High resolution, high etendue LVF monochromator using a freeform corrector

Abstract

High resolution optical spectrometers suitable for low illumination environments notwithstanding the inverse relationship between spectral resolution and etendue are described. The techniques described herein rely on freeform lenses designed to eliminate (or at least reduce) spectral broadening caused by the finite range of incidence angles generated by a beam with high etendue. Performance of a spectrometer can be improved by compensating blue shift effects with a corresponding red shift——a wavelength shift in the opposite direction. This can be accomplished by steering rays having a greater angle of incidence towards regions of the filter characterized by larger wavelengths of transmission.

Description

HIGH RESOLUTION, HIGH ETENDUE LVF MONOCHROMATOR USING A FREEFORM CORRECTORRELATED APPLICATIONS

[0001] This application claims the benefit under 35 U.S.C. § 119(e) of U.S. Provisional Application No. 63 / 510,698, entitled “HIGH RESOLUTION, HIGH ETENDUE LVG MONOCHROMATOR USING A FREEFORM CORRECTOR,” filed June 28, 2023, the entire contents of which are incorporated herein by reference.BACKGROUND

[0002] Optical spectrometers are used in a variety of contexts, for example to identify and analyze the composition of a particular object. Intensity of light scattered from the object of interest is broken down into spectral bands, and the energy at each band is quantified.BRIEF SUMMARY

[0003] Some embodiments relate to an optical spectrometer, comprising: an optical filter having a first mirror, a second mirror, and an optical cavity formed between the first and second mirrors, the optical cavity having a varying thickness along a transverse axis so that a first portion of the optical cavity has a first thickness and a second portion of the optical cavity has a second thickness smaller than the first thickness, the first and second portions being offset relative to each other along the transverse axis; a lens, defining an optical axis perpendicular to the transverse axis, configured to focus incident light on the optical lens, the lens being shaped so that: rays of the incident light that emerge from the lens at a first angle relative to the optical axis are focused on the first portion of the optical cavity, and rays of the incident light that emerge from the lens at a second angle, less than the first angle, relative to the optical axis are focused on the second portion of the optical cavity; and one or more photodetectors, wherein the optical filter is disposed optically between the lens and the one or more photodetectors.

[0004] In some embodiments, the thickness of the optical cavity varies linearly along the transverse axis.

[0005] In some embodiments, the first and second mirrors comprises stacks of dielectric layers.

[0006] Some embodiments further comprise, a mechanism configured to change a relative position of the optical filter relative to the lens along the transverse axis.

[0007] In some embodiments, the one or more photodetectors form a photodetector array.

[0008] In some embodiments, the optical filter has a spectral width in transmission that is less than 30 nm.

[0009] In some embodiments, the lens defines a surface having a first surface portion and a second surface portion having opposite concavities relative to each other, the first and second surface portions being offset relative to each other along the transverse axis.

[0010] In some embodiments, the surface has a cubic term along the transverse axis.

[0011] In some embodiments, the surface has a quadratic term along a lateral axis that is perpendicular to both the transverse axis and the optical axis.

[0012] Some embodiments relate to an optical spectrometer, comprising: an optical filter having a first mirror, a second mirror, and an optical cavity formed between the first and second mirrors, the optical cavity having a varying thickness along a transverse axis so that a first portion of the optical cavity has a first thickness and a second portion of the optical cavity has a second thickness smaller than the first thickness, the first and second portions being offset relative to each other along the transverse axis; a lens, defining an optical axis perpendicular to the transverse axis, defining a surface having a first surface portion and a second surface portion having opposite concavities relative to each other, the first and second surface portions being offset relative to each other along the transverse axis; and one or more photodetectors, wherein the optical filter is disposed optically between the lens and the one or more photodetectors.

[0013] In some embodiments, the first surface portion is such that a concave portion faces the filter, and the second surface portion is such that a convex portion faces the filter; the first portion of the optical cavity is on a first side of the second portion of the optical cavity with respect to the optical axis; and the first surface portion of the lens is on the first side of the second surface portion of the lens with respect to the optical axis.

[0014] In some embodiments, the surface has a profile, along the transverse axis, defined by a polynomial having at least one odd term.

[0015] In some embodiments, the surface has a profile, along the transverse axis, defined by a polynomial having at least one cubic term.

[0016] In some embodiments, the surface has a profile, along a lateral axis that is perpendicular to both the transverse axis and the optical axis, defined by a polynomial having at least one even term.

[0017] In some embodiments, the surface has a profile, along a lateral axis that is perpendicular to both the transverse axis and the optical axis, defined by a polynomial having at least one quadratic term.

[0018] Some embodiments relate to a method of assembling an optical spectrometer, comprising: obtaining an optical filter having a first mirror, a second mirror, and an optical cavity formed between the first and second mirrors, the optical cavity having a varying thickness along a transverse axis so that a first portion of the optical cavity has a first thickness and a second portion of the optical cavity has a second thickness smaller than the first thickness, the first and second portions being offset relative to each other along the transverse axis; obtaining a lens, defining an optical axis perpendicular to the transverse axis, defining a surface having a first surface portion and a second surface portion having opposite concavities relative to each other, the first and second surface portions being offset relative to each other along the transverse axis; obtaining one or more photodetectors; and positioning the optical filter, the lens, and the one or more photodetectors so that the optical filter is disposed optically between the lens and the one or more photodetectors.

[0019] In some embodiments, the first surface portion is such that a concave portion faces the filter, and the second surface portion is such that a convex portion faces the filter; the first portion of the optical cavity is on a first side of the second portion of the optical cavity with respect to the optical axis; and the first surface portion of the lens is on the first side of the second surface portion of the lens with respect to the optical axis.

[0020] In some embodiments, the surface has a profile, along the transverse axis, defined by a polynomial having at least one odd term.

[0021] In some embodiments, the surface has a profile, along the transverse axis, defined by a polynomial having at least one cubic term.

[0022] In some embodiments, the surface has a profile, along a lateral axis that is perpendicular to both the transverse axis and the optical axis, defined by a polynomial having at least one even term.

[0023] The foregoing summary is provided by way of illustration and is not intended to be limiting.BRIEF DESCRIPTION OF DRAWINGS

[0024] The accompanying drawings are not intended to be drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures is represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing.

[0025] FIG. 1 A is a schematic diagram illustrating a filter illuminated by a beam of light having a certain spot size.

[0026] FIG. IB is a schematic diagram illustrating a filter illuminated by a beam of light having a certain divergence.

[0027] FIG. 2A is a schematic diagram illustrating a spectrometer including a photodetector array, in accordance with some embodiments.

[0028] FIG. 2B is a schematic diagram illustrating a spectrometer including a single photodetector, in accordance with some embodiments.

[0029] FIG. 3 is a schematic diagram illustrating a spectrometer having a spherical lens.

[0030] FIG. 4A is a perspective schematic diagram illustrating a spectrometer having a freeform lens, in accordance with some embodiments.

[0031] FIG. 4B is a sideview schematic diagram illustrating a spectrometer having a freeform lens, in accordance with some embodiments.

[0032] FIG. 5 is a perspective schematic diagram illustrating an example of a freeform lens, in accordance with some embodiments.DETAILED DESCRIPTIONI. Overview

[0033] Described herein are high resolution optical spectrometers suitable for low illumination environments notwithstanding the inverse relationship between spectral resolution and etendue. Military applications are an example of a low illumination environment, where passive spectroscopy is often favored over active illumination spectroscopy because it avoids compromising the position of an observer. The techniques described herein rely on freeform lenses designed to eliminate (or at least reduce) spectral broadening caused by the finite range of incidence angles generated by a beam with high etendue. The lenses described herein can be used with various types of spectrometers. One ofthese spectrometers is the so-called “LVF” or “linear variable filter” spectrometer. This type of spectrometer relies on LVF filters. LVFs are a type of optical filter that feature a continuously varying spectral response along their length. They are used to selectively transmit or block certain wavelengths of light in a controlled manner. The spectral response of the filter changes linearly along its length. This means that different sections of the filter will pass different wavelengths of light. There are various types of LVF filters, including bandpass, longpass, and shortpass filters, each designed to transmit or block specific ranges of wavelengths.

[0034] The inventor has recognized and appreciated that the performance of conventional LVF spectrometers is limited by the inverse relationship existing between spectral resolution and etendue. The spectral resolution of a spectrometer is a quantity that characterizes the system’s ability to resolve different wavelengths of the electromagnetic spectrum. Etendue is a quantity that characterizes how spread out a beam of light is in area and angle. Etendue is the product of the area of the source and the solid angle of emission. The radiance of an optical system is equal to the derivative of the radiant flux with respect to the etendue. Unfortunately, the laws of thermodynamics limit the radiance and etendue that can be obtained given a certain source. In accordance with the first law of thermodynamics, the power throughput of a passive optical system is, at best, conserved. In accordance with the second law of thermodynamics, the etendue of an optical system cannot decrease.

[0035] Having a high etendue for a given radiance ensures that enough optical energy reaches the receiver to produce an adequate signal -to-noise ratio (SNR). In LVF spectrometers, increasing etendue can be accomplished in two ways. The first approach involves increasing the spot size of the beam at the plane of the filter. An example of a beam illuminating an LVF filter is depicted in FIG. 1 A. Unfortunately, increasing the spot size results in a reduction of the spectral resolution because a larger beam spans a greater bandwidth of wavelengths. The second approach involves increasing the solid angle of the beam. An example is depicted in FIG. IB. Once again, increasing the solid angle results in a reduction of the spectral resolution. This is due to an effect known as “blue shift.” As the angle of incidence of the cone of rays that constitute the beam is increased, the effective length of the cavity defined by the filter decreases, moving the peak of the transmission response to smaller wavelengths (hence the expression blue shift). This change in the transmission peak of the filter results in a lower spectral resolution.

[0036] The inventor has recognized and appreciated that the performance of a spectrometer can be improved by compensating the blue shift effect described above with a corresponding red shift — a wavelength shift in the opposite direction. This can be accomplished by steering rays having a greater angle of incidence towards regions of the filter characterized by larger wavelengths of transmission. In this way, the rays that are most susceptible to blue shift (those having a greater angle of incidence) are forced to pass through regions of the filter corresponding to larger wavelengths. In some embodiment, the more the incident angle deviates from the optical axis, the more it is red shifted, thereby compensating the increasing blue shift effect.II. Examples of Spectrometers

[0037] FIG. 2A is an example of a filter that may be used as a part of a spectrometer, in accordance with some embodiments. In this example, filter 10 is an LVF filter. However, filters with other geometries (other than linear) may be used, as the technology described herein is not limited to any particular shape of filter 10. As for LVF filters, filters having other shapes exhibit responses that depend on the location on which the input ray hits the filter. Filter 10 may be viewed as a monochromator in that filter 10 may transmit a selected narrow band of wavelengths. Filter 10 has a pair a mirrors (14 and 16), which may be formed using stacks of dielectric materials, although other types of reflective surfaces are also possible. Cavity 12 is defined between the mirrors. Cavity 12 behaves as a Fabry-Perot resonator, the peak transmission wavelength of which depends upon the thickness of the cavity along the z-axis (referred to herein as the optical axis). In this implementation, mirrors 14 and 16 get closer and closer to each other along the y-axis (referred to herein as the transverse axis). As a result, the thickness of the cavity 12 varies along the y-axis. In this depiction, the higher portion of cavity 12 is thicker than the lower portion. For example, portion 11 is thicker than portion 13. The larger the thickness, the larger the wavelength of the peak transmission. Thus, the spectrum of the filtered light presents larger wavelengths closer to the top of the filter and smaller wavelengths closer to the bottom of the filter. The system further includes a photodetector array 20, although a single photodetector may be used in some embodiments.

[0038] Although the incident light is depicted as a plane wave spanning the entire extension of filter 10, in some embodiments, the spot size of the incident beam may be substantiallysmaller than the extension of filter 10. In some embodiments, a mechanism (not shown in FIG. 2A) is used to change the relative position of the filter relative to the input lens (by changing the absolute position of the filter, the lens, or both along the y-axis). The mechanism allows the spectrometer to sweep the spectrum of the incident light against regions of the filter having different transmission wavelengths.

[0039] FIG. 2B illustrates an alternative arrangement in which photodetector array 20 is replaced with a single photodetector 23. A lens 22 may be positioned optically upstream from filter 10 to focus light on the filter so that a significant portion (e.g., the majority) of the light emerging from the filter is captured by photodetector 23.

[0040] For reference, FIG. 3 illustrates a portion of a spectrometer including filter 10 and a conventional, spherical lens 30. FIG. 3 illustrates that some rays have greater angles of incidence and some rays have a small, near-zero angle of incidence with the linear variable filter. The angle of incidence is measured in relationship with the optical axis (z-axis). As discussed above, rays characterized by greater angles of incidence suffer from blue shift to a greater extent than rays characterized by smaller angles of incidence. This results in a reduction of the spectral resolution of the spectrometer.

[0041] This negative effect can be obviated by using a lens specifically designed to steer rays having a greater angle of incidence towards regions of the filter characterized by larger wavelengths of transmission. These lenses are referred to herein as “freeform lenses.” In some embodiment, the more an incident angle deviates from the optical axis, the more the freeform lens red shifts that ray, thereby compensating the increasing blue shift effect. An example of such a freeform lens is depicted in FIG. 4A. Lens 100 is designed so that a cone of rays with a large angle of incidence (e.g., ray 102) is focused on a portion of filter 10 characterized by larger wavelengths of transmission. In this respect, those rays are said to be red shifted. At the same time, a cone of rays with a smaller angle of incidence (e.g., ray 104) is focused on a portion of filter 10 characterized by smaller wavelengths of transmission.This concept is further illustrated in FIG. 4B. Lens 100 steers ray 102 towards the top side of filter 10. The inner rays are progressively steered to a lesser extent.III. Examples of Freeform Lenses

[0042] FIG. 5 depicts an example implementation of a freeform lens that may be used to steer rays in the manner described in connection with FIGs. 4A-4B. Lens 100 defines asurface having a concave surface portion 110 and a convex surface portion 112. The surface portions 110 and 112 are offset relative to each other along the y-axis. Center portion 111 defines the inflection point between the surface portions (the point where the second derivative with respect to y is zero). Further, in correspondence with center portion 111, the surface of the lens may be perpendicular to the z-axis (the first derivative with respect to y is zero and / or the first derivative with respect to x is zero). In some embodiments, concave surface portion 110 is such that the concave surface faces the filter, and the convex surface portion 112 is such that the convex surface faces the filter.

[0043] Portion 11 of the optical cavity may be on a first side of portion 13 with respect to the optical axis (e.g., on top of, relative to the y-axis). Similarly, concave surface portion 110 may be on the same first side of convex surface portion 112 with respect to the optical axis. As a result, the concave surface portion is positioned in correspondence with a thicker portion of the filter, and the convex surface portion is positioned in correspondence with a thinner portion of the filter. This orientation allows the lens to perform red shifting (as opposed to worsening the blue shifting effect).

[0044] Along a line parallel to the x-axis, the surface of lens 100 may be parabolic. In some embodiments, the shape of the surface of lens 100 may be defined by the following expression: z = Ayx2+ By3where A and B are numeric coefficients that may be determined using numerical optimization techniques.

[0045] Thus, the surface of lens 100 may have a profile (z), along the y-axis, defined by a polynomial having at least one cubic term (although other odd terms are also possible in addition to, or in alternative to the cubic term). Further, the surface of lens 100 may have a profile (z), along the x-axis, defined by a polynomial having at least one quadratic term (although other even terms are also possible in addition to, or in alternative to the quadratic term). In some embodiments, higher order coefficients may be added to further improve performance.

[0046] In some embodiments, multiple titled, off-axis lens components may be used in lieu of lens 100 to achieve the red shift effect discussed above. These lens components may bearranged to mimic the surface described in connection with FIG. 5. Alternatively, multiple tilted, decentered mirrors may be used in lieu of lens 100 to achieve the red shift effect discussed above. Multiple tilted and decentered surfaces (whether spherical or with other shapes) can be used to generate aberrations in addition to a focusing effect. Alternatively, a freeform distribution gradient index lens may be used in lieu of lens 100 to achieve the red shift effect discussed above. This type of lens may have a distribution of form engineered to creates an optical path length difference that produces a compensation of blue shift effects. Alternatively, lens 100 may be two aspheric surfaces with separate optical axes to achieve the red shift effect discussed above. Alternatively, lens 100 may include a freeform gradientindex medium (GRIN) to achieve the red shift effect discussed above.IV. Conclusion

[0047] Accordingly, some embodiments relate to an optical spectrometer comprising an optical filter (e.g., filter 10), a lens (e.g., lens 100) and one or more photodetectors (e.g., photodetector 23 or photodetector array 20). The optical filter may have a first mirror, a second mirror, and an optical cavity (e.g., cavity 12) formed between the first and second mirrors. The optical cavity may have a varying thickness along a transverse axis (e.g., the y- axis) so that a first portion of the optical cavity (e.g., portion 11) has a first thickness and a second portion of the optical cavity (e.g., portion 13) has a second thickness smaller than the first thickness. The first and second portions are offset relative to each other along the transverse axis. The lens defines an optical axis (e.g., the z-axis) perpendicular to the transverse axis. The lens may be configured to focus incident light on the optical lens. The lens may be shaped so that rays of the incident light (e.g., ray 102) that emerge from the lens at a first angle relative to the optical axis are focused on the first portion of the optical cavity, and rays of the incident light (e.g., ray 104) that emerge from the lens at a second angle, less than the first angle, relative to the optical axis are focused on the second portion of the optical cavity. The optical filter is disposed optically between the lens and the one or more photodetectors.

[0048] Further, some embodiments relate to another optical spectrometer an optical filter (e.g., filter 10), a lens (e.g., lens 100) and one or more photodetectors (e.g., photodetector 23 or photodetector array 20). The optical filter may have a first mirror, a second mirror, and an optical cavity (e.g., cavity 12) formed between the first and second mirrors. The opticalcavity may have a varying thickness along a transverse axis (e.g., the y-axis) so that a first portion of the optical cavity (e.g., portion 11) has a first thickness and a second portion of the optical cavity (e.g., portion 13) has a second thickness smaller than the first thickness. The first and second portions are offset relative to each other along the transverse axis. The lens defines an optical axis (e.g., the z-axis) perpendicular to the transverse axis. The lens may define a surface having a first surface portion (e.g., concave surface portion 110) and a second surface portion (e.g., convex surface portion 112) having opposite concavities relative to each other. The first and second surface portions are offset relative to each other along the transverse axis. The optical filter is disposed optically between the lens and the one or more photodetectors.

[0049] In some embodiments, the first surface portion (e.g., 110) is such that a concave portion faces the filter, and the second surface portion (112) is such that a convex portion faces the filter. The first portion of the optical cavity (e.g., portion 11) may be on a first side (e.g., on top of, relative to the y-axis) of the second portion of the optical cavity (e.g., portion 13) with respect to the optical axis. Similarly, the first surface portion of the lens (e.g., concave surface portion 110) is on the first side of the second surface portion of the lens (e.g., convex surface portion 112) with respect to the optical axis. As a result, the concave surface portion is positioned in correspondence with a thicker portion of the filter, and the convex surface portion is positioned in correspondence with a thinner portion of the filter.

[0050] Having thus described several aspects and embodiments of the technology of this application, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those of ordinary skill in the art. Such alterations, modifications, and improvements are intended to be within the spirit and scope of the technology described in the application. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described. In addition, any combination of two or more features, systems, articles, materials, and / or methods described herein, if such features, systems, articles, materials, and / or methods are not mutually inconsistent, is included within the scope of the present disclosure.

[0051] Various aspects of the present invention may be used alone, in combination, or in a variety of arrangements not specifically discussed in the embodiments described in the foregoing and is therefore not limited in its application to the details and arrangement ofcomponents set forth in the foregoing description or illustrated in the drawings. For example, aspects described in one embodiment may be combined in any manner with aspects described in other embodiments.

[0052] The invention may be embodied as a method, of which an example has been described. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include different acts than those which are described, and / or which may involve performing some acts simultaneously, even though the acts are shown as being performed sequentially in the embodiments specifically described above.

[0053] The definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference and / or ordinary meanings of the defined terms.

[0054] The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.”

[0055] The phrase “and / or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, z.e., elements that are conjunctively present in some case and disjunctively present in other cases.

[0056] As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified.

[0057] The term “substantially” may be used to mean within ±10% of a target value in some embodiments. The term “substantially” may include the target value.

[0058] Use of ordinal terms such as “first,” “second,” etc., in the claims to modify a claim element does not by itself connotate any priority, precedence, or order of one claim element over another or the temporal order in which acts of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another claimelement having a same name (but for use of the ordinal term) to distinguish the claim elements.

[0059] Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” “having,” “involving,” and variations thereof herein, is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.

[0060] What is claimed is:

Claims

CLAIMS1. An optical spectrometer, comprising: an optical filter having a first mirror, a second mirror, and an optical cavity formed between the first and second mirrors, the optical cavity having a varying thickness along a transverse axis so that a first portion of the optical cavity has a first thickness and a second portion of the optical cavity has a second thickness smaller than the first thickness, the first and second portions being offset relative to each other along the transverse axis; a lens, defining an optical axis perpendicular to the transverse axis, configured to focus incident light on the optical lens, the lens being shaped so that: rays of the incident light that emerge from the lens at a first angle relative to the optical axis are focused on the first portion of the optical cavity, and rays of the incident light that emerge from the lens at a second angle, less than the first angle, relative to the optical axis are focused on the second portion of the optical cavity; and one or more photodetectors, wherein the optical filter is disposed optically between the lens and the one or more photodetectors.

2. The optical spectrometer of claim 1, wherein the thickness of the optical cavity varies linearly along the transverse axis.

3. The optical spectrometer of claim 1, wherein the first and second mirrors comprises stacks of dielectric layers.

4. The optical spectrometer of claim 1, further comprising a mechanism configured to change a relative position of the optical filter relative to the lens along the transverse axis.

5. The optical spectrometer of claim 1, wherein the one or more photodetectors form a photodetector array.

6. The optical spectrometer of claim 1, wherein the optical filter has a spectral width in transmission that is less than 30 nm.

7. The optical spectrometer of claim 1, wherein: the lens defines a surface having a first surface portion and a second surface portion having opposite concavities relative to each other, the first and second surface portions being offset relative to each other along the transverse axis.

8. The optical spectrometer of claim 7, wherein the surface has a cubic term along the transverse axis.

9. The optical spectrometer of claim 7, wherein the surface has a quadratic term along a lateral axis that is perpendicular to both the transverse axis and the optical axis.

10. An optical spectrometer, comprising: an optical filter having a first mirror, a second mirror, and an optical cavity formed between the first and second mirrors, the optical cavity having a varying thickness along a transverse axis so that a first portion of the optical cavity has a first thickness and a second portion of the optical cavity has a second thickness smaller than the first thickness, the first and second portions being offset relative to each other along the transverse axis; a lens, defining an optical axis perpendicular to the transverse axis, defining a surface having a first surface portion and a second surface portion having opposite concavities relative to each other, the first and second surface portions being offset relative to each other along the transverse axis; and one or more photodetectors, wherein the optical filter is disposed optically between the lens and the one or more photodetectors.

11. The optical spectrometer of claim 10, wherein: the first surface portion is such that a concave portion faces the filter, and the second surface portion is such that a convex portion faces the filter; the first portion of the optical cavity is on a first side of the second portion of the optical cavity with respect to the optical axis; and the first surface portion of the lens is on the first side of the second surface portion of the lens with respect to the optical axis.

12. The optical spectrometer of claim 10, wherein the surface has a profile, along the transverse axis, defined by a polynomial having at least one odd term.

13. The optical spectrometer of claim 12, wherein the surface has a profile, along the transverse axis, defined by a polynomial having at least one cubic term.

14. The optical spectrometer of claim 12, wherein the surface has a profile, along a lateral axis that is perpendicular to both the transverse axis and the optical axis, defined by a polynomial having at least one even term.

15. The optical spectrometer of claim 12, wherein the surface has a profile, along a lateral axis that is perpendicular to both the transverse axis and the optical axis, defined by a polynomial having at least one quadratic term.

16. A method of assembling an optical spectrometer, comprising: obtaining an optical filter having a first mirror, a second mirror, and an optical cavity formed between the first and second mirrors, the optical cavity having a varying thickness along a transverse axis so that a first portion of the optical cavity has a first thickness and a second portion of the optical cavity has a second thickness smaller than the first thickness, the first and second portions being offset relative to each other along the transverse axis; obtaining a lens, defining an optical axis perpendicular to the transverse axis, defining a surface having a first surface portion and a second surface portion having opposite concavities relative to each other, the first and second surface portions being offset relative to each other along the transverse axis; obtaining one or more photodetectors; and positioning the optical filter, the lens, and the one or more photodetectors so that the optical filter is disposed optically between the lens and the one or more photodetectors.

17. The method of claim 16, wherein: the first surface portion is such that a concave portion faces the filter, and the second surface portion is such that a convex portion faces the filter;the first portion of the optical cavity is on a first side of the second portion of the optical cavity with respect to the optical axis; and the first surface portion of the lens is on the first side of the second surface portion of the lens with respect to the optical axis.

18. The method of claim 17, wherein the surface has a profile, along the transverse axis, defined by a polynomial having at least one odd term.

19. The method of claim 18, wherein the surface has a profile, along the transverse axis, defined by a polynomial having at least one cubic term.

20. The method of claim 18, the surface has a profile, along a lateral axis that is perpendicular to both the transverse axis and the optical axis, defined by a polynomial having at least one even term.

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