Abstract:
Spectrometers, particularly compact spectrometers, are constructed without sacrificing performance characteristics, but with their instrument/device volume being reduced significantly. The light propagation path, either in transparent media or in free space, of the optical beams emitting from a small input aperture/slit of a spectrometer, is caused to be two-dimensional or unilateralized (propagating within a thin layer of air or media), enabling physical sizes of any optical elements needed thereafter to construct a spectrometer to be reduced significantly in one dimension. As a result, a significant reduction of instrument/device volume is achieved, which is applicable to and beneficial to either a classical dispersion spectrometer or a compact dispersion spectrometer.

Description:
[0001]    This application claims the benefit of U.S. Provisional Patent Application No. 60/997,983 filed in the U.S. Patent and Trademark Office on Oct. 5, 2007, the entire disclosure of which is hereby incorporated herein by reference in its entirety. 
     
    
     BACKGROUND 
       [0002]    1. Field of the Invention 
         [0003]    The embodiments relate to optical spectroscopy. In particular, the embodiments relate to spectrometers. More particularly, the embodiments relate to compact spectrometers designed to reduce and minimize their dimensions and volumes based on the unilateralized optical technique described herein. 
         [0004]    2. Description of Related Art 
         [0005]    Instruments used for spectroscopic measurements and applications belong to one family that includes monochromators and spectrometers. A monochromator is an optical device that transmits a mechanically selectable narrow band of wavelengths of light chosen from a wider range of wavelengths available at the input. A spectrometer is an optical instrument for measuring and examining the spectral characteristics of the input light over some portion of the electromagnetic spectrum, where the measured variable is often the light intensity. 
         [0006]    A monochromator may be differentiated from a spectrometer in at least two aspects: (1) a monochromator has an exit slit positioned at its spectral focal plane; while a spectrometer has no exit slit, but a linear detector array mounted at its spectral image plane; and (2) a monochromator has to be equipped with a scanning mechanism driving either a dispersive grating, or a focusing mirror, or the exit slit, in order to transmit the desired monochromatic light as the output through the exit slit; while a spectrometer has no moving parts and is capable of acquiring an instant full spectrum of the input light. 
         [0007]    Nevertheless, the optical systems of such kinds of spectroscopic instruments, regardless of whether the instrument is classified as a monochromator or a spectrometer, are the same in working principle. Therefore, monochromators and spectrometers often are considered the same kind of instruments. Further, for the sake of simplicity throughout this disclosure, only a spectrometer will be referenced in this disclosure. A typical optical system of a spectrometer basically comprises an element(s) for collimating, an element(s) for dispersing and an element(s) for focusing to form spectral images. An entrance slit functions as the input interface, where an optional input optics exists. A detector converts optical signals to electronic signals. Such conventional optical technique makes a spectrometer cumbersome, i.e., complex in construction, large in body volume and heavy in weight. Further, there exist a few technical problems inherently associated with such spectroscopic instruments, particularly for a conventional spectrometer: astigmatism over the spectrum on the detector plane, and field curvature from the spectrum focused onto the detector plane, as reviewed by U.S. Pat. No. 5,880,834. 
         [0008]    As a result, it has become a challenge to design and build a spectrometer to overcome the drawbacks and technical problems mentioned above, to which, substantial efforts have been directed and numerous improvements have been published for the purposes of simplifying optics, minimizing body volume, reducing weight, and eliminating optical aberrations, mainly astigmatism and field curvature. Among those areas of concerns, constructing compact spectrometers has generated manifold attention since the trend in modern spectrometer systems is toward a compact one and it has the potential to open up for wider applications in many industries, as shown in the following. 
         [0009]    Representatives of the art can be categorized in accordance of their construction features associated with spectrometers: lens spectrometers, mirror spectrometers, spectrometers of simple construction, and compact spectrometers. 
         [0010]    Representative of the art for lens spectrometers is U.S. Pat. No. 3,572,933 (1971) to Boostrom, which discloses a monochromator of classical configuration comprising a collimating lens, a transmission grating and a focusing lens to form spectra. U.S. Pat. No. 5,497,231 (1996) to Schmidt discloses another lens monochromator of scanning feature, which relies on a reflective planar grating. U.S. Pat. No. 6,122,051 (2000) to Ansley discloses another lens spectrometer of multi slits, which uses a prism as dispersion element. U.S. Pat. No. 7,180,590 (2007) to Bastue et al. discloses another lens spectrometer of transmission path, which is independent of temperature-induced wavelength drift. 
         [0011]    Representative of the art for mirror spectrometers is U.S. Pat. No. 5,192,981 (1993) to Slutter et al., which discloses a monochromator of Czerny-Turner geometry comprising a collimating mirror, a reflective grating and a focusing mirror. This configuration is one of those typical of early prior art efforts and is a technique that is generally well known. The improvement of the disclosure comprises the use of a single toroidal collimating mirror in the system in combination with a spherical focusing mirror to minimized optical aberrations within final spectral images. 
         [0012]    Another representative of the art for mirror spectrometers is U.S. Pat. No. 6,507,398 (2003) to Arai et al., which discloses a spectrometer of crossed Czemy-Tumer geometry where the incident beam and the reflected beam from the diffraction grating cross. Cross Czemy-Tumer configuration becomes one of preferred considerations for compact spectrometer designs. 
         [0013]    Another representative of the art for mirror spectrometers is U.S. Pat. No. 4,310,244 (1982) to Perkins et al., which discloses a monochromator of Fastie-Ebert geometry comprising a big mirror for both collimating and focusing, plus a reflective planar grating. Fastie-Ebert configuration evolves from that of Czerny-Tumer by combining the two mirrors into one. It becomes a preferred choice for a design of simple construction, as disclosed by U.S. Pat. No. 6,081,331 (2000) to Teichmann, which describes a spectrometer of Fastie-Ebert geometry formed in a cylinder body of glass. U.S. Pat. No. 7,239,386 (2007) to Chrisp et al. also discloses a design of imaging spectrometer of Fastie-Ebert configuration, which is improved by a glass-immersed mirror and a glass-immersed grating. This modification provides extra optical power to compensate optical aberrations. 
         [0014]    Representative of the art for spectrometers of simple construction is U.S. Pat. No. 4,568,187 (1986) to Toshiaki et al., which discloses a spectrometer comprising a single concave grating. The concave grating is manufactured with curved grooves of varied spacing for optimum performance, and functions for both dispersing and imaging. It has become a known art that a concave grating sets the minimum number of optical elements needed in a spectrometer, leading to a simplest structure form. 
         [0015]    Another representative of the art for spectrometers of simple construction is U.S. Pat. No. 5,182,609 (1993) to Kita et al., which discloses a spectrometer of Rowland configuration, comprising a single concave grating plus a second optical element introduced in the path for flattening spectral image formed at the focal plane. 
         [0016]    Another representative of the art for spectrometers of simple construction is U.S. Pat. No. 5,233,405 (1993) to Wildnauer et al., which discloses a double-pass monochromator comprising a lens for both collimating and focusing, and a reflective planar grating for dispersing. 
         [0017]    Another representative of the art for spectrometers of simple construction is U.S. Pat. No. 5,424,826 (1995) to Kinney, which discloses an optical micro-spectrometer system. This system consists of a group of micro-spectrometers, each of which comprises an input fiber, a lens for both collimating and focusing, and a reflective planar grating for dispersing. 
         [0018]    Another representative of the art for spectrometers of simple construction is U.S. Pat. No. 5,812,262 (1998) to Ridyard et al., which discloses an apparatus of spectrometer type for UV radiation. Constructed by a single piece of waveguide carrier, it comprises a concave mirror and a reflective planar grating for focusing light from the entrance aperture means onto the radiation detector means. This configuration relies on a fixed order of the optical elements of focusing and then dispersing the light, which makes it difficult to compensate or avoid aberrations, in particular chromatic aberration. 
         [0019]    Another representative of the art for spectrometers of simple construction is U.S. Pat. No. 6,597,452 (2003) to Jiang et al., which discloses a Littrow-type spectrometer, comprising a planar mirror, a concave mirror for both collimating and focusing, and a reflective planar grating, arranged within a compact configuration. 
         [0020]    Representative of the art for compact spectrometers is U.S. Pat. No. 5,159,404 (1992) to Bittner, which discloses a compact spectrometer where the grating and the focusing mirror are combined on one side of a single transparent carrier, and the light entrance means and light detecting means are both placed on the other side of the spectrometer, making it possible to construct a compact spectrometer with a robust body. 
         [0021]    Another representative of the art for compact spectrometers is U.S. Pat. No. 5,550,375 (1996) to Peters et al., which discloses a compact spectrometer designed as infrared spectrometric sensor. It comprises two parts: single-piece shaped base mirror plate manufactured as a microstructured body, having the concave cylindrical grating formed at one end, and the entrance port and detector slit at the other end, and a thin plate mirror as top cover. The integrated spectrometer has a thin layer (less than 1 mm) of reflective hollow cavity, which is filled with the gas to be monitored, through which infrared light propagate in divergence and convergence laterally, but guided vertically by the top and bottom mirror surfaces. This structure is only suitable for infrared peak absorption measurement of gas using a single cell detector of large area. 
         [0022]    Another representative of the art for compact spectrometers is U.S. Pat. No. 6,606,156 (2003) to Ehbets, et al., which discloses a compact spectrometer comprising a concave grating, mounted on one side of the housing. The input port and the detector array are positioned opposite the concave grating, leaving a hollow cavity where the input optical beams propagate. 
         [0023]    Another representative of the art for compact spectrometers is U.S. Pat. No. 7,081,955 (2006) to Teichmann et al., which discloses a compact spectrometer comprising two parts: the main body with grating and the focusing element being formed on the top of the housing, and the bottom substrate of detector array with light entrance means. The integrated spectrometer has a hollow cavity where the input optical beams propagate. 
         [0024]    Another representative of the art for a compact spectrometer is U.S. Pat. No. 4,744,618 (1988) to Mahlein, which discloses a waveguide based device as multiplexer/demultiplexer, where light propagates based on total internal reflection through micro structures. Functioning like a compact spectrometer, it has a unilateral-type solid monolithic glass body of the Ebert-Fastie configuration, which makes it possible to build a compact device. 
         [0025]    As stated above, conventional spectrometers are cumbersome and have large volumes, including those compact spectrometers of a single concave grating, which are either constructed from a single solid block of transparent material (e.g., glass), or integrated by mechanical mounting parts and housing. In contrast, waveguide based spectrometers typically allow for smaller volumes. The difference in volume between conventional spectrometers and waveguide based spectrometers may be attributed to the fact that the former is constructed with bulky optical elements and has a light propagation path that is three-dimensional, while the later (i.e., a waveguide based spectrometer) is constructed from a thin monolithic glass substrate in which a light propagation path exists in a thin layer (e.g., approximately 10 to 100 s micrometers) of glass media that are two-dimensional, or at least substantially unilateral. It seems that waveguide based technology may becomes a promising candidate for compact spectrometers. 
         [0026]    However, from a practical perspective, the manufacturing process of waveguide products is expensive, and there are other technical issues associated with waveguide performance, including, but not limited to, high propagation loss, stray light caused by scattering at waveguide boundary, etc.. Additionally, the coupling efficiency of waveguide devices is very susceptible to misalignment at input interfaces. 
         [0027]    In general, existing spectrometers have not been an object of miniaturization as has been other technological machines and equipment because of the lack of technology in such field of endeavor. Thus, wider applications of spectrometers have not been possible for areas where miniaturization has become increasingly necessary or preferable. The embodiments of this disclosure overcome the above-identified disadvantages. 
       SUMMARY 
       [0028]    A Cartesian coordinate system denoted by XYZO is to be referenced in the discussions to follow, where the optical system of a spectrometer resides and light propagates. The coordinate system has three axes: X, Y, Z and an origin O. Two important planes are defined here: XOZ represents the horizontal plane, or the sagittal plane; YOZ represents the vertical plane, or the tangential plane. Z represents the propagation direction of light. A beam of light is considered having a three-dimensional path, as the beam of light converges, diverges, or otherwise maintains a finite collimated size in both the tangential and sagittal planes as it propagates in Z direction. A beam of light is considered having a substantially two-dimensional (substantially unilateralized) path, if the beam of light converges, diverges, or otherwise maintains a finite collimated size in either the tangential or the sagittal planes, but is confined within a thin layer in or parallel to the other plane, as it propagates in Z direction. 
         [0029]    The main object of the embodiments is to provide an optical technique that makes the propagation path, either in transparent media or in free space, of the optical beams emitting from a small input aperture/slit of a spectrometer, two-dimensional or substantially unilateralized, thereby enabling physical sizes of any optical elements needed thereafter to construct a spectrometer to be significantly reduced in one dimension. Consequently, a significant reduction of instrument/device volume can be achieved, which is applicable to and beneficial to either a classical spectrometer or a compact spectrometer. 
         [0030]    The above description broadly sets forth a summary of the present embodiments so that the following detailed description may be better understood and contributions of the present embodiments to the art may be better appreciated. Some of the embodiments may not include all of the features or characteristics listed in the above summary. There are, of course, additional features that will be described below. In this respect, before explaining any embodiment of the invention in detail, it is to be understood that the invention is not limited in its application to the details of the construction and to the arrangement of the components set forth in the following description or as illustrated in the drawings. The invention is capable of other embodiments and of being practiced and carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein are for the purpose of description and should not be regarded as limiting. 
         [0031]    In one aspect, it is an object of at least some embodiments to provide a means to manipulate the propagation properties of the optical beams separately in two independent directions, i.e., in the tangential plane and the sagittal plane, at any intersecting location(s) between optical beams and optical elements inside a spectrometer. This is accomplished by using optical elements which have cylindrical or toroidal surfaces with main optical powers only in one direction (i.e., either in the tangential plane or in the sagittal plane). The optical elements include all types of cylindrical and toroidal lenses; all types of cylindrical and toroidal mirrors; one-dimensional, transmissive or reflective gratings of planar, concave or convex, cylindrical, toroidal or spherical substrates; herein “all types” represents properties of positive and negative optical power, spherical and aspherical shapes for cross-section. 
         [0032]    One aspect of at least some embodiments is to provide an entrance aperture of small size at the entrance slit position of spectrometers, where the entrance aperture can be the core of a single mode fiber, or the core of a multi-mode fiber, or pinholes of diameters similar to those of fibers&#39; cores, or a slit of fiber core widths whose preferred height is less than a few millimeters. The optical outputs of the entrance aperture may have a symmetrical or asymmetrical cone shapes, whose propagation paths are three-dimensional. 
         [0033]    Another aspect of at least some embodiments is to provide a collimating means to collimate the optical beams emitting from the entrance aperture in the tangential plane only, making the output beams of the collimating means anamorphic, which is substantially collimated in the tangential plane, but propagates in divergence freely in the sagittal plane. The collimating means can be a cylindrical or toroidal lens, or a concave cylindrical or toroidal mirror, or a concave conic cylindrical or toroidal mirror, all of which have main optical power in the tangential plane, but have no or little optical power in the sagittal plane. The collimating means is properly positioned behind the entrance aperture in the optical train of the spectrometer&#39;s optics, closely enough that its outputs of partially collimated anamorphic beams maintain a small and finite collimated size (e.g., no more than a few millimeters) in the tangential plane, whose propagation paths are two-dimensional. 
         [0034]    Another aspect of at least some embodiments is to provide a dispersing-focusing means, which resides at a certain distance behind the collimating means in the optical train of the spectrometer&#39;s optics. The dispersing-focusing means is capable of performing at least two tasks in the sagittal plane only: (1) dispersing the input optical beams received from the collimating means; and (2) forming spectral images of the entrance aperture onto a detector surface. Several examples of the dispersing-focusing means are explained below with respect to various embodiments. The outputs of the dispersing-focusing means remain partially collimated with a small and finite collimated size in the tangential plane, but are focused into spectral images at the detector surface in the sagittal plane. The outputs have propagation paths that are at least substantially two-dimensional. 
         [0035]    Another aspect of at least some embodiments is to provide a focusing means to focus the optical beams received from the dispersing-focusing means onto the detector surface in the tangential plane only. The focusing means can be a cylindrical or toroidal lens, or a concave cylindrical or toroidal mirror, or a concave conic cylindrical or toroidal mirror, all of which have main optical power in the tangential plane, but have no or little optical power in the sagittal plane. Thus, the outputs of the focusing means may form a linear spectral image at the detector surface. The detector may be a linear array of detector pixels residing behind the focusing means, at the end of the optical train of the spectrometer&#39;s optics. 
         [0036]    One embodiment is directed to a spectrometer comprising: (1) an entrance aperture; (2) a collimating means; (3)-(5) a dispersing-focusing means; (6) a focusing means; and (7) a detector. In such an embodiment, the dispersing-focusing means may be a transmission sub-system comprising: (3) a cylindrical/toroidal lens for collimating only in the sagittal plane; (4) a transmissive grating for dispersing only in the sagittal plane; and (5) a cylindrical/toroidal lens for focusing only in the sagittal plane. The (2) collimating means and the (6) focusing means respectively collimate and focus only in the tangential plane. The propagation paths within the spectrometer from (1) to (7) are substantially two-dimensional. 
         [0037]    Another embodiment is directed to a spectrometer comprising (1) an entrance aperture; (2) a collimating means; (3)-(5) a dispersing-focusing means; (6) a focusing means; and (7) a detector. In such an embodiment, the dispersing-focusing means may be a catadioptric sub-system comprising: (3) a cylindrical/toroidal lens or mirror for collimating in the sagittal plane; (4) a transmissive or reflective grating for dispersing in the sagittal plane; and (5) a cylindrical/toroidal lens or mirror for focusing in the sagittal plane. The (2) collimating means and the (6) focusing means respectively collimate and focus only in the tangential plane. The propagation paths within the spectrometer from (1) to (7) are substantially two-dimensional. 
         [0038]    Another embodiment is directed to a spectrometer comprising (I) an entrance aperture; (2) a collimating means; (3)-(5) a dispersing-focusing means; (6) a focusing means; and (7) a detector. In such an embodiment, the dispersing-focusing means may be a reflectance sub-system comprising: (3) a cylindrical/toroidal mirror for collimating in the sagittal plane; (4) a reflective grating for dispersing in the sagittal plane; and (5) a cylindrical/toroidal mirror for focusing in the sagittal plane. The (2) collimating means and the (6) focusing means respectively collimate and focus only in the tangential plane. The propagation paths within the spectrometer from (1) to (7) are substantially two-dimensional. 
         [0039]    Another embodiment is directed to a spectrometer with Fastie-Ebert configuration comprising (1) an entrance aperture; (2) a collimating means; (3)-(4) a dispersing-focusing means; (5) a focusing means; and (6) a detector. In such an embodiment, the dispersing-focusing means may be a reflectance sub-system comprising: (3) a cylindrical/toroidal mirror for both collimating and focusing in the sagittal plane; and (4) a reflective grating for dispersing in the sagittal plane. The (2) collimating means and the (5) focusing means respectively collimate and focus only in the tangential plane. Optical means from (2) to (5) may be fabricated by a thin piece of monolithic transparent material. The propagation paths within the spectrometer from (1) to (6) are two-dimensional. 
         [0040]    Another embodiment is directed to a spectrometer with Czerny-Tumer configuration comprising (1) an entrance aperture; (2) a collimating means; (3)-(5) a dispersing-focusing means; (6) a focusing means; and (7) a detector. In such an embodiment, the dispersing-focusing means may be a reflectance sub-system comprising: (3) a cylindrical/toroidal mirror for collimating in the sagittal plane; (4) a reflective grating for dispersing in the sagittal plane; and (5) a cylindrical/toroidal mirror for focusing in the sagittal plane. The (2) collimating means and the (6) focusing means respectively collimate and focus only in the tangential plane. Optical means from (2) to (6) may be fabricated by a thin piece of monolithic transparent material. The propagation paths within the spectrometer from (1) to (7) are two-dimensional. 
         [0041]    Another embodiment is directed to a spectrometer comprising (1) an entrance aperture; (2) a collimating means; (3)-(4) a dispersing-focusing means; (5) a focusing means; and (6) a detector. In such an embodiment, the dispersing-focusing means may be a hybrid sub-system comprising: (3) a cylindrical/toroidal lens for collimating and focusing in the sagittal plane; and (4) a reflective grating for dispersing in the sagittal plane. The (2) collimating means and the (5) focusing means respectively collimate and focus only in the tangential plane. The propagation paths within the spectrometer from (1) to (6) are two-dimensional. 
         [0042]    Another embodiment is directed to a spectrometer comprising (1) an entrance aperture; (2) a collimating means; (3) a dispersing-focusing means; (4) a focusing means; and (5) a detector. In such an embodiment, the dispersing-focusing means is a concave cylindrical reflective grating for dispersing and focusing in the sagittal plane. The (2) collimating means and the (4) focusing means respectively collimate and focus only in the tangential plane. Optical means from (2) to (4) can be fabricated by a thin piece of monolithic transparent material. The propagation paths within the spectrometer from (1) to (5) are two-dimensional. 
         [0043]    One aspect of the embodiments is directed to building a spectrometer based on one of above embodiments or their modified configurations, in which the collimating means and the focusing means fulfill tasks of (1) generating images of the entrance aperture onto the detector surface in the tangential plane, and (2) making the propagation paths of optical beams within the spectrometer two-dimensional. Meanwhile, the dispersing-focusing means of the spectrometer performs at least the following functions: (i) dispersing the received optical beams into spectra in the sagittal plane; and (ii) generating spectral images of the entrance aperture onto the detector surface in the sagittal plane. In this regard, significant improvements may be achieved in at least the following two aspects: (a) sizes and dimensions of all optical elements used inside the spectrometer are significantly reduced in the Y direction, (i.e., in the vertical plane or the tangential plane), thereby significantly reducing the instrument/device volume; and (b) optical aberration of astigmatism and curvature of spectral images are well compensated. 
         [0044]    The above summary is not intended to describe each illustrated embodiment or every implementation of the present invention. The following figures and the detailed description more particularly exemplify embodiments. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0045]      FIG. 1(   a ) is a perspective view of a prior art lens spectrometer comprising a collimating lens, a transmissive grating and a focusing lens; 
           [0046]      FIG. 1(   b ) illustrates the same type of spectrometer as in  FIG. 1(   a ), but incorporating features of an embodiment of the invention; 
           [0047]      FIG. 2(   a ) is a perspective view of a prior art catadioptric spectrometer comprising a collimating mirror, a reflective grating and a focusing lens; 
           [0048]      FIG. 2(   b ) illustrates the same type spectrometer as in  FIG. 2(   a ), but incorporating features of an embodiment of the invention; 
           [0049]      FIG. 3(   a ) is a perspective view of a prior art mirror spectrometer of Czerny-Turner configuration; 
           [0050]      FIGS. 3(   b ), ( c ) and ( d ) illustrates three of the same type of spectrometers of  FIG. 3(   a ), but incorporating features of three various embodiments of the invention, specifically  FIG. 3(   c ) illustrates the same type of spectrometer built by a piece of monolithic transparent carrier body further incorporating features of an embodiment of the invention; 
           [0051]      FIG. 4(   a ) is a perspective view of a prior art mirror spectrometer of Fastie-Ebert configuration; 
           [0052]      FIGS. 4(   b ) and ( c ) illustrate two of the same type of spectrometer as in  FIG. 4(   a ), but incorporating features of various embodiments of the invention, specifically,  FIG. 4(   c ) shows the same type of spectrometer built by a piece of monolithic transparent carrier body further incorporating features of an embodiment of the invention; 
           [0053]      FIG. 5(   a ) is a perspective view of a prior art mirror spectrometer of crossed Czemy-Tumer configuration; 
           [0054]      FIGS. 5(   b ) and ( c ) illustrate two of the same type of spectrometer as in  FIG. 5(   a ), but incorporating features of various embodiments of the invention, specifically,  FIG. 5(   c ) shows the same type of spectrometer built by a piece of monolithic transparent carrier body further incorporating features of an embodiment of the invention; 
           [0055]      FIG. 6(   a ) is a perspective view of a prior art compact spectrometer comprising a lens and a reflective grating; 
           [0056]      FIGS. 6(   b ) and ( c ) illustrate the same type of spectrometer as in  FIG. 6(   a ), but incorporating features of various embodiments of the invention; 
           [0057]      FIG. 7(   a ) is a perspective view of a prior art compact spectrometer comprising a concave grating only; 
           [0058]      FIGS. 7(   b ) to ( e ) illustrate the same type of spectrometer as in  FIG. 7(   a ), but incorporating features of various embodiments of the invention, specifically,  FIGS. 7(   c ) and ( e ) show the same spectrometer built by a piece of monolithic transparent carrier body, respectively further incorporating features of at least one embodiment of the invention; 
           [0059]      FIG. 8(   a ) to  FIG. 8(   e ) show schematic views of five embodiments of a collimating means usable with embodiments of the invention. These embodiments also may be utilized in the focusing means for constructing compact spectrometers; and 
           [0060]      FIG. 9(   a ) and  FIG. 9(   b ) show schematic views of two embodiments of the optical path configuration for the collimating means and the focusing means, for constructing compact spectrometers according to embodiments of the invention. 
       
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
       [0061]    Referring now to the drawings, to the following detailed description, and to the incorporated materials, detailed information about aspects of the invention is provided including the description of specific embodiments. The detailed description serves to explain principles of the invention. The embodiments may be susceptible to modifications and alternative forms. Embodiments are not limited to the particular forms disclosed. Rather, the embodiments cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention. 
         [0062]    Referring to  FIG. 1(   a ), a prior art lens spectrometer  100  is illustrated in ray-trace form. The optics of spectrometer  100  comprises an entrance aperture  104  that is the core of the optical fiber  102  for input signal delivery, a collimating lens  108 , a transmissive diffraction grating  112  and a focusing lens  116 . For the spectrometer  100 , the input light  106  emits from the entrance aperture  104  and propagates in divergence towards the collimating lens  108 , which collimates the divergent light  106  into the collimated light  110 . The collimated light  110  propagates and is incident upon the grating  112 , which disperses the light  110  into the dispersive collimated light  114 . The focusing lens  116  subsequently focuses the light  114  into the convergent light  118  thereby forming spectral images  120  on the detector  122 . As shown in  FIG. 1(   a ), the propagation paths for the divergent light  106 , the collimated light  110 , the dispersive light  114 , and the convergent light  118  are all three-dimensional. The three key optical elements within the spectrometer  100 , i.e., the collimating lens  108 , the grating  112  and the focusing lens  116 , must have finite working apertures large enough to accept and to manipulate the light  106 ,  110 ,  114  and  118  without truncating them at any locations. Consequently, the overall dimensional volume necessary to construct the spectrometer  100  is three-dimensional. Such a spectrometer is generally large and very difficult to be reduced without sacrificing its performance characteristics. 
         [0063]    In  FIG. 1(   b ), one embodiment of the same type of lens spectrometer  150  is illustrated in ray-trace form. The  FIG. 1(   b ) embodiment incorporates aspects of the invention. The optics of spectrometer  150  comprises an entrance aperture  153  that is the core of the optical fiber  152  for input signal delivery (e.g., the input “signal” is white light), a first cylindrical lens  155 , a second cylindrical lens  158 , a transmissive diffraction grating  162 , a third cylindrical lens  166  and a fourth cylindrical lens  169 . For the spectrometer  150 , the input light  154  emits from the entrance aperture  153  and propagates in divergence over a very short distance, then is intercepted by the first cylindrical lens  155 , which collimates the divergent light  154  only in the tangential plane (only in the YOZ plane), converting it into a partially collimated light, i.e., the anamorphic light  156 , which is collimated in the tangential plane, but remains divergent in the sagittal (XOZ plane). The light  156  propagates and is transmitted through the second cylindrical lens  158 , which collimates it only in the sagittal plane, converting it into the fully collimated light  160 . The light  160  continues to propagate and is incident upon the grating  162 , which disperses the light  160  into dispersive collimated light  164 . Upon being transmitted through the third cylindrical lens  166 , the light  164  is partially focused in the sagittal plane into the light  168 , which is further partially focused by the fourth cylindrical lens  169  in the tangential plane into the fully convergent light  170  to form spectral images  171  on the detector  172 . As shown in  FIG. 1(   b ), the propagation paths for the anamorphic light  156 , collimated light  160 , dispersive light  164 , and the anamorphic light  168  are all substantially two-dimensional. The five key optical elements within the spectrometer  150 , i.e., the first cylindrical lens  155 , the second cylindrical lens  158 , the grating  162 , the third cylindrical lens  166  and the fourth cylindrical lens  169 , have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but otherwise very small aperture dimensions in the tangential direction (i.e., vertical). That is, the spectrometer requires only dimensions sufficient to accept and to manipulate light (i.e.,  154 ,  156 ,  160 ,  164 ,  168  and  170 ) without truncating the light (i.e.,  154 ,  156 ,  160 ,  164 ,  168  and  170 ) at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical elements needed may become small fractions of their original values in the prior art for similar types of devices, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. Thus, the overall dimensional volume necessitated to construct the spectrometer  150  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its similar type of prior art spectrometer. Additionally, such a reduction in size does not sacrifice the spectrometer performance characteristics. 
         [0064]    Next, referring to  FIG. 2(   a ), a prior art catadioptric spectrometer  200  is illustrated in ray-trace form. The optics of spectrometer  200  comprises an entrance aperture  204  that is the core of the optical fiber  202  for input signal delivery, a collimating mirror  208 , a reflective diffraction grating  212  and a focusing lens  216 . For the spectrometer  200 , the input light  206  emits from the entrance aperture  204  and propagates in divergence towards the collimating mirror  208 , which collimates the divergent light  206  into the collimated light  210 . The collimated light  210  propagates and is incident upon the grating  212 , which disperses, in a reflective manner, the light  210  into the dispersive collimated light  214 . Thereafter, the focusing lens  216  focuses the light  214  into the convergent light  218  to form spectral images  220  on the detector  222 . As shown in  FIG. 2(   a ), the propagation paths for the divergent light  206 , the collimated light  210 , the dispersive light  214 , and the convergent light  218  are all three-dimensional. The three key optical elements within the spectrometer  200 , (i.e., the collimating mirror  208 , the grating  212  and the focusing lens  216 ), must have finite working apertures large enough to accept and manipulate the light (i.e.,  206 ,  210 ,  214  and  218 ) without truncating such light at any location. Generally, the overall dimensional volume necessitated to construct the spectrometer  200  is three-dimensional. Such an overall dimensional volume is generally large and very difficult to be reduced without sacrificing its performance characteristics. 
         [0065]    In  FIG. 2(   b ), a catadioptric spectrometer  250  incorporating features of one embodiment of the invention is illustrated in ray-trace form. Such a catadioptric spectrometer  250  is of the same type of spectrometer  200  illustrated in  FIG. 2(   a ). The optics of spectrometer  250  comprises an entrance aperture  253  that is the core of the optical fiber  252  for input signal delivery, a first cylindrical lens  255 , a cylindrical mirror  258 , a reflective diffraction grating  262 , a second cylindrical lens  266  and a third cylindrical lens  269 . For the spectrometer  250 , the input light  254  emits from the entrance aperture  253  and propagates in divergence over a very short distance, then is transmitted through the first cylindrical lens  255 , which collimates the divergent light  254  only in the tangential plane, thereby converting such light into a partially collimated light, (i.e., the anamorphic light  256 ), which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  256  propagates and is reflected by the cylindrical mirror  258 , which collimates it only in the sagittal plane, converting it into the fully collimated light  260 . The light  260  continues to propagate and is incident upon the grating  262 , which disperses, in a reflective manner, the light  260  into dispersive collimated light  264 . Upon being intercepted by the second cylindrical lens  266 , the light  264  is partially focused in the sagittal plane into the light  268 , which is further partially focused by the third cylindrical lens  269  in the tangential plane into the fully convergent light  270  to form spectral images  271  on the detector  272 . As shown in  FIG. 2(   b ), the propagation paths for the anamorphic light  256 , the collimated light  260 , the dispersive light  264 , and the anamorphic light  268  are all substantially two-dimensional. The five key optical elements within the spectrometer  250 , i.e., the first cylindrical lens  255 , the cylindrical mirror  258 , the grating  262 , the second cylindrical lens  266  and the third cylindrical lens  269 , have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions in the tangential direction (i.e., vertical), in order to accept and manipulate light  254 ,  256 ,  260 ,  264 ,  268  and  270  without truncating them at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical elements may become a small fraction of those dimensions in the same type of prior art spectrometer, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. Consequently, the overall dimensional volume necessitated to construct the spectrometer  250  is substantially two-dimensional, or unilateral, which is significantly reduced compared with that of its prior art spectrometer without sacrificing its performance characteristics. 
         [0066]    Next, referring to  FIG. 3(   a ), a prior art mirror spectrometer  300  of Czemy-Turner geometry is illustrated in ray-trace form. Its optics comprises an entrance aperture  304  that is the core of the optical fiber  302  for input signal delivery, a collimating mirror  308 , a reflective diffraction grating  312  and a focusing mirror  316 . For the spectrometer  300 , the input light  306  emits from the entrance aperture  304  and propagates in divergence toward the collimating mirror  308 , which collimates the divergent light  306  into the collimated light  310 . The collimated light  310  propagates and is incident upon the grating  312 , which disperses, in a reflective manner, the light  310  into the dispersive collimated light  314 , and then the focusing mirror  316  focuses the light  314  into the convergent light  318  to form spectral images  320  on the detector  322 . As shown in  FIG. 3(   a ), the propagation paths for the divergent light  306 , the collimated light  310 , the dispersive light  314 , and the convergent light  318  are all three-dimensional. The three key optical elements within the spectrometer  300 , i.e., the collimating mirror  308 , the grating  312  and the focusing mirror  316 , must have finite working apertures large enough to accept and to manipulate the light  306 ,  310 ,  314  and  318  without truncating such light at any locations. As a result, the overall dimensional volume necessitated to construct the spectrometer  300  is three-dimensional. Such an overall dimensional volume is generally large and very difficult to be reduced without sacrificing its performance characteristics. 
         [0067]    In  FIG. 3(   b ), a mirror spectrometer  330  of Czerny-Turner geometry incorporating features of one embodiment of the invention is shown in ray-trace form. The mirror spectrometer  330  is the same type of mirror spectrometer of Czemy-Turner geometry as that shown in  FIG. 3(   a ). The optics of mirror spectrometer  330  comprises an entrance aperture  333  that is the core of the optical fiber  332  for input signal delivery, a first cylindrical lens  335 , a first cylindrical mirror  338 , a reflective diffraction grating  342 , a second cylindrical mirror  346  and a second cylindrical lens  349 . For the spectrometer  330 , the input light  334  emits from the entrance aperture  333  and propagates in divergence over a very short distance, then is transmitted through the first cylindrical lens  335 , which collimates the divergent light  334  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  336 , which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  336  propagates and is reflected by the first cylindrical mirror  338 , which collimates it only in the sagittal plane, converting it into the fully collimated light  340 . The light  340  continues to propagate and is incident upon the grating  342 , which disperses, in a reflective manner, the light  340  into dispersive collimated light  344 . Upon being reflected by the second cylindrical mirror  346 , the light  344  is partially focused in the sagittal plane into the light  348 , which is further partially focused by the second cylindrical lens  349  in the tangential plane into the fully convergent light  350  to form spectral images  351  on the detector  352 . As shown in  FIG. 3(   b ), the propagation paths for the anamorphic light  336 , the collimated light  340 , the dispersive light  344 , and the anamorphic light  348  are all substantially two-dimensional. The five key optical elements within the spectrometer  330 , i.e., the first cylindrical lens  335 , the first cylindrical mirror  338 , the grating  342 , the second cylindrical mirror  346  and the second cylindrical lens  349 , must have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions are needed in the tangential direction (i.e., vertical), in order to accept and to manipulate light  334 ,  336 ,  340 ,  344 ,  348  and  350  without truncating such light at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical elements needed may become small fractions of their original values in the same prior art, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. Thus, the overall dimensional volume necessitated to construct the spectrometer  330  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer. Additionally, such a reduction in size does not sacrifice its performance characteristics. 
         [0068]    In  FIG. 3(   c ), a mirror spectrometer  360  of Czemy-Tumer geometry incorporating features of one embodiment of the invention is illustrated in ray-trace form. Such a mirror spectrometer is of the same type as those shown in  FIGS. 3(   a ) and ( b ). The spectrometer  360  is constructed by combining the five key optical elements in the spectrometer  330  together with a single piece of monolithic transparent carrier. The optics of spectrometer  360  comprises an entrance aperture  363  that is the core of the optical fiber  362  for input signal delivery, a first cylindrical surface  365 , a first cylindrical mirror  367 , a reflective diffraction grating  370 , a second cylindrical mirror  373  and a second cylindrical surface  376 . For the spectrometer  360 , the input light  364  emits from the entrance aperture  363  and propagates in divergence over a very short distance, then is transmitted through the first cylindrical surface  365 , which collimates the divergent light  364  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  366 , which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  366  propagates and is reflected by the first cylindrical mirror  367 , which collimates it only in the sagittal plane, converting it into the fully collimated light  368 . The light  368  continues to propagate and is incident upon the grating  370 , which disperses, in a reflective manner, the light  368  into the dispersive collimated light  372 . Upon being reflected by the second cylindrical mirror  373 , the light  372  is partially focused in the sagittal plane into the light  374 , which is further partially focused by the second cylindrical surface  376  in the tangential plane into the fully convergent light  377  to form spectral images  378  on the detector  379 . As shown in  FIG. 3(   c ), the propagation paths for the anamorphic light  366 , the collimated light  368 , the dispersive light  372 , and the anamorphic light  374  are all substantially two-dimensional. The five key optical surfaces within the spectrometer  360 , i.e., the first cylindrical surface  365 , the first cylindrical mirror  367 , the grating  370 , the second cylindrical mirror  373  and the second cylindrical surface  376 , must have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions needed in the tangential direction (i.e., vertical), in order to accept and to manipulate light  364 ,  366 ,  368 ,  372 ,  374  and  377  without truncating such light at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical surfaces needed may become small fractions of their original values in the same prior art, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. Thus, the overall dimensional volume necessitated to construct the spectrometer  360  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer. Additionally, such a reduction in size does not sacrifice its performance characteristics. Thus it is possible, based on the embodiment, to construct a spectrometer fabricated with a single piece of thin transparent carrier, which is robust and of very compact volume. 
         [0069]    In  FIG. 3(   d ), a mirror spectrometer  390  incorporating the features of another embodiment of the invention is illustrated. Such a mirror spectrometer is of the same type of mirror spectrometer shown in  FIGS. 3(   a ) and ( b ) in ray-trace form. The spectrometer  390  may be modified from the spectrometer  330  shown in  FIG. 3(   b ) by combining the two reflective mirrors into one mirror vertically and properly positioning the reflective grating. As a result, the overall optical paths within the spectrometer  390  have been folded three times by the single concave cylindrical mirror and the reflective grating, leading to a small instrument volume, which is very compact compared to that of the prior art spectrometer  300  shown in  FIG. 3(   a ). Such a reduction in instrument volume is achieved without sacrificing its performance characteristics. 
         [0070]      FIG. 4(   a ) represents another prior art mirror spectrometer  400 , but of Fastie-Ebert geometry, which is very similar to that of Czemy-Tumer geometry shown in  FIG. 3(   a ). A Fastie-Ebert spectrometer may be constructed from a Czemy-Turner spectrometer by properly combining the two cylindrical mirrors together into one big cylindrical mirror, which functions for both collimating and focusing. Apart from this structural difference, the respective working principles are the same.  FIG. 4(   b ) shows a mirror spectrometer  430  incorporating features of one embodiment of the invention. Such an embodiment is of the same type of spectrometer shown in  400 , and is a counterpart of  FIG. 3(   b ). In other words, both spectrometers work in the same way. Thus, the overall dimensional volume necessitated to construct the spectrometer  430  in  FIG. 4(   b ) is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer. Additionally, such a reduction in overall dimensional volume does not sacrifice its performance characteristics.  FIG. 4(   c ) shows another embodiment of the type of spectrometer shown in  FIG. 4(   a ) to which aspects of the invention have been applied. Such a spectrometer  460  is a counterpart of spectrometer  360  shown in  FIG. 3(   c ), and they both work in the same way. The spectrometer  460  in  FIG. 4(   c ) may be fabricated with a single piece of thin transparent carrier, which is robust and of very compact volume. 
         [0071]    Next, referring to  FIG. 5(   a ), another prior art mirror spectrometer  500  of crossed Czemy-Turner geometry is illustrated in ray-trace form. The spectrometer  500  is modified from the spectrometer  300  in  FIG. 3(   a ), with respect to where the incident beam and the reflected beam from the diffraction grating cross. The optics of such a spectrometer  500  comprises an entrance aperture  504  that may be the core of the optical fiber  502  for input signal delivery, a collimating mirror  508 , a reflective diffraction grating  512  and a focusing mirror  516 . For the spectrometer  500 , the input light  506  emits from the entrance aperture  504  and propagates in divergence toward the collimating mirror  508 , which collimates the divergent light  506  into the collimated light  510 . The collimated light  510  propagates and thereafter may be incident upon the grating  512 , which disperses, in a reflective manner, the light  510  into the dispersive collimated light  514 , and then the focusing mirror  516  focuses the light  514  into the convergent light  518  to form spectral images  520  on the detector  522 . As shown in  FIG. 5(   a ), the propagation paths for the divergent light  506 , the collimated light  510 , the dispersive light  514 , and the convergent light  518  are all three-dimensional. The three key optical elements within the spectrometer  500 , i.e., the collimating mirror  508 , the grating  512  and the focusing mirror  516 , must have finite working apertures large enough to accept and to manipulate the light  506 ,  510 ,  514  and  518  without truncating such light at any locations. Consequently, the overall dimensional volume necessitated to construct the spectrometer  500  is three-dimensional. Such an overall dimensional volume is generally large and very difficult to be reduced without sacrificing its performance characteristics. 
         [0072]      FIG. 5(   b ) shows a mirror spectrometer  550  of crossed Czemy-Tumer geometry incorporating the features of an embodiment of the invention. Such a mirror spectrometer  550  is of the same type of spectrometer as that shown in  FIG. 5(   a ). The optics of spectrometer  550  comprises an entrance aperture  553  that may be the core of the optical fiber  552  for input signal delivery, a first cylindrical lens  555 , a first cylindrical mirror  558 , a reflective diffraction grating  562 , a second cylindrical mirror  566  and a second cylindrical lens  569 . For the spectrometer  550 , the input light  554  emits from the entrance aperture  553  and propagates in divergence over a very short distance, then is transmitted through the first cylindrical lens  555 , which collimates the divergent light  554  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  556 , which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  556  propagates and is reflected by the first cylindrical mirror  558 , which collimates it only in the sagittal plane, converting it into the fully collimated light  560 . The light  560  continues to propagate and is incident upon the grating  562 , which disperses, in a reflective manner, the light  560  into dispersive collimated light  564 . Upon being reflected by the second cylindrical mirror  566 , the light  564  is partially focused in the sagittal plane into the light  568 , which is further partially focused by the second cylindrical lens  569  in the tangential plane into the fully convergent light  570  to form spectral images  571  on the detector  572 . As shown in  FIG. 5(   b ), the propagation paths for the anamorphic light  556 , the collimated light  560 , the dispersive light  564 , and the anamorphic light  568  are all substantially two-dimensional. The five key optical elements within the spectrometer  550 , i.e., the first cylindrical lens  555 , the first cylindrical mirror  558 , the grating  562 , the second cylindrical mirror  566  and the second cylindrical lens  569 , must have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions needed in the tangential direction (i.e., vertical), in order to accept and to manipulate light  554 ,  556 ,  560 ,  564 ,  568  and  570  without truncating such light at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical elements needed become small fractions of their original values in the same prior art, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. Thus, the overall dimensional volume necessitated to construct the spectrometer  550  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer. Additionally, such a reduction in overall dimensional volume does not sacrifice its performance characteristics. 
         [0073]      FIG. 5(   c ) shows, in ray-trace form, a mirror spectrometer  580  of crossed Czemy-Turner geometry incorporating features of an embodiment of the invention. Such a mirror spectrometer is of the same type of spectrometer as those shown in  FIG. 5(   a ) and (b). The spectrometer  580  may be constructed by combining the five key optical elements in the spectrometer  550  together with a single piece of monolithic transparent carrier. The optics of such a spectrometer comprises an entrance aperture  583  that is the core of the optical fiber  582  for input signal delivery, a first cylindrical surface  586 , a first cylindrical mirror  588 , a reflective diffraction grating  591 , a second cylindrical mirror  594  and a second cylindrical surface  596 . For the spectrometer  580 , the input light  584  emits from the entrance aperture  583  and propagates in divergence over a very short distance, then may be transmitted through the first cylindrical surface  586 , which collimates the divergent light  584  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  587 , which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  587  propagates and is reflected by the first cylindrical mirror  588 , which collimates it only in the sagittal plane, converting it into the fully collimated light  590 . The light  590  continues to propagate and is incident upon the grating  591 , which disperses, in a reflective manner, the light  590  into the dispersive collimated light  592 . Upon being reflected by the second cylindrical mirror  594 , the light  592  may be partially focused in the sagittal plane into the light  595 , which is further partially focused by the second cylindrical surface  596  in the tangential plane into the fully convergent light  597  to form spectral images  598  on the detector  599 . As shown in  FIG. 5(   c ), the propagation paths for the anamorphic light  587 , the collimated light  590 , the dispersive light  592 , and the anamorphic light  595  are all substantially two-dimensional. The five key optical surfaces within the spectrometer  580 , i.e., the first cylindrical surface  586 , the first cylindrical mirror  588 , the grating  591 , the second cylindrical mirror  594  and the second cylindrical surface  596 , must have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions needed in the tangential direction (i.e., vertical) in order to accept and to manipulate light  584 ,  587 ,  590 ,  592 ,  595  and  597  without truncating such light at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical surfaces needed become small fractions of their original values in the same prior art, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. Thus, the overall dimensional volume necessitated to construct the spectrometer  580  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer. Additionally, such a reduction in size does not sacrifice the spectrometer&#39;s performance characteristics. Thus it is possible to construct a spectrometer fabricated with a single piece of thin transparent carrier of pentagon shape, which is robust and of very compact volume. 
         [0074]    Next referring to  FIG. 6(   a ), a prior art compact spectrometer  600  is illustrated in ray-trace form. The optics of spectrometer  600  comprises an entrance aperture  604  that may be the core of the optical fiber  602  for input signal delivery, a lens  608  for both collimating and focusing, and a reflective diffraction grating  612 . For the spectrometer  600 , the input light  606  emits from the entrance aperture  604  and propagates in divergence toward the lens  608 , which collimates the divergent light  606  into the collimated light  610 . The collimated light  610  propagates and may be incident upon the grating  612 , which disperses, in a reflective manner, the light  610  into the dispersive collimated light  614 , and then the same lens  608  focuses the light  614  into the convergent light  618  to form spectral images  620  on the detector  622 . As shown in  FIG. 6(   a ), the propagation paths for the divergent light  606 , the collimated light  610 , the dispersive light  614 , and the convergent light  618  are all three-dimensional. The two key optical elements within the spectrometer  600 , i.e., the lens  608  and the grating  612 , must have finite working apertures large enough to accept and to manipulate the light  606 ,  610 ,  614  and  618  without truncating such light at any locations. As a result, the overall dimensional volume necessitated to construct the spectrometer  600  is three-dimensional. Such a spectrometer is generally large and very difficult to be reduced without sacrificing its performance characteristics. 
         [0075]      FIG. 6(   b ) shows, in ray-trace form, a compact spectrometer  650  incorporating features of an embodiment of the invention. Such spectrometer is of the same type as that shown in  FIG. 6(   a ). The optics of spectrometer  650  comprises an entrance aperture  653  that may be the core of the optical fiber  652  for input signal delivery, a first cylindrical lens  655 , a second cylindrical lens  658 , and a reflective diffraction grating  662 . For the spectrometer  650 , the input light  654  emits from the entrance aperture  653  and propagates in divergence over a very short distance, then is transmitted through the first cylindrical lens  655 , which collimates the divergent light  654  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  656 , which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  656  propagates and is transmitted through the second cylindrical lens  658 , which collimates it only in the sagittal plane, converting it into the fully collimated light  660 . The light  660  continues to propagate and is incident upon the grating  662 , which disperses, in a reflective manner, the light  660  into dispersive collimated light  664 . Upon being transmitted through the same cylindrical lens  658 , the light  664  is partially focused in the sagittal plane into the light  668 , which is further partially focused by the cylindrical lens  655  in the tangential plane into the fully convergent light  670  to form spectral images  671  on the detector  672 . As shown in  FIG. 6(   b ), the propagation paths for the anamorphic light  656 , collimated light  660 , dispersive light  664 , and the anamorphic light  668  are all substantially two-dimensional. The three key optical elements within the spectrometer  650 , i.e., the first cylindrical lens  655 , the second cylindrical lens  658  and the grating  662 , must have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions needed in the tangential direction (i.e., vertical), in order to accept and to manipulate light  654 ,  656 ,  660 ,  664 ,  668  and  670  without truncating such light at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical elements needed become small fractions of their original values in the same prior art, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. As a result, the overall dimensional volume necessitated to construct the spectrometer  650  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer. Additionally, such a reduction in size does not sacrifice the spectrometer performance characteristics. 
         [0076]      FIG. 6(   c ) shows, in ray-trace form, a compact spectrometer  680 . Such a spectrometer is of the same type as that shown in  FIG. 6(   a ). The optics of spectrometer  680  comprises an entrance aperture  683  that may be the core of the optical fiber  682  for input signal delivery, a first cylindrical surface  686 , a second cylindrical surface  688 , a reflective diffraction grating  691  and a third cylindrical surface  695 . For the spectrometer  680 , the input light  684  emits from the entrance aperture  683  and propagates in divergence over a very short distance, then is transmitted through the first cylindrical lens  686 , which collimates the divergent light  684  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  687 , which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  687  propagates and is transmitted through the second cylindrical lens  688 , which collimates it only in the sagittal plane, converting it into the fully collimated light  690 . The light  690  continues to propagate and is incident upon the grating  691 , which disperses, in a reflective manner, the light  690  into dispersive collimated light  692 . Upon being transmitted through the same cylindrical surface  688 , the light  692  is partially focused in the sagittal plane into the light  694 , which is further partially focused by the third cylindrical surface  695  in the tangential plane into the fully convergent light  696  to form spectral images  698  on the detector  699 . As shown in  FIG. 6(   c ), the propagation paths for the anamorphic light  687 , collimated light  690 , dispersive light  692 , and the ariamorphic light  694  are all substantially two-dimensional. The four key optical elements/surfaces within the spectrometer  680 , i.e., the first cylindrical surface  686 , the second cylindrical surface  688 , the grating  691  and the third cylindrical surface  695 , must have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions needed in the tangential direction (i.e., vertical), in order to accept and to manipulate light  684 ,  687 ,  690 ,  692 ,  694  and  696  without truncating such light at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical elements/surface needed become small fractions of their original values in the same prior art, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. Thus, the overall dimensional volume necessitated to construct the spectrometer  680  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer. Additionally, such a reduction in size does not sacrifice the spectrometer performance characteristics. 
         [0077]    Next referring to  FIG. 7(   a ), a prior art compact spectrometer  700  is illustrated in ray-trace form. The optics of spectrometer  700  comprises an entrance aperture  702  that may be the core of the optical fiber  701  for input signal delivery, and a concave diffraction grating  706 . For the spectrometer  700 , the input light  705  emits from the entrance aperture  702  and propagates in divergence towards the concave grating  706 , which disperses, in a reflective manner, the divergent light  705  and focuses it into the convergent light  707  to form spectral images  710  on the detector  711 . As shown in  FIG. 7(   a ), the propagation paths for the divergent light  705  and the convergent light  707  are all three-dimensional. The single key optical element within the spectrometer  700 , i.e., the concave grating  706 , must have finite working apertures large enough to accept and to manipulate the light  705  and  707  without truncating them at any locations. As a result, the overall dimensional volume needed to construct the spectrometer  700  is three-dimensional. Such a spectrometer is generally still large for many applications and very difficult to be reduced without sacrificing its performance characteristics. 
         [0078]      FIG. 7(   b ) shows, in ray-trace form, a compact spectrometer  720  that incorporates features of an embodiment of the invention. Such a spectrometer is of the same type as that shown in  FIG. 7(   a ). The optics of spectrometer  720  comprises an entrance aperture  722  that may be the core of the optical fiber  721  for input signal delivery, a first cylindrical lens  724 , a concave cylindrical grating  626 , and a second cylindrical lens  728 . For the spectrometer  720 , the input light  723  emits from the entrance aperture  722  and propagates in divergence over a very short distance, then is transmitted through the first cylindrical lens  724 , which collimates the divergent light  723  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  725 , which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  725  propagates and is reflected by the concave cylindrical grating  726 , which disperses, in a reflective manner, the light  725  and focuses it only in the sagittal plane into the anamorphic light  727 , which remains collimated in the tangential plane, but is dispersed and convergent in the sagittal plane. Upon being transmitted through the second cylindrical lens  728 , the light  727  is focused in the tangential plane into the fully convergent light  729  to form spectral images  730  on the detector  731 . As shown in  FIG. 7(   b ), the propagation paths for the anamorphic light  725 , and the dispersed anamorphic light  727  are all substantially two-dimensional. The three key optical elements within the spectrometer  720 , i.e., the first cylindrical lens  724 , the grating  726 , and the second cylindrical lens  728 , must have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions are needed in the tangential direction (i.e., vertical), in order to accept and to manipulate light  723 ,  725 ,  727  and  729  without truncating such light at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical elements needed become small fractions of their original values in the same prior art, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. As a result, the overall dimensional volume necessitated to construct the spectrometer  720  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer without sacrificing its performance characteristics. 
         [0079]      FIG. 7(   c ) shows, in ray-trace form, a compact spectrometer  740  that incorporates features of an embodiment of the invention. Such a spectrometer is of the same type as those shown in  FIGS. 7(   a ) and ( b ). The spectrometer  740  is constructed by combining the three key optical elements in the spectrometer  720  together with a single piece of monolithic transparent carrier. The optics of spectrometer  740  comprises an entrance aperture  742  that may be the core of the optical fiber  741  for input signal delivery, a first cylindrical surface  744 , a concave cylindrical grating  746 , and a second cylindrical surface  748 . For the spectrometer  740 , the input light  743  emits from the entrance aperture  742  and propagates in divergence over a very short distance, then is transmitted through the first cylindrical surface  744 , which collimates the divergent light  743  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  745 , which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  745  propagates in the transparent medium and is intercepted by the concave cylindrical grating  746 , which disperses, in a reflective manner, the light  745  and focuses it only in the sagittal plane into the anamorphic light  747 , which remains collimated in the tangential plane, but is dispersed and convergent in the sagittal plane. Upon being transmitted through the second cylindrical surface  748 , the light  747  is focused in the tangential plane into the fully convergent light  749  to form spectral images  750  on the detector  751 . As shown in  FIG. 7(   c ), the propagation paths for the anamorphic light  745  and the anamorphic light  747  are all substantially two-dimensional. The three key optical surfaces within the spectrometer  740 , i.e., the first cylindrical surface  744 , the concave cylindrical grating  746  and the second cylindrical surface  748 , must have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions are needed in the tangential direction (i.e., vertical), in order to accept and manipulate light  743 ,  745 ,  747  and  749  without truncating such light at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical surfaces needed may become small fractions of their original values in the same prior art, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. Thus, the overall dimensional volume necessitated to construct the spectrometer  740  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer. Additionally, such a reduction in the overall dimensional volume does not sacrifice the spectrometer performance characteristics. Thus it is possible to easily construct a spectrometer fabricated with a single piece of thin transparent carrier, which is robust and of very compact volume. 
         [0080]      FIG. 7(   d ) shows, in ray-trace form, a compact spectrometer  760  that incorporates features of an embodiment of the invention. Such a spectrometer is of the same type as that shown in  FIG. 7(   a ). The optics of spectrometer  760  comprises an entrance aperture  762  that may be the core of the optical fiber  761  for input signal delivery, a cylindrical lens  764  and a concave cylindrical grating  766 . For the spectrometer  760 , the input light  763  emits from the entrance aperture  762  and propagates in divergence over a very short distance, then is transmitted through the cylindrical lens  764 , which collimates the divergent light  763  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  765 , which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  765  propagates and is intercepted by the concave cylindrical grating  766 , which disperses, in a reflective manner, the light  765  and focuses it only in the sagittal plane into the anamorphic light  767 , which remains collimated in the tangential plane, but is dispersed and convergent in the sagittal plane. Upon being transmitted through the same cylindrical lens  764 , the light  767  is focused in the tangential plane into the fully convergent light  769  to form spectral images  770  on the detector  771 . As shown in  FIG. 7(   d ), the propagation paths for the anamorphic light  765 , and the dispersed anamorphic light  767  are all substantially two-dimensional. The two key optical elements within the spectrometer  760 , i.e., the first cylindrical lens  764 , and the grating  766 , must have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions needed in the tangential direction (i.e., vertical), in order to accept and to manipulate light  763 ,  765 ,  767  and  769  without truncating such light at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical elements needed may become small fractions of their original values in the same prior art, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. Thus, the overall dimensional volume needed to construct the spectrometer  760  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer. Additionally, such a reduction in the overall dimensional volume does not sacrifice the spectrometer performance characteristics. 
         [0081]      FIG. 7(   e ) shows, in ray-trace form, a compact spectrometer  780  that incorporates features of an embodiment of the invention. Such a spectrometer is of the same type as those shown in  FIGS. 7(   a ) and ( d ). The spectrometer  780  is constructed by combining the two key optical elements in the spectrometer  760  together with a single piece of monolithic transparent carrier. The optics of spectrometer  780  comprises an entrance aperture  782  that may be the core of the optical fiber  781  for input signal delivery, a cylindrical surface  784  and a concave cylindrical grating  786 . For the spectrometer  780 , the input light  783  emits from the entrance aperture  782  and propagates in divergence over a very short distance, then is transmitted through the cylindrical surface  784 , which collimates the divergent light  783  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  785 , which is collimated in the tangential plane, but remains divergent in the sagittal plane. The light  785  propagates in the transparent medium and is reflected by the concave cylindrical grating  786 , which disperses, in a reflective manner, the light  785  and focuses it only in the sagittal plane into the anamorphic light  787 , which remains collimated in the tangential plane, but is dispersed and convergent in the sagittal plane. Upon being transmitted through the same cylindrical surface  784 , the light  787  is focused in the tangential plane into the fully convergent light  789  to form spectral images  790  on the detector  791 . As shown in  FIG. 7(   e ), the propagation paths for the anamorphic light  785  and the anamorphic light  787  are all substantially two-dimensional. The two key optical surfaces within the spectrometer  780 , i.e., the first cylindrical surface  784  and the concave cylindrical grating  786 , must have finite working aperture dimensions large enough only in the sagittal direction (i.e., horizontal), but very small aperture dimensions needed in the tangential direction (i.e., vertical), in order to accept and to manipulate light  783 ,  785 ,  787  and  789  without truncating such light at any locations. In practice, the tangential dimensions (i.e., vertical) of those key optical surfaces needed may become small fractions of their original values in the same prior art, for example, around ⅕˜ 1/10 (i.e., an approximate reduction in size of 80% to 90% may be achieved) or even better. Consequently, the overall dimensional volume needed to construct the spectrometer  780  is substantially two-dimensional, or substantially unilateral, which is significantly reduced compared with that of its prior art spectrometer. Additionally, such a reduction in the overall dimensional volume does not sacrifice the spectrometer performance characteristics. Thus it is possible to easily construct a spectrometer fabricated with a single piece of thin transparent carrier, which is robust and of very compact volume. 
         [0082]    In  FIGS. 8(   a ) to ( e ), five embodiments of a collimating means (collimator) based on embodiments are represented. Each collimator serves the same purpose and any of them may be chosen in building a specific spectrometer incorporating features of an embodiment of the invention. The functionalities of such collimating means have been described in those associated embodiments shown in  FIG. 1  to  FIG. 7 . Thus, for purposes of brevity, only  FIG. 8(   a ) will be described in detail. In  FIG. 8(   a ), the input light  804  emits from the entrance aperture  803  that may be the core of the optical fiber  802  for input signal delivery, and propagates in divergence over a very short distance, then is intercepted by the cylindrical lens  806 , which, as the light passes through the lens, collimates the divergent light  804  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  808 , which is collimated in the tangential plane, but remains divergent in the sagittal plane, leading to a propagation path that is substantially two-dimensional. Without using this cylindrical lens  806 , the propagation path of light  804  would follow dashed-line  807 , which is three-dimensional. The cylindrical lens  806  may be made from transparent optical materials. Either of its curved surfaces may be generally toroidal, i.e., having the major optical power in the tangential plane, but have little or no optical power in the sagittal plane. Its section profile in the tangential plane may take any of the following forms: plano-convex, bi-convex, or meniscus, where curved profiles can be spherical, aspherical, or conic curves. The cylindrical lens  806  also can be used as the focusing means (focusing optics) for constructing the same compact spectrometer based on the present invention. 
         [0083]    In  FIG. 8(   b ), another embodiment for the collimating means is presented, which shows a cylindrical surface  816  formed on a monolithic transparent carrier, by which the compact spectrometer is constructed. The cylindrical surface  816  may be generally toroidal, i.e., having the major optical power in the tangential plane, but have little or no optical power in the sagittal plane. Its section profile in the tangential plane may take any of the following forms: spherical, aspherical, or conic curves. The cylindrical surface  816  also can be used as the focusing means for constructing the same compact spectrometer based on the present invention. 
         [0084]    In  FIG. 8(   c ), another embodiment for the collimating means is presented, which shows a concave cylindrical mirror  826  at 45°. The mirror  826  may be generally toroidal, i.e., having the major optical power in the tangential plane, but have little or no optical power in the sagittal plane. Its section profile in the tangential plane may take any of the following forms: spherical, aspherical, or conic curves. The concave cylindrical mirror  826  also may be used as the focusing means for constructing the same compact spectrometer based on the present invention. 
         [0085]    In  FIG. 8(   d ), another embodiment for the collimating means is presented, which shows a concave cylindrical mirror  836  at 45° working with a folding mirror  835  at 45°. For the purposes of easy manufacturing, mounting and aligning, these two mirrors may be constructed on the same piece of transparent material. The concave cylindrical mirror  836  may be generally toroidal, i.e., having the major optical power in the tangential plane, but have little or no optical power in the sagittal plane. Its section profile in the tangential plane may take any of the following forms: spherical, aspherical, or conic curves. The assembly made by the concave cylindrical mirror  836  plus the 45° folding mirror  835  also may be used as the focusing means for constructing the same compact spectrometer incorporating features of embodiments of the invention. 
         [0086]    In  FIG. 8(   e ), another embodiment for the collimating means is presented, which shows a concave cylindrical mirror  846  formed at 45° on a monolithic transparent carrier, on which the compact spectrometer is constructed. The concave cylindrical mirror  846  may be generally toroidal, i.e., having the major optical power in the tangential plane, but have little or no optical power in the sagittal plane. Its section profile in the tangential plane may take any of the following forms: spherical, aspherical, or conic curves. The concave cylindrical mirror  846  also can be used as the focusing means for constructing the same compact spectrometer based on aspects of the present invention. 
         [0087]    In  FIG. 9(   a ), one embodiment of the optical path configuration for the collimating means and the focusing means based on aspects of the present invention is represented. The functionalities of such collimating means and focusing means have been fully described in the embodiments shown in  FIG. 1  to  FIG. 7 , thus only the light path properties will be explained in detail. In  FIG. 9(   a ), the input light  906  emits from the entrance aperture  904  that may be the core of the optical fiber  902  for input signal delivery, and propagates in divergence over a very short distance, then is intercepted by the cylindrical lens  908 , which, as the light passes through the lens, collimates the divergent light  906  only in the tangential plane, converting it into a partially collimated light, i.e., the anamorphic light  910  which is substantially collimated in the tangential plane, but remains divergent in the sagittal plane, leading to a propagation path that is two-dimensional. In reality, the collimation of the anamorphic light  910  in the tangential plane is merely an approximation. Precisely speaking, the anamorphic light  910  also has a very small amount of divergence in the tangential plane because of the finite height of the entrance aperture  904  that is the core of the optical fiber  902 . The “speed” of the divergence is a function of the aperture height (here, it is the fiber core), the numerical aperture (NA) of the input beams  906 , the focal length and the clear aperture of the cylindrical lens  908  in the tangential plane. Being affected by these parameters, as the anamorphic light  910  propagates a certain distance (through other optical means not shown here) into the anamorphic light  912 , its width in the tangential plane is slowly increasing. As the anamorphic light  912  is transmitted through the cylindrical lens  914 , it will be focused into the convergent light  916  in the tangential plane to form the spectral images  918  on the detector  920 . Truncation on the light  912  by the lens  914  in the tangential plane will not happen unless the width of light  912  is larger than the clear aperture of lens  914  in the tangential plane. An exemplary calculation shows that, for a typical application scenario where the fiber core equals to 50 micron, NA is 0.22, the two cylindrical lenses  908  and  914  have the same focal length of 8 mm, the maximum separating distance allowable between the two cylindrical lenses  908  and  914  is approximately 240 mm in air without any beam truncation, if they both have the same clear aperture of 5 mm in the tangential plane, which is only approximately 2.1% of the path length. This makes it possible to construct a spectrometer of long optical path with small dimensional volume. 
         [0088]    In  FIG. 9(   b ), another embodiment of the optical path configuration for the collimating means and the focusing means according to aspects of the invention is represented. In  FIG. 9(   b ), the input light  936  emits from the entrance aperture  934  that may be the core of the optical fiber  932  for input signal delivery, and propagates in divergence over a very short distance, then is transmitted through the cylindrical lens  938 , which converts the divergent light  936  into an anamorphic light  940 , which is slowly convergent in the tangential plane, but remains divergent in the sagittal plane, leading to a propagation path that is substantially two-dimensional. As the anamorphic light  940  continues to propagate, it will form an intermediate focus at position  942  in the tangential plane. After passing point  942  (through other optical means not shown here), the anamorphic light  940  becomes another form of anamorphic light  944 , which is slowly divergent in the tangential plane. As the anamorphic light  944  is transmitted through the cylindrical lens  946 , it will be focused into the convergent light  948  in the tangential plane to form the spectral images  950  on the detector  952 . In fact, the height of the entrance aperture  934 , the middle point  942  and the spectral image  950  represent three optically conjugated positions of the input object, the intermediate image and the final image. The two cylindrical lenses  938  and  946  function as an image relay system of 1:1 magnification in the tangential plane. The same exemplary calculation results for no beam truncation as  FIG. 9(   a ) are achieved for the same application scenario. This optical path configuration is applicable to all embodiments of spectrometers shown in  FIG. 1  to  FIG. 9  based on the present invention, and it still works well in the cases of  FIGS. 7(   b )˜( e ), even when the concave cylindrical grating in each embodiment is replaced by a concave spherical grating, because of the symmetrical properties of this configuration. 
         [0089]    The embodiments provided above and other potential embodiments with modifications based on this invention are particularly beneficial to compact spectrometers of small volumes. The associated optical technique has driven the merit of performance-volume of such a kind of spectrometer to its limit, as determined by the following parameter: (1) the input focal length of the spectrometer optics ƒ′ IFL ; (2) the height of the entrance aperture φ, (3) the F/Number (or equivalently NA) of the input beam; (4) the co-efficient n for the total optical path length which is optical configuration dependent. The term “Dimension Improving Ratio (DIR)” may be used to show the significance of volume reduction for a spectrometer, which is defined as: DIR=A :B, where A is the reduced height of the spectrometer optics based on the technique of this invention, B is the height originally needed for the same spectrometer by existing technologies. The explicit expression of DIR is: 
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         [0090]    Here another embodiment of exemplary calculation is shown in details. It starts from the specifications of input parameters of: (1) the input focal length of the spectrometer optics: ƒ′ IFL =65 mm; (2) the height of the entrance aperture: φ=0.05 mm; (3) the F/Number of the input beam: F/2.27 (NA 0.22); (4) the co-efficient n for the total optical path length: n=4 for Czerny-Turner type or Fastie-Ebert type. The optimum focal length of the cylindrical element for the first collimating optics (same for the second focusing optics) is governed by: ƒ′ 1st ≈(n×φ×ƒ′ IFL ×F/#) 1/2 =5.43 mm. The smallest volumes of the optics of spectrometers possibly to achieve is: 65×65×4.8 mm 3  (an 84% reduction compared with original volume of 65×65×29.4 mm 3 ) with DIR=0.16. For a spectrometer of single concave grating configuration, n=2, leading to an optimum value for ƒ′ 1st =3.84 mm, and the smallest volumes of the optics of spectrometers possibly to achieve is 65×42×3.3 mm 3  (a 89% reduction compared with original volume of 65×42×29.4 mm 3 ) with DIR=0.11. 
         [0091]    For purposes of this disclosure, an optical element is a component that performs at least one optical function. An optical member includes at least one optical element and performs at least one optical function. However, an optical member may include a plurality of optical elements that are integrated to perform a plurality of optical functions. For example,  FIG. 1(   b ) illustrates optical elements  155 ,  158 ,  162 ,  166  and  169  that each perform a single optical function (i.e., first collimating, second collimating, dispersing, first focusing, and second focusing). Based on the configuration of the embodiment illustrated in  FIG. 1(   b ), each optical element corresponds to a single optical member. In contrast,  FIG. 6(   b ) illustrates optical elements  655 ,  658 , and  662 . However, as described above for  FIG. 6(   b ), optical elements  655  and  658  perform two optical functions, respectively. Specifically, optical element  655  performs the first collimating and the second focusing. Optical element  658  performs the second collimating and the first focusing. Further, the embodiment illustrated in  FIG. 6(   b ) includes at least three optical members— 655 ,  658  and  662 . Each optical member includes a single optical element, but as discussed above, the optical members corresponding to the optical elements  655  and  658  perform a plurality of optical functions. Additionally,  FIG. 6(   c ) illustrates an embodiment that includes four optical elements and two optical members. In other words, the first optical member includes the optical elements  686 ,  688  and  695  and the second optical member includes  691 . As discussed above with respect to  FIG. 6(   c ), the optical element  688  performs two functions—the second collimating and the first focusing.