Optical gratings are well known in the art and are used to disperse optical spectra spatially. Such gratings are commonly used in optical spectrometers to analyze the spectral composition of an optical beam. There is always a tradeoff between the length of an optical spectrometer and its resolution. Thus, if a higher wavelength resolution is required, the length required is also longer. Consider an example of a typical 1-meter long grating spectrometer in the market, which has a typical wavelength resolution of about ΔλSM=0.1 nm at a center operating wavelength λSM=1000 nm or ΔλSM/λSM=10−4. The dimensionless quantity for the length of the spectrometer LSM is LSM/λSM and LSM/λSM=106 in this example. The dimensionless product of the relative resolution ΔλSM/λSM and the relative physical size LSM/λSM of the spectrometer is dependent on the design of the spectrometer.
Resolution Vs Size/Area Factor
More precisely, consider a curved grating spectrometer device 1000 shown in FIG. 1 with a curved grating CG 1010 and a wave or beam propagating region between the input slit SLI1 1201 and the grating or the grating and the output slit SLO1 1401 called grating-propagating region GPR 1020. If the spectrometer has a width WSM 1002 and a length LSM 1003, then the dimensionless product of interest is:RSSM=(ΔλSM/λSM)*[(WSM/λSM)*(LSM/λSM)]0.5,  (1)where “*” denotes numerical multiplication. This factor (RSSM) is referred to here as the “resolution vs size” factor of the spectrometer. The square of RSSM factor is called “resolution vs area” factor:RASM=(RSSM)2.  (2)The factors RASM and RSSM basically measure the compactness of a spectrometer for a given spectral resolution power. In this example, if WSM=LSM=1 m, then RSSM=100 and RASM=104. The smaller the RSSM and RASM value, the more compact is the spectrometer.
Only a few conventional spectrometers have RASM factor less than about 100 (or RSSM<10). This is primarily because of the various limitations in the prior arts (as will be described below). In many photonic integrated circuit, electronic-photonic integrated circuit, and fiber-optic applications, RASM factors of smaller than about RASM=1 down to RASM=0.01 are desirable. These cannot be achieved via the methods in the prior arts.
Furthermore, for many applications in photonic integrated circuits, electronic-photonic integrated circuits, and fiber-optics, the background extinction of the spectral power outside the wavelength range that is supposed to be detected is important, which measures the relative optical power from outside the detection wavelength range that is scattered into the output slit or photodetector of the spectrometer.
Spectral Output Power Efficiency
A parameter that measures such unwanted background scattering is the “adjacent-channel power extinction ratio”. As shown by FIGS. 1 and 1B, assuming the input light at an input slit labeled as slit “SLI1” 1201 for which “I1” labels “input” “1” has an optical spectrum with optical power spectral density PSI1(λ) 1131 at wavelength λ. Let λI1-O1 1321 be the center spectral wavelength that goes from the spectrometer's input slit SLI1 to the spectrometer's output slit labeled as output slit “SLO1” 1401. Let the optical power spectrum entering or being detected at output slit SLO1 1401 be PSI1-O1(λ) 1331. PSI1-O1(λ) 1331 is then given in terms of the spectral density of the input beam PSI1(λ) by:PSI1-O1(λ)=ηeffI1-O1(λ)*PSI1(λ).  (3)The factor ηeffI1-O1(λM) is the efficiency of passing or detecting the power of the input beam from input slit SLI1 at the output slit SLO1 at wavelength λM, assuming that the input beam is basically a monochromatic light source at λM (e.g. a narrow-bandwidth laser light). The factor ηeffI1-O1(λM) is called the “spectral power output efficiency”.
As shown in FIG. 1B, the power at input slit SL 1201 over a small spectral bandwidth Δλ centered at wavelength λA (small comparing to the spectral bandwidth of PSI1-O1(λ) at λA or more precisely, small enough so that PSI1-O1(λ) at λA does not change much over the wavelength bandwidth Δλ), denoted as PI1(λA; Δλ) 1131DL, is given by:PI1(λA;Δλ)=PSI1(λA)*Δλ.  (4A)Let the total optical power entering or being detected at output slit SLO1 1401 over a small spectral bandwidth Δλ centered at wavelength λA for the beam from input slit SLI1 be PI1-O1(λA; Δλ) 1331DL. PI1-O1(λA; Δλ) is then given by PSI1-O1(λA)*Δλ=ηeffI1-O1(λ)*PSI1(λ)Δλ according to Eq. (3). It is also given by ηeffI1-O1(λA)*(λA; Δλ). PI1-O1(λA; Δλ) is thus related to the spectral density of the input beam PSI1(λA) by:PI1-O1(λA;Δλ)=PSI1-O1(λA)*Δλ=ηeffI1-O1(λA)*PSI1(λA)*Δλ.  (4B)In the situation that Δλ is large, Eqs. (4A) and (4B) should be more precisely converted to an integration of PSI1-O1(λ) with respect to wavelength λ over wavelength bandwidth Δλ centered at wavelength λ=λA given by:
                                              ⁢                                            P                              I                ⁢                                                                  ⁢                1                                      ⁡                          (                                                λ                  A                                ;                Δλ                            )                                =                                    ∫                                                λ                  A                                -                                  Δλ                  2                                                                              λ                  A                                +                                  Δλ                  2                                                      ⁢                                                            PS                                      I                    ⁢                    1                                                  ⁡                                  (                  λ                  )                                            ⁢                                                          ⁢              d              ⁢                                                          ⁢              λ                                                          (                  5          ⁢          A                )                                                      P                                          I                ⁢                1                            -                              O                ⁢                1                                              ⁡                      (                                          λ                A                            ;              Δλ                        )                          =                                            ∫                                                λ                  A                                -                                  Δλ                  2                                                                              λ                  A                                +                                  Δλ                  2                                                      ⁢                                                            PS                                                            I                      ⁢                      1                                        -                                          O                      ⁢                      1                                                                      ⁡                                  (                  λ                  )                                            ⁢                                                          ⁢                              ⅆ                λ                                              =                                    ∫                                                λ                  A                                -                                  Δλ                  2                                                                              λ                  A                                +                                  Δλ                  2                                                      ⁢                                                            η                                                            effI                      ⁢                      1                                        -                                          O                      ⁢                      1                                                                      ⁡                                  (                  λ                  )                                            *                                                PS                                      I                    ⁢                    1                                                  ⁡                                  (                  λ                  )                                            ⁢                                                          ⁢              d              ⁢                                                          ⁢              λ                                                          (                  5          ⁢          B                )            Total Power and Power Spectrum
To define the “adjacent-channel power extinction ratio”, let the spectral resolution bandwidth for a beam at wavelength λI1-O1 1321 from input slit SLI1 1201 to output slit SLO1 1401 be ΔλRes-I1-O1 1381Res, as shown in FIG. 1B. Assuming the input power spectrum at an input slit of the spectrometer PSI1(λ) 1131 is approximately a constant for the wavelength λ around λ=λI1-O1 1321. Let the optical power from the input slit SLI1 1201 entering or being detected at an output slit of the spectrometer SLO1 1401 over the spectral bandwidth given by the spectral resolution bandwidth ΔλRes-I1-O1 1381Res be PI1-O1(λI1-O1; ΔλRes-I1-O1), where:
                                          P                                          I                ⁢                1                            -                              O                ⁢                1                                              =                                                    P                                                      I                    ⁢                    1                                    -                                      O                    ⁢                    1                                                              ⁡                              (                                                      λ                                                                  I                        ⁢                        1                                            -                                              O                        ⁢                        1                                                                              ;                                      Δλ                                          Res                      -                                              I                        ⁢                        1                                            -                                              O                        ⁢                        1                                                                                            )                                      =                                          ∫                                                      λ                                                                  I                        ⁢                        1                                            -                                              O                        ⁢                        1                                                                              -                                                            Δλ                                              Res                        -                                                  I                          ⁢                          1                                                -                                                  O                          ⁢                          1                                                                                      2                                                                                        λ                                                                  I                        ⁢                        1                                            -                                              O                        ⁢                        1                                                                              +                                                            Δλ                                              Res                        -                                                  I                          ⁢                          1                                                -                                                  O                          ⁢                          1                                                                                      2                                                              ⁢                                                                    η                                                                  effI                        ⁢                        1                                            -                                              O                        ⁢                        1                                                                              ⁡                                      (                    λ                    )                                                  *                                                      PS                                          I                      ⁢                      1                                                        ⁡                                      (                    λ                    )                                                  ⁢                                                                  ⁢                d                ⁢                                                                  ⁢                λ                                                    ,                            (                  6          ⁢          A                )            Then at an adjacent wavelength that is separated from λI1-O1 1321 by “one spectral resolution bandwidth” ΔλRes-I1-O1 1381Res away, the power detected over the same spectral resolution bandwidth ΔλRes-I1-O1 1381Res given by:
                                                        P                              I1                -                                  O                  ⁢                  1                                                      ⁡                          (                                                                    λ                                                                  I                        ⁢                        1                                            -                                              O                        ⁢                        1                                                                              +                                      Δλ                                          Res                      -                                              I                        ⁢                        1                                            -                                              O                        ⁢                        1                                                                                            ;                                  Δλ                                      Res                    -                                          I                      ⁢                      1                                        -                                          O                      ⁢                      1                                                                                  )                                =                                    ∫                                                λ                                                            I                      ⁢                      1                                        -                                          O                      ⁢                      1                                                                      +                                  Δλ                                      Res                    -                                          I                      ⁢                      1                                        -                                          O                      ⁢                      1                                                                      -                                                      Δλ                                          Res                      -                                              I                        ⁢                        1                                            -                                              O                        ⁢                        1                                                                              2                                                                              λ                                                            I                      ⁢                      1                                        -                                          O                      ⁢                      1                                                                      +                                  Δλ                                      Res                    -                                          I                      ⁢                      1                                        -                                          O                      ⁢                      1                                                                      +                                                      ΔλΔλ                                          Res                      -                                              I                        ⁢                        1                                            -                                              O                        ⁢                        1                                                                              2                                                      ⁢                                                            η                                                            effI                      ⁢                      1                                        -                                          O                      ⁢                      1                                                                      ⁡                                  (                  λ                  )                                            *                                                PS                                      I                    ⁢                    1                                                  ⁡                                  (                  λ                  )                                            ⁢                                                          ⁢              d              ⁢                                                          ⁢              λ                                      ,                            (                  6          ⁢          B                )            should ideally be zero at the output slit (as the spectral power output efficiency ηeffI1-O1(λ) shall be approximately zero when λ=λI1-O1+ΔλRes-I1-O1 is not at the supposed output detection/passing wavelength λI1-O1 1321 of output slot SLO1 and is at one resolution bandwidth ΔλRes-I1-O1 1381Res away from λI1-O1 1321).Adjacent-Channel Extinction Ratio
The ratio of the power at λI1-O1 1321 and the power at λI1-O1+ΔλRes-I1-O1 given by:ηace(λI1-O1)=PI1-O1(λI1-O1;ΔλRes-I1-O1)/PI1-O1(λI1-O1+ΔλRes-I1-O1;ΔλRes-I1-O1),  (7)is called the “adjacent-channel power extinction ratio” or “adjacent-wavelength power extinction ratio” or “adjacent wavelength-channel extinction ratio”, or “adjacent channel extinction ratio” for the beam from input slit SLI1 1201 to output slit SLO1 1401. It is also related to what is known to those skilled in the art as “adjacent-channel crosstalk rejection” or “adjacent-channel crosstalk extinction”. These terminologies will thus be used interchangeably below.Adjacent Channel Extinction Ratio of a Spectrometer
Let ηace(λSM) denotes the adjacent channel extinction ratio for the spectrometer at its center operating wavelength λSM. It is called the “adjacent channel extinction ratio of a spectrometer”. This adjacent-channel extinction ratio ηace(λSM) is typically higher than about 100, especially when the size of the spectrometer is small (i.e. when the RASM factor is smaller than about 104). For many applications in photonic integrated circuits, electronic-photonic integrated circuits, and fiber-optics, ηace(λSM) higher than about 100, and RASM factor smaller than about 104 are desirable but they are largely not reachable via the methods in the prior arts.
Furthermore, for many applications in photonic integrated circuits, electronic-photonic integrated circuits, and fiber-optics, it is desirable that at the wavelength range of interest, the efficiency of passing or detecting the input spectral power be efficient, which is given by the spectral power output efficiency factor ηeffI1-O1(λI1-O1) defined above.
Spectral Power Output Efficiency of a Spectrometer
Let ηeff(λSM) denote the spectral power output efficiency ηeffI1-O1(λI1-O1) when the output slit wavelength λI1-O1 1321 is at the center operating wavelength λSM of the spectrometer. In an ideal situation, ηeff(λSM) shall be equal to unity (i.e. the number “1” so that the output power is equal to the input power or it has 100% passing or detection efficiency) or at least near unity. In many applications ηace(λSM) higher than about 100 with RASM factor smaller than about 104 and ηeff(λSM) higher than about 0.5 or having a higher than 50% passing/detection efficiency is desirable. In some other applications, ηace(λSM) higher than about 100 with RASM factor smaller than about 104, and ηeff(λSM) higher than about 0.1 (i.e. having a higher than 10% passing/detection efficiency) are desirable. These capabilities are desirable for the spectrometer but they are largely not reachable via the methods in the prior arts.
Conventional Rowland Gratings and their General Limitations
Although a small input slit width is required to achieve high spectral resolution for a spectrometer, a smaller input slit width WI1 1291W (see FIG. 1A), however, leads to a larger beam divergence full angle θdvdf-BI1-IP % 1141 (see FIG. 2) from the input slit defined as the beam angular span that encompasses IP %=95% of the beam energy. It can be shown that the conventional Rowland design in the prior art works reasonably well only up to (θdvdf-BI1-95%)/2<4° (for a grating in air for which the beam propagation region has a refractive index of ngr=1). More specifically, θdvdf-BI1-95% is the full divergence angle of the input beam due to spatial diffraction that we refer to as the “divergence-diffraction” (dvdf) angle. The 95% refers to the angle being defined by 95% of the beam power to be within the divergence angle. Note that the 4° is half the divergence angle of θdvdf-BI1-95%. If the grating cannot intercept or refocus a large percentage of the beam power well, then it will result in an unnecessarily large beam spot size at the output slit, and hence a worse wavelength resolution at the output slit.
Let the input beam be a monochromatic beam with wavelength λI1-O1 1321 (see FIG. 2) and the focal spot size for this beam at the output slit SLO1 1401 be WBI1-O1-IP % 1381W (see FIG. 2). When θdvdf-BI1-95%/2>4°, the Rowland design could not give a sharp enough focus at the output slit SLO1 1401, thereby limiting the resolution of the spectrometer.
A half-diffraction angle of θdvdf-BI1-95%/2=4° corresponds to a slit width of about 25 microns (for an input beam wavelength around λI1-O1=1000 nm and n=1). In the current art, it is typically difficult to make slit width smaller than about 25 microns, and Rowland design is adequate for most present spectrometers with slit widths larger than about 25 microns (for a grating in air for which the beam propagation region has a refractive index of ngr=1).
Slit size (n = 3.4)θD(>95% energy)10μm 6°6μm10°2μm24°1μm46°0.5μm70°0.3μm90°
Focusing Aberration Limitation:
In the case of the Rowland design, when (θdvdf-BI1-95%)/2>4 Degree(s) (DEG), serious aberration in the refocusing beam will occur to limit wavelength resolution (θdvdf-BI1-95% 1141 is the full divergence angle of the input beam due to spatial diffraction that we refer to as the “divergence-diffraction” (dvdf) angle. The 95% refers to the angle being defined by 95% of the beam power to be within the divergence angle). This is shown in FIG. 4, illustrating the ray tracing for the typical Roland-Echelle design at 4, 8, and 16 DEG diffraction. The ray tracing will allow us to see potential focusing distortion or aberration at the exit slit. In the figure, we show the focusing behavior for two sets of rays with wavelengths separated by 0.4 nm. Let the “half angle” θdiv=(θdvdf-BI1-95%)/2. From the figure, we see that their focused spots are clearly separated when θdiv=4 DEG. However, when θdiv=8 DEG, the focused spots began to smear out. FIG. 4A shows the cases of θdiv=16 DEG, and θdiv=50 DEG, for which the focused spots smear out even more. Thus, there is substantial distortion for the focusing rays when θdiv>4 DEG. Further simulations based on numerical solutions to Maxwell's wave equations using finite-difference time-domain (FDTD) method also show similar onset of focused spot size distortion at θdiv>4 DEG. In short, the current designs are close to their resolution-area (RASM) limits at θdiv of about 4 DEG and cannot be made substantially more compact without losing wavelength resolution.