3, the coating duty cycle is 80% at the thick end of the wedge, and decreases to a coating duty cycle of 30% at the narrow end of the wedge.

With reference to FIG. 4, it is contemplated to arrange a plurality of shadow masks 30 on the shadow mask wafer 22, e.g. three shadow masks 30 on the shadow mask wafer 22 in the illustrative example of FIG. 4. In FIG. 4 the stroke of the reciprocation is denoted ds, and it will be appreciated that the end portions of the shadow masks 30 of length equal to the stroke ds will not provide reliable results because during part of the stroke the mask moves “out of range”.

With continuing reference to FIG. 4 and with further reference to FIG. 5, the three shadow masks 30 of the configuration of the shadow mask wafer 22 shown in FIG. 4 produce a corresponding three interference filters 50 after deposition. As noted in FIG. 5, each interference filter 50 has a thickness gradient (and hence a gradient in its optical bandpass wavelength or optical notch) that varies along the optical gradient direction d_G.

In diagrammatic FIG. 1, the separation between the substrate 10 and the shadow mask 22 is relatively large. In general, reducing this separation reduces leakage and provides more precise thickness control, so that the shadow mask is preferably positioned close to the surface of the substrate 10. However, if it is too close there is increased possibility of particulates coming off the shadow mask 22 and contaminating the optical interference filter, so that some finite separation is usually advantageous.

With returning reference to FIG. 1, in a typical interference filter manufacturing sequence, the substrate 10 is loaded into the deposition system, and various preparatory operations are optionally performed such as evacuating the vacuum chamber, backfilling with any desired ambient, turning on a substrate heater to bring the substrate 10 to a desired deposition temperature, and so forth. Both shutters 16, 18 are initially closed, and any preparation of the deposition sources 12, 14 is performed (for example, starting the resistive heaters or electron beams to raise the sources to a temperature at which they produce source material evaporation, or setting up the sputtering beam, et cetera). The reciprocation actuator 42 is started to generate reciprocation between the shadow mask 22 and the substrate 10. Thereafter, the deposition is performed by alternating between the states: (1) shutter 16 open and shutter 18 closed; and (2) shutter 16 closed and shutter 18 open. Each such alternation produces one period of the interference filter, which typically includes (without loss of generality) N periods. The stroke and frequency of the reciprocation is typically held constant throughout the interference filter evaporation—as a consequence, each constituent layer has the same percentage gradient in its thickness over the length of the interference filter in the optical gradient direction d_G. For example, referring back to the illustrative example of FIG. 3, if the thickness of a layer “A” deposited by source 12 would be 100 nm in the absence of the shadow mask 30, then layer “A” will vary from 80 nm at the wide end of the wedge to 30 nm at the narrow end of the wedge; while, if the thickness of a layer “B” deposited by source 12 would be 150 nm in the absence of the shadow mask 30, then layer “B” will vary from 120 nm at the wide end of the wedge to 45 nm at the narrow end of the wedge.

It should be noted that the reciprocation can have various characteristics. For example, the reciprocation can be sinusoidal in nature, e.g. moving fastest at the mid-point of the stroke and slowing down to zero velocity at each end of the stroke, which is advantageous from a mechanical reliability/lifetime standpoint. While the detailed stroke timing may affect the effective coating duty cycle, this is suitably accounted for during calibration as described next.

The detailed shape of the openings of the shadow mask 30 can be designed as follows. A calibration shadow mask is made with wedges 32 having straight edges running from very wide to very narrow, and a calibration deposition is performed using this calibration shadow mask and only the deposition source 12 (single layer deposition). The layer thickness is then profiled along the optical gradient direction d_G (for example, using ellipsometry or another optical technique, or mechanical profilometry along the layer edge, or so forth) to determine the time-averaged deposition rate versus coating duty cycle provided by the shadow mask. The same calibration is run for the deposition source 14. (As previously noted, the percentage gradient should be the same for both layers, but the deposition rate may be different, and so it may be advantageous to perform separate calibrations for both sources 12, 14 as just mentioned). In parallel, the interference filter is designed using conventional optical filter design methods thereby providing the desired thickness as a function of position along the optical gradient direction d_G. For a linear variable filter, this thickness as a function of position should be linear; more generally, however, the disclosed manufacturing techniques enable fabrication of one-dimensionally varying optical filter in which the thickness may vary non-linearly along the optical gradient direction d_G. Given the thickness versus position along the optical gradient direction d_G (from the filter design) and the time-averaged deposition rate versus coating duty cycle provided by the shadow mask calibrations, it is straightforward to design the wedge width along the reciprocation direction d_R as a function of position along the optical gradient direction d_G in order to achieve the target thickness versus position along the optical gradient direction d_G.

The illustrative wedge openings 32 of the embodiment of FIGS. 2-4 are one suitable geometry for the openings of the shadow mask. However, other opening designs can be used. In general, the goal is to provide a shadow mask having an opening fraction along the direction of reciprocation d_R that varies along the transverse optical gradient direction d_G with the desired profile. The mask should be periodic with mask period T_M smaller than the reciprocation stroke ds, so that the reciprocation produces time-averaged coating over a reciprocation cycle corresponding to the opening fraction of the mask.

With reference to FIGS. 6-8, some other illustrative shadow masks are shown. In a “digital” shadow mask 60 shown in FIG. 6, openings 61, 62, 63, 64 are each strips oriented along the optical gradient direction d_G of successively longer length, with the longest strip openings 64 extending over most of the shadow mask 60 along the optical gradient direction d_G. The resulting coating duty cycle versus position along optical gradient direction d_G is plotted to the left of the shadow mask 60. While the illustrated coating duty cycle exhibits discrete “jumps”, these abrupt transitions along the optical gradient direction d_G may be washed out in an actual deposition by spreading of the deposition beam 20 and other factors. While the illustrative digital shadow mask 60 has strip openings 61, 62, 63, 64 of four lengths in the mask period T_M, it will be appreciated that strips of more different lengths may be provided to fine-tune the gradient.

With reference to FIG. 7, a shadow mask 70 includes “double-wedge” openings 72 in which the widest part of each wedge meets in the middle of the mask. This results in a thickness versus position profile along the optical gradient direction d_G as shown to the left of the mask 70 in FIG. 7. This illustrates that the thickness-versus-position profile does not need to be monotonic.

With reference to FIG. 8, a shadow mask 80 includes openings 82 which have strongly non-linear right-hand edges, producing a strongly non-linear thickness versus position profile along the optical gradient direction d_G as shown to the left of the mask 80 in FIG. 8.

The illustrative fabrication techniques for manufacturing a one-dimensionally varying optical filter can be used for a wide range of filter types. In the case of transmission filters, the substrate 10 is typically a glass substrate or other substrate of transparent material (at the wavelength range of interest). In an illustrative example, the layers are amorphous silicon and silicon dioxide, although a wide range of dielectric, semiconducting, metal, or other optical-grade materials may be used. The disclosed techniques can also be used to generate reflective interference filters (sometimes referred to as Bragg reflectors)—in this case the substrate may be optically opaque, translucent, or transparent, and the layers may be various materials that, in a stacked arrangement, produce the desired spectrally selective reflection.

In another contemplated approach, the openings of the shadow mask do not include the linear gradient, e.g. the wedge-shaped openings 32 of the shadow mask 30 are replaced by rectangular openings. In such a case, no optical gradient along the optical gradient direction d_G is defined (since there is no variation in the opening “width” along the direction do with rectangular openings). However, such a mask can still be useful, for example, by adjusting the duty cycle of the reciprocation in the direction of reciprocation d_R during deposition of successive filters (or filter sections). This has the effect of producing a multiplicity of filters having similar profiles but differing duty cycles and wavelength characteristics fabricated via a single deposition. For example, the first filter in the stack of filters may use a duty cycle giving thicker layers corresponding to a first wavelength, while the second filter may use a duty cycle giving thinner layers corresponding to a second wavelength, and so forth. This duty cycle variation concept can also be used in conjunction with a mask such as the mask 30 of FIG. 1 having wedge-shaped openings or other openings that vary along a direction d_G to produce an optical gradient along direction do as well as different layer thicknesses over the height of the layer stack due to variation of the reciprocation duty cycle.

It will be appreciated that various of the above-disclosed and other features and functions, or alternatives thereof, may be desirably combined into many other different systems or applications. It will be further appreciated that various presently unforeseen or unanticipated alternatives, modifications, variations or improvements therein may be subsequently made by those skilled in the art which are also intended to be encompassed by the following claims.

Claims

1. A method of manufacturing a one-dimensionally varying optical filter, the method comprising: providing a shadow mask wafer having a plurality of shadow masks disposed thereon, at least one of the plurality of shadow masks having an opening fraction along a direction of reciprocation that varies along a direction transverse to the direction of reciprocation;coating a substrate through the at least one shadow mask to form a stack of layers of two or more different types; andduring the coating, generating reciprocation between the shadow mask wafer and the substrate in the direction of reciprocation.

2. The method of claim 1 further comprising: during the coating, rotating the substrate and the shadow mask wafer together as a unit.

3. The method of claim 1 wherein the providing comprises: providing the shadow mask wafer having the plurality of shadow masks disposed thereon, wherein the at least one of the plurality of shadow masks having a one-dimensional array of wedge-shaped openings along the direction of reciprocation with each wedge extending from a narrow end to a wide end along the direction transverse to the direction of reciprocation.

4. The method of claim 1 wherein the providing comprises: providing the shadow mask wafer having the plurality of shadow masks disposed thereon, wherein the at least one of the plurality of shadow masks having a density of openings along the direction of reciprocation that varies along the direction transverse to the direction of reciprocation.

5. The method of claim 1 wherein the variation of the opening fraction along the direction transverse to the direction of reciprocation is effective to deposit the one-dimensionally varying optical filter as a linear variable filter.

6. The method of claim 1 wherein the opening fraction of the at least one shadow mask along the direction of reciprocation varies non-linearly along the direction transverse to the direction of reciprocation.

7. The method of claim 1 wherein the coating comprises one of sputtering, electron-beam evaporation, and thermal evaporation.

8. The method of claim 1 wherein: the providing comprises providing the at least one of the plurality of shadow masks on the shadow mask wafer having a repeating pattern of one or more openings along the direction of reciprocation; andthe generated reciprocation has a stroke equal to or greater than the period of the repeating pattern of one or more openings along the direction of reciprocation.

9. The method of claim 1 wherein the operation of generating reciprocation between the shadow mask wafer and the substrate consists of one of: reciprocating the shadow mask wafer in the direction of reciprocation while not reciprocating the substrate; andreciprocating the substrate in the direction of reciprocation while not reciprocating the shadow mask wafer.

10. A one-dimensionally varying optical filter manufactured by a method as set forth in claim 1.

11. A method of manufacturing a one-dimensionally varying optical filter, the method comprising: coating a substrate through at least one of a plurality of shadow masks on a shadow mask wafer to form a stack of layers of two or more different types; andduring the coating, generating reciprocation between the shadow mask wafer and the substrate wherein the at least one of the plurality of shadow masks has openings configured to, in combination with the generated reciprocation, define a coating duty cycle that varies along a direction transverse to the direction of reciprocation.

12. The method of claim 11 further comprising: during the coating, rotating the substrate and the shadow mask wafer together as a unit.

13. The method of claim 11 wherein the at least one of the plurality of shadow masks has a one-dimensional array of wedge-shaped openings along the direction of reciprocation with each wedge extending from a narrow end to a wide end along the direction transverse to the direction of reciprocation.

14. The method of claim 11 wherein the at least one of the plurality of shadow masks has a density of openings along the direction of reciprocation that varies along the direction transverse to the direction of reciprocation.

15. The method of claim 11 wherein the openings in combination with the generated reciprocation are configured to define a coating duty cycle variation along the direction transverse to the direction of reciprocation that is effective to deposit the one-dimensionally varying optical filter as a linear variable filter.

16. The method of claim 11 wherein the coating comprises one of sputtering, electron-beam evaporation, and thermal evaporation.

17. The method of claim 11 wherein: the at least one of the plurality of shadow masks has a repeating pattern of one or more openings along the direction of reciprocation defining a mask period along the direction of reciprocation, andthe generated reciprocation has a stroke equal to or greater than the mask period along the direction of reciprocation.

18. The method of claim 11 wherein the operation of generating reciprocation between the shadow mask wafer and the substrate consists of one of: reciprocating the shadow mask wafer in the direction of reciprocation while not reciprocating the substrate; andreciprocating the substrate in the direction of reciprocation while not reciprocating the shadow mask wafer.

19. A linear variable filter manufactured by a method as set forth in claim 11.

20. A method of manufacturing a one-dimensionally varying optical filter, the method comprising: coating a substrate to form a stack of layers of two or more different types; andduring the coating, varying the time-averaged deposition rate along an optical gradient direction by generating reciprocation between a shadow mask wafer having a plurality of shadow masks disposed thereon and the substrate in a reciprocation direction that is transverse to the optical gradient direction.

21. The method of claim 20 further comprising: during the coating, rotating the substrate and the shadow mask wafer together as a unit.

22. A method of claim 21 further comprising rotating the rotating unit of the substrate and the shadow mask wafer about a second axis as a planet in a planetary configuration.

23. The method of claim 20 wherein the coating comprises one of sputtering, electron-beam evaporation, and thermal evaporation.

24. The method of claim 20 wherein: at least one of the plurality of shadow masks is periodic with a mask period defined along the direction of reciprocation, andthe generated reciprocation has a stroke equal to or greater than the mask period along the direction of reciprocation.

25. The method of claim 20 wherein the operation of generating reciprocation between the shadow mask wafer and the substrate consists of one of: reciprocating the shadow mask wafer in the direction of reciprocation while not reciprocating the substrate; andreciprocating the substrate in the direction of reciprocation while not reciprocating the shadow mask wafer.

26. A method of manufacturing a one-dimensionally varying optical filter, the method comprising: providing a shadow mask wafer having a plurality of shadow masks disposed therein, wherein at least one of the plurality of shadow masks having openings;coating a substrate through the at least one shadow mask to form a stack of layers of two or more different types; andduring the coating, generating reciprocation between the shadow mask wafer and the substrate in the direction of reciprocation;wherein the duty cycle of the reciprocating is varied during the coating in order to produce a multiplicity of filters having similar profiles but differing wavelength characteristics via a single deposition.

27. The method of claim 26 wherein the openings of the at least one of the plurality of shadow masks do not have an opening fraction along a direction of reciprocation that varies along a direction transverse to the direction of reciprocation.