1. Introduction
Spectral filters play a crucial role in the operation of mode-locked fiber lasers. By adjusting the bandwidth (BW) of spectral filters in the fiber lasers, various types of mode-locking operations can be achieved, including dissipative soliton (DS) pulses and self-similar (SS) pulses [
1,
2]. Spectral filters can be implemented in various ways, such as bandpass filters [
3], birefringent filters [
4,
5,
6], and grating–collimator filters. In particular, the grating–collimator filter—featuring a lens–fiber configuration, commonly referred to as a collimator, to couple grating-diffracted light into a fiber—presents an appealing filter design for a mode-locked fiber laser since the filter utilizes a fiber [
7,
8,
9].
Recently, Ryu et al. [
10] derived an analytical expression for diffraction grating––collimator spectral filters, demonstrating that the spectral filtering effect arises due to deviations from the ideal fiber coupling—where ideal coupling occurs when grating-diffracted light at the center wavelength is optimally coupled into the fiber. According to the spectral filter BW equation, the BW decreases with increasing input beam diameter and grating density, as these factors reduce the coupling efficiency, thereby enhancing the spectral filtering effect.
Nevertheless, there are practical limitations to arbitrarily selecting beam diameter and grating density. In the previous study [
10], a BW of ~4 nm at 1030 nm (Ytterbium (Yb)-doped fiber laser case) was achieved using a beam diameter of 1 mm with a 300 lines/mm grating. A higher grating density cannot be used, as it would result in a very narrow spectral BW inhibiting mode-locking. Although a 300 lines/mm grating can generate ~4 nm spectral BW for SS fiber lasers, achieving a larger BW is challenging since a lower density grating would lead to higher-order diffractions, thereby reducing the overall efficiency. Similarly, at 1550 nm (Erbium (Er)-doped fiber laser case), a BW of 4 nm was achieved using a beam diameter of 0.5 mm and a 600 lines/mm grating, while such a small beam size makes optical alignment more challenging [
7].
These limitations indicate that combining an optical element with weaker diffractive effects with a larger input beam diameter is preferable. A promising alternative is the dispersive prism, which inherently exhibits a weaker spatial dispersion compared to a diffraction grating. Moreover, prisms are advantageous over diffraction gratings in building mode-locked fiber lasers for three key reasons: (1) unlike gratings, prisms do not produce higher-order diffractions or scatter light due to artifacts (e.g., groove defects), enabling high transmission over a broad wavelength range; (2) prisms are easier to maintain, while gratings are prone to degradation over time due to environmental factors such as humidity and dust; (3) the group-delay dispersion (GDD) of prism spectral filters is in the order of 10−6 of that of grating spectral filters, owing to their inherently weaker spatial dispersion. Therefore, the prism spectral filter has a minimal impact on the pulse propagation in the fiber cavity, making it a desirable choice.
In this manuscript, we demonstrate that prism-based spectral filters are promising alternatives to diffraction grating-based spectral filters. An analytical approach for prism-based spectral filters, derived using the ABCDEF matrix formalism, is presented. By utilizing multiple SF11 prisms, spectral filters with various BWs can be created. Furthermore, we experimentally demonstrate the operation of mode-locked fiber lasers by integrating these filters, achieves stable mode-locking in both DS and SS fiber lasers.
2. Analysis of Prism-Based Spectral Filters
The schematic of the prism–collimator spectral filter is shown in
Figure 1, where
denotes the separation distance between the prism and the lens,
the focal length of the lens, and
the Brewster angle. As the beam passes through the prism, an arbitrary wavelength
experiences a different refractive index
, resulting in beam refraction within the prism. After being refracted by the prism, each wavelength is focused onto a different point on the tip of a single-mode fiber (SMF), leading to both translational and angular offsets relative to
, which induce the spectral filtering effect [
10]. We refer to this lens–SMF configuration as the collimator throughout this paper.
The ABCD matrix, also known as the ray transfer matrix, is commonly used to describe the effect of optical elements on the beam [
11]. To account for both translational error
and angular error
, this formalism is extended to the ABCDEF matrix, as shown by Equation (1) [
12]. Equation (2) lists the ABCDEF matrices corresponding to free space, lenses, and prisms.
The ABCDEF matrix allows us to compute the final position and angle of a beam, accounting for errors accumulated during propagation through the optical system. In this configuration, the prism introduces an angular error owing to refraction, which can be expressed as an offset relative to the ideal deviation angle.
We utilize the Brewster angle to minimize the loss. When the angles of incidence and emergence are equal, as shown in
Figure 2, the deviation angle
can be calculated using Equation (3), where
denotes the refractive index of the prism and
is the apex angle of the prism. SF11 flint glass was selected as the prism material in this study due to its relatively high refractive index among commercially available optical glasses.
The angular error
introduced by the prism can be approximated using a Taylor expansion around
, as given by Equation (4), where
is the derivative of
with respect to
. Since the second-order term contributes less than 1% of the first-order term within the considered spectral filter’s BW range (
), higher-order terms can be neglected in this approximation.
By incorporating this angular error
into the prism matrix
and multiplying the matrices corresponding to the configuration shown in
Figure 1, the final matrix at the tip of the fiber
can be obtained using Equation (5) [
10].
Assuming that a Gaussian input beam is focused onto the single-mode fiber (SMF) with the translational error
and the angular error
, the SMF coupling efficiency is given by Equation (6) [
13], where
denotes the input beam radius and
the mode field radius of the SMF.
Here,
, the total transmission of the prism-based spectral filter, indicates a Gaussian spectral filter shape. Accordingly, the full width at half maximum (FWHM), representing the BW, of the SF11 prism-based spectral filter (hereafter, the SF11 filter) is calculated using Equation (7) [
10], where
denotes the input beam diameter and
the mode field diameter (MFD) of the SMF.
We investigated a typical case of a 1030 nm Yb-doped fiber laser design. It is assumed that an input beam with a diameter of
propagates over a free-space distance
, and is then focused by a lens with a focal length
into an HI1060 fiber, whose mode field diameter is
at
.
Figure 3a illustrates the BW of an SF11 filter as a function of the separation distance
, calculated using Equation (6) with the assumed parameters. By utilizing two prisms in the filter, the spectral filter BW can be further reduced. Each curve in
Figure 3 corresponds to a configuration involving either (a) one or (b) two SF11 prisms, hereafter referred to as the single- and dual-SF11 filters, respectively.
The BW of the single-SF11 filter is approximately 8 nm and does not vary significantly with changes in . A narrower spectral filter is clearly achieved by introducing a second SF11 prism, placed after the first and aligned at the Brewster angle. Using the same calculation procedure as in the single-SF11 case, except that includes additional free-space and prism ABCDEF matrices, the dual-SF11 filter yields the BW of approximately 4 nm.
Figure 4 presents the experimental results for the (a) single- and (b) dual-SF11 filters, along with corresponding Gaussian fits. The transmission was calculated by comparing the spectra of a mode-locked pulse before and after propagation through the SF11 filters.
From the transmission data, the FWHM (BW) of the single- and dual-SF11 filters are approximately 8.7 nm and 4.3 nm, respectively. Overall, the experimental spectral profiles exhibit good agreement with the Gaussian fits, consistent with the assumed Gaussian filtering effect. It is notable that multiple prisms can be incorporated into the spectral filter with minimal loss, provided that the polarization of the incident beam is properly aligned and the prisms are oriented at the Brewster angles.
3. Design of Mode-Locked Fiber Lasers
We confirmed that the single-SF11 filter can provide a BW of ~8 nm, while the dual-SF11 filter achieves a narrower BW of ~4 nm. Furthermore, the single- and dual-SF11 filters satisfy requirements for spectral filter BW ranges for DS (~8 nm) and SS (~4 nm) pulse operation, respectively [
14].
Figure 5 shows the schematic of a DS fiber laser design (outlined in orange) and SS fiber laser design (outlined in yellow), incorporating the single- and dual-SF11 filters, respectively. In both designs, the total fiber cavity length is approximately 4 m, comprising 2.4 m of HI1060 fiber (SMF 1 and the WDM section), 0.6 m of Yb-doped fiber (Coractive, YB 401) pumped via the WDM with a 980 nm pump diode, and another 1 m of HI1060 fiber (SMF 2). Correspondingly,
Figure 6 shows the experimental configuration of the SS fiber laser, with the beam path indicated by yellow arrows.
In our configurations, the group-delay dispersion (GDD) introduced by the SF11 filters is on the order of −10
−8 ps
2, which is negligible compared to that of grating–collimator filters (on the order of −10
−2 ps
2) [
10]. This minimal dispersion ensures that the overall GDD in our laser designs remains well within the normal dispersion regime.
In the operation of lasers, the SF11 filters function as a spectral filter (SF). A half-wave plate placed in front of the SF11 prism is used to align the polarization of the incident beam to the transverse magnetic (TM) field with respect to the prism. Additionally, quarter-wave plates, half-wave plates, and polarizers along the beam path act as a saturable absorber (SA) by manipulating the nonlinear polarization evolution (NPE).